Linear integral equations -rainer kress

Applied Mathematical Sciences

Rainer Kress

Linear Integral Equations Third Edition

Applied Mathematical Sciences

Volume 82

Founding EditorsFritz John, Joseph P. LaSalle and Lawrence Sirovich

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P.J. [email protected]

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AdvisorsL. GreengardJ. KeenerR.V. KohnB. MatkowskyR. PegoC. PeskinA. SingerA. StevensA. Stuart

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Rainer Kress

Linear Integral Equations

Third Edition

123

Rainer KressInstitut fur Numerische und AngewandteGeorg-August-Universitat GottingenGottingen, Germany

ISSN 0066-5452 ISSN 2196-968X (electronic)ISBN 978-1-4614-9592-5 ISBN 978-1-4614-9593-2 (eBook)DOI 10.1007/978-1-4614-9593-2Springer New York Heidelberg Dordrecht London

Library of Congress Control Number: 2013954393

Mathematics Subject Classification (2010): 35J05, 35P25, 35R25, 35R30, 45A05, 65M30, 65R20,65R30, 76Q05, 78A45

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To the Memory of My Parents

Preface to the Third Edition

In the fourteen years since the second edition of this book appeared, linear integralequations have continued to be an active area of mathematics and they have revealedmore of their power and beauty to me. Therefore I am pleased to have the opportu-nity to make adjustments and additions to the book’s contents in this third edition.In the spirit of the two preceding editions, I have kept the balance between theory,applications and numerical methods. To preserve the character of an introduction asopposed to a research monograph, I have made no attempts to include most of therecent developments.

In addition to making corrections and additions throughout the text and updatingthe references, the following topics have been added. In order to make the introduc-tion to the basic functional analytic tools more complete the Hahn–Banach exten-sion theorem and the Banach open mapping theorem are now included in the text.The treatment of boundary value problems in potential theory has been extended bya more complete discussion of integral equations of the first kind in the classicalHolder space setting and of both integral equations of the first and second kind inthe contemporary Sobolev space setting. In the numerical solution part of the book,I included a new collocation method for two-dimensional hypersingular boundaryintegral equations and the collocation method for the Lippmann–Schwinger equa-tion based on fast Fourier transform techniques due to Vainikko. The final chapterof the book on inverse boundary value problems for the Laplace equation has beenlargely rewritten with special attention to the trilogy of decomposition, iterative andsampling methods.

Some of the additions to this third edition were written when I was visiting theInstitut Mittag-Leffler, Djursholm, Sweden, in spring 2013 during the scientific pro-gram on Inverse Problems and Applications. I gratefully acknowledge the hospital-ity and the support.

vii

viii Preface to the Third Edition

Over the years most of the thirty-one PhD students that I supervised wrote theirthesis on topics related to integral equations. Their work and my discussions withthem have had significant influence on my own perspective on integral equations aspresented in this book. Therefore, I take this opportunity to thank my PhD studentsas a group without listing them individually. A special note of thanks is given to myfriend David Colton for reading over the new parts of the book and helping me withthe English language.

I hope that this new edition of my book continues to attract readers to the field ofintegral equations and their applications.

Gottingen, Germany Rainer Kress

Preface to the Second Edition

In the ten years since the first edition of this book appeared, integral equations andintegral operators have revealed more of their mathematical beauty and power to me.Therefore, I am pleased to have the opportunity to share some of these new insightswith the readers of this book. As in the first edition, the main motivation is to presentthe fundamental theory of integral equations, some of their main applications, andthe basic concepts of their numerical solution in a single volume. This is done frommy own perspective of integral equations; I have made no attempt to include all ofthe recent developments.

In addition to making corrections and adjustments throughout the text and updat-ing the references, the following topics have been added: In Section 4.3 the presen-tation of the Fredholm alternative in dual systems has been slightly simplified and inSection 5.3 the short presentation on the index of operators has been extended. Thetreatment of boundary value problems in potential theory now includes proofs of thejump relations for single- and double-layer potentials in Section 6.3 and the solutionof the Dirichlet problem for the exterior of an arc in two dimensions (Section 7.8).The numerical analysis of the boundary integral equations in Sobolev space settingshas been extended for both integral equations of the first kind in Section 13.4 andintegral equations of the second kind in Section 12.4. Furthermore, a short outlineon fast O(n log n) solution methods has been added in Section 14.4. Because inverseobstacle scattering problems are now extensively discussed in the monograph [32],in the concluding Chapter 18 the application to inverse obstacle scattering problemshas been replaced by an inverse boundary value problem for Laplace’s equation.

I would like to thank Peter Hahner and Andreas Vogt for carefully reading themanuscript and for a number of suggestions for improving it. Thanks also go to thosereaders who helped me by letting me know the errors and misprints they found inthe first edition.

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x Preface to the Second Edition

I hope that this book continues to attract mathematicians and scientists to thefield of integral equations and their applications.


Preface to the First Edition

I fell in love with integral equations about twenty years ago when I was working onmy thesis, and I am still attracted by their mathematical beauty. This book will tryto stimulate the reader to share this love with me.

Having taught integral equations a number of times I felt a lack of a text whichadequately combines theory, applications and numerical methods. Therefore, in thisbook I intend to cover each of these fields with the same weight. The first part pro-vides the basic Riesz–Fredholm theory for equations of the second kind with com-pact operators in dual systems including all functional analytic concepts necessaryfor developing this theory. The second part then illustrates the classical applicationsof integral equation methods to boundary value problems for the Laplace and theheat equation as one of the main historical sources for the development of integralequations, and also introduces Cauchy type singular integral equations. The thirdpart is devoted to describing the fundamental ideas for the numerical solution ofintegral equations. Finally, in a fourth part, ill-posed integral equations of the firstkind and their regularization are studied in a Hilbert space setting.

In order to make the book accessible not only to mathematicians but also to physi-cists and engineers I have planned it as self-contained as possible by requiring onlya solid foundation in differential and integral calculus and, for parts of the book, incomplex function theory. Some background in functional analysis will be helpful,but the basic concepts of the theory of normed spaces will be briefly reviewed, andall functional analytic tools which are relevant in the study of integral equations willbe developed in the book. Of course, I expect the reader to be willing to accept thefunctional analytic language for describing the theory and the numerical solution ofintegral equations. I hope that I succeeded in finding the adequate compromise be-tween presenting integral equations in the proper modern framework and the dangerof being too abstract.

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xii Preface to the First Edition

An introduction to integral equations cannot present a complete picture of allclassical aspects of the theory and of all recent developments. In this sense, thisbook intends to tell the reader what I think appropriate to teach students in a two-semester course on integral equations. I am willing to admit that the choice of afew of the topics might be biased by my own preferences and that some importantsubjects are omitted.

I am indebted to Dipl.-Math. Peter Hahner for carefully reading the book, forchecking the solutions to the problems and for a number of suggestions for valuableimprovements. Thanks also go to Frau Petra Trapp who spent some time assistingme in the preparation of the LATEX version of the text. And a particular note ofthanks is given to my friend David Colton for reading over the book and helping mewith the English language. Part of the book was written while I was on sabbaticalleave at the Department of Mathematics at the University of Delaware. I gratefullyacknowledge the hospitality.


Contents

1 Introduction and Basic Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . 11.1 Abel’s Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Convergence and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Scalar Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6 Best Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Bounded and Compact Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Bounded Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 The Dual Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 Neumann Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5 Compact Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Riesz Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.1 Riesz Theory for Compact Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Spectral Theory for Compact Operators . . . . . . . . . . . . . . . . . . . . . . . . 403.3 Volterra Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4 Dual Systems and Fredholm Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1 Dual Systems via Bilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 Dual Systems via Sesquilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 484.3 The Fredholm Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.4 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

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5 Regularization in Dual Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.1 Regularizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2 Normal Solvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.3 Index and Fredholm Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6 Potential Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.1 Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.2 Boundary Value Problems: Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . 836.3 Surface Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.4 Boundary Value Problems: Existence . . . . . . . . . . . . . . . . . . . . . . . . . . 916.5 Nonsmooth Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7 Singular Boundary Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037.1 Holder Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037.2 The Cauchy Integral Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087.3 The Riemann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1167.4 Integral Equations with Cauchy Kernel . . . . . . . . . . . . . . . . . . . . . . . . 1187.5 Cauchy Integral and Logarithmic Potential . . . . . . . . . . . . . . . . . . . . . 1257.6 Boundary Integral Equations in Holder Spaces . . . . . . . . . . . . . . . . . . 1297.7 Boundary Integral Equations of the First Kind . . . . . . . . . . . . . . . . . . 1327.8 Logarithmic Single-Layer Potential on an Arc . . . . . . . . . . . . . . . . . . . 135Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

8 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1418.1 The Sobolev Space Hp[0, 2π] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1418.2 The Sobolev Space Hp(Γ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1518.3 Weak Solutions to Boundary Value Problems . . . . . . . . . . . . . . . . . . . 159Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

9 The Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1719.1 Initial Boundary Value Problem: Uniqueness . . . . . . . . . . . . . . . . . . . . 1719.2 Heat Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1749.3 Initial Boundary Value Problem: Existence . . . . . . . . . . . . . . . . . . . . . 179Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

10 Operator Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18310.1 Approximations via Norm Convergence . . . . . . . . . . . . . . . . . . . . . . . . 18310.2 Uniform Boundedness Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18510.3 Collectively Compact Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19010.4 Approximations via Pointwise Convergence . . . . . . . . . . . . . . . . . . . . 19110.5 Successive Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

Contents xv

11 Degenerate Kernel Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19911.1 Degenerate Operators and Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19911.2 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20111.3 Trigonometric Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20411.4 Degenerate Kernels via Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . 21011.5 Degenerate Kernels via Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . 215Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

12 Quadrature Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21912.1 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21912.2 Nystrom’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22412.3 Weakly Singular Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22912.4 Nystrom’s Method in Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 237Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

13 Projection Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24113.1 The Projection Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24113.2 Projection Methods for Equations of the Second Kind . . . . . . . . . . . . 24613.3 The Collocation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24913.4 Collocation Methods for Equations of the First Kind . . . . . . . . . . . . . 25513.5 A Collocation Method for Hypersingular Equations . . . . . . . . . . . . . . 26313.6 The Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26713.7 The Lippmann–Schwinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 272Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

14 Iterative Solution and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27914.1 Stability of Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27914.2 Two-Grid Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28314.3 Multigrid Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28714.4 Fast Matrix-Vector Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

15 Equations of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29715.1 Ill-Posed Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29715.2 Regularization of Ill-Posed Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 30115.3 Compact Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30315.4 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30915.5 Regularization Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

16 Tikhonov Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32316.1 Weak Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32316.2 The Tikhonov Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32416.3 Quasi-Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

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16.4 Minimum Norm Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33416.5 Classical Tikhonov Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33816.6 Ill-Posed Integral Equations in Potential Theory . . . . . . . . . . . . . . . . . 343Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

17 Regularization by Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35117.1 Projection Methods for Ill-Posed Equations . . . . . . . . . . . . . . . . . . . . . 35117.2 The Moment Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35817.3 Hilbert Spaces with Reproducing Kernel . . . . . . . . . . . . . . . . . . . . . . . 36017.4 Moment Collocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

18 Inverse Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36518.1 An Inverse Problem for the Laplace Equation . . . . . . . . . . . . . . . . . . . 36518.2 Decomposition Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36818.3 Differentiability with Respect to the Boundary . . . . . . . . . . . . . . . . . . 37818.4 Iterative Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38318.5 Sampling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

Chapter 1Introduction and Basic Functional Analysis

The topic of this book is linear integral equations of which

∫ b

aK(x, y)ϕ(y) dy = f (x), x ∈ [a, b], (1.1)

and

ϕ(x) −∫ b

aK(x, y)ϕ(y) dy = f (x), x ∈ [a, b], (1.2)

are typical examples. In these equations the function ϕ is the unknown, and theso-called kernel K and the right-hand side f are given functions. Solving these in-tegral equations amounts to determining a function ϕ such that (1.1) or (1.2), re-spectively, are satisfied for all a ≤ x ≤ b. The term integral equation was first usedby du Bois-Reymond [45] in 1888. The equations (1.1) and (1.2) carry the name ofFredholm because of his contributions to the field and are called Fredholm integralequations of the first and second kind, respectively. In the first equation the unknownfunction only occurs under the integral whereas in the second equation it also ap-pears outside the integral. Later on we will see that this is more than just a formaldifference between the two types of equations. A first impression on the differencecan be obtained by considering the special case of a constant kernel K(x, y) = c � 0for all x, y ∈ [a, b]. On one hand, it is easily seen that the equation of the secondkind (1.2) has a unique solution given by

ϕ = f +c

1 − c(b − a)

∫ b

af (y) dy

if c(b−a) � 1. If c(b−a) = 1 then (1.2) is solvable if and only if∫ b

af (y) dy = 0 and

the general solution is given by ϕ = f + γ with an arbitrary constant γ. On the otherhand, the equation of the first kind (1.1) is solvable if and only if f is a constant,f (x) = α for all x ∈ [a, b] with an arbitrary constant α. In this case every function ϕ

with∫ b

aϕ(y) dy = α/c is a solution.

R. Kress, Linear Integral Equations, Applied Mathematical Sciences 82,DOI 10.1007/978-1-4614-9593-2 1, © Springer Science+Business Media New York 2014

1

2 1 Introduction and Basic Functional Analysis

Of course, the integration domains in (1.1) and (1.2) are not restricted to aninterval [a, b]. In particular, the integration can be over multi-dimensional domainsor surfaces and for the integral equation of the first kind the domain where the equa-tion is required to be satisfied need not coincide with the integration domain.

We will regard the integral equations (1.1) and (1.2) as operator equations

Aϕ = f

andϕ − Aϕ = f

of the first and second kind, respectively, in appropriate normed function spaces.The symbol A : X → Y will mean a single-valued mapping whose domain of

definition is a set X and whose range is contained in a set Y, i.e., for every ϕ ∈ Xthe mapping A assigns a unique element Aϕ ∈ Y. The range A(X) is the set A(X) :={Aϕ : ϕ ∈ X} of all image elements. We will use the terms mapping, function, andoperator synonymously.

Existence and uniqueness of a solution to an operator equation can be equiva-lently expressed by the existence of the inverse operator. If for each f ∈ A(X) thereis only one element ϕ ∈ X with Aϕ = f , then A is said to be injective and to havean inverse A−1 : A(X)→ X defined by A−1 f := ϕ. The inverse mapping has domainA(X) and range X. It satisfies A−1A = I on X and AA−1 = I on A(X), where I de-notes the identity operator mapping each element into itself. If A(X) = Y, then themapping is said to be surjective. The mapping is called bijective if it is injective andsurjective, i.e., if the inverse mapping A−1 : Y → X exists.

In the first part of the book we will present the Riesz–Fredholm theory for com-pact operators which, in particular, answers the question of existence and uniquenessof solutions to integral equations of the second kind with sufficiently smooth ker-nels. In order to develop the theory, we will assume that the reader is familiar withthe elementary properties of linear spaces, normed spaces, and bounded linear oper-ators. For convenience and to introduce notations, in this chapter, we briefly recall afew basic concepts from the theory of normed spaces, omitting most of the proofs.For a more detailed study, see Aubin [12], Brezis [21], Heuser [94], Kantorovic andAkilov [116], Rudin [209], and Taylor [229] among others.

1.1 Abel’s Integral Equation

As an appetizer we consider Abel’s integral equation that occurred as one of thefirst integral equations in mathematical history. A tautochrone is a planar curve forwhich the time taken by an object sliding without friction in uniform gravity to itslowest point is independent of its starting point. The problem to identify this curvewas solved by Huygens in 1659 who, using geometrical tools, established that thetautochrone is a cycloid.

1.1 Abel’s Integral Equation 3

In 1823 Abel [1] attacked the more general problem of determining a planarcurve such that the time of descent for a given starting height y coincides with thevalue f (y) of a given function f . The tautochrone then reduces to the special casewhen f is a constant. Following Abel we describe the curve by x = ψ(y) (withψ(0) = 0) and, using the principle of conservation of energy, for the velocity v atheight 0 ≤ η ≤ y we obtain

12v2 + gη = gy

where g denotes the earth’s gravity. Therefore, denoting arc length by s, the totaltime f (y) required for the object to fall from P = (ψ(y), y) to P0 = (0, 0) is given by

f (y) =∫ P0

P

dsv=

∫ y

0

√1 + [ψ′(η)]2

2g(y − η)dη.

Form this, setting

ϕ :=

√1 + [ψ′]2

2g

we obtain

f (y) =∫ y

0

ϕ(y)√y − η dη, y > 0, (1.3)

which is known as Abel’s integral equation. Given the shape function ϕ, the fallingtime f is obtained by simply evaluating the integral on the right-hand side of (1.3).Conversely, given the function f , finding ϕ requires the solution of the integral equa-tion (1.3) which certainly is a more challenging task.

For any solution ϕ of (1.3) that is continuous on some interval (0, a] and satisfies|ϕ(y)| ≤ C/

√y for all y ∈ (0, a] and some constant C, by interchanging the order of

integration, we obtain that∫ z

0

f (y)√z − y dy =

∫ z

0

1√z − y

∫ y

0

ϕ(η)√y − η dη dy

=

∫ z

0ϕ(η)

∫ z

η

dy√(z − y)(y − η)

dη

= π

∫ z

0ϕ(η) dη.

Here, we substituted y = z − (z − η) cos2 t with the result

∫ z

η

dy√(z − y)(y − η)

= 2∫ π/2

0dt = π.


Therefore any solution of (1.3) has the form

ϕ(z) =1π

ddz

∫ z

0

f (y)√z − y dy, z ∈ (0, a]. (1.4)

Assuming that f is continuously differentiable on [0, a], by partial integration weobtain ∫ z

0

f (y)√z − y dy = 2

√z f (0) + 2

∫ z

0

√z − y f ′(y) dy

and this transforms (1.4) into

ϕ(z) =1π

{f (0)√

z+

∫ z

0

f ′(y)√z − y dy

}, z ∈ (0, a]. (1.5)

Inserting (1.5) in (1.3) (after renaming the variables) and interchanging the order ofintegration as above shows that (1.4) indeed is a solution of (1.3).

For the special case of a constant f = π√

a/2g with a > 0 one obtains from (1.5)that

ϕ(y) =√

a2gy

, 0 < y ≤ a. (1.6)

Note that the restriction y ≤ a is a consequence of 2g ϕ2 = 1+ [ψ′]2 ≥ 1. For the arclength s, we have

dsdy=

√1 + [ψ′]2 =

√2g ϕ

and from (1.6) it follows that

s(y) = 2√

ay, 0 ≤ y ≤ a.

For a convenient parameterization we set

y(t) =a2

(1 − cos t), 0 ≤ t ≤ π, (1.7)

and obtain firsts(t) = 2a sin

t2, 0 ≤ t ≤ π,

and then with the aid of some trigonometric identities

x(t) =a2

(t + sin t), 0 ≤ t ≤ π. (1.8)

Hence, the tautochrone as given by the parameterization (1.7) and (1.8) is the cycloidgenerated as the trajectory described by a point on the circle of radius a/2 when thecircle is rolling along the straight line y = a. This property of the cycloid, togetherwith the fact that the involute of a cycloid again is a cycloid, was exploited byHuygens to build a cycloidal pendulum for which the frequency of the oscillationsdoes not depend on the amplitude in contrast to the circular pendulum.

1.2 Convergence and Continuity 5

1.2 Convergence and Continuity

Definition 1.1. Let X be a complex (or real) linear space (vector space). A function‖ · ‖ : X → IR with the properties

(N1) ‖ϕ‖ ≥ 0, (positivity)

(N2) ‖ϕ‖ = 0 if and only if ϕ = 0, (definiteness)

(N3) ‖αϕ‖ = |α| ‖ϕ‖, (homogeneity)

(N4) ‖ϕ + ψ‖ ≤ ‖ϕ‖ + ‖ψ‖, (triangle inequality)

for all ϕ, ψ ∈ X, and all α ∈ C (or IR) is called a norm on X. A linear space Xequipped with a norm is called a normed space.

As a consequence of (N3) and (N4) we note the second triangle inequality

| ‖ϕ‖ − ‖ψ‖ | ≤ ‖ϕ − ψ‖. (1.9)

For two elements in a normed space ‖ϕ−ψ‖ is called the distance between ϕ and ψ.

Definition 1.2. A sequence (ϕn) of elements of a normed space X is called conver-gent if there exists an element ϕ ∈ X such that limn→∞ ‖ϕn − ϕ‖ = 0, i.e., if for everyε > 0 there exists an integer N(ε) such that ‖ϕn − ϕ‖ < ε for all n ≥ N(ε). Theelement ϕ is called the limit of the sequence (ϕn), and we write

limn→∞ ϕn = ϕ or ϕn → ϕ, n→ ∞.

Note that by (N4) the limit of a convergent sequence is uniquely determined. Asequence that does not converge is called divergent.

Definition 1.3. A function A : U ⊂ X → Y mapping a subset U of a normed spaceX into a normed space Y is called continuous at ϕ ∈ U if limn→∞ Aϕn = Aϕ for everysequence (ϕn) from U with limn→∞ ϕn = ϕ. The function A : U ⊂ X → Y is calledcontinuous if it is continuous for all ϕ ∈ U.

An equivalent definition is the following: A function A : U ⊂ X → Y is contin-uous at ϕ ∈ U if for every ε > 0 there exists δ > 0 such that ‖Aϕ − Aψ‖ < ε for allψ ∈ U with ‖ϕ − ψ‖ < δ. Here we have used the same symbol ‖ · ‖ for the normson X and Y. The function A is called uniformly continuous if for every ε > 0 thereexists δ > 0 such that ‖Aϕ − Aψ‖ < ε for all ϕ, ψ ∈ U with ‖ϕ − ψ‖ < δ.

Note that by (1.9) the norm is a continuous function.In our study of integral equations the basic example of a normed space will be

the linear space C[a, b] of continuous real- or complex-valued functions ϕ definedon an interval [a, b] ⊂ IR furnished either with the maximum norm

‖ϕ‖∞ := maxx∈[a,b]

|ϕ(x)|


or the mean square norm

‖ϕ‖2 :=

(∫ b

a|ϕ(x)|2dx

)1/2

.

Convergence of a sequence of continuous functions in the maximum norm is equiv-alent to uniform convergence, and convergence in the mean square norm is calledmean square convergence. Throughout this book, unless stated otherwise, we alwaysassume that C[a, b] (or C(G), i.e., the space of continuous real- or complex-valuedfunctions on compact subsets G ⊂ IRm) is equipped with the maximum norm.

Definition 1.4. Two norms on a linear space are called equivalent if they have thesame convergent sequences.

Theorem 1.5. Two norms ‖ · ‖a and ‖ · ‖b on a linear space X are equivalent if andonly if there exist positive numbers c and C such that

c‖ϕ‖a ≤ ‖ϕ‖b ≤ C‖ϕ‖afor all ϕ ∈ X. The limits with respect to the two norms coincide.

Proof. Provided that the conditions are satisfied, from ‖ϕn − ϕ‖a → 0, n → ∞, itfollows ‖ϕn − ϕ‖b → 0, n→ ∞, and vice versa.

Conversely, let the two norms be equivalent and assume that there is no C > 0such that ‖ϕ‖b ≤ C‖ϕ‖a for all ϕ ∈ X. Then there exists a sequence (ϕn) satisfying‖ϕn‖a = 1 and ‖ϕn‖b ≥ n2. Now, the sequence (ψn) with ψn := n−1ϕn converges tozero with respect to ‖ · ‖a, whereas with respect to ‖ · ‖b it is divergent because of‖ψn‖b ≥ n. �

Theorem 1.6. On a finite-dimensional linear space all norms are equivalent.

Proof. In a linear space X with finite dimension m and basis f1, . . . , fm every elementcan be expressed in the form

ϕ =

m∑k=1

αk fk.

As is easily verified,‖ϕ‖∞ := max

k=1,...,m|αk | (1.10)

defines a norm on X. Let ‖ · ‖ denote any other norm on X. Then, by the triangleinequality we have

‖ϕ‖ ≤ C‖ϕ‖∞for all ϕ ∈ X, where

C :=m∑

k=1

‖ fk‖.

Assume that there is no c > 0 such that c‖ϕ‖∞ ≤ ‖ϕ‖ for all ϕ ∈ X. Then thereexists a sequence (ϕn) with ‖ϕn‖ = 1 such that ‖ϕn‖∞ ≥ n. Consider the sequence

1.2 Convergence and Continuity 7

(ψn) with ψn := ‖ϕn‖−1∞ ϕn and write

ψn =

m∑k=1

αkn fk.

Because of ‖ψn‖∞ = 1 each of the sequences (αkn), k = 1, . . . ,m, is bounded in C.Hence, by the Bolzano–Weierstrass theorem we can select convergent subsequencesαk,n( j) → αk, j → ∞, for each k = 1, . . . ,m. This now implies ‖ψn( j) − ψ‖∞ → 0,j→ ∞, where

ψ :=m∑

k=1

αk fk,

and ‖ψn( j) − ψ‖ ≤ C‖ψn( j) − ψ‖∞ → 0, j → ∞. But on the other hand we have‖ψn‖ = 1/‖ϕn‖∞ → 0, n → ∞. Therefore, ψ = 0, and consequently ‖ψn( j)‖∞ → 0,j→ ∞, which contradicts ‖ψn‖∞ = 1 for all n. �

For an element ϕ of a normed space X and a positive number r the set B(ϕ; r) :={ψ ∈ X : ‖ψ − ϕ‖ < r} is called the open ball of radius r and center ϕ, the setB[ϕ; r] := {ψ ∈ X : ‖ψ − ϕ‖ ≤ r} is called a closed ball.

Definition 1.7. A subset U of a normed space X is called open if for each elementϕ ∈ U there exists r > 0 such that B(ϕ; r) ⊂ U.

Obviously, open balls are open.

Definition 1.8. A subset U of a normed space X is called closed if it contains alllimits of convergent sequences of U.

A subset U of a normed space X is closed if and only if its complement X \ U isopen. Obviously, closed balls are closed. In particular, using the norm (1.10), it canbe seen that finite-dimensional subspaces of a normed space are closed.

Definition 1.9. The closure U of a subset U of a normed space X is the set of alllimits of convergent sequences of U. A set U is called dense in another set V ifV ⊂ U, i.e., if each element in V is the limit of a convergent sequence from U.

A subset U is closed if and only if it coincides with its closure. By the Weierstrassapproximation theorem (see [40]) the linear subspace P of polynomials is dense inC[a, b] with respect to the maximum norm and the mean square norm.

Definition 1.10. A subset U of a normed space X is called bounded if there exists apositive number C such that ‖ϕ‖ ≤ C for all ϕ ∈ U.

Convergent sequences are bounded.


1.3 Completeness

Definition 1.11. A sequence (ϕn) of elements of a normed space is called a Cauchysequence if for each ε > 0 there exists an integer N(ε) such that ‖ϕn − ϕm‖ < ε forall n,m ≥ N(ε), i.e., if limn,m→∞ ‖ϕn − ϕm‖ = 0.

Every convergent sequence is a Cauchy sequence, whereas the converse in gen-eral is not true. This gives rise to the following definition.

Definition 1.12. A subset U of a normed space X is called complete if every Cauchysequence of elements of U converges to an element in U. A normed space X is calleda Banach space if it is complete.

Note that in a complete set we can decide on the convergence of a sequence with-out having to know its limit element. Complete sets are closed, and closed subsetsof a complete set are complete. Since the Cauchy criterion is sufficient for conver-gence of sequences of complex numbers, using the norm (1.10), we observe thatfinite-dimensional normed spaces are Banach spaces. The Cauchy criterion is alsosufficient for uniform convergence of a sequence of continuous functions toward acontinuous limit function. Therefore, the space C[a, b] is complete with respect tothe maximum norm. As can be seen from elementary counterexamples, C[a, b] isnot complete with respect to the mean square norm.

Each normed space can be completed in the following sense.

Theorem 1.13. For each normed space X there exists a Banach space X such that Xis isomorphic and isometric to a dense subspace of X, i.e., there is a linear bijectivemapping I from X into a dense subspace of X such that ‖Iϕ‖X = ‖ϕ‖X for all ϕ ∈ X.The space X is uniquely determined up to isometric isomorphisms, i.e., to any twosuch Banach spaces there exists a linear bijective mapping between the two spacesleaving the norms invariant.

For a proof of this concept of completion we refer to any introduction to func-tional analysis. Using Lebesgue integration theory, it can be seen that the completionof C[a, b] with respect to the mean square norm yields the complete space L2[a, b]of measurable and Lebesgue square-integrable functions, or to be more precise, ofequivalence classes of such functions that coincide almost everywhere with respectto the Lebesgue measure (see [12, 209, 229]).

1.4 Compactness

Definition 1.14. A subset U of a normed space X is called compact if every opencovering of U contains a finite subcovering, i.e., if for every family V j, j ∈ J, (forsome index set J) of open sets with the property

U ⊂⋃j∈J

V j

1.4 Compactness 9

there exists a finite subfamily V j(k), j(k) ∈ J, k = 1, . . . , n, such that

U ⊂n⋃

k=1

V j(k).

A subset U is called sequentially compact if every sequence of elements from Ucontains a subsequence that converges to an element in U.

A subset U of a normed space is called totally bounded if for each ε > 0 thereexists a finite number of elements ϕ1, . . . , ϕn in U such that

U ⊂n⋃

j=1

B(ϕ j; ε),

i.e., each element ϕ ∈ U has a distance less than ε from at least one of the el-ements ϕ1, . . . , ϕn. Note that each sequentially compact set U is totally bounded.Otherwise there would exist a positive ε and a sequence (ϕn) in U with the property‖ϕn − ϕm‖ ≥ ε for all n � m. This would imply that the sequence (ϕn) does notcontain a convergent subsequence, which contradicts the sequential compactness ofU. For each totally bounded set U, letting ε = 1/m, m = 1, 2, . . . , and collectingthe corresponding finite systems of elements ϕ1, . . . , ϕn depending on m, we obtaina sequence that is dense in U.

Theorem 1.15. A subset of a normed space is compact if and only if it is sequentiallycompact.

Proof. Let U be compact and assume that it is not sequentially compact. Then thereexists a sequence (ϕn) in U that does not contain a convergent subsequence withlimit in U. Consequently, for each ϕ ∈ U there exists an open ball B(ϕ; r) withcenter ϕ and radius r(ϕ) containing at most a finite number of the elements of thesequence (ϕn). The set of these balls clearly is an open covering of U, and since Uis compact, it follows that U contains only a finite number of the elements of thesequence (ϕn). This is a contradiction.

Conversely, let U be sequentially compact and let V j, j ∈ J, be an open coveringof U. First, we show that there exists a positive number ε such that for every ϕ ∈ Uthe ball B(ϕ; ε) is contained in at least one of the sets V j. Otherwise there wouldexist a sequence (ϕn) in U such that the ball B(ϕn; 1/n) is not contained in one ofthe V j. Since U is sequentially compact, this sequence (ϕn) contains a convergentsubsequence (ϕn(k)) with limit ϕ ∈ U. The element ϕ is contained in some V j, andsince V j is open, using the triangle inequality, we see that B(ϕn(k); 1/n(k)) ⊂ V j forsufficiently large k. This is a contradiction. Now, since the sequentially compact setU is totally bounded, there exists a finite number of elements ϕ1, . . . , ϕn in U suchthat the balls B(ϕk; ε), k = 1, . . . , n, cover U. But for each of these balls there ex-ists a set V j(k), j(k) ∈ J, such that B(ϕk; ε) ⊂ V j(k). Hence, the finite family V j(k),k = 1, . . . , n, covers U. �


In particular, from Theorem 1.15 we observe that compact sets are bounded,closed, and complete.

Definition 1.16. A subset of a normed space is called relatively compact if its clo-sure is compact.

As a consequence of Theorem 1.15, a set U is relatively compact if and onlyif each sequence of elements from U contains a convergent subsequence. Hence,analogous to compact sets, relatively compact sets are totally bounded.

Theorem 1.17. A bounded and finite-dimensional subset of a normed space is rela-tively compact.

Proof. This follows from the Bolzano–Weierstrass theorem using the norm (1.10). �

For the compactness in C(G), the space of continuous real- or complex-valuedfunctions defined on a compact set G ⊂ IRm, furnished with the maximum norm

‖ϕ‖∞ := maxx∈G|ϕ(x)|,

we have the following criterion.

Theorem 1.18 (Arzela–Ascoli). A set U ⊂ C(G) is relatively compact if and onlyif it is bounded and equicontinuous, i.e., if there exists a constant C such that

|ϕ(x)| ≤ C

for all x ∈ G and all ϕ ∈ U, and for every ε > 0 there exists δ > 0 such that

|ϕ(x) − ϕ(y)| < εfor all x, y ∈ G with |x − y| < δ and all ϕ ∈ U. (Here and henceforth by |z| :=(z2

1 + · · · + z2m)1/2 we denote the Euclidean norm of a vector z = (z1, . . . , zm) ∈ IRm.)

Proof. Let U be bounded and equicontinuous and let (ϕn) be a sequence in U. Wechoose a sequence (xi) from the compact and, consequently, totally bounded setG that is dense in G. Since the sequence (ϕn(xi)) is bounded in C for each xi, bythe standard diagonalization procedure we can choose a subsequence (ϕn(k)) suchthat (ϕn(k)(xi)) converges in C as k → ∞ for each xi. More precisely, since (ϕn(x1))is bounded, we can choose a subsequence (ϕn1(k)) such that (ϕn1(k)(x1)) convergesas k → ∞. The sequence (ϕn1(k)(x2)) again is bounded and we can choose a sub-sequence (ϕn2(k)) of (ϕn1(k)) such that (ϕn2(k)(x2)) converges as k → ∞. Repeatingthis process of selecting subsequences, we arrive at a double array (ϕni(k)) such thateach row (ϕni(k)) is a subsequence of the previous row (ϕni−1(k)) and each sequence(ϕni(k)(xi)) converges as k → ∞. For the diagonal sequence ϕn(k) := ϕnk(k) we havethat (ϕn(k)(xi)) converges as k → ∞ for all xi.

Since the set (xi), i = 1, 2, . . . , is dense in G, given ε > 0, the balls B(xi; δ),i = 1, 2, . . . , cover G. Since G is compact we can choose i ∈ IN such that each point

1.5 Scalar Products 11

x ∈ G has a distance less than δ from at least one element x j of the set x1, . . . , xi.Next choose N(ε) ∈ IN such that

|ϕn(k)(x j) − ϕn(l)(x j)| < εfor all k, l ≥ N(ε) and all j = 1, . . . , i. From the equicontinuity we obtain

|ϕn(k)(x) − ϕn(l)(x)| ≤ |ϕn(k)(x) − ϕn(k)(x j)| + |ϕn(k)(x j) − ϕn(l)(x j)|

+|ϕn(l)(x j) − ϕn(l)(x)| < 3ε

for all k, l ≥ N(ε) and all x ∈ G. This establishes the uniform convergence of thesubsequence (ϕn(k)), i.e., convergence in the maximum norm. Hence U is relativelycompact.

Conversely, let U be relatively compact. Then U is totally bounded, i.e., givenε > 0 there exist functions ϕ1, . . . , ϕi ∈ U such that

minj=1,...,i

‖ϕ − ϕ j‖∞ < ε

3

for all ϕ ∈ U. Since each of the ϕ1, . . . , ϕi is uniformly continuous on the compactset G, there exists δ > 0 such that

|ϕ j(x) − ϕ j(y)| < ε

3

for all x, y ∈ G with |x − y| < δ and all j = 1, . . . , i. Then for all ϕ ∈ U, choosing j0such that

‖ϕ − ϕ j0‖∞ = minj=1,...,i

‖ϕ − ϕ j‖∞,

we obtain

|ϕ(x) − ϕ(y)| ≤ |ϕ(x) − ϕ j0 (x)| + |ϕ j0 (x) − ϕ j0 (y)| + |ϕ j0 (y) − ϕ(y)| < εfor all x, y ∈ G with |x − y| < δ. Therefore U is equicontinuous. Finally, U isbounded, since relatively compact sets are bounded. �

1.5 Scalar Products

We now consider the important special case where the norm is given in terms of ascalar product.


Definition 1.19. Let X be a complex (or real) linear space. Then a function (· , ·) :X × X → C (or IR) with the properties

(H1) (ϕ, ϕ) ≥ 0, (positivity)

(H2) (ϕ, ϕ) = 0 if and only if ϕ = 0, (definiteness)

(H3) (ϕ, ψ) = (ψ, ϕ), (symmetry)

(H4) (αϕ + βψ, χ) = α(ϕ, χ) + β(ψ, χ), (linearity)

for all ϕ, ψ, χ ∈ X, and α, β ∈ C (or IR) is called a scalar product, or an inner product,on X. (By the bar we denote the complex conjugate.)

As a consequence of (H3) and (H4) we note the antilinearity

(ϕ, αψ + βχ) = α(ϕ, ψ) + β(ϕ, χ). (1.11)

Theorem 1.20. A scalar product satisfies the Cauchy–Schwarz inequality

|(ϕ, ψ)|2 ≤ (ϕ, ϕ)(ψ, ψ)

for all ϕ, ψ ∈ X, with equality if and only if ϕ and ψ are linearly dependent.

Proof. The inequality is trivial for ϕ = 0. For ϕ � 0 it follows from

(αϕ + βψ, αϕ + βψ) = |α|2(ϕ, ϕ) + 2 Re{αβ(ϕ, ψ)

}+ |β|2(ψ, ψ)

= (ϕ, ϕ)(ψ, ψ) − |(ϕ, ψ)|2,

where we have set α = −(ϕ, ϕ)−1/2 (ϕ, ψ) and β = (ϕ, ϕ)1/2. Since (· , ·) is positivedefinite, this expression is nonnegative and is equal to zero if and only if αϕ+βψ = 0.In the latter case ϕ and ψ are linearly dependent because β � 0. �

Theorem 1.21. A scalar product (· , ·) on a linear space X defines a norm by

‖ϕ‖ := (ϕ, ϕ)1/2

for all ϕ ∈ X. If X is complete with respect to this norm it is called a Hilbert space;otherwise it is called a pre-Hilbert space.

Proof. We leave it as an exercise to verify the norm axioms. The triangle inequalityfollows from the Cauchy–Schwarz inequality. �

Note that L2[a, b] is a Hilbert space with the scalar product given by

(ϕ, ψ) :=∫ b

aϕ(x)ψ(x) dx.

1.6 Best Approximation 13

Definition 1.22. Two elements ϕ and ψ of a pre-Hilbert space X are called orthogo-nal if

(ϕ, ψ) = 0.

Two subsets U and V of X are called orthogonal if each pair of elements ϕ ∈ U andψ ∈ V are orthogonal. For two orthogonal elements or subsets we write ϕ ⊥ ψ andU ⊥ V, respectively. A subset U of X is called an orthogonal system if (ϕ, ψ) = 0 forall ϕ, ψ ∈ U with ϕ � ψ. An orthogonal system U is called an orthonormal systemif ‖ϕ‖ = 1 for all ϕ ∈ U. The set

U⊥ := {ψ ∈ X : ψ ⊥ U}is called the orthogonal complement of the subset U.

1.6 Best Approximation

Definition 1.23. Let U ⊂ X be a subset of a normed space X and let ϕ ∈ X. Anelement v ∈ U is called a best approximation to ϕ with respect to U if

‖ϕ − v‖ = infu∈U ‖ϕ − u‖,

i.e., v ∈ U has smallest distance from ϕ.

Theorem 1.24. Let U be a finite-dimensional subspace of a normed space X. Thenfor every element of X there exists a best approximation with respect to U.

Proof. Let ϕ ∈ X and choose a minimizing sequence (un) for ϕ, i.e., un ∈ U satisfies

‖ϕ − un‖ → d := infu∈U ‖ϕ − u‖, n→ ∞.

Because of ‖un‖ ≤ ‖ϕ − un‖ + ‖ϕ‖, the sequence (un) is bounded. Since U has finitedimension, it is closed and by Theorem 1.17 the sequence (un) contains a convergentsubsequence (un(k)) with limit v ∈ U. Then

‖ϕ − v‖ = limk→∞‖ϕ − un(k)‖ = d

completes the proof. �

Theorem 1.25. Let U be a linear subspace of a pre-Hilbert space X. An element vis a best approximation to ϕ ∈ X with respect to U if and only if

(ϕ − v, u) = 0

for all u ∈ U, i.e., if and only if ϕ − v ⊥ U. To each ϕ ∈ X there exists at most onebest approximation with respect to U.


Proof. This follows from the equality

‖(ϕ − v) + αu‖2 = ‖ϕ − v‖2 + 2αRe(ϕ − v, u) + α2‖u‖2,which is valid for all v, u ∈ U and all α ∈ IR. �

Theorem 1.26. Let U be a complete linear subspace of a pre-Hilbert space X. Thento every element of X there exists a unique best approximation with respect to U.

Proof. Let ϕ ∈ X and choose a sequence (un) with

‖ϕ − un‖2 ≤ d2 +1n, n ∈ IN, (1.12)

where d := infu∈U ‖ϕ − u‖. Then

‖(ϕ − un) + (ϕ − um)‖2 + ‖un − um‖2 = 2‖ϕ − un‖2 + 2‖ϕ − um‖2 ≤ 4d2 +2n+

2m

for all n,m ∈ IN, and since12

(un + um) ∈ U, it follows that

‖un − um‖2 ≤ 4d2 +2n+

2m− 4

∥∥∥∥∥ϕ − 12

(un + um)∥∥∥∥∥

2

≤ 2n+

2m.

Hence, (un) is a Cauchy sequence, and because U is complete, there exists an ele-ment v ∈ U such that un → v, n → ∞. Passing to the limit n → ∞ in (1.12) showsthat v is a best approximation of ϕ with respect to U. �

We wish to extend Theorem 1.25 to the case of convex subsets of a pre-Hilbertspace. A subset U of a linear space X is called convex if

λϕ1 + (1 − λ)ϕ2 ∈ U

for all ϕ1, ϕ2 ∈ U and all λ ∈ (0, 1).

Theorem 1.27. Let U be a convex subset of a pre-Hilbert space X. An element v ∈ Uis a best approximation to ϕ ∈ X with respect to U if and only if

Re(ϕ − v, u − v) ≤ 0

for all u ∈ U. To each ϕ ∈ X there exists at most one best approximation with respectto U.

Proof. This follows from the equality

‖ϕ − [(1 − λ)v + λu]‖2 = ‖ϕ − v‖2 − 2λRe(ϕ − v, u − v) + λ2‖u − v‖2,which is valid for all u, v ∈ U and all λ ∈ (0, 1). �

1.6 Best Approximation 15

It is left to the reader to carry Theorem 1.26 over from the case of a linear sub-space to the case of a convex subset.

For a subset U of a linear space X we denote the set spanned by all linear com-binations of elements of U by span U.

Theorem 1.28. Let {un : n ∈ IN} be an orthonormal system in a pre-Hilbert spaceX. Then the following properties are equivalent:

(a) span{un : n ∈ IN} is dense in X.(b) Each ϕ ∈ X can be expanded in a Fourier series

ϕ =

∞∑n=1

(ϕ, un)un.

(c) For each ϕ ∈ X we have Parseval’s equality

‖ϕ‖2 =∞∑

n=1

|(ϕ, un)|2.

The properties (a)–(c) imply that

(d) ϕ = 0 is the only element in X with (ϕ, un) = 0 for all n ∈ IN,

and (a), (b), (c), and (d) are equivalent if X is a Hilbert space. An orthonormalsystem with the property (a) is called complete.

Proof. What has to be understood by a series is explained in Problem 1.5.(a)⇒ (b): By Theorems 1.25 and 1.26, the partial sum

ϕn =

n∑k=1

(ϕ, uk)uk

is the best approximation to ϕ with respect to span{u1, . . . , un}. Since by assumptionspan{un : n ∈ IN} is dense in X, the sequence of the best approximations converges,i.e., ϕn → ϕ, n→ ∞.(b)⇒ (c): This follows by taking the scalar product of the Fourier series with ϕ.(c)⇒ (a): This follows from

∥∥∥∥∥∥∥ϕ −n∑

k=1

(ϕ, uk)uk

∥∥∥∥∥∥∥2

= ‖ϕ‖2 −n∑

k=1

|(ϕ, uk)|2.

(c)⇒ (d): This is trivial.(d)⇒ (a): In the case of a Hilbert space X, we set U := span{un : n ∈ IN} and assumethat X � U. Then there exists ϕ ∈ X with ϕ � U. Since X is complete, U is alsocomplete. Therefore, by Theorems 1.25 and 1.26 the best approximation v to ϕ withrespect to U exists and satisfies (v − ϕ, un) = 0 for all n ∈ IN. Hence, we have thecontradiction ϕ = v ∈ U. �


Problems

1.1. Show that finite-dimensional subspaces of normed spaces are closed and complete.

1.2. A norm ‖ · ‖a on a linear space X is called stronger than a norm ‖ · ‖b if every sequenceconverging with respect to the norm ‖ · ‖a also converges with respect to the norm ‖ · ‖b. The samefact is also expressed by saying that the norm ‖ · ‖b is weaker than the norm ‖ · ‖a. Show that ‖ · ‖ais stronger than ‖ · ‖b if and only if there exists a positive number C such that ‖ϕb‖ ≤ C‖ϕ‖a for allϕ ∈ X. Show that on C[a, b] the maximum norm is stronger than the mean square norm. Constructa counterexample to demonstrate that these two norms are not equivalent.

1.3. Show that a continuous real-valued function on a compact subset of a normed space assumesits supremum and its infimum and that it is uniformly continuous.

1.4. Construct a counterexample to demonstrate that C[a, b] is not complete with respect to themean square norm.

1.5. Let (ϕn) be a sequence of elements of a normed space X. The series

∞∑k=1

ϕk

is called convergent if the sequence (Sn) of partial sums

Sn :=n∑

k=1

ϕk

converges. The limit S = limn→∞ Sn is called the sum of the series. Show that in a Banach space Xthe convergence of the series

∞∑k=1

‖ϕk‖

is a sufficient condition for the convergence of the series∑∞

k=1 ϕk and that∥∥∥∥∥∥∥∞∑

k=1

ϕk

∥∥∥∥∥∥∥ ≤∞∑

k=1

‖ϕk‖.

Hint: Show that (Sn) is a Cauchy sequence.

Chapter 2Bounded and Compact Operators

In this chapter we briefly review the basic properties of bounded linear operatorsin normed spaces and then introduce the concept of compact operators that is offundamental importance in the study of integral equations.

2.1 Bounded Operators

Recall that an operator A : X → Y mapping a linear space X into a linear space Y iscalled linear if

A(αϕ + βψ) = αAϕ + βAψ

for all ϕ, ψ ∈ X and all α, β ∈ C (or IR).

Definition 2.1. A linear operator A : X → Y from a normed space X into a normedspace Y is called bounded if there exists a positive number C such that

‖Aϕ‖ ≤ C‖ϕ‖for all ϕ ∈ X. Each number C for which this inequality holds is called a bound forthe operator A. (Again we use the same symbol ‖ · ‖ for the norms on X and Y.)

Theorem 2.2. A linear operator A : X → Y from a normed space X into a normedspace Y is bounded if and only if

‖A‖ := sup‖ϕ‖=1‖Aϕ‖ < ∞. (2.1)

The number ‖A‖ is the smallest bound for A. It is called the norm of A and can alsobe expressed by

‖A‖ = sup‖ϕ‖≤1‖Aϕ‖.


17

18 2 Bounded and Compact Operators

Proof. For a bounded linear operator A with bound C we have sup‖ϕ‖=1 ‖Aϕ‖ ≤ C.Conversely, for all ψ ∈ X we can estimate

‖Aψ‖ = ‖ψ‖∥∥∥∥∥∥A

(1‖ψ‖ ψ

)∥∥∥∥∥∥ ≤ sup‖ϕ‖=1‖Aϕ‖ ‖ψ‖

i.e., A has the bound sup‖ϕ‖=1 ‖Aϕ‖. The equality sup‖ϕ‖=1 ‖Aϕ‖ = sup‖ϕ‖≤1 ‖Aϕ‖ alsois a consequence of the last inequality. �

Hence a linear operator is bounded if and only if it maps bounded sets in Xinto bounded sets in Y. Recall the Definition 1.3 for the continuity of an operatorA : X → Y mapping a normed space X into a normed space Y.

Theorem 2.3. For a linear operator A : X → Y mapping a normed space X into anormed space Y the following properties are equivalent:

1. A is continuous at one element.2. A is continuous.3. A is bounded.

Proof. 1. ⇒ 2.: Let A be continuous at ϕ0 ∈ X. Then for every ϕ ∈ X and everysequence (ϕn) with ϕn → ϕ, n→ ∞, we have

Aϕn = A(ϕn − ϕ + ϕ0) + A(ϕ − ϕ0)→ A(ϕ0) + A(ϕ − ϕ0) = A(ϕ), n→ ∞,since ϕn − ϕ + ϕ0 → ϕ0, n→ ∞. Therefore, A is continuous at all ϕ ∈ X.2. ⇒ 3.: Let A be continuous and assume there is no C > 0 such that ‖Aϕ‖ ≤ C‖ϕ‖for all ϕ ∈ X. Then there exists a sequence (ϕn) in X with ‖ϕn‖ = 1 and ‖Aϕn‖ ≥ n.Consider the sequence ψn := ‖Aϕn‖−1ϕn. Then ψn → 0, n → ∞, and since A iscontinuous Aψn → A(0) = 0, n→ ∞. This is a contradiction to ‖Aψn‖ = 1 for all n.3. ⇒ 1.: Let A be bounded and let (ϕn) be a sequence in X with ϕn → 0, n → ∞.Then from ‖Aϕn‖ ≤ C‖ϕn‖ it follows that Aϕn → 0, n → ∞. Thus, A is continuousat ϕ = 0. �

Theorem 2.4. Each linear operator A : X → Y from a finite-dimensional normedspace X into a normed space Y is bounded.

Proof. The mapping ϕ → ‖Aϕ‖ + ‖ϕ‖ is a norm on X. By Theorem 1.6 on the finite-dimensional linear space X all norms are equivalent, i.e., there exists a C > 0 suchthat ‖Aϕ‖ ≤ ‖Aϕ‖ + ‖ϕ‖ ≤ C‖ϕ‖ for all ϕ ∈ X. �

Remark 2.5. Let X, Y, and Z be normed spaces and let A : X → Y and B : Y → Zbe bounded linear operators. Then the product BA : X → Z, defined by (BA)ϕ :=B(Aϕ) for all ϕ ∈ X, is a bounded linear operator with ‖BA‖ ≤ ‖A‖ ‖B‖.Proof. This follows from ‖(BA)ϕ‖ = ‖B(Aϕ)‖ ≤ ‖B‖ ‖A‖ ‖ϕ‖. �

Every linear combination of bounded linear operators again is a bounded linearoperator, i.e., the set L(X, Y) of bounded linear operators from X into Y forms alinear space.

2.1 Bounded Operators 19

Theorem 2.6. The linear space L(X, Y) of bounded linear operators from a normedspace X into a normed space Y is a normed space with the norm (2.1). If Y is aBanach space then L(X, Y) also is a Banach space.

Proof. The proof consists in carrying over the norm axioms and the completenessfrom Y onto L(X, Y). For the second part, let (An) be a Cauchy sequence in L(X, Y),i.e., ‖Am−An‖ → 0, m, n→ ∞. Then for each ϕ ∈ X the sequence (Anϕ) is a Cauchysequence in Y and converges, since Y is complete. Then Aϕ := limn→∞ Anϕ definesa bounded linear operator A : X → Y, which is the limit of the sequence (An), i.e.,‖An − A‖ → 0, n→ ∞. �

Definition 2.7. Let X and Y be normed spaces. A sequence (An) of operators fromL(X, Y) is called norm convergent to an operator A ∈ L(X, Y) if An → A, n → ∞, inthe norm of L(X, Y), i.e., if ‖An − A‖ → 0, n→ ∞. It is called pointwise convergentto an operator A ∈ L(X, Y) if Anϕ→ Aϕ, n→ ∞, for every ϕ ∈ X.

Theorem 2.8. Norm convergence of bounded linear operators implies pointwiseconvergence, but the converse is not true.

Proof. The first statement is evident from ‖Anϕ − Aϕ‖ ≤ ‖An − A‖ ‖ϕ‖ for all ϕ ∈ X.That the converse is not true follows from the following counterexample with X =C[0, 1] and Y = IR. For the bounded linear operators given by

Qϕ :=∫ 1

0ϕ(x) dx and Qnϕ :=

1n

n∑k=1

ϕ

(kn

)

the definition of the Riemann integral implies pointwise convergence Qnϕ → Qϕ,n → ∞, for all ϕ ∈ C[0, 1] . For 0 < ε < 1/2n we choose a function ψε ∈ C[0, 1]with 0 ≤ ψε(x) ≤ 1 for all x ∈ [0, 1] such that ψε(x) = 1 if mink=1,...,n |x− xk | ≥ ε andψε(xk) = 0, k = 1, . . . , n. Then we have

‖Qn − Q‖∞ = sup‖ϕ‖∞=1

|(Qn − Q)ϕ| ≥ |(Qn − Q)ψε| = |Qψε| ≥ 1 − 2nε.

Passing to the limit ε→ 0 it follows that ‖Qn − Q‖∞ ≥ 1 for all n, i.e., the sequence(Qn) is not norm convergent. �

Definition 2.9. A bijective bounded linear operator A : X → Y from a normed spaceX onto a normed space Y is called an isomorphism if its inverse A−1 : Y → X isbounded. Two normed spaces are called isomorphic if there exists an isomorphismbetween them.

Two isomorphic normed spaces do not differ in their linear and topological struc-ture. In particular, they have the same convergent sequences. Equivalent norms cre-ate isomorphic spaces.


2.2 The Dual Space

It is an important principle of functional analysis to connect the investigation of anormed space X with that of its dual space defined as the space of bounded linearoperators X∗ := L(X,C) (or X∗ := L(X, IR)). The elements of X∗ are called boundedlinear functionals on X. By Theorem 2.6 the dual space of a normed space X is aBanach space with the norm

‖F‖ := sup‖ϕ‖=1|F(ϕ)|

for F ∈ X∗.The following extension theorem ensures the existence of nontrivial bounded

linear functionals for each normed space.

Theorem 2.10 (Hahn–Banach). Let U be a subspace of a normed space X and Fa bounded linear functional on U. Then there exists a bounded linear functional Gon X with the properties

G(ϕ) = F(ϕ), ϕ ∈ U,

and‖G‖ = ‖F‖.

Proof. We first consider a real normed space X. Assume that X is spanned by U andan element ϕ0 ∈ X with ϕ0 � U, that is,

X = span{U, ϕ0} = {ϕ + αϕ0 : ϕ ∈ U, α ∈ IR}.If for any real number γ we define G : X → IR by

G(ϕ + αϕ0) := F(ϕ) + γα, ϕ ∈ U, α ∈ IR,

then G is a linear functional on X that is an extension of F, that is, G(ϕ) = F(ϕ)for all ϕ ∈ U. We will now show how to choose γ such that G is bounded with‖G‖ ≤ ‖F‖. Then also ‖G‖ = ‖F‖ is satisfied.

For all pairs ψ, χ ∈ U we can estimate

F(χ) − F(ψ) = F(χ − ψ) ≤ ‖F‖ ‖χ − ψ‖ ≤ ‖F‖ (‖χ + ϕ0‖ + ‖ψ + ϕ0‖)whence

−F(ψ) − ‖F‖ ‖ψ + ϕ0‖ ≤ −F(χ) + ‖F‖ ‖χ + ϕ0‖follows. Therefore there exists γ ∈ IR with the property

supψ∈U

[−F(ψ) − ‖F‖ ‖ψ + ϕ0‖ ] ≤ γ ≤ infχ∈U

[−F(χ) + ‖F‖ ‖χ + ϕ0‖ ] . (2.2)

Now if α > 0 we choose χ = ϕ/α and, multiplying by α, we obtain from (2.2) that

G(ϕ + αϕ0) = F(ϕ) + γα ≤ ‖F‖ ‖ϕ + αϕ0‖.

2.2 The Dual Space 21

If α < 0 we choose ψ = ϕ/α and obtain from (2.2) that again

G(ϕ + αϕ0) = F(ϕ) + γα ≤ ‖F‖ ‖ϕ + αϕ0‖.Hence G(ϕ) ≤ ‖F‖ ‖ϕ‖ for all ϕ ∈ X and replacing ϕ by −ϕ we also have that−G(ϕ) ≤ ‖F‖ ‖ϕ‖ for ϕ ∈ X. Consequently ‖G‖ ≤ ‖F‖.

Now we consider the family of bounded linear functionals G on subspaces of Xthat are extensions of F such that the norms of G and F coincide. We make thisfamily into a partially ordered family be defining G1 � G2 to mean that G1 is anextension of G2. Then Zorn’s lemma (among others, see [40]) ensures the existenceof a maximal extension G of F. We have to show that the domain of definition of Gcoincides with X itself. Assume to the contrary that the domain V of G is a propersubspace of X. Then, taking V as U in the above, we obtain an extension of F in asubspace of X that contains V as a proper subspace which contradicts the maximalityof G.

Finally, we can consider a complex normed space X also as a real linear space inwhich for each ϕ ∈ X the elements ϕ and iϕ are linearly independent. If we define

Fr(ϕ) := Re F(ϕ), ϕ ∈ U,

then Fr : U → IR is a real bounded linear functional satisfying ‖Fr‖ ≤ ‖F‖ and

F(ϕ) = Fr(ϕ) − iFr(iϕ) (2.3)

for all ϕ ∈ U. Hence, there exists an extension Gr : X → IR of Fr with ‖Gr‖ = ‖Fr‖.We set

G(ϕ) := Gr(ϕ) − iGr(iϕ), ϕ ∈ X.

Then G : X → C is linear since

G(iϕ) = Gr(iϕ) − iGr(−ϕ) = i[Gr(ϕ) − iGr(iϕ)] = iG(ϕ)

for all ϕ ∈ X and in view of (2.3) it is an extension of F. Further, abbreviatingθ = arg(G(ϕ)), we have the inequality

|G(ϕ)| = e−iθ G(ϕ) = G(e−iθ ϕ) = Gr(e−iθ ϕ) ≤ ‖Gr‖ ‖e−iθ ϕ‖ = ‖Gr‖ ‖ϕ‖ ≤ ‖F‖ ‖ϕ‖for all ϕ ∈ X whence ‖G‖ ≤ ‖F‖ follows. This completes the proof. �

We finish the section with two important corollaries of the Hahn–Banach exten-sion theorem.

Corollary 2.11. For each element ϕ0 � 0 in a normed space X there exists abounded linear functional G with the properties

G(ϕ0) = ‖ϕ0‖ and ‖G‖ = 1.

Proof. Apply the Hahn–Banach theorem to U = span{ϕ0} and F(αϕ0) := α ‖ϕ0‖. �


Corollary 2.12. Each element ϕ of a normed space X satisfies

‖ϕ‖ = supF∈X∗ , ‖F‖=1

|F(ϕ)|.

Proof. Because of |F(ϕ)| ≤ ‖F‖ ‖ϕ‖ clearly sup‖F‖=1 |F(ϕ)| ≤ ‖ϕ‖. For ϕ � 0, byCorollary 2.11, choose a functional F0 such that F0(ϕ) = ‖ϕ‖ and ‖F0‖ = 1. Thensup‖F‖=1 |F(ϕ)| ≥ |F0(ϕ)| = ‖ϕ‖. �

2.3 Integral Operators

Now we want to introduce integral operators. We assume that the reader is familiarwith the Riemann integral for real- and complex-valued functions in IRm. A set G inIRm is called Jordan measurable if the characteristic function χG, given by χG(x) = 1for x ∈ G and χG(x) = 0 for x � G, is Riemann integrable. The Jordan measure |G| isthe integral of χG. For each Jordan measurable set G the closure G and the boundary∂G also are Jordan measurable with |G| = |G| and |∂G| = 0. If G is compact andJordan measurable, then each function f ∈ C(G) is Riemann integrable. In addition,we assume that G is the closure of an open set or, equivalently, that G coincides withthe closure of its interior. This ensures that each nonnegative function f ∈ C(G)satisfying

∫G

f (x) dx = 0 must vanish identically, i.e., f (x) = 0 for all x ∈ G.

Theorem 2.13. Let G ⊂ IRm be a nonempty compact and Jordan measurable setthat coincides with the closure of its interior. Let K : G × G → C be a continuousfunction. Then the linear operator A : C(G)→ C(G) defined by

(Aϕ)(x) :=∫

GK(x, y)ϕ(y) dy, x ∈ G, (2.4)

is called an integral operator with continuous kernel K. It is a bounded linear oper-ator with

‖A‖∞ = maxx∈G

∫G|K(x, y)| dy.

Proof. Clearly (2.4) defines a linear operator A : C(G) → C(G). For each ϕ ∈ C(G)with ‖ϕ‖∞ ≤ 1 we have

|(Aϕ)(x)| ≤∫

G|K(x, y)| dy, x ∈ G,

and thus

‖A‖∞ = sup‖ϕ‖∞≤1

‖Aϕ‖∞ ≤ maxx∈G

∫G|K(x, y)| dy.

Since K is continuous, there exists x0 ∈ G such that∫

G|K(x0, y)| dy = max

x∈G

∫G|K(x, y)| dy.

2.4 Neumann Series 23

For ε > 0 choose ψ ∈ C(G) by setting

ψ(y) :=K(x0, y)|K(x0, y)| + ε , y ∈ G.

Then ‖ψ‖∞ ≤ 1 and

‖Aψ‖∞ ≥ |(Aψ)(x0)| =∫

G

|K(x0, y)|2|K(x0, y)| + ε dy

≥∫

G

|K(x0, y)|2 − ε2

|K(x0, y)| + ε dy =∫

G|K(x0, y)| dy − ε|G|.

Hence

‖A‖∞ = sup‖ϕ‖∞≤1

‖Aϕ‖∞ ≥ ‖Aψ‖∞ ≥∫

G|K(x0, y)| dy − ε|G|,

and because this holds for all ε > 0, we have

‖A‖∞ ≥∫

G|K(x0, y)| dy = max

x∈G

∫G|K(x, y)| dy.

This concludes the proof. �

Theorem 2.13 can be extended to the integral operator A : C(G) → C(M)given by

(Aϕ)(x) :=∫

GK(x, y)ϕ(y) dy, x ∈ M, (2.5)

where K : M×G → C is continuous, M ⊂ IRn is a compact set and n can be differentfrom m.

2.4 Neumann Series

For operator equations of the second kind

ϕ − Aϕ = f

existence and uniqueness of a solution can be established by the Neumann seriesprovided that A is a contraction, i.e., ‖A‖ < 1.

Theorem 2.14. Let A : X → X be a bounded linear operator on a Banach spaceX with ‖A‖ < 1 and let I : X → X denote the identity operator. Then I − A has abounded inverse on X that is given by the Neumann series

(I − A)−1 =

∞∑k=0

Ak


and satisfies

‖(I − A)−1‖ ≤ 11 − ‖A‖ .

(The iterated operators Ak are defined recursively by A0 := I and Ak := AAk−1 fork ∈ IN.)

Proof. Since ‖A‖ < 1, in view of Remark 2.5, we have absolute convergence

∞∑k=0

‖Ak‖ ≤∞∑

k=0

‖A‖k = 11 − ‖A‖

in the Banach space L(X, X), and therefore, by Problem 1.5, the Neumann seriesconverges in the operator norm and defines a bounded linear operator

S :=∞∑

k=0

Ak

with

‖S ‖ ≤ 11 − ‖A‖ .

The operator S is the inverse of I − A, as can be seen from

(I − A)S = (I − A) limn→∞

n∑k=0

Ak = limn→∞(I − An+1) = I

and

S (I − A) = limn→∞

n∑k=0

Ak(I − A) = limn→∞(I − An+1) = I,

since ‖An+1‖ ≤ ‖A‖n+1 → 0, n→ ∞. �

Obviously, the partial sums

ϕn :=n∑

k=0

Ak f

of the Neumann series satisfy ϕn+1 = Aϕn + f for n ≥ 0. Hence, the Neumann seriesis related to successive approximations by the following theorem.

Theorem 2.15. Under the assumptions of Theorem 2.14 for each f ∈ X the succes-sive approximations

ϕn+1 := Aϕn + f , n = 0, 1, 2, . . . , (2.6)

with arbitrary ϕ0 ∈ X converge to the unique solution ϕ of ϕ − Aϕ = f .

2.5 Compact Operators 25

Proof. By induction it can be seen that

ϕn = Anϕ0 +

n−1∑k=0

Ak f , n = 1, 2, . . . ,

whence

limn→∞ϕn =

∞∑k=0

Ak f = (I − A)−1 f

follows. �

Corollary 2.16. Let K be a continuous kernel satisfying

maxx∈G

∫G|K(x, y)| dy < 1.

Then for each f ∈ C(G) the integral equation of the second kind

ϕ(x) −∫

GK(x, y)ϕ(y) dy = f (x), x ∈ G,

has a unique solution ϕ ∈ C(G). The successive approximations

ϕn+1(x) :=∫

GK(x, y)ϕn(y) dy + f (x), n = 0, 1, 2, . . . ,

with arbitrary ϕ0 ∈ C(G) converge uniformly to this solution.

The method of successive approximations has two drawbacks. First, the Neu-mann series ensures existence of solutions to integral equations of the second kindonly for sufficiently small kernels, and second, in general, it cannot be summed inclosed form. Later in the book we will have more to say about using successiveapproximations to obtain approximate solutions (see Section 10.5).

2.5 Compact Operators

To provide the tools for establishing the existence of solutions to a wider class ofintegral equations we now turn to the introduction and investigation of compactoperators.

Definition 2.17. A linear operator A : X → Y from a normed space X into a normedspace Y is called compact if it maps each bounded set in X into a relatively compactset in Y.

Since by Definition 1.16 and Theorem 1.15 a subset U of a normed space Y isrelatively compact if each sequence in U contains a subsequence that converges inY, we have the following equivalent condition for an operator to be compact.


Theorem 2.18. A linear operator A : X → Y is compact if and only if for eachbounded sequence (ϕn) in X the sequence (Aϕn) contains a convergent subsequencein Y.

We proceed by establishing the basic properties of compact operators.

Theorem 2.19. Compact linear operators are bounded.

Proof. This is obvious, since relatively compact sets are bounded (see Theorem1.15). �

Theorem 2.20. Linear combinations of compact linear operators are compact.

Proof. Let A, B : X → Y be compact linear operators and let α, β ∈ C. Then for eachbounded sequence (ϕn) in X, since A and B are compact, we can select a subsequence(ϕn(k)) such that both sequences (Aϕn(k)) and (Bϕn(k)) converge. Hence (αA+βB)ϕn(k)

converges, and therefore αA + βB is compact. �

Theorem 2.21. Let X, Y, and Z be normed spaces and let A : X → Y and B : Y → Zbe bounded linear operators. Then the product BA : X → Z is compact if one of thetwo operators A or B is compact.

Proof. Let (ϕn) be a bounded sequence in X. If A is compact, then there exists asubsequence (ϕn(k)) such that Aϕn(k) → ψ ∈ Y, k → ∞. Since B is bounded andtherefore continuous, we have B(Aϕn(k)) → Bψ ∈ Z, k → ∞. Hence BA is compact.If A is bounded and B is compact, the sequence (Aϕn) is bounded in Y, since boundedoperators map bounded sets into bounded sets. Therefore, there exists a subsequence(ϕn(k)) such that (BA)ϕn(k) = B(Aϕn(k)) → χ ∈ Z, k → ∞. Hence, again BA iscompact. �

Theorem 2.22. Let X be a normed space and Y be a Banach space. Let the sequenceAn : X → Y of compact linear operators be norm convergent to a linear operatorA : X → Y, i.e., ‖An − A‖ → 0, n→ ∞. Then A is compact.

Proof. Let (ϕm) be a bounded sequence in X, i.e., ‖ϕm‖ ≤ C for all m ∈ IN and someC > 0. Because the An are compact, by the standard diagonalization procedure (seethe proof of Theorem 1.18), we can choose a subsequence (ϕm(k)) such that (Anϕm(k))converges for every fixed n as k → ∞. Given ε > 0, since ‖An − A‖ → 0, n → ∞,there exists n0 ∈ IN such that ‖An0 −A‖ < ε/3C. Because (An0ϕm(k)) converges, thereexists N(ε) ∈ IN such that

‖An0ϕm(k) − An0ϕm(l)‖ < ε

3

for all k, l ≥ N(ε). But then we have

‖Aϕm(k) − Aϕm(l)‖ ≤ ‖Aϕm(k) − An0ϕm(k)‖ + ‖An0ϕm(k) − An0ϕm(l)‖

+‖An0ϕm(l) − Aϕm(l)‖ < ε.Thus (Aϕm(k)) is a Cauchy sequence, and therefore it is convergent in the Banachspace Y. �


Theorem 2.23. Let A : X → Y be a bounded linear operator with finite-dimensionalrange A(X). Then A is compact.

Proof. Let U ⊂ X be bounded. Then the bounded operator A maps U into thebounded set A(U) contained in the finite-dimensional space A(X). By the Bolzano–Weierstrass Theorem 1.17 the set A(U) is relatively compact. Therefore A iscompact. �

Lemma 2.24 (Riesz). Let X be a normed space, U ⊂ X a closed subspace withU � X, and α ∈ (0, 1). Then there exists an element ψ ∈ X with ‖ψ‖ = 1 such that‖ψ − ϕ‖ ≥ α for all ϕ ∈ U.

Proof. Because U � X, there exists an element f ∈ X with f � U, and because U isclosed, we have

β := infϕ∈U ‖ f − ϕ‖ > 0.

We can choose g ∈ U such that

β ≤ ‖ f − g‖ ≤ β

α.

Now we define

ψ :=f − g‖ f − g‖ .

Then ‖ψ‖ = 1, and for all ϕ ∈ U we have

‖ψ − ϕ‖ = 1‖ f − g‖ ‖ f − {g + ‖ f − g‖ϕ} ‖ ≥

β

‖ f − g‖ ≥ α,

since g + ‖ f − g‖ϕ ∈ U. �

Theorem 2.25. The identity operator I : X → X is compact if and only if X hasfinite dimension.

Proof. Assume that I is compact and X is not finite-dimensional. Choose an ar-bitrary ϕ1 ∈ X with ‖ϕ1‖ = 1. Then U1 := span{ϕ1} is a finite-dimensional andtherefore closed subspace of X. By Lemma 2.24 there exists ϕ2 ∈ X with ‖ϕ2‖ = 1and ‖ϕ2 −ϕ1‖ ≥ 1/2. Now consider U2 := span{ϕ1, ϕ2}. Again by Lemma 2.24 thereexists ϕ3 ∈ X with ‖ϕ3‖ = 1 and ‖ϕ3−ϕ1‖ ≥ 1/2, ‖ϕ3−ϕ2‖ ≥ 1/2. Repeating this pro-cedure, we obtain a sequence (ϕn) with the properties ‖ϕn‖ = 1 and ‖ϕn−ϕm‖ ≥ 1/2,n � m. This implies that the bounded sequence (ϕn) does not contain a convergentsubsequence. Hence we have a contradiction to the compactness of I. Therefore, ifthe identity operator is compact, X has finite dimension. The converse statement isan immediate consequence of Theorem 2.23. �

This theorem, in particular, implies that the converse of Theorem 2.19 is false. Italso justifies the distinction between operator equations of the first and second kind,because obviously for a compact operator A the operators A and I −A have differentproperties. In particular, the following theorem immediately follows from Theorems2.21 and 2.25.


Theorem 2.26. A compact linear operator A : X → Y cannot have a boundedinverse unless X has finite dimension.

Theorem 2.27. Integral operators with continuous kernel are compact linear oper-ators on C(G).

Proof. Let U ⊂ C(G) be bounded, i.e., ‖ϕ‖∞ ≤ C for all ϕ ∈ U and some C > 0.Then for the integral operator A defined by (2.4) we have that

|(Aϕ)(x)| ≤ C|G|maxx,y∈G|K(x, y)|

for all x ∈ G and all ϕ ∈ U, i.e., A(U) is bounded. Since K is uniformly continuouson the compact set G ×G, for every ε > 0 there exists δ > 0 such that

|K(x, z) − K(y, z)| < ε

C|G|for all x, y, z ∈ G with |x − y| < δ. Then

|(Aϕ)(x) − (Aϕ)(y)| < εfor all x, y ∈ G with |x − y| < δ and all ϕ ∈ U, i.e., A(U) is equicontinuous. Hence Ais compact by the Arzela–Ascoli Theorem 1.18. �

We wish to mention that the compactness of the integral operator with continuouskernel also can be established by finite-dimensional approximations using Theorems2.22 and 2.23 in the Banach space C(G). In this context note that the proofs of The-orems 2.22 and 1.18 are similar in structure. The finite-dimensional operators canbe obtained by approximating either the continuous kernel by polynomials throughthe Weierstrass approximation theorem or the integral through a Riemann sum (see[31]).

For substantial parts of this book we will consider integral operators as operatorsin classical spaces of continuous functions. However, for certain topics we will findit necessary and convenient to consider integral operators also in Hilbert spaces.Therefore we carry the previous theorem over to the L2 space of measurable andLebesgue square integrable functions.

Theorem 2.28. Integral operators with continuous kernel are compact linear oper-ators on L2(G).

Proof. For the integral operator with continuous kernel A defined by (2.4), using theCauchy–Schwarz inequality and proceeding as in the proof of Theorem 2.27 it canbe established that each bounded set U ⊂ L2(G) is mapped into a set A(U) ⊂ C(G)that is bounded with respect to the maximum norm and equicontinuous. Hence,A : L2(G) → C(G) is compact and from this the statement of the theorem followssince the maximum norm is stronger than the mean squares norm and C(G) is densein L2(G). �


We note that Theorems 2.27 and 2.28 can be extended to the integral operatorgiven by (2.5) where for the L2 case the set M is required to be Lebesgue measurable.

Now we extend our investigation to integral operators with a weakly singularkernel, i.e., the kernel K is defined and continuous for all x, y ∈ G ⊂ IRm, x � y, andthere exist positive constants M and α ∈ (0,m] such that

|K(x, y)| ≤ M|x − y|α−m, x, y ∈ G, x � y. (2.7)

Theorem 2.29. Integral operators with weakly singular kernel are compact linearoperators on C(G).

Proof. The integral in (2.4) defining the operator A exists as an improper integral,since

|K(x, y)ϕ(y)| ≤ M‖ϕ‖∞|x − y|α−m

and ∫G|x − y|α−m dy ≤ ωm

∫ d

0ρα−mρm−1 dρ =

ωm

αdα,

where we have introduced polar coordinates with origin at x, d is the diameter of G,and ωm denotes the surface area of the unit sphere in IRm.

Now we choose a continuous function h : [0,∞) → IR with 0 ≤ h(t) ≤ 1 for allt ≥ 0 such that h(t) = 0 for 0 ≤ t ≤ 1/2 and h(t) = 1 for t > 1. For n ∈ IN we definecontinuous kernels Kn : G ×G → C by

Kn(x, y) :=

⎧⎪⎪⎪⎨⎪⎪⎪⎩h(n|x − y|)K(x, y), x � y,

0, x = y.

The corresponding integral operators An : C(G) → C(G) are compact by Theorem2.27. We have the estimate

|(Aϕ)(x) − (Anϕ)(x)| =∣∣∣∣∣∣∫

G∩B[x;1/n]{1 − h(n|x − y)}K(x, y)ϕ(y) dy

∣∣∣∣∣∣

≤ M‖ϕ‖∞ωm

∫ 1/n

0ρα−mρm−1 dρ = M‖ϕ‖∞ ωm

αnα, x ∈ G.

From this we observe that Anϕ→ Aϕ, n→ ∞, uniformly, and therefore Aϕ ∈ C(G).Furthermore it follows that

‖A − An‖∞ ≤ Mωm

αnα→ 0, n→ ∞,

and thus A is compact by Theorem 2.22. �

For the compactness of integral operators with weakly singular kernel on L2(G)see Problem 4.5.


Finally, we want to expand the analysis to integral operators defined on surfacesin IRm. Having in mind applications to boundary value problems, we will confineour attention to surfaces that are boundaries of smooth domains in IRm. A boundedopen domain D ⊂ IRm with boundary ∂D is said to be of class Cn, n ∈ IN, if theclosure D admits a finite open covering

D ⊂p⋃

q=1

Vq

such that for each of those Vq that intersect with the boundary ∂D we have theproperties: The intersection Vq ∩ D can be mapped bijectively onto the half-ballH := {x ∈ IRm : |x| < 1, xm ≥ 0} in IRm, this mapping and its inverse are n timescontinuously differentiable, and the intersection Vq ∩ ∂D is mapped onto the diskH ∩ {x ∈ IRm : xm = 0}.

In particular, this implies that the boundary ∂D can be represented locally by aparametric representation

x(u) = (x1(u), . . . , xm(u))

mapping an open parameter domain U ⊂ IRm−1 bijectively onto a surface patch Sof ∂D with the property that the vectors

∂x∂ui

, i = 1, . . . ,m − 1,

are linearly independent at each point x of S . Such a parameterization we call aregular parametric representation. The whole boundary ∂D is obtained by matchinga finite number of such surface patches.

On occasion, we will express the property of a domain D to be of class Cn alsoby saying that its boundary ∂D is of class Cn.

The vectors ∂x/∂ui, i = 1, . . . ,m − 1, span the tangent plane to the surface at thepoint x. The unit normal ν is the unit vector orthogonal to the tangent plane. It isuniquely determined up to two opposite directions. The surface element at the pointx is given by

ds =√g du1 · · · dum−1,

where g is the determinant of the positive definite matrix with entries

gi j :=∂x∂ui· ∂x∂u j

, i, j = 1, . . . ,m − 1.

In this book, for two vectors a = (a1, . . . , am) and b = (b1, . . . , bm) in IRm (or Cm) wedenote by a · b = a1b1 + · · · + ambm the dot product.

Assume that ∂D is the boundary of a bounded open domain of class C1. In theBanach space C(∂D) of real- or complex-valued continuous functions defined on


the surface ∂D and equipped with the maximum norm

‖ϕ‖∞ := maxx∈∂D|ϕ(x)|,

we consider the integral operator A : C(∂D)→ C(∂D) defined by

(Aϕ)(x) :=∫∂D

K(x, y)ϕ(y) ds(y), x ∈ ∂D, (2.8)

where K is a continuous or weakly singular kernel. According to the dimension ofthe surface ∂D, a kernel K is said to be weakly singular if it is defined and continuousfor all x, y ∈ ∂D, x � y, and there exist positive constants M and α ∈ (0,m− 1] suchthat

|K(x, y)| ≤ M|x − y|α−m+1, x, y ∈ ∂D, x � y. (2.9)

Analogously to Theorems 2.27 and 2.29 we can prove the following theorem.

Theorem 2.30. Integral operators with continuous or weakly singular kernel arecompact linear operators on C(∂D) if ∂D is of class C1.

Proof. For continuous kernels the proof of Theorem 2.27 essentially remains unal-tered. For weakly singular kernels the only major difference in the proof comparedwith the proof of Theorem 2.29 arises in the verification of the existence of the inte-gral in (2.8). Since the surface ∂D is of class C1, the normal vector ν is continuouson ∂D. Therefore, we can choose R ∈ (0, 1] such that

ν(x) · ν(y) ≥ 12

(2.10)

for all x, y ∈ ∂D with |x−y| ≤ R. Furthermore, we can assume that R is small enoughsuch that the set S [x; R] := {y ∈ ∂D : |y − x| ≤ R} is connected for each x ∈ ∂D.Then the condition (2.10) implies that S [x; R] can be projected bijectively onto thetangent plane to ∂D at the point x. By using polar coordinates in the tangent planewith origin in x, we now can estimate

∣∣∣∣∣∣∫

S [x;R]K(x, y)ϕ(y) ds(y)

∣∣∣∣∣∣ ≤ M‖ϕ‖∞∫

S [x;R]|x − y|α−m+1 ds(y)

≤ 2M‖ϕ‖∞ωm−1

∫ R

0ρα−m+1ρm−2 dρ

= 2M‖ϕ‖∞ωm−1Rα

α.

Here we have used the facts that |x − y| ≥ ρ, that the surface element

ds(y) =ρm−2dρdων(x) · ν(y)

,


expressed in polar coordinates on the tangent plane, can be estimated with the aidof (2.10) by ds(y) ≤ 2ρm−2dρdω, and that the projection of S [x; R] onto the tangentplane is contained in the disk of radius R and center x. Furthermore, we have

∣∣∣∣∣∣∫∂D\S [x;R]

K(x, y)ϕ(y) ds(y)

∣∣∣∣∣∣ ≤ M‖ϕ‖∞∫∂D\S [x;R]

Rα−m+1 ds(y)

≤ M‖ϕ‖∞Rα−m+1|∂D|.Hence, for all x ∈ ∂D the integral (2.8) exists as an improper integral. For thecompactness of A, we now can adopt the proof of Theorem 2.29. �

Problems

2.1. Let A : X → Y be a bounded linear operator from a normed space X into a normed space Y andlet X and Y be the completions of X and Y, respectively. Then there exists a uniquely determinedbounded linear operator A : X → Y such that Aϕ = Aϕ for all ϕ ∈ X. Furthermore, ‖A‖ = ‖A‖. Theoperator A is called the continuous extension of A. (In the sense of Theorem 1.13 the space X isinterpreted as a dense subspace of its completion X.)Hint: For ϕ ∈ X define Aϕ = limn→∞ Aϕn, where (ϕn) is a sequence from X with ϕn → ϕ, n→∞.

2.2. Show that Theorem 2.15 remains valid for operators satisfying ‖Ak‖ < 1 for some k ∈ IN.

2.3. Write the proofs for the compactness of the integral operator with continuous kernel in C(G)using finite-dimensional approximations as mentioned after the proof of Theorem 2.27.

2.4. Show that the result of Theorem 2.13 for the norm of the integral operator remains valid forweakly singular kernels.Hint: Use the approximations from the proof of Theorem 2.29.

2.5. Let X be a Hilbert space and {un : n ∈ IN} a complete orthonormal system in X. A linearoperator A : X → X is called a Hilbert–Schmidt operator if the series

∞∑n=1

‖Aun‖2

converges. Show that Hilbert–Schmidt operators are compact and that linear integral operators withcontinuous kernel A : L2[a, b] → L2[a, b] are Hilbert–Schmidt operators.Hint: Define operators with finite-dimensional range by

Anϕ := An∑

k=1

(ϕ, uk)uk

and use Theorems 2.22 and 2.23.

Chapter 3Riesz Theory

We now present the basic theory for an operator equation

ϕ − Aϕ = f (3.1)

of the second kind with a compact linear operator A : X → X on a normed spaceX. This theory was developed by Riesz [205] and initiated through Fredholm’s [55]work on integral equations of the second kind.

In his work from 1916 Riesz interpreted an integral equation as a special case of(3.1). The notion of a normed space was not yet available in 1916. Riesz set his workup in the function space of continuous real-valued functions on the interval [0, 1].He called the maximum of the absolute value of a function f on [0, 1] the norm of fand confirmed its properties that we now know as the norm axioms of Definition 1.1.Only these axioms, and not the special meaning as the maximum norm, were usedby Riesz. The concept of a compact operator also was not yet available in 1916.However, using the notion of compactness (see Definition 1.14) as introduced byFrechet in 1906, Riesz formulated that the integral operator A defined by (2.4) forthe special case G = [0, 1] maps bounded subsets of C[0, 1] into relatively compactsets, i.e., A is a compact operator in the sense of Definition 2.17.

What is fascinating about the work of Riesz is that his proofs are still usable and,as we shall do, can almost literally be transferred from the case of an integral oper-ator on the space of continuous functions to the general case of a compact operatoron a normed space.

3.1 Riesz Theory for Compact Operators

Given a compact linear operator A : X → X on a normed space X, we define

L := I − A,

where I denotes the identity operator.


33

34 3 Riesz Theory

Theorem 3.1 (First Riesz Theorem). The nullspace of the operator L, i.e.,

N(L) := {ϕ ∈ X : Lϕ = 0},is a finite-dimensional subspace.

Proof. The nullspace of the bounded linear operator L is a closed subspace of X,since for each sequence (ϕn) with ϕn → ϕ, n → ∞, and Lϕn = 0 we have thatLϕ = 0. Each ϕ ∈ N(L) satisfies Aϕ = ϕ, and therefore the restriction of A to N(L)coincides with the identity operator on N(L). The operator A is compact on X andtherefore also compact from N(L) into N(L), since N(L) is closed. Hence N(L) isfinite-dimensional by Theorem 2.25. �

Theorem 3.2 (Second Riesz Theorem). The range of the operator L, i.e.,

L(X) := {Lϕ : ϕ ∈ X},is a closed linear subspace.

Proof. The range of the linear operator L is a linear subspace. Let f be an element ofthe closure L(X). Then there exists a sequence (ϕn) in X such that Lϕn → f , n→ ∞.By Theorem 1.24 to each ϕn we choose a best approximation χn with respect toN(L), i.e.,

‖ϕn − χn‖ = infχ∈N(L)

‖ϕn − χ‖.The sequence defined by

ϕn := ϕn − χn, n ∈ IN,

is bounded. We prove this indirectly, i.e., we assume that the sequence (ϕn) is notbounded. Then there exists a subsequence (ϕn(k)) such that ‖ϕn(k)‖ ≥ k for all k ∈ IN.Now we define

ψk :=ϕn(k)

‖ϕn(k)‖ , k ∈ IN.

Since ‖ψk‖ = 1 and A is compact, there exists a subsequence (ψk( j)) such that

Aψk( j) → ψ ∈ X, j→ ∞.Furthermore,

‖Lψk‖ = ‖Lϕn(k)‖‖ϕn(k)‖ ≤

‖Lϕn(k)‖k

→ 0, k → ∞,

since the sequence (Lϕn) is convergent and therefore bounded. Hence Lψk( j) → 0 asj→ ∞. Now we obtain

ψk( j) = Lψk( j) + Aψk( j) → ψ, j→ ∞,

3.1 Riesz Theory for Compact Operators 35

and since L is bounded, from the two previous equations we conclude that Lψ = 0.But then, because χn(k) + ‖ϕn(k)‖ψ ∈ N(L) for all k in IN, we find

‖ψk − ψ‖ = 1‖ϕn(k)‖ ‖ϕn(k) − {χn(k) + ‖ϕn(k)‖ψ}‖

≥ 1‖ϕn(k)‖ inf

χ∈N(L)‖ϕn(k) − χ‖ = 1

‖ϕn(k)‖ ‖ϕn(k) − χn(k)‖ = 1.

This contradicts the fact that ψk( j) → ψ, j→ ∞.Therefore the sequence (ϕn) is bounded, and since A is compact, we can select a

subsequence (ϕn(k)) such that (Aϕn(k)) converges as k → ∞. In view of Lϕn(k) → f ,k → ∞, from ϕn(k) = Lϕn(k) + Aϕn(k) we observe that ϕn(k) → ϕ ∈ X, k → ∞. Butthen Lϕn(k) → Lϕ ∈ X, k → ∞, and therefore f = Lϕ ∈ L(X). Hence L(X) = L(X),and the proof is complete. �

For n ≥ 1 the iterated operators Ln can be written in the form

Ln = (I − A)n = I − An,

where

An =

n∑k=1

(−1)k−1

(nk

)Ak

is compact by Theorems 2.20 and 2.21. Therefore by Theorem 3.1 the nullspacesN(Ln) are finite-dimensional subspaces, and by Theorem 3.2 the ranges Ln(X) areclosed subspaces.

Theorem 3.3 (Third Riesz Theorem). There exists a uniquely determined nonneg-ative integer r, called the Riesz number of the operator A, such that

{0} = N(L0)⊂� N(L1)

⊂� · · · ⊂� N(Lr) = N(Lr+1) = · · · , (3.2)

andX = L0(X)

⊃� L1(X)

⊃� · · · ⊃� Lr(X) = Lr+1(X) = · · · . (3.3)

Furthermore, we have the direct sum

X = N(Lr) ⊕ Lr(X),

i.e., for each ϕ ∈ X there exist uniquely determined elements ψ ∈ N(Lr) andχ ∈ Lr(X) such that ϕ = ψ + χ.

Proof. Our proof consists of four steps:1. Because each ϕ with Lnϕ = 0 satisfies Ln+1ϕ = 0, we have

{0} = N(L0) ⊂ N(L1) ⊂ N(L2) ⊂ · · · .

36 3 Riesz Theory

Now assume that{0} = N(L0)

⊂� N(L1)

⊂� N(L2)

⊂� · · · .

Since by Theorem 3.1 the nullspace N(Ln) is finite-dimensional, the Riesz Lemma2.24 implies that for each n ∈ IN there exists ϕn ∈ N(Ln+1) such that ‖ϕn‖ = 1 and

‖ϕn − ϕ‖ ≥ 12

for all ϕ ∈ N(Ln). For n > m we consider

Aϕn − Aϕm = ϕn − (ϕm + Lϕn − Lϕm).

Then ϕm + Lϕn − Lϕm ∈ N(Ln), because

Ln(ϕm + Lϕn − Lϕm) = Ln−m−1Lm+1ϕm + Ln+1ϕn − Ln−mLm+1ϕm = 0.

Hence

‖Aϕn − Aϕm‖ ≥ 12

for n > m, and thus the sequence (Aϕn) does not contain a convergent subsequence.This is a contradiction to the compactness of A.

Therefore in the sequence N(Ln) there exist two consecutive nullspaces that areequal. Define

r := min{k : N(Lk) = N(Lk+1)

}.

Now we prove by induction that

N(Lr) = N(Lr+1) = N(Lr+2) = · · · .Assume that we have proven N(Lk) = N(Lk+1) for some k ≥ r. Then for eachϕ ∈ N(Lk+2) we have Lk+1Lϕ = Lk+2ϕ = 0. This implies that Lϕ ∈ N(Lk+1) = N(Lk).Hence Lk+1ϕ = LkLϕ = 0, and consequently ϕ ∈ N(Lk+1). Therefore, N(Lk+2) ⊂N(Lk+1), and we have established that

{0} = N(L0)⊂� N(L1)

⊂� · · · ⊂� N(Lr) = N(Lr+1) = · · · .

2. Because for each ψ = Ln+1ϕ ∈ Ln+1(X) we can write ψ = LnLϕ, we have

X = L0(X) ⊃ L1(X) ⊃ L2(X) ⊃ · · · .Now assume that

X = L0(X)⊃� L1(X)

⊃� L2(X)

⊃� · · · .

Since by Theorem 3.2 the range Ln(X) is a closed subspace, the Riesz Lemma 2.24implies that for each n ∈ IN there exist ψn ∈ Ln(X) such that ‖ψn‖ = 1 and

‖ψn − ψ‖ ≥ 12


for all ψ ∈ Ln+1(X). We write ψn = Lnϕn and for m > n we consider

Aψn − Aψm = ψn − (ψm + Lψn − Lψm).

Then ψm + Lψn − Lψm ∈ Ln+1(X), because

ψm + Lψn − Lψm = Ln+1(Lm−n−1ϕm + ϕn − Lm−nϕm).

Hence

‖Aψn − Aψm‖ ≥ 12

for m > n, and we can derive the same contradiction as before.Therefore in the sequence Ln(X) there exist two consecutive ranges that are equal.

Defineq := min

{k : Lk(X) = Lk+1(X)

}.

Now we prove by induction that

Lq(X) = Lq+1(X) = Lq+2(X) = · · · .Assume that we have proven Lk(X) = Lk+1(X) for some k ≥ q. Then for eachψ = Lk+1ϕ ∈ Lk+1(X) we can write Lkϕ = Lk+1ϕ for some ϕ ∈ X, because Lk(X) =Lk+1(X). Hence ψ = Lk+2ϕ ∈ Lk+2(X), and therefore Lk+1(X) ⊂ Lk+2(X), i.e., wehave proven that

X = L0(X)⊃� L1(X)

⊃� · · · ⊃� Lq(X) = Lq+1(X) = · · · .

3. Now we show that r = q. Assume that r > q and let ϕ ∈ N(Lr). Then, becauseLr−1ϕ ∈ Lr−1(X) = Lr(X), we can write Lr−1ϕ = Lrϕ for some ϕ ∈ X. Since Lr+1ϕ =Lrϕ = 0, we have ϕ ∈ N(Lr+1) = N(Lr), i.e., Lr−1ϕ = Lrϕ = 0. Thus ϕ ∈ N(Lr−1),and hence N(Lr−1) = N(Lr). This contradicts the definition of r.

On the other hand, assume that r < q and let ψ = Lq−1ϕ ∈ Lq−1(X). BecauseLψ = Lqϕ ∈ Lq(X) = Lq+1(X), we can write Lψ = Lq+1ϕ for some ϕ ∈ X. ThereforeLq(ϕ − Lϕ) = Lψ − Lq+1ϕ = 0, and from this we conclude that Lq−1(ϕ − Lϕ) = 0,because N(Lq−1) = N(Lq). Hence ψ = Lqϕ ∈ Lq(X), and consequently Lq−1(X) =Lq(X). This contradicts the definition of q.4. Let ψ ∈ N(Lr) ∩ Lr(X). Then ψ = Lrϕ for some ϕ ∈ X and Lrψ = 0. ThereforeL2rϕ = 0, whence ϕ ∈ N(L2r) = N(Lr) follows. This implies ψ = Lrϕ = 0.

Let ϕ ∈ X be arbitrary. Then Lrϕ ∈ Lr(X) = L2r(X) and we can writeLrϕ = L2rϕ for some ϕ ∈ X. Now define ψ := Lrϕ ∈ Lr(X) and χ := ϕ − ψ.Then Lrχ = Lrϕ− L2rϕ = 0, i.e., χ ∈ N(Lr). Therefore the decomposition ϕ = χ+ψproves the direct sum X = N(Lr) ⊕ Lr(X). �

Operators for which the sequence of the nullspaces of the iterated operators hastwo subsequent elements that coincide are called operators of finite ascent. Anal-ogously, operators for which the sequence of the ranges of the iterated operatorshas two subsequent elements that coincide are called operators of finite descent.

38 3 Riesz Theory

Using this terminology the third Riesz theorem states that for a compact linear oper-ator A the operator I−A has finite ascent and descent. We note that only for these twostatements of Theorem 3.3 the proofs make use of the compactness of A whereas allthe other statements are just linear algebra.

We are now ready to derive the following fundamental result of the Riesz theory.

Theorem 3.4. Let A : X → X be a compact linear operator on a normed space X.Then I−A is injective if and only if it is surjective. If I−A is injective (and thereforealso bijective), then the inverse operator (I − A)−1 : X → X is bounded, i.e., I − A isan isomorphism.

Proof. By (3.2) injectivity of I − A is equivalent to r = 0, and by (3.3) surjectivityof I − A is also equivalent to r = 0. Therefore injectivity of I − A and surjectivity ofI − A are equivalent.

It remains to show that L−1 is bounded when L = I − A is injective. Assume thatL−1 is not bounded. Then there exists a sequence ( fn) in X with ‖ fn‖ = 1 such that‖L−1 fn‖ ≥ n for all n ∈ IN. Define

gn :=fn

‖L−1 fn‖ and ϕn :=L−1 fn‖L−1 fn‖ , n ∈ IN.

Then gn → 0, n → ∞, and ‖ϕn‖ = 1 for all n. Since A is compact, we can select asubsequence (ϕn(k)) such that Aϕn(k) → ϕ ∈ X, k → ∞. Then, since ϕn − Aϕn = gn,we observe that ϕn(k) → ϕ, k → ∞, and ϕ ∈ N(L). Hence ϕ = 0, and this contradicts‖ϕn‖ = 1 for all n ∈ IN. �

If B : Y → Z is a bounded linear operator mapping a Banach space Y bijectivelyonto a Banach space Z, then by the Banach open mapping Theorem 10.8 the inverseoperator B−1 : Z → Y is bounded. For the Riesz theory we do not need to use thisdeep result from functional analysis and also do not require completeness of X.

We can rewrite Theorems 3.1 and 3.4 in terms of the solvability of an operatorequation of the second kind as follows.

Corollary 3.5. Let A : X → X be a compact linear operator on a normed space X.If the homogeneous equation

ϕ − Aϕ = 0 (3.4)

only has the trivial solution ϕ = 0, then for each f ∈ X the inhomogeneous equation

ϕ − Aϕ = f (3.5)

has a unique solution ϕ ∈ X and this solution depends continuously on f .If the homogeneous equation (3.4) has a nontrivial solution, then it has only a

finite number m ∈ IN of linearly independent solutions ϕ1, . . . , ϕm and the inhomo-geneous equation (3.5) is either unsolvable or its general solution is of the form

ϕ = ϕ +

m∑k=1

αkϕk,


where α1, . . . , αm are arbitrary complex numbers and ϕ denotes a particular solutionof the inhomogeneous equation.

The main importance of the Riesz theory for compact operators lies in the factthat it reduces the problem of establishing the existence of a solution to (3.5) to thegenerally much simpler problem of showing that (3.4) has only the trivial solutionϕ = 0.

It is left to the reader to formulate Theorem 3.4 and its Corollary 3.5 for integralequations of the second kind with continuous or weakly singular kernels.

Corollary 3.6. Theorem 3.4 and its Corollary 3.5 remain valid when I − A is re-placed by S − A, where S : X → Y is a bounded linear operator that has a boundedinverse S −1 : Y → X, i.e., S : X → Y is an isomorphism, and A : X → Y is acompact linear operator from a normed space X into a normed space Y.

Proof. This follows immediately from the fact that we can transform the equation

Sϕ − Aϕ = f

into the equivalent formϕ − S −1Aϕ = S −1 f ,

where S −1A : X → X is compact by Theorem 2.21. �

The decomposition X = N(Lr) ⊕ Lr(X) of Theorem 3.3 generates an operatorP : X → N(Lr) that maps ϕ ∈ X onto Pϕ := ψ defined by the unique decompositionϕ = ψ + χ with ψ ∈ N(Lr) and χ ∈ Lr(X). This operator is called a projectionoperator, because it satisfies P2 = P (see Chapter 13). We conclude this sectionwith the following lemma on this projection operator that we are going to use in thefollowing chapter. (Note that the proof only uses the finite dimension of N(Lr) andthe fact that Lr(X) is closed.)

Lemma 3.7. The projection operator P : X → N(Lr) defined by the decompositionX = N(Lr) ⊕ Lr(X) is compact.

Proof. Assume that P is not bounded. Then there exists a sequence (ϕn) in X with‖ϕn‖ = 1 such that ‖Pϕn‖ ≥ n for all n ∈ IN. Define

ψn :=ϕn

‖Pϕn‖ , n ∈ IN.

Then ψn → 0, n → ∞, and ‖Pψn‖ = 1 for all n ∈ IN. Since N(Lr) is finite-dimensional and (Pψn) is bounded, by Theorem 1.17 there exists a subsequence(ψn(k)) such that Pψn(k) → χ ∈ N(Lr), k → ∞. Because ψn → 0, n → ∞,we also have Pψn(k) − ψn(k) → χ, k → ∞. This implies that χ ∈ Lr(X), sincePψn(k) − ψn(k) ∈ Lr(X) for all k and Lr(X) is closed. Hence χ ∈ N(Lr) ∩ Lr(X), andtherefore χ = 0, i.e., Pψn(k) → 0, k → ∞. This contradicts ‖Pψn‖ = 1 for all n ∈ IN.Hence P is bounded, and because P has finite-dimensional range P(X) = N(Lr), byTheorem 2.23 it is compact. �

40 3 Riesz Theory

3.2 Spectral Theory for Compact Operators

We continue by formulating the results of the Riesz theory in terms of spectralanalysis.

Definition 3.8. Let A : X → X be a bounded linear operator on a normed spaceX. A complex number λ is called an eigenvalue of A if there exists an elementϕ ∈ X, ϕ � 0, such that Aϕ = λϕ. The element ϕ is called an eigenelement of A.A complex number λ is called a regular value of A if (λI − A)−1 : X → X existsand is bounded. The set of all regular values of A is called the resolvent set ρ(A) andR(λ; A) := (λI − A)−1 is called the resolvent of A. The complement of ρ(A) in C iscalled the spectrum σ(A) and

r(A) := supλ∈σ(A)

|λ|

is called the spectral radius of A.

For the spectrum of a compact operator we have the following properties.

Theorem 3.9. Let A : X → X be a compact linear operator on an infinite-dimensional normed space X. Then λ = 0 belongs to the spectrum σ(A) andσ(A) \ {0} consists of at most a countable set of eigenvalues with no point of ac-cumulation except, possibly, λ = 0.

Proof. Theorem 2.26 implies that λ = 0 belongs to the spectrum σ(A). For λ � 0 wecan apply the Riesz theory to λI − A. Either N(λI − A) = {0} and (λI − A)−1 existsand is bounded by Corollary 3.6 or N(λI − A) � {0}, i.e., λ is an eigenvalue. Thuseach λ � 0 is either a regular value or an eigenvalue of A.

It remains to show that for each R > 0 there exist only a finite number of eigenval-ues λ with |λ| ≥ R. Assume, on the contrary, that we have a sequence (λn) of distincteigenvalues satisfying |λn| ≥ R. Choose eigenelements ϕn such that Aϕn = λnϕn forn = 0, 1, . . . , and define finite-dimensional subspaces

Un := span{ϕ0, . . . , ϕn}.It is readily verified that eigenelements corresponding to distinct eigenvalues arelinearly independent. Hence, we have Un−1 ⊂ Un and Un−1 � Un for n = 1, 2, . . . .Therefore, by the Riesz Lemma 2.24 we can choose a sequence (ψn) of elementsψn ∈ Un such that ‖ψn‖ = 1 and

‖ψn − ψ‖ ≥ 12

for all ψ ∈ Un−1 and n = 1, 2, . . . . Writing

ψn =

n∑k=0

αnkϕk

3.3 Volterra Integral Equations 41

we obtain

λnψn − Aψn =

n−1∑k=0

(λn − λk)αnkϕk ∈ Un−1.

Therefore, for m < n we have

Aψn − Aψm = λnψn − (λnψn − Aψn + Aψm) = λn(ψn − ψ),

where ψ := λ−1n (λnψn − Aψn + Aψm) ∈ Un−1. Hence

‖Aψn − Aψm‖ ≥ |λn|2≥ R

2

for m < n, and the sequence (Aψn) does not contain a convergent subsequence. Thiscontradicts the compactness of A. �

3.3 Volterra Integral Equations

Integral equations of the form∫ x

aK(x, y)ϕ(y) dy = f (x), x ∈ [a, b], (3.6)

and

ϕ(x) −∫ x

aK(x, y)ϕ(y) dy = f (x), x ∈ [a, b], (3.7)

with variable limits of integration are called Volterra integral equations of the firstand second kind, respectively. Equations of this type were first investigated byVolterra [242]. One can view Volterra equations as special cases of Fredholm equa-tions with K(x, y) = 0 for y > x, but they have some special properties. In particular,Volterra integral equations of the second kind are always uniquely solvable.

Theorem 3.10. For each right-hand side f ∈ C[a, b] the Volterra integral equationof the second kind (3.6) with continuous kernel K has a unique solution ϕ ∈ C[a, b].

Proof. We extend the kernel onto [a, b] × [a, b] by setting K(x, y) := 0 for y > x.Then K is continuous for x � y and

|K(x, y)| ≤ M := maxa≤y≤x≤b

|K(x, y)|

for all x � y. Hence, K is weakly singular with α = 1.Now let ϕ ∈ C[a, b] be a solution to the homogeneous equation

ϕ(x) −∫ x

aK(x, y)ϕ(y) dy = 0, x ∈ [a, b].

42 3 Riesz Theory

By induction we show that

|ϕ(x)| ≤ ‖ϕ‖∞ Mn(x − a)n

n!, x ∈ [a, b], (3.8)

for n = 0, 1, 2, . . . . This certainly is true for n = 0. Assume that the inequality (3.8)is proven for some n ≥ 0. Then

|ϕ(x)| =∣∣∣∣∣∫ x

aK(x, y)ϕ(y) dy

∣∣∣∣∣ ≤ ‖ϕ‖∞ Mn+1(x − a)n+1

(n + 1)!.

Passing to the limit n → ∞ in (3.8) yields ϕ(x) = 0 for all x ∈ [a, b]. The statementof the theorem now follows from Theorems 2.29 and 3.4. �

In terms of spectral theory we can formulate the last result as follows: A Volterraintegral operator with continuous kernel has no spectral values different from zero.

Analogous to the above estimate (3.8), the integral operator A with the Volterrakernel satisfies

|(Anϕ)(x)| ≤ ‖ϕ‖∞ Mn(x − a)n

n!, x ∈ [a, b],

for n = 0, 1, 2, . . . . Consequently

‖An‖∞ ≤ Mn(b − a)n

n!< 1

for sufficiently large n and from Problem 2.2 we can conclude that successive ap-proximations always converge for Volterra integral equations of the second kindwith continuous kernels. We will meet Volterra integral equations again in Chap-ter 9 where we also will consider weakly singular kernels.

To illustrate the close relation between Volterra integral equations and initialvalue problems for ordinary differential equations we now consider the equationof the second order

u′′ + a1u′ + a2u = v (3.9)

on the interval [a, b] with coefficients a1, a2, v ∈ C[a, b] subject to the inititial con-ditions

u(a) = u0, u′(a) = u′0 (3.10)

for some constants u0 and u′0. After setting

ϕ := u′′ (3.11)

two integrations yield

u′(x) = u′0 +∫ x

aϕ(y) dy (3.12)

3.3 Volterra Integral Equations 43

and

u(x) = u0 + (x − a) u′0 +∫ x

a(x − y)ϕ(y) dy. (3.13)

Now let u be a solution to the initial value problem. Then, by combining (3.11)–(3.13) it can be seen that ϕ = u′′ satisfies the integral equation (3.7) with the kernelgiven by

K(x, y) := −a1(x) − a2(x)(x − y)

and the right-hand side given by

f (x) := v(x) − a1(x)u′0 − a2(x)[u0 + (x − a) u′0].

Conversely, for any solution ϕ ∈ C[0, 1] the function u defined by (3.13) providesa solution to the initial value problem. Thus Theorem 3.10 establishes the Picard–Lindelof theorem on existence and uniqueness of a solution for the special case ofthe linear initial value problem (3.9) and (3.10). We note that the proof of the Picard–Lindelof theorem for nonlinear ordinary differential equations relies on a nonlinearVolterra integral equation.

Despite the fact that, in general, integral equations of the first kind are moredelicate than integral equations of the second kind, in some cases Volterra integralequations of the first kind can be treated by reducing them to equations of the secondkind. Consider ∫ x

aK(x, y)ϕ(y) dy = f (x), x ∈ [a, b], (3.14)

and assume that the derivatives Kx = ∂K/∂x and f ′ exist and are continuous and thatK(x, x) � 0 for all x ∈ [a, b]. Then differentiating with respect to x reduces (3.14) to

ϕ(x) +∫ x

a

Kx(x, y)K(x, x)

ϕ(y) dy =f ′(x)

K(x, x), x ∈ [a, b]. (3.15)

Equations (3.14) and (3.15) are equivalent if f (a) = 0. If Ky = ∂K/∂y exists and iscontinuous and again K(x, x) � 0 for all x ∈ [a, b], then there is a second method toreduce the equation of the first kind to one of the second kind. In this case, setting

ψ(x) :=∫ x

aϕ(y) dy, x ∈ [a, b],

and performing an integration by parts in (3.14) yields

ψ(x) −∫ x

a

Ky(x, y)

K(x, x)ψ(y) dy =

f (x)K(x, x)

, x ∈ [a, b]. (3.16)

Abel’s integral equation (1.3) is an example of a Volterra integral equation of thefirst kind for which the above assumptions are not satisfied.

We leave it as an exercise to extend this short discussion of Volterra integralequations to the case of Volterra integral equations for functions of more than oneindependent variable.

44 3 Riesz Theory

Problems

3.1. Let A : X → Y be a compact linear operator from a normed space X into a normed space Y.The continuous extension A : X → Y of A is compact with A(X) ⊂ Y (see Problem 2.1).

3.2. Let X be a linear space, let A, B : X → X be linear operators satisfying AB = BA, and letAB have an inverse (AB)−1 : X → X. Then A and B have inverse operators A−1 = B(AB)−1 andB−1 = A(AB)−1.

3.3. Prove Theorem 3.4 under the assumption that An is compact for some n ≥ 1.Hint: Use Problem 3.2 to prove that the set (σ(A))n := {λn : λ ∈ σ(A)} is contained in the spectrumσ(An). Then use Theorem 3.9 to show that there exists an integer m ≥ n such that each of theoperators

Lk := exp2πik

mI − A, k = 1, . . . ,m − 1,

has a bounded inverse. Then the equations R(I − A)ϕ = R f and (I − A)ϕ = f , where R :=∏m−1

k=1 Lk,are equivalent.

3.4. Let Xi, i = 1, . . . , n, be normed spaces. Show that the Cartesian product X := X1 × · · · × Xn ofn-tuples ϕ = (ϕ1 , . . . , ϕn) is a normed space with the maximum norm

‖ϕ‖∞ := maxi=1,...,n

‖ϕi‖.

Let Aik : Xk → Xi, i, k = 1, . . . , n, be linear operators. Show that the matrix operator A : X → Xdefined by

(Aϕ)i :=n∑

k=1

Aikϕk

is bounded or compact if and only if each of its components Aik : Xk → Xi is bounded or compact,respectively. Formulate Theorem 3.4 for systems of operator and integral equations of the secondkind.

3.5. Show that the integral operator with continuous kernel

K(x, y) :=∞∑

k=0

1(k + 1)2

{cos(k + 1)x sin ky − sin(k + 1)x cos ky}

on the interval [0, 2π] has no eigenvalues.

Chapter 4Dual Systems and Fredholm Alternative

In the case when the homogeneous equation has nontrivial solutions, the Riesz the-ory, i.e., Theorem 3.4 gives no answer to the question of whether the inhomogeneousequation for a given inhomogeneity is solvable. This question is settled by the Fred-holm alternative, which we shall develop in this chapter. Rather than presenting itin the context of the Riesz–Schauder theory with the adjoint operator in the dualspace we will consider the Fredholm theory for compact adjoint operators in dualsystems generated by non-degenerate bilinear or sesquilinear forms. This symmet-ric version is better suited for applications to integral equations and contains theRiesz–Schauder theory as a special case.

4.1 Dual Systems via Bilinear Forms

Throughout this chapter we tacitly assume that all linear spaces under considerationare complex linear spaces; the case of real linear spaces can be treated analogously.

Definition 4.1. Let X, Y be linear spaces. A mapping

〈· , ·〉 : X × Y → C

is called a bilinear form if

〈α1ϕ1 + α2ϕ2, ψ〉 = α1〈ϕ1, ψ〉 + α2〈ϕ2, ψ〉,

〈ϕ, β1ψ1 + β2ψ2〉 = β1〈ϕ, ψ1〉 + β2〈ϕ, ψ2〉for all ϕ1, ϕ2, ϕ ∈ X, ψ1, ψ2, ψ ∈ Y, and α1, α2, β1, β2 ∈ C. The bilinear form is callednon-degenerate if for every ϕ ∈ X with ϕ � 0 there exists ψ ∈ Y such that 〈ϕ, ψ〉 � 0;and for every ψ ∈ Y with ψ � 0 there exists ϕ ∈ X such that 〈ϕ, ψ〉 � 0.

Definition 4.2. Two normed spaces X and Y equipped with a non-degenerate bilin-ear form 〈· , ·〉 : X × Y → C are called a dual system and denoted by 〈X, Y〉.


45

46 4 Dual Systems and Fredholm Alternative

Theorem 4.3. Each normed space X together with its dual space X∗ forms thecanonical dual system 〈X, X∗〉 with the bilinear form

〈ϕ, F〉 := F(ϕ), ϕ ∈ X, F ∈ X∗.

Proof. The bilinearity is obvious. By the Hahn–Banach Corollary 2.11, for eachϕ � 0 in X there exists F ∈ X∗ with F(ϕ) � 0. For each F � 0 in X∗ trivially thereexists ϕ ∈ X with F(ϕ) � 0. �

Theorem 4.4. Let G ⊂ IRm be as in Theorem 2.13. Then 〈C(G),C(G)〉 is a dualsystem with the bilinear form

〈ϕ, ψ〉 :=∫

Gϕ(x)ψ(x) dx, ϕ, ψ ∈ C(G). (4.1)

Proof. The bilinearity is obvious and the non-degenerateness is a consequence of∫G|ϕ(x)|2dx > 0 for all ϕ ∈ C(G) with ϕ � 0. �

Definition 4.5. Let 〈X1, Y1〉 and 〈X2, Y2〉 be two dual systems. Then two operatorsA : X1 → X2, B : Y2 → Y1 are called adjoint (with respect to these dual systems) if

〈Aϕ, ψ〉 = 〈ϕ, Bψ〉for all ϕ ∈ X1, ψ ∈ Y2. (We use the same symbol 〈· , ·〉 for the bilinear forms on〈X1, Y1〉 and 〈X2, Y2〉.)Theorem 4.6. Let 〈X1, Y1〉 and 〈X2, Y2〉 be two dual systems. If an operator A :X1 → X2 has an adjoint B : Y2 → Y1, then B is uniquely determined, and A and Bare linear.

Proof. Suppose that there exist two adjoint operators to A and denote these by B1

and B2. Let B := B1 − B2. Then

〈ϕ, Bψ〉 = 〈ϕ, B1ψ〉 − 〈ϕ, B2ψ〉 = 〈Aϕ, ψ〉 − 〈Aϕ, ψ〉 = 0

for all ϕ ∈ X1 and ψ ∈ Y2. Hence, because 〈· , ·〉 is non-degenerate, we have Bψ = 0for all ψ ∈ Y2, i.e., B1 = B2.

To show that B is linear we observe that

〈ϕ, β1Bψ1 + β2Bψ2〉 = β1〈ϕ, Bψ1〉 + β2〈ϕ, Bψ2〉

= β1〈Aϕ, ψ1〉 + β2〈Aϕ, ψ2〉

= 〈Aϕ, β1ψ1 + β2ψ2〉

= 〈ϕ, B(β1ψ1 + β2ψ2)〉for all ϕ ∈ X1, ψ1, ψ2 ∈ Y2, and β1, β2 ∈ C, i.e., β1Bψ1 + β2Bψ2 = B(β1ψ1 + β2ψ2).In a similar manner, it is seen that A is linear. �

4.1 Dual Systems via Bilinear Forms 47

Theorem 4.7. Let K be a continuous or a weakly singular kernel. Then in the dualsystem 〈C(G),C(G)〉 the (compact) integral operators defined by

(Aϕ)(x) :=∫

GK(x, y)ϕ(y) dy, x ∈ G,

and

(Bψ)(x) :=∫

GK(y, x)ψ(y) dy, x ∈ G,

are adjoint.

Proof. The theorem follows from

〈Aϕ, ψ〉 =∫

G(Aϕ)(x)ψ(x) dx

=

∫G

(∫G

K(x, y)ϕ(y) dy

)ψ(x) dx

=

∫Gϕ(y)

(∫G

K(x, y)ψ(x) dx

)dy

=

∫Gϕ(y)(Bψ)(y) dy = 〈ϕ, Bψ〉.

In the case of a weakly singular kernel, interchanging the order of integration is jus-tified by the fact that Anϕ → Aϕ, n → ∞, uniformly on G, where An is the integraloperator with continuous kernel Kn introduced in the proof of Theorem 2.29. �

The operator A : C[0, 1] → C[0, 1] defined by Aϕ := ϕ(1) provides an exampleof a compact operator that does not have an adjoint operator with respect to the dualsystem 〈C[0, 1],C[0, 1]〉 of Theorem 4.4. To the contrary let B : C[0, 1] → C[0, 1]

be an adjoint of A and choose a function ψ ∈ C[0, 1] with∫ 1

0ψ(x) dx = 1. With the

aid of the Cauchy–Schwarz inequality we estimate

1 = |ϕ(1)| = |〈Aϕ, ψ〉| = |〈ϕ, Bψ〉| ≤ ‖ϕ‖2‖Bψ‖2for all ϕ ∈ C[0, 1] with ϕ(1) = 1. Considering this inequality for the sequence (ϕn)with ϕn(x) := xn we arrive at a contradiction, since the right-hand side tends to zeroas n→ ∞.

Theorem 4.8. Let A : X → Y be a bounded linear operator mapping a normedspace X into a normed space Y. Then the adjoint operator A∗ : Y∗ → X∗ withrespect to the canonical dual systems 〈X, X∗〉 and 〈Y, Y∗〉 exists. It is given by

A∗F := FA, F ∈ Y∗,

and is called the dual operator of A. It is bounded with norm ‖A‖ = ‖A∗‖.


Proof. Since F and A both are linear and bounded, A∗F = FA : X → C also is linearand bounded, i.e., A∗ : Y∗ → X∗ with

‖A∗F‖ = ‖FA‖ ≤ ‖F‖ ‖A‖. (4.2)

The operators A und A∗ are adjoint since

〈Aϕ, F〉 = F(Aϕ) = (A∗F)(ϕ) = 〈ϕ, A∗F〉.Furthermore A∗ : Y∗ → X∗ is linear by Theorem 4.6 and bounded with ‖A∗‖ ≤ ‖A‖by (4.2). By Corollary 2.12 we conclude that

‖Aϕ‖ = sup‖F‖=1

|F(Aϕ)| = sup‖F‖=1

|(A∗F)(ϕ)| ≤ ‖A∗‖ ‖ϕ‖,

i.e., we also have that ‖A‖ ≤ ‖A∗‖ and from this finally ‖A‖ = ‖A∗‖ follows. �

As a result due to Schauder [215] (see also [94, 112, 116, 229]) it can be shownthat under the assumptions of Theorem 4.8 the adjoint operator A∗ is compact if andonly if A is compact, provided that Y is a Banach space.

4.2 Dual Systems via Sesquilinear Forms

Definition 4.9. Let X, Y be linear spaces. A mapping (· , ·) : X × Y → C is called asesquilinear form if

(α1ϕ1 + α2ϕ2, ψ) = α1(ϕ1, ψ) + α2(ϕ2, ψ),

(ϕ, β1ψ1 + β2ψ2) = β1(ϕ, ψ1) + β2(ϕ, ψ2)

for all ϕ1, ϕ2, ϕ ∈ X, ψ1, ψ2, ψ ∈ Y, and α1, α2, β1, β2 ∈ C. (Here, the bar indicatesthe complex conjugate.)

We leave it as an exercise to formulate Definition 4.5 and Theorem 4.6 in dualsystems generated by non-degenerate sesquilinear forms. Sesquilinear forms differfrom bilinear forms by their anti-linearity with respect to the second space. However,there is a close relation between them. Assume there exists a mapping ∗ : Y → Ywith the properties (β1ψ1 +β2ψ2)∗ = β1ψ

∗1 + β2ψ

∗2 and (ψ∗)∗ = ψ for all ψ1, ψ2, ψ ∈ Y

and β1, β2 ∈ C. Such a mapping is called an involution and provides a one-to-onecorrespondence between bilinear and sesquilinear forms by (ϕ, ψ) = 〈ϕ, ψ∗〉. In thespace C(G) the natural involution is given by ψ∗(x) := ψ(x) for all x ∈ G and allψ ∈ C(G). Again we leave it as an exercise to formulate Theorems 4.4 and 4.7 forthe corresponding sesquilinear form given by the scalar product

(ϕ, ψ) :=∫

Gϕ(x)ψ(x) dx, ϕ, ψ ∈ C(G). (4.3)

4.2 Dual Systems via Sesquilinear Forms 49

The operator A : C[0, 1]→ C[0, 1] considered in the example on p. 47, of course,also serves as an example of a compact operator that does not have an adjoint oper-ator with respect to the dual system (C[0, 1],C[0, 1]) generated by (4.3).

In the sequel, we will demonstrate that in Hilbert spaces for bounded linear oper-ators the adjoint operators always exist. From Definition 1.19 we observe that eachscalar product on a linear space X may be considered as a non-degenerate sesquilin-ear form that is symmetric, i.e., (ϕ, ψ) = (ψ, ϕ) for all ϕ, ψ ∈ X, and positive definite,i.e., (ϕ, ϕ) > 0 for all ϕ ∈ X with ϕ � 0. Thus each pre-Hilbert space canonically isa dual system.

Theorem 4.10 (Riesz). Let X be a Hilbert space. Then for each bounded linearfunctional F : X → C there exists a unique element f ∈ X such that

F(ϕ) = (ϕ, f ) (4.4)

for all ϕ ∈ X. The norms of the element f and the linear function F coincide, i.e.,

‖ f ‖ = ‖F‖. (4.5)

Proof. Uniqueness follows from the fact that because of the positive definiteness ofthe scalar product, f = 0 is the only representer of the zero function F = 0 in thesense of (4.4). For F � 0 choose w ∈ X with F(w) � 0. Since F is continuous, thenullspace N(F) = {ϕ ∈ X : F(ϕ) = 0} can be seen to be a closed, and consequentlycomplete, subspace of the Hilbert space X. By Theorem 1.26 there exists the bestapproximation v to w with respect to N(F), and by Theorem 1.25 it satisfies

w − v ⊥ N(F).

Then for g := w − v we have that

(F(g)ϕ − F(ϕ)g, g) = 0, ϕ ∈ X,

because F(g)ϕ − F(ϕ)g ∈ N(F) for all ϕ ∈ X. Hence,

F(ϕ) =

⎛⎜⎜⎜⎜⎝ϕ, F(g)g‖g‖2

⎞⎟⎟⎟⎟⎠for all ϕ ∈ X, which completes the proof of (4.4).

From (4.4) and the Cauchy–Schwarz inequality we have that

|F(ϕ)| ≤ ‖ f ‖ ‖ϕ‖, ϕ ∈ X,

whence ‖F‖ ≤ ‖ f ‖ follows. On the other hand, inserting f into (4.4) yields

‖ f ‖2 = F( f ) ≤ ‖F‖ ‖ f ‖,and therefore ‖ f ‖ ≤ ‖F‖. This concludes the proof of the norm equality (4.5). �


This theorem establishes the existence of a bijective anti-linear mapping betweena Hilbert space and its dual space that is isometric in the sense that it preserves thenorms. In view of (4.4), this mapping J : X → X∗ is given by

(J( f ))(ϕ) = (ϕ, f ) (4.6)

for all ϕ, f ∈ X.

Theorem 4.11. Let X and Y be Hilbert spaces, and let A : X → Y be a boundedlinear operator. Then there exists a uniquely determined linear operator A∗ : Y → Xwith the property

(Aϕ, ψ) = (ϕ, A∗ψ)

for all ϕ ∈ X and ψ ∈ Y, i.e., A and A∗ are adjoint with respect to the dual systems(X, X) and (Y, Y) generated by the scalar products on X and Y. The operator A∗is bounded and ‖A∗‖ = ‖A‖. (Again we use the same symbol (· , ·) for the scalarproducts on X and Y.)

Proof. For each ψ ∈ Y the mapping ϕ → (Aϕ, ψ) clearly defines a bounded lin-ear function on X, since |(Aϕ, ψ)| ≤ ‖A‖ ‖ϕ‖ ‖ψ‖. By Theorem 4.10 we can write(Aϕ, ψ) = (ϕ, f ) for some f ∈ X. Therefore, setting A∗ψ := f we define an oper-ator A∗ : Y → X that is an adjoint of A. By Theorem 4.6, the adjoint is uniquelydetermined and linear. Using the Cauchy–Schwarz inequality, we derive

‖A∗ψ‖2 = (A∗ψ, A∗ψ) = (AA∗ψ, ψ) ≤ ‖A‖ ‖A∗ψ‖ ‖ψ‖for all ψ ∈ Y. Hence, A∗ is bounded with ‖A∗‖ ≤ ‖A‖. Conversely, since A is theadjoint of A∗, we also have ‖A‖ ≤ ‖A∗‖. Hence ‖A∗‖ = ‖A‖. �

Using the anti-linear isometric isomorphism J between the Hilbert space andits dual space, an alternative proof of Theorem 4.11 can also be obtained as anapplication of the corresponding Theorem 4.8 for the dual operator.

Theorem 4.12. Let X and Y be Hilbert spaces and let A : X → Y be a compactlinear operator. Then the adjoint operator A∗ : Y → X is also compact.

Proof. Let (ψn) be a bounded sequence in Y, i.e, ‖ψn‖ ≤ C for all n ∈ IN andsome C > 0. By Theorem 4.11 the adjoint operator A∗ : Y → X is bounded and,consequently, the operator AA∗ : Y → Y is compact by Theorem 2.21. Hence thereexists a subsequence (ψn(k)) such that (AA∗ψn(k)) converges in Y. But then from

‖A∗(ψn(k) − ψn( j))‖2 = (AA∗(ψn(k) − ψn( j)), ψn(k) − ψn( j))

≤ 2C‖AA∗(ψn(k) − ψn( j))‖we observe that (A∗ψn(k)) is a Cauchy sequence, and therefore it converges in theHilbert space X. �

4.2 Dual Systems via Sesquilinear Forms 51

The following theorem is due to Lax [156] and provides a useful tool to extendresults on the boundedness and compactness of linear operators from a given normto a weaker scalar product norm.

Theorem 4.13 (Lax). Let X and Y be normed spaces, both of which have a scalarproduct (· , ·), and assume that there exists a positive constant c such that

|(ϕ, ψ)| ≤ c‖ϕ‖ ‖ψ‖ (4.7)

for all ϕ, ψ ∈ X. Let U be a subspace of X and let A : X → Y and B : Y → X bebounded linear operators satisfying

(Aϕ, ψ) = (ϕ, Bψ) (4.8)

for all ϕ ∈ U and ψ ∈ Y. Then A : U → Y is bounded with respect to the norms ‖ · ‖sinduced by the scalar products and

‖A‖2s ≤ ‖A‖ ‖B‖. (4.9)

Proof. Consider the bounded operator M : U → X given by M := BA with ‖M‖ ≤‖B‖ ‖A‖. Then, as a consequence of (4.8), M is self-adjoint, i.e.,

(Mϕ, ψ) = (ϕ, Mψ)

for all ϕ, ψ ∈ U. Therefore, using the Cauchy–Schwarz inequality, we obtain

‖Mnϕ‖2s = (Mnϕ, Mnϕ) = (ϕ, M2nϕ) ≤ ‖M2nϕ‖sfor all ϕ ∈ U with ‖ϕ‖s ≤ 1 and all n ∈ IN. From this, by induction, it follows that

‖Mϕ‖s ≤ ‖M2nϕ‖2−n

s .

By (4.7) we have ‖ψ‖s ≤ √c ‖ψ‖ for all ψ ∈ X. Hence,

‖Mϕ‖s ≤{√

c ‖M2nϕ‖

}2−n

≤{√

c ‖ϕ‖ ‖M‖2n }2−n

={√

c ‖ϕ‖}2−n

‖M‖.Passing to the limit n→ ∞ now yields

‖Mϕ‖s ≤ ‖M‖for all ϕ ∈ U with ‖ϕ‖s ≤ 1. Finally, for all ϕ ∈ U with ‖ϕ‖s ≤ 1, we have from theCauchy–Schwarz inequality that

‖Aϕ‖2s = (Aϕ, Aϕ) = (ϕ, Mϕ) ≤ ‖Mϕ‖s ≤ ‖M‖.From this the statement follows. �


For an example of an application of Lax’s theorem let X = Y = C(G) be equippedwith the maximum norm and the L2 scalar product (4.3). Using the approximationsfrom the proof of Theorem 2.29, without any further analysis, from Theorems 2.22and 4.13 it can be seen that integral operators with weakly singular kernels arecompact in the completion of C(G) with respect to the scalar product (4.3), i.e.,in L2(G) (see Problem 4.5). Here, completeness is required for the application ofTheorem 2.22 on the compactness of the limit of a norm convergent sequence ofcompact operators.

4.3 The Fredholm Alternative

Now we proceed to develop the Fredholm theory for compact operators, which wewill write for a dual system generated by a bilinear form. The analysis for the caseof a sesquillinear form is analogous. We begin with the following technical lemma.

Lemma 4.14. Let 〈X, Y〉 be a dual system. Then to every set of linearly independentelements ϕ1, . . . , ϕn ∈ X there exists a set ψ1, . . . , ψn ∈ Y such that

〈ϕi, ψk〉 = δik, i, k = 1, . . . , n,

where δik = 1 for i = k and δik = 0 for i � k. The same statement holds with theroles of X and Y interchanged.

Proof. For one linearly independent element the lemma is true, since 〈· , ·〉 is non-degenerate. Assume that the assertion of the lemma has been proven for n ≥ 1 lin-early independent elements. Let ϕ1, . . . , ϕn+1 be n+1 linearly independent elements.Then, by our induction assumption, for every m = 1, . . . , n + 1, to the set

ϕ1, . . . , ϕm−1, ϕm+1, . . . , ϕn+1

of n elements in X there exists a set of n elements

ψ(m)1 , . . . , ψ(m)

m−1, ψ(m)m+1, . . . , ψ

(m)n+1

in Y such that

〈ϕi, ψ(m)k 〉 = δik, i, k = 1, . . . , n + 1, i, k � m. (4.10)

Since 〈· , ·〉 is non-degenerate, there exists χm ∈ Y such that

αm :=

⟨ϕm, χm −

n+1∑k=1k�m

ψ(m)k 〈ϕk, χm〉

⟩=

⟨ϕm −

n+1∑k=1k�m

〈ϕm, ψ(m)k 〉ϕk, χm

⟩� 0,

4.3 The Fredholm Alternative 53

because otherwise

ϕm −n+1∑k=1k�m

〈ϕm, ψ(m)k 〉ϕk = 0,

which is a contradiction to the linear independence of the ϕ1, . . . , ϕn+1. Define

ψm :=1αm

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩χm −

n+1∑k=1k�m

ψ(m)k 〈ϕk, χm〉

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭.

Then 〈ϕm, ψm〉 = 1, and for i � m we have

〈ϕi, ψm〉 = 1αm

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩〈ϕi, χm〉 −

n+1∑k=1k�m

〈ϕi, ψ(m)k 〉〈ϕk, χm〉

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭= 0

because of (4.10). Hence we obtain ψ1, . . . , ψn+1 such that

〈ϕi, ψk〉 = δik, i, k = 1, . . . , n + 1,

and the lemma is proven. �

Theorem 4.15 (First Fredholm Theorem). Let 〈X, Y〉 be a dual system and A :X → X, B : Y → Y be compact adjoint operators. Then the nullspaces of theoperators I − A and I − B have the same finite dimension.

Proof. By the first Riesz Theorem 3.1 we have

m := dim N(I − A) < ∞and

n := dim N(I − B) < ∞.We assume that m < n. Then we choose a basis ϕ1, . . . , ϕm for N(I−A) (if m > 0) anda basis ψ1, . . . , ψn for N(I − B). By Lemma 4.14 there exist elements a1, . . . , am ∈ Y(if m > 0) and b1, . . . , bn ∈ X such that

〈ϕi, ak〉 = δik, i, k = 1, . . . ,m,

〈bi, ψk〉 = δik, i, k = 1, . . . , n.

Define a linear operator T : X → X by

Tϕ :=m∑

i=1

〈ϕ, ai〉bi, ϕ ∈ X, (4.11)


if m > 0 and T := 0 if m = 0. Recall the compact projection operator

P : X → N[(I − A)r]

from Lemma 3.7. Since T : N[(I − A)r] → X is bounded by Theorem 2.4, fromTheorem 2.21 we have that T P : X → X is compact. Therefore, in view of Theorem2.20, we can apply the Riesz theory to the operator A − T P.

For m > 0, from

〈ϕ − Aϕ + T Pϕ, ψk〉 = 〈ϕ, ψk − Bψk〉 + 〈T Pϕ, ψk〉 = 〈T Pϕ, ψk〉we find that

〈ϕ − Aϕ + T Pϕ, ψk〉 =⎧⎪⎪⎪⎨⎪⎪⎪⎩〈Pϕ, ak〉, k = 1, . . . ,m,

0, k = m + 1, . . . , n.(4.12)

Now let ϕ ∈ N(I − A + T P). Then from (4.12) we see that

〈Pϕ, ak〉 = 0, k = 1, . . . ,m, (4.13)

and therefore T Pϕ = 0. Hence ϕ ∈ N(I − A) and, consequently, we can write

ϕ =m∑

i=1

αiϕi,

where αi = 〈ϕ, ai〉 for i = 1, . . . ,m. Now from Pϕ = ϕ for ϕ ∈ N(I − A) and (4.13)we conclude that ϕ = 0. Thus we have proven that I − A + T P is injective. This, ofcourse, is also true in the case when m = 0.

Since I − A + T P is injective, by Theorem 3.4 the inhomogeneous equation

ϕ − Aϕ + T Pϕ = bn

has a unique solution ϕ. Now, with the aid of the second line of (4.12) (which is alsotrue for m = 0) we arrive at the contradiction

1 = 〈bn, ψn〉 = 〈ϕ − Aϕ + T Pϕ, ψn〉 = 0.

Therefore m ≥ n. Interchanging the roles of A and B shows that n ≥ m. Hencem = n. �

Theorem 4.16 (Second Fredholm Theorem). Let 〈X, Y〉 be a dual system and A :X → X, B : Y → Y be compact adjoint operators. Then

(I − A)(X) = { f ∈ X : 〈 f , ψ〉 = 0, ψ ∈ N(I − B)}and

(I − B)(Y) = {g ∈ Y : 〈ϕ, g〉 = 0, ϕ ∈ N(I − A)}.


Proof. It suffices to carry out the proof for the range of I − A, and by Theorems 3.4and 4.15 we only need to consider the case where m > 0. Let f ∈ (I − A)(X), i.e,f = ϕ − Aϕ for some ϕ ∈ X. Then

〈 f , ψ〉 = 〈ϕ − Aϕ, ψ〉 = 〈ϕ, ψ − Bψ〉 = 0

for all ψ ∈ N(I − B).Conversely, assume that f satisfies 〈 f , ψ〉 = 0 for all ψ ∈ N(I − B). From the

proof of the previous theorem we know that the equation

ϕ − Aϕ + T Pϕ = f

has a unique solution ϕ. Then, in view of (4.12) and the condition on f , we have

〈Pϕ, ak〉 = 〈ϕ − Aϕ + T Pϕ, ψk〉 = 〈 f , ψk〉 = 0, k = 1, . . . ,m.

Hence T Pϕ = 0, and thus ϕ also satisfies ϕ − Aϕ = f . �

We now summarize our results in the so-called Fredholm alternative.

Theorem 4.17. Let A : X → X, B : Y → Y be compact adjoint operators in a dualsystem 〈X, Y〉. Then either I − A and I − B are bijective or I − A and I − B havenontrivial nullspaces with finite dimension

dim N(I − A) = dim N(I − B) ∈ IN

and the ranges are given by

(I − A)(X) = { f ∈ X : 〈 f , ψ〉 = 0, ψ ∈ N(I − B)}and

(I − B)(Y) = {g ∈ Y : 〈ϕ, g〉 = 0, ϕ ∈ N(I − A)}.We explicitly note that this theorem implies that for the first of the two alterna-

tives each one of the four properties I − A injective, I − A surjective, I − B injective,and I − B surjective implies the three other ones.

Choosing the dual system introduced in Theorem 4.4 and the integral operatorswith continuous or weakly singular kernels considered in Theorem 4.7, our resultsinclude the classical Fredholm alternative for integral equations of the second kindthat was first obtained by Fredholm [55] and which we now state as a corollary.

Corollary 4.18. Let G ⊂ IRm be as in Theorem 4.4 and let K be a continuous orweakly singular kernel. Then either the homogeneous integral equations

ϕ(x) −∫

GK(x, y)ϕ(y) dy = 0, x ∈ G,

and

ψ(x) −∫

GK(y, x)ψ(y) dy = 0, x ∈ G,


only have the trivial solutions ϕ = 0 and ψ = 0 and the inhomogeneous integralequations

ϕ(x) −∫

GK(x, y)ϕ(y) dy = f (x), x ∈ G,

and

ψ(x) −∫

GK(y, x)ψ(y) dy = g(x), x ∈ G,

have a unique solution ϕ ∈ C(G) and ψ ∈ C(G) for each f ∈ C(G) and g ∈ C(G),respectively, or the homogeneous integral equations have the same finite numberm ∈ IN of linearly independent solutions and the inhomogeneous integral equationsare solvable if and only if the right-hand sides satisfy

∫G

f (x)ψ(x) dx = 0

for all solutions ψ of the homogeneous adjoint equation and∫

Gϕ(x)g(x) dx = 0

for all solutions ϕ of the homogeneous equation, respectively.

The original proof by Fredholm for this result for continuous kernels is based onconsidering the integral equations as a limiting case of systems of finite-dimensionallinear equations by approximating the integrals by Riemann sums. In Cramer’s rulefor this linear system Fredholm passes to the limit by using Koch’s theory of infinitedeterminants from 1896 and Hadamard’s inequality for determinants from 1893 (see[112]). The idea to view integral equations as the limiting case of linear systemshad already been used by Volterra in 1896, but it was Fredholm who completed itsuccessfully. We wish to mention that this original proof by Fredholm is shorter thanour more general approach, however, it is restricted to the case of integral equationswith continuous kernels.

Our results also include the so-called Riesz–Schauder theory as developed bySchauder [215] (see also [94, 112, 116, 229]) by taking Y = X∗, the dual spaceof X. Our more general form of the Fredholm alternative seems to be more appro-priate for the discussion of integral equations because of its symmetric structure.In particular, in our setting the adjoint of an integral equation in a function spaceagain is an integral equation in a function space, whereas in the Schauder theory theadjoint equation is an equation in the dual space of bounded linear functionals. Inthe case of C[0, 1], for example, the elements of the dual space are functions withbounded variation, i.e., are no longer necessarily continuous functions. Hence, theSchauder theory does not immediately include the classical results of Fredholm onintegral equations with continuous kernels. Furthermore, the above presentation ofthe Fredholm alternative does not make use of the Hahn–Banach theorem.

The Fredholm alternative in dual systems was first proven by Wendland [245,247] under the assumption that the bilinear or sesquilinear form is bounded. For a


pair of normed spaces X and Y, a bilinear form 〈· , ·〉 : X × Y → C is called boundedit there exists a positive real number C such that

|〈ϕ, ψ〉| ≤ C‖ϕ‖ ‖ψ‖for all ϕ ∈ X and ψ ∈ Y. Boundedness for a sesquilinear form is defined analo-gously. An elementary proof, which does not use the Hahn–Banach theorem, i.e.,Zorn’s lemma, and which also does not require the bilinear or sesquilinear form tobe bounded, was first given in [31] (see also [136]). The current version of the proofof Theorem 4.15, which differs slightly from the proof in [31] and in the first editionof this book, is due to Martensen [163]. For a history of the Fredholm alternative indual systems we refer to [139].

Example 4.19. Consider the integral equation

ϕ(x) −∫ b

aex−yϕ(y) dy = f (x), x ∈ [a, b]. (4.14)

As the simple integral equation on p. 1, this equation belongs to the class of equa-tions with so-called degenerate kernels which can be reduced to solving a finite-dimensional linear system. We shall meet degenerate kernels again later in Chap-ter 11 on the numerical solution of integral equations. Obviously a solution of (4.14)must be of the form

ϕ(x) = f (x) + cex, (4.15)

where c is a constant. Inserting (4.15) into (4.14), we observe that ϕ solves theintegral equation provided that c satisfies

c{1 − (b − a)} =∫ b

ae−y f (y) dy. (4.16)

Now either b − a � 1 or b − a = 1. In the first case (4.16) has a unique solutionleading to the unique solution

ϕ(x) = f (x) +1

1 − (b − a)

∫ b

ae−y f (y) dy ex

of the integral equation. In the second case, (4.16) has a solution if and only if

∫ b

ae−y f (y) dy = 0, (4.17)

and then any c satisfies (4.16). Note that, for b − a = 1, the function

ψ(x) = e−x


is the solution of the homogeneous adjoint equation

ψ(x) −∫ b

aey−xψ(y) dy = 0, x ∈ [a, b],

and therefore (4.17) coincides with the solvability condition of the Fredholm alter-native. �

Theorem 4.20. Let U be a subspace of a linear space X and assume that U andX are normed spaces carrying two different norms. Let A : X → X be a linearoperator such that A(U) ⊂ U and A : U → U and A : X → X both are compact.Further let Y be a normed space and let 〈· , ·〉 : X × Y → C be a non-degeneratebilinear form such that its restriction 〈· , ·〉 : U × Y → C is also non-degenerate andlet B : Y → Y be a compact operator that is adjoint to A with respect to this bilinearform. Then the nullspaces of I − A : U → U and I − A : X → X coincide.

Proof. By the first Fredholm Theorem 4.15, applied in the dual system < U, Y >,the dimensions of the nullspaces of I − A : U → U and I − B : Y → Y are finiteand coincide. Again by the first Fredholm Theorem 4.15, applied in the dual system< X, Y >, the dimensions of the nullspaces of I − A : X → X and I − B : Y → Y arealso finite and coincide. Thus the nullspaces of I − A : U → U and I − A : X → Xmust be the same. �

As an example for the application of Theorem 4.20 we consider the integral op-erator A with weakly singular kernel given by (2.4) both in C(G) and in L2(G). ByTheorem 2.30 and Problem 4.5 we have compactness of A in both spaces. There-fore, Theorem 4.20 implies that all the eigenfunctions, i.e., the eigenelements of anintegral operator with a weakly singular kernel are continuous. This idea to use theFredholm alternative in two different dual systems for showing that the eigenspacesfor weakly singular integral operators in the space of continuous functions and inthe L2 space coincide is due to Hahner [83].

4.4 Boundary Value Problems

Following Jorgens [112], we conclude this chapter by giving some flavor of the useof the Riesz–Fredholm theory to solve boundary value problems by considering theordinary differential equation of the second order

u′′ + a1u′ + a2u = v (4.18)

on the interval [0, 1] with coefficients a1, a2, v ∈ C[0, 1] subject to the boundaryconditions

u(0) = u0, u(1) = u1. (4.19)

4.4 Boundary Value Problems 59

The general idea in the application of integral equations in the treatment of boundaryvalue problems is to equivalently transform the boundary value problem into anintegral equation and then solve the integral equation.

Let u be twice continuously differentiable and set ϕ := −u′′. Then, by partialintegration we find the relations

u(x) = u(0) + u′(0)x −∫ x

0(x − y)ϕ(y) dy,

u(x) = u(1) − u′(1)(1 − x) +∫ 1

x(x − y)ϕ(y) dy,

0 =∫ 1

0ϕ(y) dy + u′(1) − u′(0).

Multiplying the first equation by (1 − x), the second by x, the third by (1 − x)x, andthen adding, we obtain

u(x) = u(0)(1 − x) + u(1)x +∫ x

0(1 − x)yϕ(y) dy +

∫ 1

xx(1 − y)ϕ(y) dy. (4.20)

Differentiating this equation yields

u′(x) = u(1) − u(0) −∫ x

0yϕ(y) dy +

∫ 1

x(1 − y)ϕ(y) dy. (4.21)

Now let u be a solution to the boundary value problem. Then, using the differ-ential equation (4.18) and the boundary condition (4.19), from (4.20) and (4.21) wededuce that ϕ = −u′′ satisfies the integral equation

ϕ(x) −∫ 1

0K(x, y)ϕ(y) dy = f (x), x ∈ [0, 1], (4.22)

with the kernel

K(x, y) :=

⎧⎪⎪⎪⎨⎪⎪⎪⎩y{a2(x)(1 − x) − a1(x)}, 0 ≤ y < x ≤ 1,

(1 − y){a2(x)x + a1(x)}, 0 ≤ x < y ≤ 1,

and the right-hand side

f (x) := (u1 − u0)a1(x) + {u0(1 − x) + u1x}a2(x) − v(x), x ∈ [0, 1].

Conversely, let ϕ ∈ C[0, 1] be a solution to the integral equation (4.22) and definea twice continuously differentiable function u by

u(x) := u0(1 − x) + u1x +∫ x

0(1 − x)yϕ(y) dy +

∫ 1

xx(1 − y)ϕ(y) dy, x ∈ [0, 1].


Then u(0) = u0, u(1) = u1, and by construction of the integral equation we have−u′′ = ϕ = a1u′ + a2u − v. Therefore, the boundary value problem and the integralequation are equivalent. In particular, via

w(x) :=∫ x

0(1 − x)yϕ(y) dy +

∫ 1

x(1 − y)xϕ(y) dy, x ∈ [0, 1],

the homogeneous boundary value problem

w′′ + a1w′ + a2w = 0, w(0) = w(1) = 0 (4.23)

and the homogeneous integral equation

ϕ(x) −∫ 1

0K(x, y)ϕ(y) dy = 0, x ∈ [0, 1], (4.24)

are equivalent. Note that w � 0 implies that ϕ � 0 and vice versa.Because the kernel K is continuous and bounded on 0 ≤ y < x ≤ 1 and on

0 ≤ x < y ≤ 1, it is weakly singular with α = 1. Hence, the Fredholm alternativeis valid: Either the inhomogeneous integral equation (4.22) is uniquely solvablefor each right-hand side f ∈ C[0, 1] and therefore the boundary value problemitself also is uniquely solvable for all inhomogeneities v ∈ C[0, 1] and all boundaryvalues u0 and u1, or the homogeneous integral equation (4.24), and consequently thehomogeneous boundary value problem, have nontrivial solutions.

The homogeneous boundary value problem (4.23) has at most one linearly in-dependent solution. To show this, let w1, w2 be two solutions of (4.23). Then thereexist constants λ1, λ2 such that λ1w

′1(0) + λ2w

′2(0) = 0 and |λ1| + |λ2| > 0. From

the homogeneous boundary conditions we also have that λ1w1(0) + λ2w2(0) = 0.Now the Picard–Lindelof uniqueness theorem for initial value problems implies thatλ1w1 + λ2w2 = 0 on [a, b], i.e., two solutions to the homogeneous boundary valueproblem (4.23) are linearly dependent. Therefore, if (4.23) has nontrivial solutions,it has one linearly independent solution and the homogeneous integral equation andits adjoint equation both also have one linearly independent solution.

Let ψ be a solution to the homogeneous adjoint equation

ψ(x) −∫ 1

0K(y, x)ψ(y) dy = 0, x ∈ [0, 1],

i.e.,

ψ(x) =∫ x

0(1 − x){a2(y)y + a1(y)}ψ(y) dy +

∫ 1

xx{a2(y)(1 − y) − a1(y)}ψ(y) dy.

Then ψ(0) = ψ(1) = 0 and for the derivative we find

ψ′(x) − a1(x)ψ(x) = −∫ 1

0{a2(y)y + a1(y)}ψ(y) dy +

∫ 1

xa2(y)ψ(y) dy,

4.4 Boundary Value Problems 61

whence(ψ′ − a1ψ)′ + a2ψ = 0

follows. By the Picard–Lindelof theorem this homogeneous adjoint boundary valueproblem again admits at most one linearly independent solution. Therefore, in thecase when the homogeneous boundary value problem has a nontrivial solution,the homogeneous adjoint integral equation and the homogeneous adjoint bound-ary value problem are equivalent. By the Fredholm alternative, the inhomogeneousintegral equation (4.22) and therefore the inhomogeneous boundary value problem(4.18) and (4.19) are solvable if and only if

∫ 1

0f (x)ψ(x) dx = 0 (4.25)

is satisfied for the solutions ψ of the homogeneous adjoint boundary value problem.Using the differential equation and the boundary condition for ψ, we find

∫ 1

0{(u1 − u0)a1(x) + [u0(1 − x) + u1x]a2(x)}ψ(x) dx = u0ψ

′(0) − u1ψ′(1).

Hence, the condition (4.25) and

∫ 1

0v(x)ψ(x) dx = u0ψ

′(0) − u1ψ′(1)

are equivalent, and we can summarize our results in the following form.

Theorem 4.21. Either the inhomogeneous boundary value problem

u′′ + a1u′ + a2u = v, u(0) = u0, u(1) = u1,

is uniquely solvable for all right-hand sides v and boundary values u0, u1 or thehomogeneous boundary value problem

w′′ + a1w′ + a2w = 0, w(0) = w(1) = 0,

and the homogeneous adjoint boundary value problem

(ψ′ − a1ψ)′ + a2ψ = 0, ψ(0) = ψ(1) = 0,

each have one linearly independent solution w and ψ, respectively. In the latter case,the inhomogeneous boundary value problem is solvable if and only if

∫ 1

0v(x)ψ(x) dx = u0ψ

′(0) − u1ψ′(1).


Problems

4.1. Let 〈· , ·〉 : C[a, b] × C[a, b] → IR be a degenerate bilinear form. Then there exists a functionf ∈ C[a, b] such that 〈 f , ψ〉 = 0 for all ψ ∈ C[a, b]. Since f � 0, without loss of generality we mayassume that f (a) = 1. The compact operators A, B : C[a, b]→ C[a, b] defined by Aϕ := ϕ(a) f andBψ := 0 are adjoint with respect to 〈· , ·〉. By showing that N(I − A) = span{ f } and N(I − B) = {0}demonstrate that for the validity of the Fredholm alternative the bilinear form necessarily must benon-degenerate.

4.2. Let X be the linear space of all functions ϕ ∈ C(0, 1] for which positive numbers M and α(depending on ϕ) exist such that |ϕ(x)| ≤ Mxα−1/2 for all x ∈ (0, 1]. Then X is a normed space withthe norm

‖ϕ‖ := supx∈(0,1]

√x |ϕ(x)|,

and 〈X, X〉 is a dual system with the bilinear form

〈ϕ, ψ〉 :=∫ 1

0ϕ(x)ψ(x) dx.

Show that the integral operators A, B : X → X with continuous kernel K defined as in Theorem4.7 are compact and adjoint. By using the sequence (ϕn) given by ϕn(x) := x1/n−1/2 show that thebilinear form is not bounded.

4.3. Formulate and prove the Fredholm alternative for a pair of operators S − A and T − B, whereS and T each have a bounded inverse and A and B are compact.

4.4. Show that under the assumptions of Theorem 4.17 the operators A and B both have Riesznumber one if and only if for each pair of basis ϕ1, . . . , ϕm and ψ1, . . . , ψm of the nullspaces N(I−A)and N(I − B) the matrix 〈ϕi, ψk〉, i, k = 1, . . . ,m, is nonsingular.

4.5. Use Lax’s Theorem 4.13 to show that the integral operator with weakly singular kernel ofTheorem 2.29 is a compact operator from L2(G) into L2(G) (see also Problem 2.3).

Chapter 5Regularization in Dual Systems

In this chapter we will consider equations that are singular in the sense that they arenot of the second kind with a compact operator. We will demonstrate that it is stillpossible to obtain results on the solvability of singular equations provided that theycan be regularized, i.e., they can be transformed into equations of the second kindwith a compact operator.

5.1 Regularizers

The following definition will say more precisely what we mean by regularizing abounded linear operator.

Definition 5.1. Let X1, X2 be normed spaces and let K : X1 → X2 be a boundedlinear operator. A bounded linear operator R� : X2 → X1 is called a left regularizerof K if

R�K = I − A�, (5.1)

where A� : X1 → X1 is compact; a bounded linear operator Rr : X2 → X1 is called aright regularizer of K if

KRr = I − Ar, (5.2)

where Ar : X2 → X2 is compact. A bounded linear operator R : X2 → X1 is called aregularizer of K if

RK = I − A� and KR = I − Ar, (5.3)

where A� : X1 → X1 and Ar : X2 → X2 are compact.

The difference between a left and a right regularizer is compact, since by multi-plying (5.1) from the right by Rr and (5.2) from the left by R� and then subtractingwe obtain Rr − R� = A�Rr − R�Ar, which is compact by Theorems 2.20 and 2.21.Again by Theorem 2.21, we observe that adding a compact operator to a regularizerpreserves the regularizing property. Therefore, provided that there exist a left and aright regularizer we may always assume that R� = Rr = R.


63

64 5 Regularization in Dual Systems

Let us first consider regularizing from the left. Any solution to the original equa-tion

Kϕ = f (5.4)

also solves the regularized equation

ϕ − A�ϕ = R� f . (5.5)

Therefore, by regularizing from the left we do not lose any solutions. Conversely,let ϕ be a solution of the regularized equation (5.5). Then

R�(Kϕ − f ) = 0,

and ϕ solves the original equation (5.4) provided that N(R�) = {0}. We call a leftregularizer an equivalent left regularizer if the original and the regularized equationhave the same solutions. Then, we have the following theorem.

Theorem 5.2. A left regularizer is an equivalent left regularizer if and only if it isinjective.

Now let us treat regularizing from the right. Here we have to compare the originalequation (5.4) and the regularized equation

ψ − Arψ = f . (5.6)

Provided that ψ solves the regularized equation, ϕ := Rrψ is a solution to the originalequation (5.4). Therefore, by regularizing from the right we do not create additionalsolutions. Conversely, to each solution ϕ of the original equation (5.4) there corre-sponds a solution ψ of the regularized equation (5.6) with Rrψ = ϕ provided thatRr(X2) = X1. We call a right regularizer an equivalent right regularizer if it mapsthe solutions of the regularized equation onto the solutions of the original equation.Then, we have the following theorem.

Theorem 5.3. A right regularizer is an equivalent right regularizer if and only if itis surjective.

From Theorems 5.2 and 5.3 we conclude that in a situation where we can estab-lish the existence of an injective left regularizer or a surjective right regularizer theresults of the Riesz theory partially carry over to the singular equation Kϕ = f . Inparticular, if K is injective and has an equivalent left regularizer, then K is surjectiveby Theorem 3.4. If K is surjective and has an equivalent right regularizer, then Kis injective. The transformation of a Volterra equation of the first kind into Volterraequations of the second kind as described in Section 3.3 may serve as a first example(see Problem 5.1).

5.2 Normal Solvability 65

5.2 Normal Solvability

In this section we want to demonstrate that it is also possible to obtain solvabilityresults by regularization that is not equivalent. To this end, we first prove thefollowing lemma.

Lemma 5.4. Under the assumptions of Theorem 4.15 let I − A have a nontrivialnullspace. Then the Riesz number r(A) of A and the Riesz number r(B) of B coincide,i.e., r(A) = r(B) = r ∈ IN. The corresponding projection operators

P : X → N[(I − A)r] and P′ : Y → N[(I − B)r]

defined by the direct sums

X = N[(I − A)r] ⊕ (I − A)r(X) and Y = N[(I − B)r] ⊕ (I − B)r(Y),

respectively, are adjoint. The elements a1, . . . , am and b1, . . . , bm entering in thedefinition (4.11) of the operator T can be chosen such that the operator S :=(I − A + T P) : X → X and its inverse S −1 : X → X both have adjoint opera-tors S ′ : Y → Y and [S −1]′ : Y → Y, respectively. The adjoint S ′ has an inverseand [S −1]′ = [S ′]−1.

Proof. By the Fredholm Theorem 4.15, for q ∈ IN, the two nullspaces N[(I − A)q]and N[(I−B)q] have the same finite dimension . Therefore, in view of Theorem 3.3,the Riesz numbers of A and B must coincide.

As a consequence of Theorem 4.16, applied to (I − A)r and (I − B)r, we have〈Pϕ, ψ − P′ψ〉 = 0 for ϕ ∈ X and ψ ∈ Y, since Pϕ ∈ N[(I − A)r] and ψ − P′ψ ∈(I − B)r(Y). Therefore

〈Pϕ, ψ〉 = 〈Pϕ, P′ψ + ψ − P′ψ〉 = 〈Pϕ, P′ψ〉for all ϕ ∈ X and ψ ∈ Y. Analogously, we have 〈ϕ, P′ψ〉 = 〈Pϕ, P′ψ〉, and conse-quently 〈Pϕ, ψ〉 = 〈ϕ, P′ψ〉 for all ϕ ∈ X and ψ ∈ Y.

Let ϕ ∈ N[(I−A)r] such that 〈ϕ, ψ〉 = 0 for all ψ ∈ N[(I−B)r]. Then, by Theorem4.16 we also have ϕ ∈ (I−A)r(X), and therefore ϕ = 0. Analogously,ψ ∈ N[(I−B)r]and 〈ϕ, ψ〉 = 0 for all ϕ ∈ N[(I−A)r] implies that ψ = 0. Therefore, the bilinear formgenerating the dual system 〈X, Y〉 is non-degenerate on N[(I − A)r] × N[(I − B)r].Using Lemma 4.14, this implies that we can choose the elements entering in thedefinition of T such that a1, . . . , am ∈ N[(I − B)r] and b1, . . . , bm ∈ N[(I − A)r], andconsequently PT = T .

The operator T ′ : Y → Y defined by

T ′ψ :=m∑

i=1

〈bi, ψ〉ai, ψ ∈ Y,


clearly is the adjoint of T and P′T ′ = T ′. Then we have

〈T Pϕ, ψ〉 = 〈PT Pϕ, ψ〉 = 〈ϕ, P′T ′P′ψ〉 = 〈ϕ, T ′P′ψ〉for all ϕ ∈ X and ψ ∈ Y, i.e., the operators T P and T ′P′ are adjoint. Analogous toT P, the operator T ′P′ is also compact. Therefore, by the Fredholm alternative, thebijectivity of S := I − A+ T P implies bijectivity of S ′ := I − B+ T ′P′. Now the laststatement of the lemma follows from

〈S −1 f , g〉 = 〈S −1 f , S ′[S ′]−1g〉 = 〈S S −1 f , [S ′]−1g〉 = 〈 f , [S ′]−1g〉for all f ∈ X and g ∈ Y, i.e., [S −1]′ = [S ′]−1. �

Theorem 5.5. Let X1, X2 be normed spaces, let K : X1 → X2 be a bounded linearoperator and let R : X2 → X1 be a regularizer of K. Then K and R have finite-dimensional nullspaces.

Proof. By assumption we have that RK = I−A� and KR = I−Ar, where A� and Ar arecompact. Let Kϕ = 0. Then RKϕ = 0, and therefore N(K) ⊂ N(I−A�). By Theorem3.1, N(I − A�) has finite dimension. Therefore dim N(K) ≤ dim N(I − A�) < ∞.Analogously N(R) ⊂ N(I − Ar) implies dim N(R) ≤ dim N(I − Ar) < ∞. �

Theorem 5.6. Assume that 〈X1, Y1〉 and 〈X2, Y2〉 are two dual systems and let K :X1 → X2, K′ : Y2 → Y1 and R : X2 → X1, R′ : Y1 → Y2 be two pairs of boundedand adjoint operators such that R is a regularizer of K and R′ is a regularizer of K′.Then the nullspaces of K and K′ have finite dimension, and the ranges are given by

K(X1) = { f ∈ X2 : 〈 f , ψ〉 = 0, ψ ∈ N(K′)}and

K′(Y2) = {g ∈ Y1 : 〈ϕ, g〉 = 0, ϕ ∈ N(K)}.Proof. The finite dimension of the nullspaces follows from Theorem 5.5. By symme-try it suffices to prove the statement on the ranges for the operator K. By assumptionwe have

RK = I − A, K′R′ = I − B,

where A : X1 → X1 and B : Y1 → Y1 are compact and adjoint.Let f ∈ K(X1), i.e., f = Kϕ for some ϕ ∈ X1. Then

〈 f , ψ〉 = 〈Kϕ, ψ〉 = 〈ϕ,K′ψ〉 = 0

for all ψ ∈ N(K′), and therefore

K(X1) ⊂ { f ∈ X2 : 〈 f , ψ〉 = 0, ψ ∈ N(K′)}.Conversely, assume that f ∈ X2 satisfies 〈 f , ψ〉 = 0 for all ψ ∈ N(K′). Then

〈R f , χ〉 = 〈 f ,R′χ〉 = 0

5.2 Normal Solvability 67

for all χ ∈ N(I − B), since K′R′χ = χ − Bχ = 0. Therefore, by Theorem 4.17, theregularized equation

ϕ − Aϕ = R f (5.7)

is solvable. If the regularizer R is injective, then by Theorem 5.2 each solution of(5.7) also satisfies the original equation Kϕ = f , and therefore in this case the proofis complete.

If R is not injective, by Theorem 5.5 its nullspace is finite-dimensional. Let n :=dim N(R) and choose a basis h1, . . . , hn of N(R). By Lemma 4.14 there exist elementsc1, . . . , cn ∈ Y2 such that

〈hi, ck〉 = δik, i, k = 1, . . . , n.

In the the case where I − A is injective, for the solution ϕ = (I − A)−1R f of (5.7) wecan write

Kϕ − f =n∑

k=1

βkhk,

whereβk = 〈K(I − A)−1R f − f , ck〉 = 〈 f , gk〉

andgk := R′(I − B)−1K′ck − ck

for k = 1, . . . , n. For all ψ ∈ X1 we have

〈ψ,K′gk〉 = 〈Kψ, gk〉 = 〈K(I − A)−1RKψ − Kψ, ck〉 = 0,

whence K′gk = 0 follows for k = 1, . . . , n. This implies 〈 f , gk〉 = 0 for k = 1, . . . , n,and therefore Kϕ = f , i.e., the proof is also complete in this case.

It remains to consider the case where both R and I−A have a nontrivial nullspace.Let m := dim N(I−A), choose a basis ϕ1, . . . , ϕm of N(I − A), and recall the analysisof Theorems 4.15 and 4.16 and Lemma 5.4. The general solution of the regularizedequation (5.7) has the form

ϕ =m∑

i=1

αiϕi + (I − A + T P)−1R f (5.8)

with complex coefficients α1, . . . , αm. Since for each solution ϕ of (5.7) we have thatKϕ − f ∈ N(R), as above we can write

Kϕ − f =n∑

k=1

βkhk,

where

βk =

m∑i=1

αi〈Kϕi, ck〉 + 〈K(I − A + T P)−1R f − f , ck〉, k = 1, . . . , n.


Therefore the solution (5.8) of the regularized equation solves the original equationif and only if the coefficients α1, . . . , αm solve the linear system

m∑i=1

αi〈Kϕi, ck〉 = 〈 f − K(I − A + T P)−1R f , ck〉, k = 1, . . . , n. (5.9)

If the matrix 〈Kϕi, ck〉 of this linear system has rank n, then (5.9) is solvable. Oth-erwise, i.e., for rank p < n, from elementary linear algebra we know that the condi-tions that are necessary and sufficient for the solvability of (5.9) can be expressed inthe form

n∑k=1

ρik〈 f − K(I − A + T P)−1R f , ck〉 = 0, i = 1, . . . , n − p,

for some matrix ρik, or in short form

〈 f , gi〉 = 0, i = 1, . . . , n − p, (5.10)

where

gi :=n∑

k=1

ρik

{ck − R′(I − B + T ′P′)−1K′ck

}, i = 1, . . . , n − p.

Now the proof will be completed by showing that gi ∈ N(K′)For ψ ∈ X1, observing that ψ − Aψ = RKψ, analogously to (5.8) we can write

ψ =m∑

i=1

αiϕi + (I − A + T P)−1RKψ

for some α1, . . . , αm ∈ C, and consequently

m∑i=1

αiKϕi + K(I − A + T P)−1RKψ − Kψ = 0.

This implies that the linear system (5.9) with f replaced by Kψ has a solution.Therefore, since the conditions (5.10) are necessary for the solvability,

〈ψ,K′gi〉 = 〈Kψ, gi〉 = 0, i = 1, . . . , n − p,

for all ψ ∈ X1. This now implies that K′gi = 0, i = 1, . . . , n − p. �

Comparing Theorem 5.6 with the Fredholm alternative for compact operators wenote that the only difference lies in the fact that the dimensions of the nullspaces,in general, are no longer the same. In particular, this implies that from injectivity ofK, in general, we cannot conclude surjectivity of K as in the Riesz theory. This alsowill give rise to the introduction of the index of an operator in the following section.

5.3 Index and Fredholm Operators 69

As in Chapter 4 our analysis includes the special case of the canonical dualsystems 〈X1, X∗1〉 and 〈X2, X∗2〉 with the dual spaces X∗1 and X∗2 of X1 and X2. In thissetting the solvability conditions of Theorem 5.6 usually are referred to as normalsolvability of the operator K, and in this case the results of Theorem 5.6 were firstobtained by Atkinson [9]. For the reasons already mentioned in Chapter 4, our moregeneral setting again seems to be more appropriate for the discussion of integralequations. This will become obvious from the various examples discussed in thefollowing chapters, which will include the classical results by Noether [185] onsingular integral equations with Hilbert kernels. For convenience, we reformulateTheorem 5.6 in terms of solvability conditions.

Corollary 5.7. Under the assumptions of Theorem 5.6 each of the homogeneousequations

Kϕ = 0 and K′ψ = 0

has at most a finite number of linearly independent solutions. The inhomogeneousequations

Kϕ = f and K′ψ = g

are solvable if and only if the conditions

〈 f , ψ〉 = 0 and 〈ϕ, g〉 = 0

are satisfied for all solutions ψ and ϕ of the homogeneous equations

K′ψ = 0 and Kϕ = 0,

respectively.

5.3 Index and Fredholm Operators

We conclude this chapter by introducing the concept of the index of an operator. LetU and V be subspaces of a linear space X such that

X = U ⊕ V.

Let n := dim V < ∞ and assume that W is another subspace W with the propertyX = U ⊕ W. Choose a basis v1, . . . , vn of V and let w1, . . . , wn+1 be n + 1 elementsfrom W. Then, since X = U ⊕ V , there exist elements u1, . . . , un+1 from U and amatrix ρik such that

wi = ui +

n∑k=1

ρikvk, i = 1, . . . , n + 1.


The homogeneous linear system

n+1∑i=1

ρikλi = 0, k = 1, . . . , n,

of n equations for n + 1 unknowns has a nontrivial solution λ1, . . . , λn+1. Then

n+1∑i=1

λiwi =

n+1∑i=1

λiui,

and consequentlyn+1∑i=1

λiwi = 0,

since U ∩ W = {0}. Hence the elements w1, . . . , wn+1 are linearly dependent, andtherefore dim W ≤ dim V . Interchanging the roles of V and W we also have dim V ≤dim W, whence dim V = dim W follows.

Therefore, the codimension of a subspace U of X is well defined by setting

codim U := dim V

if there exists a finite-dimensional subspace V such that X = U⊕V and codim U = ∞otherwise. In particular, codim X = 0. Obviously, the codimension is a measure forthe deviation of the subspace U from the whole space X.

Assume that U1 and U2 are subspaces of X such that codim U1 < ∞ and U1 ⊂ U2.Then X = U1 ⊕ V1, where V1 is a finite-dimensional subspace. For the finite-dimensional subspace V1∩U2 of V1 there exists another finite-dimensional subspaceV2 such that V1 = (V1 ∩ U2) ⊕ V2. Now let ϕ ∈ U2 ∩ V2. Then, since V2 ⊂ V1, wehave ϕ ∈ V1 ∩ U2. Therefore ϕ = 0, since (V1 ∩ U2) ∩ V2 = {0}. Now let ϕ ∈ Xbe arbitrary. Then, since X = U1 ⊕ V1, we have ϕ = u1 + v1 for some u1 ∈ U1 andv1 ∈ V1. Furthermore, since V1 = (V1 ∩ U2) ⊕ V2, we have v1 = w2 + v2 for somew2 ∈ V1 ∩ U2 and v2 ∈ V2. Then ϕ = u2 + v2, where u2 := u1 + w2 ∈ U2. Hencewe have proven that X = U2 ⊕ V2, and this implies that codim U2 ≤ codim U1 ifU1 ⊂ U2.

Definition 5.8. A linear operator K : X → Y from a linear space X into a linearspace Y is said to have finite defect if its nullspace has finite dimension and its rangehas finite codimension. The number

ind K := dim N(K) − codim K(X)

is called the index of the operator K.

Note that in the proof of the following theorem we do not use an adjoint operator,i.e., the analysis is based only on the Riesz theory.


Theorem 5.9. Let A be a compact operator on a normed space X. Then I − A hasindex zero.

Proof. Since by Theorem 3.4 the statement is obvious when I − A is injective, weonly need to consider the case where m := dim N(I − A) ∈ IN. From the direct sumX = N(I − A)r ⊕ (I − A)r(X) and (I − A)r(X) ⊂ (I − A)(X) we conclude that thereexists a finite-dimensional subspace U of X such that X = (I − A)(X) ⊕ U. We needto show that dim U = m.

Let n := dim U and note that n � 0 by Theorem 3.4. We choose bases ϕ1, . . . , ϕm

of N(I − A) and b1, . . . , bn of U. In the case where r > 1 we choose additionalelements ϕm+1, . . . , ϕmr with mr > m such that ϕ1, . . . , ϕm, ϕm+1, . . . , ϕmr is a basis ofN(I − A)r. Then we define a linear operator T : N(I − A)r → U by prescribing

T : ϕk →⎧⎪⎪⎪⎨⎪⎪⎪⎩

bk, k ≤ min(m, n),

0, k > min(m, n).(5.11)

By Theorem 2.4 the operator T is bounded. Consequently, since the projectionoperator P : X → N(I − A)r is compact, the operator T P : X → U is compact.

Now assume that m < n and let

ϕ − Aϕ + T Pϕ = 0.

Then, since (I − A)(X) ∩ U = {0}, we have ϕ − Aϕ = 0 and T Pϕ = 0. Therefore wecan write

ϕ =

m∑k=1

αkϕk,

and the definition (5.11) of T implies that

m∑k=1

αkbk = T Pϕ = 0.

From this we conclude that αk = 0, k = 1, . . . ,m, since the bk are linearlyindependent. Hence ϕ = 0, and therefore the operator I − A + T P is injective, andconsequently surjective by Theorem 3.4. Therefore the equation

ϕ − Aϕ + T Pϕ − bn = 0

is uniquely solvable. For its solution ϕ we conclude that T Pϕ = bn, since we have(I − A)(X) ∩ U = {0}. In view of the definition (5.11) of T this is a contradiction tothe linear independence of b1, . . . , bn.

Now assume that m > n. Since X = (I − A)(X) ⊕ U and T P(X) ⊂ U, we canrepresent each f ∈ X in the form

f = (I − A + T P)ϕ +n∑

k=1

αkbk


for some ϕ ∈ X and complex coefficients α1, . . . , αn. From (5.11) we conclude that(I − A + T P)ϕk = Tϕk = bk, k = 1, . . . , n, and therefore

f = (I − A + T P)

⎛⎜⎜⎜⎜⎜⎝ϕ +n∑

k=1

αkϕk

⎞⎟⎟⎟⎟⎟⎠ ,

i.e., the operator I − A + T P is surjective, and consequently injective by Theorem3.4. Since m > n, from (5.11) we have (I−A+T P)ϕm = Tϕm = 0, and the injectivityof I − A + T P leads to the contradiction ϕm = 0. �

Operators with closed range and finite defect are called Fredholm operators. Fora detailed analysis of Fredholm operators we refer to [94, 112]. From Theorem 3.3and Theorem 5.7 we have that for a compact operator A : X → X the operator I − Ais a Fredholm operator.

Theorem 5.10. Under the assumptions of Theorem 5.6 the operators K and K′ havefinite defect with index

ind K = dim N(K) − dim N(K′) = − ind K′.

Proof. Let n := dim N(K′) and choose a basis ψ1, . . . , ψn of N(K′) if n > 0. ByLemma 4.14 there exist elements b1, . . . , bn ∈ X2 such that

〈bi, ψk〉 = δik, i, k = 1, . . . , n. (5.12)

DefineU := span{b1, . . . , bn}.

Then K(X1)∩U = {0} by Theorem 5.6 and (5.12). Furthermore, for f ∈ X2 we have

u :=n∑

k=1

〈 f , ψk〉bk ∈ U

and f − u ∈ K(X1) by Theorem 5.6. Hence X2 = K(X1) ⊕ U, and thereforecodim K(X1) = dim U = dim N(K′). �

From Theorems 5.6 and 5.10 we observe that under the additional assumption ofboundedness for the two bilinear forms that create the dual systems the operators Kand K′ are Fredholm operators.

Theorem 5.11. For two operators K1 and K2 satisfying the assumptions of Theorem5.6 we have

ind K1K2 = ind K1 + ind K2.

Proof. For the sake of notational brevity we confine ourselves to the case of a dualsystem 〈X, Y〉 and two operators K1,K2 : X → X, which satisfy the assumptions ofTheorems 5.6 and 5.10, i.e., together with their adjoint operators K′1,K

′2 : Y → Y


they possess regularizers. Then K1K2 : X → X and its adjoint K′2K′1 : Y → Y satisfythe assumptions of Theorems 5.6 and 5.10.

Denote m j := dim N(Kj), m′j := dim N(K′j), and choose bases of the nullspaces

N(Kj) = span{ϕ j,1, . . . , ϕ j,mj}, N(K′j) = span{ψ j,1, . . . , ψ j,m′j}for j = 1, 2. Let ϕ ∈ N(K1K2). Then K2ϕ ∈ N(K1), i.e.,

K2ϕ =

m1∑i=1

αiϕ1,i.

By Theorem 5.6 this equation is solvable if and only if

m1∑i=1

αi〈ϕ1,i, ψ2,k〉 = 0, k = 1, . . . ,m′2. (5.13)

By p we denote the rank of the m1 × m′2 matrix 〈ϕ1,i, ψ2,k〉. Then the solution spaceof (5.13) has dimension m1 − p. Therefore

dim N(K1K2) = dim N(K2) + m1 − p = m2 + m1 − p.

Similarly, let ψ ∈ N(K′2K′1). Then K′1ψ ∈ N(K′2), i.e.,

K′1ψ =m′2∑k=1

βkψ2,k.

This equation is solvable if and only if

m′2∑k=1

βk〈ϕ1,i, ψ2,k〉 = 0, i = 1, . . . ,m1. (5.14)

The solution space of (5.14) has dimension m′2 − p. Therefore

dim N(K′2K′1) = dim N(K′1) + m′2 − p = m′1 + m′2 − p.

In view of Theorem 5.10, combining the two results yields

ind K1K2 = (m1 − m′1) + (m2 − m′2) = ind K1 + ind K2,

and the proof is complete. �

Corollary 5.12. Under the assumptions of Theorem 5.6 the index is stable with re-spect to compact perturbations, i.e., for compact operators C with a compact adjointC′ we have

ind(K +C) = ind K.


Proof. Let K and K′ be adjoint operators with adjoint regularizers R and R′. Then,since RK = I − A, where A is compact, Theorems 5.9 and 5.11 imply that

ind R + ind K = ind RK = ind(I − A) = 0,

i.e., ind K = − ind R. For a compact operator C the operator R also regularizes K+Cand the operator R′ regularizes K′ + C′. Therefore

ind(K +C) = − ind R = ind K,


For the history of the development of the notion of the index of an operator werefer to [44].

Of course, this chapter can provide only a first glance into the theory of singularoperators. For a detailed study, in the canonical dual system 〈X, X∗〉, we refer to themonograph by Mikhlin and Prossdorf [170].

Problems

5.1. Show that the transformations of the Volterra integral equation of the first kind (3.14) into theVolterra equations of the second kind (3.15) and (3.16) can be interpreted as regularizations fromthe left and from the right, respectively.Hint: Use the space C1[a, b] of continuously differentiable functions furnished with the norm‖ϕ‖1 := ‖ϕ‖∞ + ‖ϕ′‖∞.

5.2. Convince yourself where in the proof of Theorem 5.6 use is made of the fact that the operatorsK and K′ possess regularizers from the left and from the right.

5.3. Use Theorem 5.9 for an alternative proof of Theorem 4.17.

5.4. Let X1, X2 be Banach spaces, let K : X1 → X2 be a bounded operator, and let R : X2 → X1 bea left (right) regularizer of K. Show that for all operators C : X1 → X2 with ‖C‖ < ‖R‖ the operatorK +C has a left (right) regularizer.

5.5. Use Problem 5.4 to show that in Banach spaces under the assumptions of Theorem 5.6 theindex is stable with respect to small perturbations, i.e., there exists a positive number γ (dependingon K and K′) such that

ind(K +C) = ind K

for all operators C with adjoint C′ satisfying max(‖C‖, ‖C′‖) < γ (see [9, 43]).

Chapter 6Potential Theory

The solution of boundary value problems for partial differential equations is one ofthe most important fields of applications for integral equations. In the second half ofthe 19th century the systematic development of the theory of integral equations wasinitiated by the treatment of boundary value problems and there has been an ongo-ing fruitful interaction between these two areas of applied mathematics. It is the aimof this chapter to introduce the main ideas of this field by studying the basic bound-ary value problems of potential theory. For the sake of simplicity we shall confineour presentation to the case of two and three space dimensions. The extension tomore than three dimensions is straightforward. As we shall see, the treatment of theboundary integral equations for the potential theoretic boundary value problems de-livers an instructive example for the application of the Fredholm alternative, sinceboth its cases occur in a natural way. This chapter covers the classical approachto boundary integral equations of the second kind in the space of continuous func-tions. The treatment of boundary integral equations of the first and of the secondkind in Holder spaces and in Sobolev spaces will be the topic of the two subsequentchapters.

6.1 Harmonic Functions

We begin with a brief outline of the basic properties of harmonic functions go-ing back to the early development of potential theory at the beginning of the19th century with contributions by Dirichlet, Gauss, Green, Riemann and Weier-strass. For further study of potential theory we refer to Constanda [33], Courantand Hilbert [36], Folland [54], Helms [89], Kanwal [118], Kellogg [119], Marten-sen [162], and Mikhlin [169].

Definition 6.1. A twice continuously differentiable real-valued function u, definedon a domain D ⊂ IRm, m = 2, 3, is called harmonic if it satisfies Laplace’s equation

Δu = 0 in D,


75

76 6 Potential Theory

where

Δu :=m∑

j=1

∂2u

∂x2j

.

Harmonic functions describe time-independent temperature distributions, poten-tials of electrostatic and magnetostatic fields, and velocity potentials of incompress-ible irrotational fluid flows.

There is a close connection between harmonic functions in IR2 and holomorphicfunctions in C. From the Cauchy–Riemann equations we readily observe that boththe real and imaginary parts of a holomorphic function f (z) = u(x1, x2) + iv(x1, x2),z = x1 + ix2, are harmonic functions.

Most of the basic properties of harmonic functions can be deduced from thefundamental solution that is introduced in the following theorem. Recall that by |x|we denote the Euclidean norm of a vector x ∈ IRm.

Theorem 6.2. The function

Φ(x, y) :=

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

12π

ln1|x − y| , m = 2,

14π

1|x − y| , m = 3,

defined for all x � y in IRm is called the fundamental solution of Laplace’s equation.For fixed y ∈ IRm it is harmonic in IRm \ {y}.Proof. This follows by straightforward differentiation. �

For n ∈ IN, by Cn(D) we denote the linear space of real- or complex-valuedfunctions defined on the domain D, which are n times continuously differentiable.By Cn(D) we denote the subspace of all functions in Cn(D), which with all theirderivatives up to order n can be extended continuously from D into the closure D. Inthis chapter, we mostly deal with real-valued functions but with proper interpretationour results remain valid for complex-valued functions. From p. 30 we recall whatis meant by saying a bounded domain D or its boundary ∂D belong to class Cn forn ∈ IN.

One of the basic tools in studying harmonic functions is provided by Green’sintegral theorems. Recall that for two vectors a = (a1, . . . , am) and b = (b1, . . . , bm)in IRm we denote by a · b = a1b1 + · · · + ambm the dot product.

Theorem 6.3 (Green’s Theorem). Let D be a bounded domain of class C1 and letν denote the unit normal vector to the boundary ∂D directed into the exterior of D.Then, for u ∈ C1(D) and v ∈ C2(D) we have Green’s first theorem

∫D

(uΔv + grad u · grad v) dx =∫∂D

u∂v

∂νds (6.1)

6.1 Harmonic Functions 77

and for u, v ∈ C2(D) we have Green’s second theorem

∫D

(uΔv − vΔu) dx =∫∂D

(u∂v

∂ν− v ∂u

∂ν

)ds. (6.2)

Proof. We apply Gauss’ divergence theorem∫

Ddiv A dx =

∫∂Dν · A ds

to the vector field A ∈ C1(D) defined by A := u grad v and use

div(u grad v) = grad u · grad v + u div grad v

to establish (6.1). To obtain (6.2) we interchange u and v and then subtract. �

Note that our regularity assumptions on D are sufficient conditions for the validityof Gauss’ and Green’s theorems and can be weakened. In particular, the boundarycan be allowed to have edges and corners. For a detailed study see, for example,[169, 177].

Corollary 6.4. Let v ∈ C2(D) be harmonic in D. Then∫∂D

∂v

∂νds = 0. (6.3)

Proof. This follows by choosing u = 1 in (6.1). �

Theorem 6.5 (Green’s Formula). Let D be as in Theorem 6.3 and let u ∈ C2(D)be harmonic in D. Then

u(x) =∫∂D

{∂u∂ν

(y)Φ(x, y) − u(y)∂Φ(x, y)∂ν(y)

}ds(y), x ∈ D. (6.4)

Proof. For x ∈ D we choose a sphere Ω(x; r) := {y ∈ IRm : |y − x| = r} of radiusr such that Ω(x; r) ⊂ D and direct the unit normal ν to Ω(x; r) into the interior ofΩ(x; r). Now we apply Green’s second theorem (6.2) to the harmonic functions uand Φ(x, ·) in the domain {y ∈ D : |y − x| > r} to obtain

∫∂D∪Ω(x;r)

{u(y)

∂Φ(x, y)∂ν(y)

− ∂u∂ν

(y)Φ(x, y)

}ds(y) = 0.

Since on Ω(x; r) we have

grady Φ(x, y) =ν(y)

ωmrm−1, (6.5)


where ω2 = 2π, ω3 = 4π, a straightforward calculation, using the mean value theo-rem and (6.3), shows that

limr→0

∫Ω(x;r)

{u(y)

∂Φ(x, y)∂ν(y)

− ∂u∂ν

(y)Φ(x, y)

}ds(y) = u(x),

whence (6.4) follows. �

From Green’s formula we can conclude that harmonic functions are analyticfunctions of their independent variables.

Theorem 6.6. Harmonic functions are analytic, i.e., each harmonic function has alocal power series expansion.

Proof. For f ∈ C(∂D), we show that the function

u(x) :=∫∂DΦ(x, y) f (y) ds(y), x ∈ D,

is analytic in D and confine ourselves to the two-dimensional case. To this end wefix an arbitrary x0 ∈ D and choose R > 0 such that the disk B[x0; 3R] ⊂ D and defineV[x0; R] := {(x, y) : |x − x0| ≤ R, |y − x0| ≥ 3R}. With the aid of

|x − y|2 = |x0 − y|2{

1 + 2(x0 − y) · (x − x0)|x0 − y|2 +

|x − x0|2|x0 − y|2

}

we obtain the series

ln1|x − y| = ln

1|x0 − y| +

12

∞∑k=1

(−1)k

ktk

= ln1

|x0 − y| +12

∞∑k=1

(−1)k

k

k∑j=0

(kj

)ξ jηk− j,

(6.6)

where t := ξ + η with

ξ := 2(x0 − y) · (x − x0)|x0 − y|2 and η :=

|x − x0|2|x0 − y|2 .

For (x, y) ∈ V[x0; R] and k ∈ IN we can estimate∣∣∣∣∣∣∣∣(−1)k

k

k∑j=0

(kj

)ξ jηk− j

∣∣∣∣∣∣∣∣ ≤k∑

j=0

(kj

)|ξ| j|η|k− j = (|ξ| + |η|)k ≤ 7k

9k.

Therefore the series (6.6) has a convergent majorant and consequently it is abso-lutely and uniformly convergent in V[x0; R]. Hence we can reorder the series and


collect powers of ξ2 and η resulting in

ln1|x − y| = ln

1|x0 − y| +

∞∑k=1

⎧⎪⎪⎪⎨⎪⎪⎪⎩ξk∑

j=0

αk jξ2 jηk− j +

k∑j=0

βk jξ2 jηk− j

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (6.7)

with real coefficients αk j and βk j. Since the two inner sums in this expansion arehomogeneous polynomials of degree 2k + 1 and 2k, respectively, we can rewrite(6.7) in the form

ln1|x − y| = ln

1|x0 − y| +

∞∑k=1

∑j1+ j2=k

γ j1 j2 (x0, y)(x1 − x01) j1(x2 − x02) j2 (6.8)

with real coefficients γ j1 j2 depending on x0 = (x01, x02) and y and uniform conver-gence in V[x0; R]. Thus we may integrate the series (6.8) term by term and obtain

u(x) =∞∑

k=0

∑j1+ j2=k

c j1 j2 (x0)(x1 − x01) j1(x2 − x02) j2

with real coefficients c j1 j2 depending on x0 and uniform convergence for all x =(x1, x2) ∈ B[x0; R].

The proofs for the second term in Green’s formula and the three-dimensionalcase are analogous. �

An alternative proof of Theorem 6.6 makes use of the fact that each holomor-phic function of several complex variables, i.e., a function satisfying the Cauchy–Riemann equations with respect to each of the complex variables is also analyticand vice versa (see [64]). Then the theorem follows from the observation that thefundamental solution Φ(x, y) is an analytic function of the Cartesian coordinates x j,j = 1, . . . ,m, of x and the fact that the integrands in (6.4) and their derivatives withrespect to x are continuous with respect to y if x is contained in a compact subset ofD. Therefore the Cauchy–Riemann equations for u can be verified by differentiatingwith respect to x under the integral.

From Theorem 6.6 it follows that a harmonic function that vanishes in an opensubset of its domain of definition must vanish identically.

The following theorem is a special case of a more general result for partial dif-ferential equations known as Holmgren’s theorem.

Theorem 6.7. Let D be as in Theorem 6.5 and let u ∈ C2(D) ∩ C1(D) be harmonicin D such that

u =∂u∂ν= 0 on Γ (6.9)

for some open subset Γ ⊂ ∂D. Then u vanishes identically in D.


Proof. In view of (6.4), we use Green’s representation formula to extend thedefinition of u by setting

u(x) :=∫∂D\Γ

{∂u∂ν

(y)Φ(x, y) − u(y)∂Φ(x, y)∂ν(y)

}ds(y)

for x ∈ (IRm \ D) ∪ Γ. Then, by Green’s second integral theorem (6.2), applied to uand Φ(x, ·), we have u = 0 in IRm \ D. By G we denote a component of IRm \ D withΓ ∩ ∂G � ∅. Clearly u solves the Laplace equation in (IRm \ ∂D) ∪ Γ and thereforeu = 0 in D, since D and G are connected through the gap Γ in ∂D. �

Theorem 6.8 (Mean Value Theorem). Let u be harmonic in an open ball B(x; r) ={y ∈ IRm : |y − x| < r} with boundary Ω(x; r) and continuous in the closure B[x; r].Then

u(x) =m

ωmrm

∫B[x;r]

u(y) dy =1

ωmrm−1

∫Ω(x;r)

u(y) ds(y), (6.10)

i.e., the value of u at the center of the ball is equal to the integral mean values overboth the ball and its boundary surface (ω2 = 2π, ω3 = 4π).

Proof. For each 0 < ρ < r we have u ∈ C2(B[x; ρ]) and can apply (6.3) and (6.4)with the result

u(x) =1

ωmρm−1

∫|y−x|=ρ

u(y) ds(y), (6.11)

whence the second mean value formula follows by passing to the limit ρ → r.Multiplying (6.11) by ρm−1 and integrating with respect to ρ from 0 to r we obtainthe first mean value formula. �

Theorem 6.9 (Maximum-Minimum Principle). A harmonic function on a domaincannot attain its maximum or its minimum unless it is constant.

Proof. It suffices to carry out the proof for the maximum. Let u be a harmonic func-tion in the domain D and assume that it attains its maximum value in D, i.e., theset DM := {x ∈ D : u(x) = M} where M := supx∈D u(x) is not empty. Since u iscontinuous, DM is closed relative to D. Let x be any point in DM and apply the meanvalue Theorem 6.8 to the harmonic function M−u in a ball B(x; r) with B[x; r] ⊂ D.Then

0 = M − u(x) =m

ωmrm

∫B[x;r]{M − u(y)} dy,

so that u = M in B(x; r). Therefore DM is open relative to D. Hence D = DM , i.e., uis constant in D. �

Corollary 6.10. Let D be a bounded domain and let u be harmonic in D and con-tinuous in D. Then u attains both its maximum and its minimum on the boundary.

For the study of exterior boundary value problems we also need to investigatethe asymptotic behavior of harmonic functions as |x| → ∞. To this end we extendGreen’s formula to unbounded domains.


Theorem 6.11. Assume that D is a bounded domain of class C1 with a connectedboundary ∂D and outward unit normal ν and let u ∈ C2(IRm \ D) be a boundedharmonic function. Then

u(x) = u∞ +∫∂D

{u(y)

∂Φ(x, y)∂ν(y)

− ∂u∂ν

(y)Φ(x, y)

}ds(y) (6.12)

for x ∈ IRm \ D and some constant u∞. For m = 2, in addition,∫∂D

∂u∂ν

ds = 0 (6.13)

and the mean value property at infinity

u∞ =1

2πr

∫|y|=r

u(y) ds(y) (6.14)

for sufficiently large r is satisfied.

Proof. Without loss of generality we may assume that the origin x = 0 is containedin D. Since u is bounded, there exists a constant M > 0 such that |u(x)| ≤ M for allx ∈ IRm \ D. Choose R0 large enough to ensure that y ∈ IRm \ D for all |y| ≥ R0/2.Then for a fixed x with |x| ≥ R0 we can apply the mean value Theorem 6.8 to thecomponents of grad u. From this and Gauss’ integral theorem we obtain

grad u(x) =m

ωmrm

∫B[x;r]

grad u(y) dy = − mωmrm

∫Ω(x;r)

ν(y)u(y) ds(y),

where ν is the unit normal to Ω(x; r) directed into the interior of Ω(x; r) and wherewe choose the radius to be r = |x|/2. Then we can estimate

| grad u(x)| ≤ mMr=

2mM|x| (6.15)

for all |x| ≥ R0.For m = 2, we choose r large enough such that Ωr := Ω(0; r) is contained in

IR2 \ D and apply Green’s second theorem (6.2) to u and Φ(0, ·) in the annulusr < |y| < R and use (6.5) to obtain

1r

∫Ωr

u ds − ln1r

∫Ωr

∂u∂ν

ds =1R

∫ΩR

u ds − ln1R

∫ΩR

∂u∂ν

ds.

(Note that ν is the interior normal toΩr andΩR.) From this, with the aid of Corollary6.4 applied in the annulus between ∂D and Ωr , we find

1r

∫Ωr

u ds + ln1r

∫∂D

∂u∂ν

ds =1R

∫ΩR

u ds + ln1R

∫∂D

∂u∂ν

ds. (6.16)


Since the first term on the right-hand side is bounded by 2πM, letting R → ∞ in(6.16) implies that the integral in (6.13) must be zero. Note that (6.13) only holds inthe two-dimensional case and is a consequence of the fact that in IR2 the fundamentalsolution is not bounded at infinity.

For x ∈ IRm \ D, m = 2, 3, we now choose r large enough such that D ⊂ B(x; r).Then by Green’s formula (6.4), applied in the domain between ∂D and Ω(x; r), wehave that

u(x) =∫∂D∪Ω(x;r)

{u(y)

∂Φ(x, y)∂ν(y)

− ∂u∂ν

(y)Φ(x, y)

}ds(y). (6.17)

With the aid of Corollary 6.4 we find∫Ω(x;r)

∂u∂ν

(y)Φ(x, y) ds(y) =1

4πr

∫∂D

∂u∂ν

(y) ds(y)→ 0, r → ∞,

if m = 3, and∫Ω(x;r)

∂u∂ν

(y)Φ(x, y) ds(y) =1

2πln

1r

∫∂D

∂u∂ν

(y) ds(y) = 0

if m = 2, where we have made use of (6.13). With the aid of (6.5) we can write∫Ω(x;r)

u(y)∂Φ(x, y)∂ν(y)

ds(y) =1

ωmrm−1

∫|y−x|=r

u(y) ds(y).

From the mean value theorem we have

u(x + y) − u(y) = grad u(y + θx) · xfor some θ ∈ [0, 1], and using (6.15) we can estimate

|u(x + y) − u(y)| ≤ 2mM |x||y| − |x|

provided that |y| is sufficiently large. Therefore

1ωmrm−1

∣∣∣∣∣∣∫|y−x|=r

u(y) ds(y) −∫|y|=r

u(y) ds(y)

∣∣∣∣∣∣ ≤Cr

for some constant C > 0 depending on x and all sufficiently large r. Now choose asequence (rn) of radii with rn → ∞. Since the integral mean values

μn :=1

ωmrm−1n

∫|y|=rn

u(y) ds(y)

are bounded through |μn| ≤ M, n ∈ IN, by the Bolzano–Weierstrass theorem we mayassume that the sequence (μn) converges, i.e., μn → u∞, n → ∞, for some u∞ ∈ IR.

6.2 Boundary Value Problems: Uniqueness 83

From this, in view of the above estimates, we now have that∫Ω(x;rn)

{u(y)

∂Φ(x, y)∂ν(y)

− ∂u∂ν

(y)Φ(x, y)

}ds(y)→ u∞, n→ ∞.

Hence (6.12) follows by setting r = rn in (6.17) and passing to the limit n → ∞.Finally, (6.14) follows by setting R = rn in (6.16), passing to the limit n → ∞, andusing (6.13). �

From (6.12), using the asymptotic behavior of the fundamental solution

Φ(x, y) :=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

12π

ln1|x| + O

(1|x|

), m = 2,

O

(1|x|

), m = 3,

(6.18)

and∂Φ(x, y)∂x j

= O

(1|x|m−1

),

∂2Φ(x, y)∂x j∂xk

= O

(1|x|m

)(6.19)

for |x| → ∞, which holds uniformly for all directions x/|x| and all y ∈ ∂D, and theproperty (6.13) if m = 2, we can deduce that bounded harmonic functions in anexterior domain satisfy

u(x) = u∞ + O

(1|x|

), grad u(x) = O

(1|x|m−1

), |x| → ∞, (6.20)

uniformly for all directions.

6.2 Boundary Value Problems: Uniqueness

Green’s formula (6.4) represents any harmonic function in terms of its boundaryvalues and its normal derivative on the boundary, the so-called Cauchy data. In thesubsequent analysis we shall see that a harmonic function is already completelydetermined by either its boundary values or, up to a constant, its normal derivativealone. In the sequel, let D ⊂ IRm be a bounded domain of class C2. For the sake ofsimplicity for the rest of this chapter we assume that the boundary ∂D is connected.Again by νwe denote the unit normal of ∂D directed into the exterior domain IRm\D.

Interior Dirichlet Problem. Find a function u that is harmonic in D, is continuousin D, and satisfies the boundary condition

u = f on ∂D,

where f is a given continuous function.


Interior Neumann Problem. Find a function u that is harmonic in D, is continuousin D, and satisfies the boundary condition

∂u∂ν= g on ∂D

in the senselim

h→+0ν(x) · grad u(x − hν(x)) = g(x), x ∈ ∂D,

of uniform convergence on ∂D, where g is a given continuous function.

Exterior Dirichlet Problem. Find a function u that is harmonic in IRm \ D, is con-tinuous in IRm \ D, and satisfies the boundary condition

u = f on ∂D,

where f is a given continuous function. For |x| → ∞ it is required that

u(x) = O (1), m = 2, and u(x) = o (1), m = 3,

uniformly for all directions.

Exterior Neumann Problem. Find a function u that is harmonic in IRm \ D, iscontinuous in IRm \ D, and satisfies the boundary condition

∂u∂ν= g on ∂D

in the sense of uniform convergence on ∂D, where g is a given continuous function.For |x| → ∞ it is required that u(x) = o (1) uniformly for all directions.

Note that for the exterior problems we impose that u∞ = 0, with the exception ofthe Dirichlet problem in IR2, where u is only required to be bounded.

These boundary value problems carry the names of Dirichlet, who made impor-tant contributions to potential theory, and Neumann, who gave the first rigorousexistence proof (see Problem 6.5). From the numerous applications we mention:(1) Determine the stationary temperature distribution in a heat-conducting body

from the temperature on the boundary or from the heat flux through the boundary.(2) Find the potential of the electrostatic field in the exterior of a perfect conductor.(3) Find the velocity potential of an incompressible irrotational flow around an

obstacle.Our aim is to establish that each of the above potential theoretic boundary value

problems has a unique solution depending continuously on the given boundary data,i.e., they are well-posed in the sense of Hadamard (see Section 15.1).

In our uniqueness proofs we need to apply Green’s Theorem 6.3. Since forsolutions to the boundary value problems we do not assume differentiability up tothe boundary, we introduce the concept of parallel surfaces. These are described by

∂Dh := {z = x + hν(x) : x ∈ ∂D},

6.2 Boundary Value Problems: Uniqueness 85

with a real parameter h. Because ∂D is assumed to be of class C2, we observe that∂Dh is of class C1. For m = 3, let x(u) = (x1(u), x2(u), x3(u)), u = (u1, u2), be aregular parametric representation of a surface patch of ∂D . Then straightforwarddifferential geometric calculations show that the determinants

g(u) := det

[∂x∂ui· ∂x∂u j

]and g(u; h) := det

[∂z∂ui· ∂z∂u j

]

are related by

g(u; h) = g(u){1 − 2hH(u) + h2K(u)

}2,

where H and K denote the mean and Gaussian curvature of ∂D, respectively (see[162, 177]). This verifies that the parallel surfaces are well defined provided theparameter h is sufficiently small to ensure that 1−2hH+h2K remains positive. Thisalso ensures that in a sufficiently small neighborhood of ∂D each point z can beuniquely represented in the form z = x + hν(x), where x ∈ ∂D and h ∈ IR.

In particular, the surface elements ds on ∂D and dsh on ∂Dh are related by

dsh(z) ={1 − 2hH + h2K

}ds(x). (6.21)

Since ν(x) · ν(x) = 1, we have

∂ν(x)∂ui

· ν(x) = 0, i = 1, 2,

for all x ∈ ∂D, and therefore the tangential vectors

∂z∂ui=∂x∂ui+ h

∂ν(x)∂ui

, i = 1, 2,

for all (sufficiently small) h lie in the tangent plane to ∂D at the point x, i.e., thenormal vector νh(z) of the parallel surface ∂Dh coincides with the normal vectorν(x) of ∂D for all x ∈ ∂D. Hence, in view of (6.21), Theorems 6.5 and 6.11 remainvalid for harmonic functions u ∈ C(D) and u ∈ C(IRm \ D), respectively, providedthey have a normal derivative in the sense of uniform convergence.

Note that in two dimensions the equation (6.21) has to be replaced by dsh(z) =(1− κh)ds(x), where κ denotes the curvature of ∂D, i.e., for the representation ∂D ={x(s) : s0 ≤ s ≤ s1} in terms of the arc length we have κ = ν · x′′.Theorem 6.12. Both the interior and the exterior Dirichlet problems have at mostone solution.

Proof. The difference u := u1 − u2 of two solutions to the Dirichlet problem is a har-monic function that is continuous up to the boundary and satisfies the homogeneousboundary condition u = 0 on ∂D. Then, from the maximum-minimum principle ofCorollary 6.10 we obtain u = 0 in D for the interior problem, and observing thatu(x) = o (1), |x| → ∞, we also obtain u = 0 in IR3 \ D for the exterior problem inthree dimensions.


For the exterior problem in two dimensions, by the maximum-minimum principleTheorem 6.9 the supremum and the infimum of the bounded harmonic function u areeither attained on the boundary or equal to u∞. When the maximum and minimumare both attained on the boundary then from the homogeneous boundary conditionwe immediately have u = 0 in IRm \ D. If the supremum is equal to u∞, then fromu(x) ≤ u∞ for all x ∈ IR2 \ D and the mean value property (6.14) we observe thatu = u∞ in the exterior of some circle. Now we can apply the maximum principle tosee that u = u∞ in all of IR2 \ D and the homogeneous boundary condition finallyimplies u = 0 in IR2 \D. The case where the infimum is equal to u∞ is settled by thesame argument. �

Theorem 6.13. Two solutions of the interior Neumann problem can differ only by aconstant. The exterior Neumann problem has at most one solution.

Proof. The difference u := u1 − u2 of two solutions for the Neumann problem isa harmonic function continuous up to the boundary satisfying the homogeneousboundary condition ∂u/∂ν = 0 on ∂D in the sense of uniform convergence. For theinterior problem, suppose that u is not constant in D. Then there exists some closedball B contained in D such that

∫B| grad u|2dx > 0. From Green’s first theorem (6.1),

applied to the interior Dh of some parallel surface ∂Dh := {x − hν(x) : x ∈ ∂D} withsufficiently small h > 0, we derive

∫B| grad u|2dx ≤

∫Dh

| grad u|2dx =∫∂Dh

u∂u∂ν

ds.

Passing to the limit h→ 0, we obtain the contradiction∫

B| grad u|2dx ≤ 0. Hence, u

must be constant.For the exterior problem, assume that grad u � 0 in IRm \ D. Then again, there

exists some closed ball B contained in IRm \ D such that∫

B| grad u|2dx > 0. From

Green’s first theorem, applied to the domain Dh,r between some parallel surface∂Dh := {x+hν(x) : x ∈ ∂D}with sufficiently small h > 0 and some sufficiently largesphere Ωr of radius r centered at the origin (with interior normal ν), we obtain

∫B| grad u|2dx ≤

∫Dh,r

| grad u|2dx = −∫Ωr

u∂u∂ν

ds −∫∂Dh

u∂u∂ν

ds.

Letting r → ∞ and h → 0, with the aid of the asymptotics (6.20), we arrive at thecontradiction

∫B| grad u|2dx ≤ 0. Therefore, u is constant in IRm \ D and the constant

must be zero, since u∞ = 0. �

From the proofs it is obvious that our uniqueness results remain valid underweaker regularity conditions on the boundary. Uniqueness for the Dirichlet prob-lem via the maximum-minimum principle needs no regularity of the boundary,and uniqueness for the Neumann problem holds for those boundaries for whichGreen’s integral theorem is valid. We have formulated the boundary value problemsfor C2 boundaries, since we shall establish the existence of solutions under theseconditions.

6.3 Surface Potentials 87

6.3 Surface Potentials

Definition 6.14. Given a function ϕ ∈ C(∂D), the functions

u(x) :=∫∂Dϕ(y)Φ(x, y) ds(y), x ∈ IRm \ ∂D, (6.22)

and

v(x) :=∫∂Dϕ(y)

∂Φ(x, y)∂ν(y)

ds(y), x ∈ IRm \ ∂D, (6.23)

are called, respectively, single-layer and double-layer potential with density ϕ. Intwo dimensions, occasionally, for obvious reasons we will call them logarithmicsingle-layer and logarithmic double-layer potential.

For fixed y ∈ IRm the fundamental solution u = Φ(· , y) represents the potentialof a unit point source located at the point y, i.e., gradx Φ(x, y) gives the force-fieldof this point source acting at the point x. The single-layer potential is obtained bydistributing point sources on the boundary ∂D. For h > 0, by the mean value theoremwe have

Φ(x, y + hν(y)) −Φ(x, y − hν(y)) = 2h ν(y) · gradΦ(x, y + θhν(y))

for some θ = θ(y) ∈ [−1, 1]. Therefore, the double-layer potential can be interpretedas the limit h → 0 of the superposition of the single-layer potentials uh and u−h

with densities ϕ/2h on ∂Dh and −ϕ/2h on ∂D−h, respectively, i.e., the double-layerpotential is obtained by distributing dipoles on the boundary ∂D.

Since for points x � ∂D we can interchange differentiation and integration, thesingle- and double-layer potentials represent harmonic functions in D and IRm \ D.For the solution of the boundary value problems we need to investigate the behav-ior of the potentials at the boundary ∂D where the integrals become singular. Theboundary behavior is expressed by the following so-called jump relations.

Theorem 6.15. Let ∂D be of class C2 and ϕ ∈ C(∂D). Then the single-layer poten-tial u with density ϕ is continuous throughout IRm. On the boundary we have

u(x) =∫∂Dϕ(y)Φ(x, y)ds(y), x ∈ ∂D, (6.24)

where the integral exists as an improper integral.

Proof. Analogous to the proofs of Theorems 2.29 and 2.30, by using the cut-offfunction h, it can be shown that the single-layer potential u is the uniform limit of asequence of functions un that are continuous in IRm. �

For the further analysis of the jump relations we need the following lemma. Theinequality (6.25) expresses the fact that the vector x − y for x close to y is almostorthogonal to the normal vector ν(x).


Lemma 6.16. Let ∂D be of class C2. Then there exists a positive constant L suchthat

|ν(x) · {x − y}| ≤ L|x − y|2 (6.25)

and|ν(x) − ν(y)| ≤ L|x − y| (6.26)

for all x, y ∈ ∂D.

Proof. We confine ourselves to the two-dimensional case. For the three-dimensionalcase we refer to [31]. Let Γ = {x(s) : s ∈ [0, s0]} be a regular parameterizationof a patch Γ ⊂ ∂D, i.e., x : [0, 1] → Γ ⊂ ∂D is injective and twice continuouslydifferentiable with x′(s) � 0 for all s ∈ [0, s0]. Then, by Taylor’s formula we have

|ν(x(t)) · {x(t) − x(τ)}| ≤ 12

max0≤s≤s0

|x′′(s)| |t − τ|2,

|ν(x(t)) − ν(x(τ))| ≤ max0≤s≤s0

∣∣∣∣∣ dds

ν(x(s))∣∣∣∣∣ |t − τ|,

|x(t)) − x(τ)| ≥ min0≤s≤s0

|x′(s)| |t − τ|.

The statement of the lemma is evident from this. �

Example 6.17. For the double-layer potential with constant density we have

2∫∂D

∂Φ(x, y)∂ν(y)

ds(y) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

−2, x ∈ D,

−1, x ∈ ∂D,

0, x ∈ IRm \ D.

(6.27)

This follows for x ∈ IRm \ D from (6.3) applied to Φ(x, ·) and for x ∈ D from (6.4)applied to u = 1 in D. The result for x ∈ ∂D is derived by excluding x from theintegration by circumscribing it with a sphere Ω(x; r) of radius r and center x withthe unit normal directed toward the center. Let H(x; r) := Ω(x; r)∩D. Then, by (6.3)applied to Φ(x, ·), we have

∫{y∈∂D:|y−x|≥r}

∂Φ(x, y)∂ν(y)

ds(y) +∫

H(x;r)

∂Φ(x, y)∂ν(y)

ds(y) = 0,

and from

limr→0

2∫

H(x;r)

∂Φ(x, y)∂ν(y)

ds(y) = limr→0

2ωmrm−1

∫H(x;r)

ds(y) = 1

the result follows. �

6.3 Surface Potentials 89

Theorem 6.18. For ∂D of class C2, the double-layer potential v with continuousdensity ϕ can be continuously extended from D to D and from IRm \ D to IRm \ Dwith limiting values

v±(x) =∫∂Dϕ(y)

∂Φ(x, y)∂ν(y)

ds(y) ± 12ϕ(x), x ∈ ∂D, (6.28)

wherev±(x) := lim

h→+0v(x ± hν(x))

and where the integral exists as an improper integral.

Proof. Because of Lemma 6.16 we have the estimate∣∣∣∣∣∂Φ(x, y)∂ν(y)

∣∣∣∣∣ = |ν(y) · {x − y}|ωm|x − y|m ≤ L

ωm|x − y|m−2, x � y, (6.29)

i.e., the integral in (6.28) has a weakly singular kernel. Therefore, by Theorem 2.30the integral exists for x ∈ ∂D as an improper integral and represents a continuousfunction on ∂D.

As pointed out on p. 85, in a sufficiently small neighborhood U of ∂D we canrepresent every x ∈ U uniquely in the form x = z + hν(z), where z ∈ ∂D andh ∈ [−h0, h0] for some h0 > 0. Then we write the double-layer potential v withdensity ϕ in the form

v(x) = ϕ(z)w(x) + u(x), x = z + hν(z) ∈ U \ ∂D,

where

w(x) :=∫∂D

∂Φ(x, y)∂ν(y)

ds(y)

and

u(x) :=∫∂D{ϕ(y) − ϕ(z)} ∂Φ(x, y)

∂ν(y)ds(y). (6.30)

For x ∈ ∂D, i.e., for h = 0, the integral in (6.30) exists as an improper integraland represents a continuous function on ∂D. Therefore, in view of Example 6.17, toestablish the theorem it suffices to show that

limh→0

u(z + hν(z)) = u(z), z ∈ ∂D,

uniformly on ∂D.From (6.25) we can conclude that

|x − y|2 ≥ 12

{|z − y|2 + |x − z|2

}


for x = z + hν(z) and h ∈ [−h0, h0] provided that h0 is sufficiently small. Therefore,writing

∂Φ(x, y)∂ν(y)

=ν(y) · {z − y}ωm|x − y|m +

ν(y) · {x − z}ωm |x − y|m ,

and again using (6.25), we can estimate

∣∣∣∣∣∂Φ(x, y)∂ν(y)

∣∣∣∣∣ ≤ C1

{1

|x − y|m−2+

|x − z|[|z − y|2 + |x − z|2]m/2

}

for some constant C1 > 0. Recalling the proof of Theorem 2.30 and denoting∂D(z; r) := ∂D ∩ B[z; r], for sufficiently small r we project onto the tangent planeand deduce that

∫∂D(z;r)

∣∣∣∣∣∂Φ(x, y)∂ν(y)

∣∣∣∣∣ ds(y) ≤ C1

{∫ r

0dρ +

∫ r

0

|x − z|ρm−2 dρ(ρ2 + |x − z|2)m/2

}

≤ C1

{r +

∫ ∞

0

λm−2 dλ(λ2 + 1)m/2

}.

(6.31)

From the mean value theorem we obtain that∣∣∣∣∣∂Φ(x, y)∂ν(y)

− ∂Φ(z, y)∂ν(y)

∣∣∣∣∣ ≤ C2|x − z||z − y|m

for some constant C2 > 0 and 2|x − z| ≤ |z − y|. Hence we can estimate∫∂D\∂D(z:r)

∣∣∣∣∣∂Φ(x, y)∂ν(y)

− ∂Φ(z, y)∂ν(y)

∣∣∣∣∣ ds(y) ≤ C3|x − z|

rm(6.32)

for some constant C3 > 0 and |x − z| ≤ r/2. Now we can combine (6.31) and (6.32)to find that

|u(x) − u(z)| ≤ C

{max|y−z|≤r

|ϕ(y) − ϕ(z)| + |x − z|rm

}

for some constant C > 0, all sufficiently small r, and |x − z| ≤ r/2. Given ε > 0 wecan choose r > 0 such that

max|y−z|≤r

|ϕ(y) − ϕ(z)| ≤ ε

2C

for all z ∈ ∂D, since ϕ is uniformly continuous on ∂D. Then, taking δ < εrm/2C, wesee that |u(x) − u(z)| < ε for all |x − z| < δ, and the proof is complete. �

Theorem 6.19. Let ∂D be of class C2. Then for the single-layer potential u withcontinuous density ϕ we have

∂u±∂ν

(x) =∫∂Dϕ(y)

∂Φ(x, y)∂ν(x)

ds(y) ∓ 12ϕ(x), x ∈ ∂D, (6.33)

6.4 Boundary Value Problems: Existence 91

where∂u±∂ν

(x) := limh→+0

ν(x) · grad u(x ± hν(x))

is to be understood in the sense of uniform convergence on ∂D and where the inte-gral exists as an improper integral.

Proof. Let v denote the double-layer potential with density ϕ and let U be as in theproof of Theorem 6.18. Then for x = z + hν(z) ∈ U \ ∂D we can write

ν(z) · grad u(x) + v(x) =∫∂D{ν(y) − ν(z)} · grady Φ(x, y) ϕ(y) ds(y),

where we have made use of gradx Φ(x, y) = − grady Φ(x, y). Using (6.26), analogousto the single-layer potential in Theorem 6.15, the right-hand side can be seen to becontinuous in U. The proof is now completed by applying Theorem 6.18. �

Theorem 6.20. Let ∂D be of class C2. Then the double-layer potential v with con-tinuous density ϕ satisfies

limh→+0

ν(x) · {grad v(x + hν(x)) − grad v(x − hν(x))}= 0 (6.34)

uniformly for all x ∈ ∂D.

Proof. We omit the rather lengthy proof, which is similar in structure to the proof ofTheorem 6.18. For a detailed proof we refer to [31]. �

6.4 Boundary Value Problems: Existence

Green’s formula shows that each harmonic function can be represented as a combi-nation of single- and double-layer potentials. For boundary value problems we tryto find a solution in the form of one of these two potentials. To this end we introducetwo integral operators K,K′ : C(∂D)→ C(∂D) by

(Kϕ)(x) := 2∫∂Dϕ(y)

∂Φ(x, y)∂ν(y)

ds(y), x ∈ ∂D, (6.35)

and

(K′ψ)(x) := 2∫∂Dψ(y)

∂Φ(x, y)∂ν(x)

ds(y), x ∈ ∂D. (6.36)

Because of (6.29) the integral operators K and K′ have weakly singular kernelsand therefore are compact by Theorem 2.30. Note that in two dimensions for C2

boundaries the kernels of K and K′ actually turn out to be continuous (see Problem6.1). As seen by interchanging the order of integration, K and K′ are adjoint withrespect to the dual system 〈C(∂D),C(∂D)〉 defined by

〈ϕ, ψ〉 :=∫∂Dϕψ ds, ϕ, ψ ∈ C(∂D).


Theorem 6.21. The operators I − K and I − K′ have trivial nullspaces

N(I − K) = N(I − K′) = {0}.The nullspaces of the operators I + K and I + K′ have dimension one and

N(I + K) = span{1}, N(I + K′) = span{ψ0}with ∫

∂Dψ0 ds = 1, (6.37)

i.e., the Riesz number is one. The function ψ0 is called the natural charge on ∂D andthe single-layer potential with density ψ0 is constant in D.

Proof. Let ϕ be a solution to the homogeneous equation ϕ − Kϕ = 0 and define adouble-layer potential v by (6.23). Then by (6.28) we have 2v− = Kϕ − ϕ = 0 andfrom the uniqueness for the interior Dirichlet problem (Theorem 6.12) it followsthat v = 0 in D. From (6.34) we see that ∂v+/∂ν = 0 on ∂D, and since v(x) = o (1),|x| → ∞, from the uniqueness for the exterior Neumann problem (Theorem 6.13)we find that v = 0 in IRm \ D. Hence, from (6.28) we deduce ϕ = v+ − v− = 0 on ∂D.Thus N(I − K) = {0} and, by the Fredholm alternative, N(I − K′) = {0}.

Now let ϕ be a solution to ϕ+Kϕ = 0 and again define v by (6.23). Then by (6.28)we have 2v+ = Kϕ+ ϕ = 0 on ∂D. Since v(x) = o (1), |x| → ∞, from the uniquenessfor the exterior Dirichlet problem it follows that v = 0 in IRm \ D. From (6.34) we seethat ∂v−/∂ν = 0 on ∂D and from the uniqueness for the interior Neumann problemwe find that v is constant in D. Hence, from (6.28) we deduce that ϕ is constant on∂D. Therefore, N(I + K) ⊂ span{1}, and since by (6.27) we have 1 + K1 = 0, itfollows that N(I + K) = span{1}.

By the Fredholm alternative, N(I+K′) also has dimension one. Hence N(I+K′) =span{ψ0} with some function ψ0 ∈ C(∂D) that does not vanish identically. Assumethat 〈1, ψ0〉 = 0 and define a single-layer potential u with density ψ0. Then by (6.24)and (6.33) we have

u+ = u−,∂u−∂ν= 0, and

∂u+∂ν= −ψ0 on ∂D (6.38)

in the sense of uniform convergence. From ∂u−/∂ν = 0 on ∂D, by the uniquenessfor the interior Neumann problem (Theorem 6.13), we conclude that u is constantin D. Assume that u is not constant in IRm \ D. Then there exists a closed ball Bcontained in IRm \ D such that

∫B| grad u|2dx > 0. By Green’s theorem (6.1), using

the jump relations (6.38), the assumption 〈1, ψ0〉 = 0 and the fact that u+ is constanton ∂D, we find

∫B| grad u|2dx ≤ −

∫Ωr

u∂u∂ν

ds −∫∂D

u+∂u+∂ν

ds

= −∫Ωr

u∂u∂ν

ds +∫∂D

u+ψ0 ds = −∫Ωr

u∂u∂ν

ds


whereΩr denotes a sphere with sufficiently large radius r centered at the origin (andinterior normal ν). With the help of

∫∂Dψ0ds = 0, using (6.18) and (6.19), it can

be seen that u has the asymptotic behavior (6.20) with u∞ = 0. Therefore, passingto the limit r → ∞, we arrive at the contradiction

∫B| grad u|2dx ≤ 0. Hence, u is

constant in IRm \ D and from the jump relation (6.38) we derive the contradictionψ0 = 0. Therefore, we can normalize such that 〈1, ψ0〉 = 1. The statement on theRiesz number is a consequence of Problem 4.4. �

Theorem 6.22. The double-layer potential

u(x) =∫∂Dϕ(y)

∂Φ(x, y)∂ν(y)

ds(y), x ∈ D, (6.39)

with continuous density ϕ is a solution of the interior Dirichlet problem providedthat ϕ is a solution of the integral equation

ϕ(x) − 2∫∂Dϕ(y)

∂Φ(x, y)∂ν(y)

ds(y) = −2 f (x), x ∈ ∂D. (6.40)

Proof. This follows from Theorem 6.18. �

Theorem 6.23. The interior Dirichlet problem has a unique solution.

Proof. The integral equation ϕ − Kϕ = −2 f of the interior Dirichlet problem isuniquely solvable by Theorem 3.4, since N(I − K) = {0}. �

From Theorem 6.15 we see that in order to obtain an integral equation of thesecond kind for the Dirichlet problem it is crucial to seek the solution in the form ofa double-layer potential rather than a single-layer potential, which would lead to anintegral equation of the first kind. Historically, this important observation goes backto Beer [16].

The double-layer potential approach (6.39) for the exterior Dirichlet problemleads to the integral equation ϕ + Kϕ = 2 f for the density ϕ. Since N(I + K′) =span{ψ0}, by the Fredholm alternative, this equation is solvable if and only if〈 f , ψ0〉 = 0. Of course, for arbitrary boundary data f we cannot expect this con-dition to be satisfied. Therefore we modify our approach as follows.

Theorem 6.24. The modified double-layer potential

u(x) =∫∂Dϕ(y)

{∂Φ(x, y)∂ν(y)

+1|x|m−2

}ds(y), x ∈ IRm \ D, (6.41)

with continuous density ϕ is a solution to the exterior Dirichlet problem providedthat ϕ is a solution of the integral equation

ϕ(x) + 2∫∂Dϕ(y)

{∂Φ(x, y)∂ν(y)

+1|x|m−2

}ds(y) = 2 f (x), x ∈ ∂D. (6.42)

Here, we assume that the origin is contained in D.


Proof. This again follows from Theorem 6.18. Observe that u has the requiredbehavior for |x| → ∞, i.e., u(x) = O (1) if m = 2 and u(x) = o (1) if m = 3. �

Theorem 6.25. The exterior Dirichlet problem has a unique solution.

Proof. The integral operator K : C(∂D)→ C(∂D) defined by

(Kϕ)(x) := 2∫∂Dϕ(y)

{∂Φ(x, y)∂ν(y)

+1|x|m−2

}ds(y), x ∈ ∂D,

is compact, since the difference K−K has a continuous kernel. Let ϕ be a solution tothe homogeneous equation ϕ+Kϕ = 0 and define u by (6.41). Then 2u = Kϕ+ϕ = 0on ∂D, and by the uniqueness for the exterior Dirichlet problem it follows that u = 0in IRm \ D. Using (6.19), we deduce the asymptotic behavior

|x|m−2u(x) =∫∂Dϕ ds + O

(1|x|

), |x| → ∞,

uniformly for all directions. From this, since u = 0 in IRm\D, we obtain∫∂Dϕ ds = 0.

Therefore ϕ + Kϕ = 0, and from Theorem 6.21 we conclude that ϕ is constant on∂D. Now

∫∂Dϕ ds = 0 implies that ϕ = 0, and the existence of a unique solution to

the integral equation (6.42) follows from Theorem 3.4. �

Theorem 6.26. The single-layer potential

u(x) =∫∂Dψ(y)Φ(x, y) ds(y), x ∈ D, (6.43)

with continuous density ψ is a solution of the interior Neumann problem providedthat ψ is a solution of the integral equation

ψ(x) + 2∫∂Dψ(y)

∂Φ(x, y)∂ν(x)

ds(y) = 2g(x), x ∈ ∂D. (6.44)

Proof. This follows from Theorem 6.19. �

Theorem 6.27. The interior Neumann problem is solvable if and only if∫∂Dg ds = 0 (6.45)

is satisfied.

Proof. The necessity of condition (6.45) is a consequence of Green’s theorem (6.3)applied to a solution u. Its sufficiency follows from the fact that by Theorem 6.21 itcoincides with the solvability condition of the Fredholm alternative for the inhomo-geneous integral equation (6.44), i.e., for ψ + K′ψ = 2g. �


Theorem 6.28. The single-layer potential

u(x) =∫∂Dψ(y)Φ(x, y) ds(y), x ∈ IRm \ D, (6.46)

with continuous density ψ is a solution of the exterior Neumann problem providedthat ψ is a solution of the integral equation

ψ(x) − 2∫∂Dψ(y)

∂Φ(x, y)∂ν(x)

ds(y) = −2g(x), x ∈ ∂D, (6.47)

and, if m = 2, also satisfies ∫∂Dψ ds = 0. (6.48)

Proof. Again this follows from Theorem 6.19. Observe that for m = 2 the additionalcondition (6.48) ensures that u has the required behavior u(x) = o (1), |x| → ∞, ascan be seen from (6.18). �

Theorem 6.29. In IR3 the exterior Neumann problem has a unique solution. In IR2

the exterior Neumann problem is uniquely solvable if and only if∫∂Dg ds = 0 (6.49)

is satisfied.

Proof. By Theorems 3.4 and 6.21 the equation ψ − K′ψ = −2g is uniquely solvablefor each right-hand side g. If (6.49) is satisfied, using the fact that 1 + K1 = 0, wefind

2〈1, ψ〉 = 〈1 − K1, ψ〉 = 〈1, ψ − K′ψ〉 = −2〈1, g〉 = 0.

Hence, the additional property (6.48) is satisfied in IR2. That condition (6.49) is nec-essary for the solvability in IR2 follows from (6.13). �

We finally show that the solutions depend continuously on the given boundarydata.

Theorem 6.30. The solutions to the Dirichlet and Neumann problems depend con-tinuously in the maximum norm on the given data.

Proof. For the Dirichlet problem the assertion follows from the maximum-minimumprinciple (Theorem 6.9). In two dimensions, for the exterior problem, from the form(6.41) of the solution u we observe that we have to incorporate the value u∞ at infin-ity through

∫∂Dϕ ds. But this integral depends continuously on the given boundary

data, since the inverse (I + K)−1 of I + K is bounded by Theorem 3.4.For the Neumann problem we first observe that for single-layer potentials u with

continuous density ψ for any closed ball B in IRm we have an estimate of the form

‖u‖∞,B ≤ ‖w‖∞,B‖ψ‖∞,∂D,


where the function

w(x) :=∫∂D|Φ(x, y)| ds(y), x ∈ IRm,

is continuous in IRm by Theorem 6.15. Then for the exterior problem choose a suffi-ciently large ball B and the continuous dependence of the solution on the boundarydata in B follows from the boundedness of the inverse (I − K′)−1 of I − K′. Inthe remaining exterior of B, continuity then follows from the maximum-minimumprinciple.

For the interior problem we can expect continuity only after making the solutionu unique by an additional condition, for example, by requiring that

∫∂D

u ds = 0.From 〈1,K′ψ〉 = 〈K1, ψ〉 = −〈1, ψ〉 we observe that K′ maps the closed subspaceC0(∂D) :=

{ψ ∈ C(∂D) :

∫∂Dψ ds = 0

}into itself. By Theorem 6.21 the operator

I + K′ has a trivial nullspace in C0(∂D). Hence, the inverse (I + K′)−1 is boundedfrom C0(∂D) onto C0(∂D), i.e., the unique solution ψ0 of ψ0 + K′ψ0 = g satisfying∫∂Dψ0 ds = 0 depends continuously on g. Therefore, as above, the corresponding

single-layer potential u0 depends continuously on g in the maximum norm. Finally,u := u0 −

∫∂D

u0 ds/|∂D| yields a solution vanishing in the integral mean on theboundary, and it depends continuously on g. �

6.5 Nonsmooth Boundaries

Despite the fact that the integral equation method of this chapter provides an ele-gant approach to constructively prove the existence of solutions for the boundaryvalue problems of potential theory we do not want to disguise its major drawback:the relatively strong regularity assumption on the boundary to be of class C2. It ispossible to slightly weaken the regularity and allow Lyapunov boundaries insteadof C2 boundaries and still remain within the framework of compact operators. Theboundary is said to satisfy a Lyapunov condition if at each point x ∈ ∂D the normalvector ν exists and there are positive constants L and α such that for the angle ϑ(x, y)between the normal vectors at x and y the estimate ϑ(x, y) ≤ L|x − y|α holds for allx, y ∈ ∂D. For the treatment of the Dirichlet and Neumann problem for Lyapunovboundaries, which does not differ essentially from that for C2 boundaries, we referto [169].

However, the situation changes considerably if the boundary is allowed to haveedges and corners. This effects the form of the integral equations and the compact-ness of the integral operators as we will demonstrate by considering the interiorDirichlet problem in a two-dimensional domain D with corners. We assume that theboundary ∂D is piecewise twice differentiable, i.e., ∂D consists of a finite numberof closed arcs Γ1, . . . , Γp that are all of class C2 and that intersect only at the cor-ners x1, . . . , xp. At the corners the normal vector is discontinuous (see Fig. 6.1 for adomain with three corners).

6.5 Nonsmooth Boundaries 97

For simplicity, we restrict our analysis to boundaries that are straight lines in aneighborhood of each of the corners. In particular, this includes the case where ∂Dis a polygon. The interior angle at the corner xi we denote by γi and assume that0 < γi < 2π, i = 1, . . . , p, i.e., we exclude cusps. For a boundary with corners,the continuity of the double-layer potential with continuous density as stated inTheorem 6.18 remains valid, but at the corners the jump relation (6.28) has to bemodified into the form

v±(xi) =∫∂Dϕ(y)

∂Φ(xi, y)∂ν(y)

ds(y) ± 12δ±i ϕ(xi), i = 1, . . . , p, (6.50)

where δ+i = γi/π and δ−i = 2 − γi/π. It is a matter of straightforward application ofGreen’s theorem as in Example 6.17 to verify (6.50) for constant densities. For arbi-trary continuous densities, the result can be obtained from the C2 case of Theorem6.18 by a superposition of two double-layer potentials on two C2 curves intersectingat the corner with the density ϕ equal to zero on the parts of the two curves lyingoutside ∂D.

Trying to find the solution to the interior Dirichlet problem in the form of adouble-layer potential with continuous density ϕ as in Theorem 6.22 reduces theboundary value problem to solving the integral equation ϕ − Kϕ = −2 f , where theoperator K : C(∂D)→ C(∂D) is given by

(Kϕ)(x) :=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

(Kϕ)(x), x � xi, i = 1, . . . , p,

(Kϕ)(x) +(γi

π− 1

)ϕ(xi), x = xi, i = 1, . . . , p.

Note that for ϕ ∈ C(∂D), in general, Kϕ is not continuous at the corners. HoweverKϕ is continuous, since it is the sum Kϕ = v++v− of the continuous boundary valuesof the double-layer potential v.

By Problem 6.1 the kernel

k(x, y) :=ν(y) · {x − y}π|x − y|2

of the integral operator K is continuous on Γi×Γi for i = 1, . . . , p. Singularities of thekernel occur when x and y approach a corner on the two different arcs intersectingat the corner.

For n ∈ IN we use the continuous cutoff function h introduced in the proof ofTheorem 2.29 to define the operators Kn : C(∂D)→ C(∂D) by

(Knϕ)(x) :=∫∂D

h(n|x − y|)k(x, y)ϕ(y) ds(y), x ∈ ∂D.

For each n ∈ IN the operator Kn is compact, since its kernel is continuous on ∂D×Γi

for i = 1, . . . , p, i.e., we can interpret Kn as the sum of p integral operators with


continuous kernels on ∂D × Γi by subdividing the integral over ∂D into a sum ofintegrals over the arcs Γi for i = 1, . . . , p.

Γ3

Γ2

Γ1

x1

x2

x3

γ1

γ2

γ3

α

xB

A

D

z

Fig. 6.1 Domain with corners

Now consider Kn := K − Kn and assume that n is large enough that for eachx ∈ ∂D the disk B[x; 1/n] = {y ∈ IR2 : |x − y| ≤ 1/n} intersects only either one or,in the vicinity of the corners, two of the arcs Γi. By our assumption on the nature ofthe corners we can assume n is large enough that in the second case the intersectionconsists of two straight lines A and B (see Fig. 6.1). Let

M := maxi=1,...,p

maxx,y∈Γi

|k(x, y)|.

Then, by projection onto the tangent line, for the first case we can estimate

|(Knϕ)(x)| ≤ M‖ϕ‖∞∫∂D∩B[x;1/n]

ds(y) ≤ M‖ϕ‖∞ 4n.

In the second case, we first note that for x ∈ B \ {xi}, by Green’s theorem (6.3)applied in the triangle with the corners at x and at the endpoints xi and z of A wehave ∫

A

∣∣∣∣∣∂Φ(x, y)∂ν(y)

∣∣∣∣∣ ds(y) =∣∣∣∣∣∫

A

∂Φ(x, y)∂ν(y)

ds(y)∣∣∣∣∣ = α(x)

2π,

where α(x) denotes the angle of this triangle at the corner x (see Fig. 6.1). Elemen-tary triangle geometry shows that α(x) + γi ≤ π, where, without loss of generality,we have assumed that γi < π. Therefore, since for x ∈ B \ {xi} we have k(x, y) = 0for all y ∈ B \ {xi}, we obtain

|(Knϕ)(x)| ≤ 2‖ϕ‖∞∣∣∣∣∣∫

A

∂Φ(x, y)∂ν(y)

ds(y)∣∣∣∣∣ ≤ α(x)

π‖ϕ‖∞ ≤

(1 − γi

π

)‖ϕ‖∞.

6.5 Nonsmooth Boundaries 99

Finally, for the corner xi at the intersection of A and B we have

(Knϕ)(xi) =(γi

π− 1

)ϕ(xi),

since k(xi, y) = 0 for all y ∈ (A ∪ B) \ {xi}. Combining these results we observe thatwe can choose n large enough that ‖Kn‖∞ ≤ q where

q := maxi=1,...,p

∣∣∣∣∣1 − γi

π

∣∣∣∣∣ < 1.

Hence, we have a decomposition I − K = I − Kn − Kn, where I − Kn has a boundedinverse by the Neumann series Theorem 2.14 and where Kn is compact. It is leftto the reader to carry over the proof for injectivity of the operator I − K from The-orem 6.21 to the case of a boundary with corners. (For the application of Green’sintegral theorem analogous to the proof of Theorem 6.13 one needs to apply thedominated convergence theorem for Lebesgue integration; for a similar situationsee [32, Lemma 3.10].) Then existence of a solution to the inhomogeneous equationϕ − Kϕ = −2 f follows by Corollary 3.6.

This idea of decomposing the integral operator into a compact operator and abounded operator with norm less than one reflecting the behavior at the cornersgoes back to Radon [201] and can be extended to the general two-dimensional caseand to three dimensions. For details we refer to Cryer [37], Kral [131], and Wend-land [246]. For a more comprehensive study of boundary value problems in domainswith corners we refer to Grisvard [73]. For integral equations of the second kind inLipschitz domains we refer to Verchota [241].

Finally, we wish to mention that the integral equations for the Dirichlet and Neu-mann problems can also be treated in the space L2(∂D) allowing boundary data inL2(∂D). This requires the boundary conditions to be understood in a weak sense,which we want to illustrate by again considering the interior Dirichlet problem. Wesay that a harmonic function u in D assumes the boundary values f ∈ L2(∂D) in theL2 sense if

limh→+0

∫∂D

[u(x − hν(x)) − f (x)]2ds(x) = 0.

To establish uniqueness under this weaker boundary condition, we choose parallelsurfaces ∂Dh := {x − hν(x) : x ∈ ∂D} to ∂D with h > 0 sufficiently small. Then,following Miranda [171], for

J(h) :=∫∂Dh

u2ds, h > 0,

we can write

J(h) =∫∂D

{1 + 2hH(x) + h2K(x)

}[u(x − hν(x))]2ds(x)


and differentiate to obtain

12

dJdh= −

∫∂Dh

u∂u∂ν

ds +∫∂D{H(x) + hK(x)} [u(x − hν(x))]2ds(x).

Hence, using Green’s theorem (6.1), we have

12

dJdh= −

∫Dh

| grad u|2dx +∫∂D{H + hK} [u(· − hν)]2ds, (6.51)

where Dh denotes the interior of the parallel surface ∂Dh. Now let u vanish on theboundary ∂D in the L2 sense and assume that grad u � 0 in D. Then there existssome closed ball B contained in D such that I :=

∫B| grad u|2dx > 0, and from

(6.51) we deduce that dJ/dh ≤ −I for all 0 < h ≤ h0 and some sufficiently smallh0 > 0. Since J is continuous on [0, h0], is continuously differentiable on (0, h0], andsatisfies J(0) = 0, we see that J(h) ≤ −Ih for all 0 < h ≤ h0. This is a contradictionto J(h) ≥ 0 for all h > 0. Therefore u must be constant in D, and from J(0) = 0 weobtain u = 0 in D.

Using the fact that, due to Theorem 6.6, on the parallel surfaces ∂Dh for h > 0there is more regularity of u, a different approach to establishing uniqueness underweaker assumptions was suggested by Calderon [23]. It is based on representing uin terms of the double-layer operator Kh on ∂Dh and establishing ‖Kh−K‖L2(∂D) → 0as h→ 0.

To prove existence of a solution for boundary conditions in the L2 sense via thesurface potential approach, it is necessary to extend the jump relations of Theorem6.15, 6.18, 6.19, and 6.20 from C(∂D) onto L2(∂D). This can be achieved quiteelegantly through the use of Lax’s Theorem 4.13 as worked out by Kersten [120].In particular, for the double-layer potential v with density ϕ ∈ L2(∂D), the jumprelation (6.28) has to be replaced by

limh→+0

∫∂D

[2v(x ± hν(x)) − (Kϕ)(x) ∓ ϕ(x)]2ds(x) = 0. (6.52)

From this, we see that the double-layer potential with density ϕ ∈ L2(∂D) solves theDirichlet problem with boundary values f ∈ L2(∂D) provided the density solvesthe integral equation ϕ − Kϕ = −2 f in the space L2(∂D). Noting that integral op-erators with weakly singular kernels are compact from L2(∂D) into L2(∂D) (seeProblem 4.5), for existence and uniqueness of a solution to this integral equationwe need to establish that the homogeneous equation admits only the trivial solution.From Theorem 6.21 we know that the operator I−K has a trivial nullspace in C(∂D).Therefore we can apply Theorem 4.20 to obtain that I−K also has a trivial nullspacein L2(∂D). Corresponding to (6.52), for later use we also note the jump relation

limh→+0

∫∂D

[2∂u∂ν

(x ± hν(x)) − (K′ϕ)(x) ± ϕ(x)

]2

ds(x) = 0 (6.53)

for the single-layer potential u with density ϕ ∈ L2(∂D).

Problems 101

We will come back to the potential theoretic boundary value problems in thenext two chapters. In Section 7.5 we will solve the interior Dirichlet and Neumannproblems in two dimensions by integral equations of the first kind in a Holder spacesetting. Then in Section 8.3, again in two dimensions, we will solve the integralequations of the first and second kind in Sobolev spaces leading to weak solutionsof the boundary value problems.

For integral equation methods for boundary value problems for the Helmholtzequation Δu + κ2u = 0, i.e., for acoustic and electromagnetic scattering problems,we refer to [31, 32].

Problems

6.1. Use a regular 2π-periodic parameterization ∂D = {x(t) = (x1(t), x2(t)) : 0 ≤ t ≤ 2π} withcounterclockwise orientation for the boundary curve to transform the integral equation (6.40) ofthe interior two-dimensional Dirichlet problem into the form

ϕ(t) −∫ 2π

0k(t, τ)ϕ(τ) dτ = −2 f (t), 0 ≤ t ≤ 2π,

where ϕ(t) := ϕ(x(t)), f (t) := f (x(t)) and the kernel is given by

k(t, τ) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

1π

[x′(τ)]⊥ · {x(t) − x(τ)}|x(t) − x(τ)|2 , t � τ,

12π

[x′(t)]⊥ · x′′(t)|x′(t)|2 , t = τ,

where [x′]⊥ := (x′2,−x′1). Show that this kernel is continuous provided ∂D is of class C2.

6.2. Show that for an ellipse with parametric representation x(t) = (a cos t, b sin t), the kernel k ofProblem 6.1 is given by

k(t, τ) = −abπ

1a2 + b2 − (a2 − b2) cos(t + τ)

.

6.3. Extend Theorem 6.21 to domains D with non-connected boundaries and, in particular, showthat dim N(I − K) = p, where p denotes the number of bounded components of IRm \ D. Forthe interior Dirichlet problem in IR3 establish existence of a solution through a modification ofthe integral equation (6.40) analogous to (6.41) by adding a point source in each of the boundedcomponents of IR3 \ D. (For the two-dimensional case we refer to Problem 18.1.)

6.4. Let D ⊂ IR2 be of class C2 and strictly convex in the sense that the curvature of the boundary∂D is strictly positive. Show that there exists a constant 0 < δ < 1 such that

∫∂D

∣∣∣∣∣∂Φ(x1, y)∂ν(y)

− ∂Φ(x2, y)∂ν(y)

∣∣∣∣∣ ds(y) ≤ 1 − δ

for all x1, x2 ∈ ∂D.Hint: Use Example 6.17, Problem 6.1, and the property that

∂Φ(x, y)∂ν(y)

=ν(y) · {x − y}

2π|x − y|2


is negative on ∂D × ∂D to verify that∣∣∣∣∣∣∫Γ

{∂Φ(x1, y)∂ν(y)

− ∂Φ(x2, y)∂ν(y)

}ds(y)

∣∣∣∣∣∣ ≤12− a|∂D|,

for each Jordan measurable subset Γ ⊂ ∂D, where

a := minx,y∈∂D

∣∣∣∣∣∂Φ(x, y)∂ν(y)

∣∣∣∣∣ > 0.

6.5. In 1870 Neumann [184] gave the first rigorous proof for the existence of a solution to thetwo-dimensional interior Dirichlet problem in a strictly convex domain of class C2. By completelyelementary means he established that the successive approximations

ϕn+1 :=12ϕn +

12

Kϕn − f , n = 0, 1, 2, . . . ,

with arbitrary ϕ0 ∈ C(∂D) converge uniformly to the unique solution ϕ of the integral equationϕ − Kϕ = −2 f . In functional analytic terms his proof amounted to showing that the operator Lgiven by L := 1

2 (I + K) is a contraction with respect to the norm

‖ϕ‖ := | supz∈∂D

ϕ(z) − infz∈∂D

ϕ(z)| + α supz∈∂D|ϕ(z)|,

where α > 0 is appropriately chosen. This norm is equivalent to the maximum norm. Derive theabove results for yourself.Hint: Use Problem 6.4 to show that ‖L‖ ≤ (2 − δ + α)/2 by writing

(Lϕ)(x) =∫∂D

[ϕ(y) − ϕ(x)]∂Φ(x, y)∂ν(y)

ds(y)

and

(Lϕ)(x1) − (Lϕ)(x2) =12

[ϕ(x1) − ϕ(x)] − 12

[ϕ(x2) − ϕ(x)]

+

∫∂D

[ϕ(y) − ϕ(x)]

{∂Φ(x1 , y)∂ν(y)

− ∂Φ(x2, y)∂ν(y)

}ds(y),

where x ∈ ∂D is chosen such that ϕ(x) ={supz∈∂D ϕ(z) + infz∈∂D ϕ(z)

}/2 (see [117]).

Chapter 7Singular Boundary Integral Equations

In this chapter we will consider one-dimensional singular integral equationsinvolving Cauchy principal values that arise from boundary value problems forholomorphic functions in the classical Holder space setting. The investigations ofthese integral equations with Cauchy kernels by Gakhov, Muskhelishvili, Vekua,and others have had a great impact on the further development of the general the-ory of singular integral equations. For our introduction to integral equations theywill provide an application of the general idea of regularizing singular operatorsas described in Chapter 5. We assume the reader is acquainted with basic complexanalysis.

For a comprehensive study of singular integral equations with Cauchy kernelswe refer to Gakhov [59], Meister [167], Mikhlin and Prossdorf [170], Muskhel-ishvili [180], Prossdorf [197], Vekua [240], and Wendland [248].

A central piece of our analysis will be provided by the Sokhotski–Plemelj jumpand regularity relations for the Cauchy integral. These will also be used at the endof the chapter to derive further regularity properties for single- and double-layerpotentials in Holder spaces and an existence analysis for the Dirichlet and Neumannproblem for the Laplace equation via integral equations of the first kind.

7.1 Holder Continuity

For our investigation of singular integrals we first need to introduce the concept ofHolder continuity.

Definition 7.1. A real- or complex- or vector-valued function ϕ defined on a setG ⊂ IRm is called uniformly Holder continuous with Holder exponent 0 < α ≤ 1 ifthere exists a constant C such that

|ϕ(x) − ϕ(y)| ≤ C|x − y|α


103

104 7 Singular Boundary Integral Equations

for all x, y ∈ G. Here, for a vector-valued function on the left-hand side the barindicates the Euclidean norm (as on the right-hand side). By C0,α(G) we denote thelinear space of all functions defined on G that are bounded and uniformly Holdercontinuous with exponent α. The space C0,α(G) is called a Holder space.

Note that each uniformly Holder continuous function is uniformly continu-ous, but the converse is not true. We illustrate by two examples that uniformlyHolder continuous functions live between continuous and differentiable functions.The function ϕ : [0, 1/2] → IR, given by ϕ(x) = 1/ ln x for x ∈ (0, 1/2] andϕ(0) = 0, is uniformly continuous but not uniformly Holder continuous. The func-tion ψ : [0, 1] → IR, given by ψ(x) =

√x, is uniformly Holder continuous with

exponent 1/2 but not continuously differentiable on [0, 1]. In general, by the meanvalue theorem, a continuously differentiable function on a convex set with boundedderivatives is uniformly Holder continuous with exponent 1.

Theorem 7.2. The Holder space C0,α(G) is a Banach space with the norm

‖ϕ‖α := supx∈G|ϕ(x)| + sup

x,y∈Gx�y

|ϕ(x) − ϕ(y)||x − y|α .

Proof. It is clear that

|ϕ|α := supx,y∈Gx�y

|ϕ(x) − ϕ(y)||x − y|α

defines a semi-norm on C0,α(G), i.e., it satisfies all norm axioms with the exceptionof the definiteness (N2). Then ‖·‖α = ‖·‖∞+|·|α is a norm, since ‖ϕ‖∞ := supx∈G |ϕ(x)|defines a norm. Convergence in the supremum norm ‖ · ‖∞ is equivalent to uniformconvergence on G. If G is compact, the supremum norm and the maximum norm onC(G) coincide.

It remains to show that C0,α(G) is complete. Let (ϕn) be a Cauchy sequence inC0,α(G). Then obviously (ϕn) also is a Cauchy sequence with respect to the supre-mum norm and, by the sufficiency of the Cauchy criterion for uniform convergence,there exists a function ϕ ∈ C(G) such that ‖ϕn − ϕ‖∞ → 0, n → ∞. Because (ϕn)is a Cauchy sequence in C0,α(G), given ε > 0, there exists N(ε) ∈ IN such that|ϕn − ϕk |α < ε for all n, k ≥ N(ε), i.e.,

|{ϕn(x) − ϕk(x)} − {ϕn(y) − ϕk(y)}| < ε|x − y|α

for all n, k ≥ N(ε) and all x, y ∈ G. Since ϕn → ϕ, n → ∞, uniformly on G, byletting k → ∞ we have

|{ϕn(x) − ϕ(x)} − {ϕn(y) − ϕ(y)}| ≤ ε|x − y|α

for all n ≥ N(ε) and all x, y ∈ G. From this we conclude that ϕ ∈ C0,α(G) and|ϕn − ϕ|α ≤ ε for all n ≥ N(ε), which implies ‖ϕn − ϕ‖α → 0, n→ ∞. �

7.1 Holder Continuity 105

Note that the product of two uniformly Holder continuous functions ϕ and ψ isagain uniformly Holder continuous with

‖ϕψ‖α ≤ ‖ϕ‖∞‖ψ‖∞ + ‖ϕ‖∞|ψ|α + ‖ψ‖∞|ϕ|α ≤ ‖ϕ‖α‖ψ‖α.By the following technical lemma we illustrate that Holder continuity is a local

property.

Lemma 7.3. Assume that the function ϕ satisfies |ϕ(x)| ≤ M for all x ∈ G and

|ϕ(x) − ϕ(y)| ≤ C|x − y|α

for all x, y ∈ G with |x − y| ≤ a and some constants a,C, M, and 0 < α ≤ 1. Thenϕ ∈ C0,α(G) with

‖ϕ‖α ≤ M +max

(C,

2Maα

).

Proof. If |x − y| > a then

|ϕ(x) − ϕ(y)| ≤ 2M ≤ 2M

( |x − y|a

)α,

whence |ϕ|α ≤ max (C, 2M/aα) follows. �

In particular, from Lemma 7.3, we see that for α < β each function ϕ ∈ C0,β(G)is also contained in C0,α(G). For this imbedding we have the following compactnessproperty.

Theorem 7.4. Let 0 < α < β ≤ 1 and let G be compact. Then the imbeddingoperators

Iβ : C0,β(G)→ C(G)

andIα,β : C0,β(G)→ C0,α(G)

are compact.

Proof. Let U be a bounded set in C0,β(G), i.e., ‖ϕ‖β ≤ C for all ϕ ∈ U and somepositive constant C. Then we have

|ϕ(x)| ≤ C

for all x ∈ G and|ϕ(x) − ϕ(y)| ≤ C|x − y|β (7.1)

for all x, y ∈ G and ϕ ∈ U, i.e., U is bounded and equicontinuous. Therefore, by theArzela–Ascoli Theorem 1.18, the set U is relatively compact in C(G), which impliesthat the imbedding operator Iβ : C0,β(G)→ C(G) is compact.


It remains to verify that U is relatively compact in C0,α(G). From (7.1) we deducethat

|{ϕ(x) − ψ(x)} − {ϕ(y) − ψ(y)}|

= |{ϕ(x) − ψ(x)} − {ϕ(y) − ψ(y)}|α/β|{ϕ(x) − ψ(x)} − {ϕ(y) − ψ(y)}|1−α/β

≤ (2C)α/β|x − y|α(2‖ϕ − ψ‖∞)1−α/β

for all ϕ, ψ ∈ U and all x, y ∈ G. Therefore

|ϕ − ψ|α ≤ (2C)α/β21−α/β‖ϕ − ψ‖1−α/β∞

for all ϕ, ψ ∈ U, and from this we can conclude that each sequence taken from Uand converging in C(G) also converges in C0,α(G). This completes the proof. �

The following theorem illustrates how the above tools can be used to estab-lish compactness of boundary integral operators in the Holder space setting andis adopted from [31].

Theorem 7.5. For 0 < α < 1 and ∂D ∈ C2, the boundary integral operators K andK′ from potential theory as defined by (6.35) and (6.36) are compact operators fromC0,α(∂D) into itself.

Proof. We abbreviate

k(x, y) := 2∂Φ(x, y)∂ν(y)

, x, y ∈ ∂D, x � y,

and from Lemma 6.16 we observe that there exists a constant M1 depending on ∂Dsuch that

|k(x, y)| ≤ M1

|x − y|m−2, x � y. (7.2)

Using the mean value theorem and the estimates (6.25) and (6.26), from the decom-positions

ωm

2{k(x1, y) − k(x2, y)} =

{1

|x1 − y|m −1

|x2 − y|m}ν(y) · {x2 − y} + ν(y) · {x1 − x2}

|x1 − y|m

andν(y) · {x1 − x2} = ν(x1) · {x1 − x2} + {ν(y) − ν(x1)} · {x1 − x2}

we find that

|k(x1, y) − k(x2, y)| ≤ M2

{ |x1 − x2||x1 − y|m−1

+|x1 − x2|2|x1 − y|m

}(7.3)

for all x1, x2, y ∈ ∂D with 2|x1 − x2| ≤ |x1 − y| and some constant M2 dependingon ∂D.

7.1 Holder Continuity 107

As in the proof of Theorem 2.30 we choose 0 < R < 1 such that ∂D(x; R) :=∂D∩B[x; R] can be projected bijectively onto the tangent plane at x for each x ∈ ∂D.Now let x1, x2 ∈ ∂D be such that 0 < |x1 − x2| < R/4 and for

r := 4|x1 − x2|set ∂D(x; r) := ∂D∩B[x; r]. Then, using (7.2), with the aid of ∂D(x1, r) ⊂ ∂D(x2, 2r)we find ∣∣∣∣∣∣

∫∂D(x1,r)

{k(x1, y) − k(x2, y)}ϕ(y) ds(y)

∣∣∣∣∣∣

≤ M1‖ϕ‖∞{∫

∂D(x1,r)

ds(y)|x1 − y|m−2

+

∫∂D(x2,2r)

ds(y)|x2 − y|m−2

}

≤ C1‖ϕ‖∞|x1 − x2|for all ϕ ∈ C(∂D) and some constant C1 depending on ∂D. From (7.3) we obtain

∣∣∣∣∣∣∫∂D(x1,R)\D(x1,r)

{k(x1, y) − k(x2, y)}ϕ(y) ds(y)

∣∣∣∣∣∣

≤ M2‖ϕ‖∞{|x1 − x2|

∫ R

r/4

dρρ+ |x1 − x2|2

∫ R

r/4

dρρ2

}

≤ C2‖ϕ‖∞|x1 − x2|α

for all ϕ ∈ C(∂D) and some constant C2 depending on ∂D and α. Here we have usedthat

|x1 − x2| ln 1|x1 − x2| ≤

11 − α |x1 − x2|α

for |x1 − x2| < 1. Finally, making again use of (7.3) we can estimate∣∣∣∣∣∣∫∂D\D(x1,R)

{k(x1, y) − k(x2, y)}ϕ(y) ds(y)

∣∣∣∣∣∣

≤ M2‖ϕ‖∞{|x1 − x2|

∫∂D\D(x1,R)

ds(y)|x1 − y|m−1

+ |x1 − x2|2∫∂D\D(x1,R)

ds(y)|x1 − y|m

}

≤ C3‖ϕ‖∞|x1 − x2|for all ϕ ∈ C(∂D) and some constant C3 depending on ∂D. Summing up the lastthree inequalities we obtain that

|(Kϕ)(x1) − (Kϕ)(x2)| ≤ C4‖ϕ‖∞|x1 − x2|α

for all x1.x2 ∈ ∂D with |x1 − x2| < R/4 and some constant C4 depending on ∂Dand α. Now Lemma 7.3 together with the boundedness of K : C(∂D) → C(∂D)


implies that ‖Kϕ‖α ≤ C‖ϕ‖∞ for all ϕ ∈ C(∂D) and some constant C. Hence,K : C(∂D) → C0,α(∂D) is bounded and the statement of the theorem follows fromTheorems 2.21 and 7.4. The proof for the operator K′ is analogous. �

Since the Definition 7.1 of Holder continuity also covers vector fields, for a setG ⊂ IRm, we can introduce the Holder space C1,α(G) of uniformly Holder contin-uously differentiable functions as the space of differentiable functions ϕ for whichgradϕ (or the surface gradient Gradϕ in the case G = ∂D) belongs to C0,α(G). Withthe norm

‖ϕ‖1,α := ‖ϕ‖∞ + ‖ gradϕ‖0,αthe Holder space C1,α(G) is again a Banach space and we also have an imbeddingtheorem corresponding to Theorem 7.4.

Theorem 7.6. For 0 < α < 1, the boundary integral operator K defined by (6.35) isa compact operator from C1,α(∂D) into itself.

Proof. This can be proven with techniques similar to those used in the proof of theprevious Theorem 7.5 (see [31]). �

Note, that for C2 boundaries ∂D we cannot expect Theorem 7.6 to also hold forthe operator K′ since the derivative of K′ϕ contains the derivative of the normalvector ν which is only continuous.

7.2 The Cauchy Integral Operator

Now let D be a bounded and simply connected domain in the complex plane. Wedenote its boundary by Γ := ∂D and assume it to be of class C2. The normal vectorν is directed into the exterior of D. For complex integration along the contour Γ weassume that the direction of integration is counterclockwise. We confine our analysisto this basic configuration and note that it can be extended, for example, to multiplyconnected domains with less regular boundaries.

The Cauchy integral

f (z) :=1

2πi

∫Γ

ϕ(ζ)ζ − z

dζ, z ∈ C \ Γ,

with density ϕ ∈ C(Γ) defines a function that is holomorphic in C \ D and in D.Obviously, for z varying in an open domain not intersecting with Γ, the integrandis continuous with respect to ζ and continuously differentiable with respect to z.Therefore, we can differentiate under the integral to verify the Cauchy–Riemannequations for f . Occasionally, we will call a function that is holomorphic in C \ Dand in D sectionally holomorphic.

As in the case of the single- and double-layer potentials of Chapter 6 we areinterested in the behavior of the Cauchy integral for points on the boundary, where

7.2 The Cauchy Integral Operator 109

the integral becomes singular. By integrating the equation (ln ln x)′ = (x ln x)−1

between 0 ≤ x ≤ 1 we observe that the Cauchy integral for points on the boundary,in general, will not exist if the density is merely continuous. Therefore we assumethe density to be uniformly Holder continuous. Note that the introduction of Holdercontinuity by Definition 7.1 also covers functions on subsets of the complex planeby identifying IR2 and C. For the remainder of this chapter we will always assumethat α ∈ (0, 1) for the Holder exponent, despite the fact that part of our results remainvalid for α = 1.

Theorem 7.7. For densities ϕ ∈ C0,α(Γ), 0 < α < 1, the Cauchy integral exists as aCauchy principal value

12πi

∫Γ

ϕ(ζ)ζ − z

dζ =1

2πilimρ→0

∫Γ\Γ(z;ρ)

ϕ(ζ)ζ − z

dζ

for all z ∈ Γ. Here, Γ(z; ρ) := {ζ ∈ Γ : |ζ − z| ≤ ρ}.Proof. Let z ∈ Γ and set H(z; ρ) := {ζ ∈ D : |ζ − z| = ρ} with counterclockwiseorientation. Then, by Cauchy’s integral theorem,

∫Γ\Γ(z;ρ)

dζζ − z

=

∫H(z;ρ)

dζζ − z

.

Writing ζ = z + ρeiϑ, dζ = iρeiϑdϑ, we find

limρ→0

∫H(z;ρ)

dζζ − z

= limρ→0

∫H(z;ρ)

idϑ = iπ,

since Γ is of class C2. Hence

12πi

∫Γ

dζζ − z

=12, z ∈ Γ, (7.4)

in the sense of a Cauchy principal value.The normal vector ν is continuous on Γ. Therefore, as in the proof of Theorem

2.30, we can choose R ∈ (0, 1] such that the scalar product of the normal vectorssatisfies

ν(z) · ν(ζ) ≥ 12

(7.5)

for all z, ζ ∈ Γ with |z − ζ | ≤ R. Furthermore, we can assume that R is small enoughsuch that Γ(z; R) is connected for each z ∈ Γ. Then the condition (7.5) implies thatΓ(z; R) can be bijectively projected onto the tangent line to Γ at the point z. The lineelements ds(ζ) on Γ and dσ on the tangent are related by

ds(ζ) =dσ

ν(z) · ν(ζ)≤ 2 dσ.


Hence, ∫Γ(z;R)

∣∣∣∣∣ϕ(ζ) − ϕ(z)ζ − z

∣∣∣∣∣ ds ≤ 2|ϕ|α∫ R

−R|σ|α−1 dσ =

4Rα

α|ϕ|α

and∫Γ\Γ(z;R)

∣∣∣∣∣ϕ(ζ) − ϕ(z)ζ − z

∣∣∣∣∣ ds ≤ |ϕ|α∫Γ\Γ(z;R)

|ζ − z|α−1 ds ≤ Rα−1|Γ| |ϕ|α.

Therefore ∫Γ

ϕ(ζ) − ϕ(z)ζ − z

dζ

exists as an improper integral, and by the decomposition∫Γ

ϕ(ζ)ζ − z

dζ =∫Γ

ϕ(ζ) − ϕ(z)ζ − z

dζ + ϕ(z)∫Γ

dζζ − z

the proof is completed. �

After establishing the existence of the Cauchy integral for points on the boundary,we want to show that the function f defined by the Cauchy integral is uniformlyHolder continuous in D and in C \ D. For this we choose h > 0 small enough suchthat in the parallel strip

Dh := {w + ηhν(w) : w ∈ Γ, η ∈ [−1, 1]}each point z is uniquely representable through projection onto Γ in the form z =w + ηhν(w) with w ∈ Γ and η ∈ [−1, 1] (see p. 84). Then we show that the functiong : Dh → C defined by

g(z) :=1

2πi

∫Γ

ϕ(ζ) − ϕ(w)ζ − z

dζ, z ∈ Dh,

is uniformly Holder continuous with

‖g‖α ≤ C‖ϕ‖α, (7.6)

where C is some constant independent of ϕ.We begin by noting that we can choose h sufficiently small such that

|w1 − w2| ≤ 2|z1 − z2| (7.7)

for each pair z1 = z1(η1) = w1 +η1hν(w1), z2 = z2(η2) = w2 +η2hν(w2). Since Γ is ofclass C2, the normal ν is continuously differentiable. Therefore, by the mean valuetheorem, there exists a constant M > 0 such that |ν(w1)− ν(w2)| ≤ M|w1 −w2| for allw1, w2 ∈ Γ. Then, from

|w1 − w2| ≤ |z1(η) − z2(η)| + h|ν(w1) − ν(w2)|, η ∈ [−1, 1],


we observe that |w1−w2| ≤ 2|z1(η)−z2(η)| for all η ∈ [−1, 1], if we restrict h ≤ 1/2M.Now (7.7) follows from the observation that for all η1, η2 ∈ [−1, 1] by elementarygeometry we have

min{|z1(1) − z2(1)|, |z1(−1) − z2(−1)|} ≤ |z1(η1) − z2(η2)|.We can use (7.7) to estimate |ϕ(ζ) − ϕ(w)| ≤ |ϕ|α|ζ − w|α ≤ 2α|ϕ|α|ζ − z|α and

then, as in the proof of Theorem 7.7, by splitting the integral into the two parts overΓ(w; R) and Γ \ Γ(w; R), we obtain

|g(z)| ≤ C0|ϕ|α (7.8)

for all z ∈ Dh and some constant C0.Now let z1, z2 ∈ Dh0 with 0 < |z1 − z2| ≤ R/4 and set ρ := 4|z1 − z2|. Then we can

estimate ∣∣∣∣∣∣∫Γ(w1;ρ)

{ϕ(ζ) − ϕ(w1)

ζ − z1− ϕ(ζ) − ϕ(w2)

ζ − z2

}dζ

∣∣∣∣∣∣

≤ 2α|ϕ|α{∫

Γ(w1;ρ)|ζ − z1|α−1 ds +

∫Γ(w2;2ρ)

|ζ − z2|α−1 ds

}

≤ C1|z1 − z2|α|ϕ|α,

(7.9)

where C1 is some constant depending on α. Here we have made use of the fact thatΓ(w1; ρ) ⊂ Γ(w2; 2ρ). For |ζ − w1| ≥ ρ = 4|z1 − z2| we have

|ζ − z1| ≤ |ζ − z2| + |z2 − z1| ≤ |ζ − z2| + 14|ζ − w1| ≤ |ζ − z2| + 1

2|ζ − z1|.

Hence, |ζ − z1| ≤ 2|ζ − z2| for |ζ − w1| ≥ ρ, and this inequality can now be used toobtain ∣∣∣∣∣∣

∫Γ\Γ(w1;ρ)

{ϕ(ζ) − ϕ(w2)}{

1ζ − z1

− 1ζ − z2

}dζ

∣∣∣∣∣∣

≤ |z1 − z2| |ϕ|α∫Γ\Γ(w1;ρ)

|ζ − w2|α|ζ − z1| |ζ − z2| ds

≤ C′2|z1 − z2| |ϕ|α{∫ R

ρ

dσσ2−α +

|Γ|R2−α

}≤ C2|z1 − z2|α|ϕ|α,

(7.10)

where C2 and C′2 are some constants depending on α and Γ. Note that for this esti-mate we need the restriction α < 1, and we have split the integral into the two partsover Γ(w1; R) \ Γ(w1; ρ) and Γ \ Γ(w1; R). Finally, since from (7.7) and the proof ofTheorem 7.7 it is obvious that

∫Γ\Γ(w1;ρ)

dζζ − z1


is bounded by 2π, we have the estimate∣∣∣∣∣∣{ϕ(w1) − ϕ(w2)}

∫Γ\Γ(w1;ρ)

dζζ − z1

∣∣∣∣∣∣ ≤ C3|z1 − z2|α|ϕ|α (7.11)

for some constant C3. Combining (7.9)–(7.11), we obtain

2π |g(z1) − g(z2)| ≤ (C1 +C2 + C3)|z1 − z2|α|ϕ|α (7.12)

for all z1, z2 ∈ Dh with |z1 − z2| ≤ R/4. Now the desired result (7.6) follows from(7.8) and (7.12) with the aid of Lemma 7.3.

By Cauchy’s integral theorem and (7.4) we have

12πi

∫Γ

dζζ − z

=

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

1, z ∈ D,

12, z ∈ Γ,

0, z ∈ C \ D.

Therefore, from the decomposition

f (z) = g(z) +ϕ(w)2πi

∫Γ

dζζ − z

, z ∈ Dh,

we observe that f ∈ C0,α(Dh ∩ D) and f ∈ C0,α(Dh ∩ (C \ D)) with boundary values

f±(z) =1

2πi

∫Γ

ϕ(ζ)ζ − z

dζ ∓ 12ϕ(z), z ∈ Γ.

Holder continuity of f in D and C \D then follows by again employing Lemma 7.3.We summarize our results in the following theorem.

Theorem 7.8 (Sokhotski–Plemelj). For densities ϕ ∈ C0,α(Γ) the holomorphicfunction f defined by the Cauchy integral

f (z) :=1

2πi

∫Γ

ϕ(ζ)ζ − z

dζ, z ∈ C \ Γ, (7.13)

can be uniformly Holder continuously extended from D into D and from C \ D intoC \ D with limiting values

f±(z) =1

2πi

∫Γ

ϕ(ζ)ζ − z

dζ ∓ 12ϕ(z), z ∈ Γ, (7.14)

where f±(z) = limh→+0

f (z ± hν(z)). Furthermore, we have the inequalities

‖ f ‖α,D ≤ C‖ϕ‖α and ‖ f ‖α,C\D ≤ C‖ϕ‖αfor some constant C depending on α and Γ.


The formula (7.14) was first derived by Sokhotski [224] in his doctoral thesis in1873. A sufficiently rigorous proof was given by Plemelj [191] in 1908.

Corollary 7.9. The Cauchy integral operator A : C0,α(Γ)→ C0,α(Γ), defined by

(Aϕ)(z) :=1πi

∫Γ

ϕ(ζ)ζ − z

dζ, z ∈ Γ,

is bounded.

Proof. We write the Sokhotski–Plemelj formula (7.14) in the short form

f± =12

Aϕ ∓ 12ϕ (7.15)

and obtain the boundedness of A from Theorem 7.8. �

As a first application of the Sokhotski–Plemelj Theorem 7.8 we solve the prob-lem of finding a sectionally holomorphic function with a given jump along the con-tour Γ.

Theorem 7.10. Let ϕ ∈ C0,α(Γ). Then there exists a unique function f that is holo-morphic in D and C \ D, which can be extended continuously from D into D andfrom C \ D into C \ D satisfying the boundary condition

f− − f+ = ϕ on Γ,

and for which f (z) → 0, z → ∞, uniformly for all directions. This function is givenby (7.13).

Proof. By Theorem 7.8 the function f given by (7.13) has the required properties. Toestablish uniqueness, from f− − f+ = 0 on Γ, by Morera’s theorem, we conclude thatf is holomorphic everywhere in C, i.e., f is an entire function. Then from f (z)→ 0,z → ∞, by Liouville’s theorem it follows that f = 0 in C. From this proof we seethat the general sectionally holomorphic function f satisfying f− − f+ = ϕ on Γ isobtained from (7.13) by adding an arbitrary entire function. �

As a second application of Theorem 7.8 we state necessary and sufficient con-ditions for the existence of a holomorphic function in D or in C \ D with givenboundary values.

Theorem 7.11. For a given function ϕ ∈ C0,α(Γ), there exists a function f that isholomorphic in D and continuous in D with boundary values f = ϕ on Γ if and onlyif ϕ is a solution of the homogeneous integral equation of the second kind

ϕ − Aϕ = 0.

The solution is given by (7.13).


Proof. Let f be holomorphic with f = ϕ on Γ. Then by Cauchy’s integral formulawe have

f (z) =1

2πi

∫Γ

f (ζ)ζ − z

dζ =1

2πi

∫Γ

ϕ(ζ)ζ − z

dζ, z ∈ D,

and from Theorem 7.8 it follows that 2ϕ = 2 f− = Aϕ + ϕ on Γ, and thereforeϕ − Aϕ = 0. Conversely, if ϕ is a solution of ϕ − Aϕ = 0, then again by Theorem7.8 the function f defined by (7.13) has boundary values 2 f− = Aϕ+ϕ = 2ϕ on Γ. �

Obviously, for the corresponding exterior problem in C \ D with f (z) → 0 asz → ∞ we have the homogeneous integral equation ϕ + Aϕ = 0. From this andTheorem 7.11 we observe that the operator A provides an example for nullspacesN(I − A) and N(I + A) with infinite dimension, since there exist infinitely manylinearly independent holomorphic functions. This, in particular, implies that A isnot compact, and it also means that I − A and I + A cannot be regularized.

We now can prove a property of the Cauchy integral operator A, which is ofcentral importance in our study of singular integral equations with Cauchy kernels,since it will allow the construction of regularizers. Note that the following theoremis a further indication that the Cauchy integral operator A is not compact.

Theorem 7.12. The Cauchy integral operator A satisfies A2 = I.

Proof. For ϕ ∈ C0,α(Γ) we define f by (7.13). Then, by Theorem 7.11, we havef+ + A f+ = 0 and f− − A f− = 0. Hence, using (7.15), we derive

A2ϕ = A( f− + f+) = f− − f+ = ϕ,

and this is the desired result. �

Theorem 7.13. The operators A and −A are adjoint with respect to the dual system〈C0,α(Γ),C0,α(Γ)〉 with the non-degenerate bilinear form

〈ϕ, ψ〉 :=∫Γ

ϕ(z)ψ(z) dz, ϕ, ψ ∈ C0,α(Γ).

Proof. It is left as an exercise to show that 〈· , ·〉 is non-degenerate on C0,α(Γ). Forϕ, ψ ∈ C0,α(Γ) define

f (z) :=1

2πi

∫Γ

ϕ(ζ)ζ − z

dζ, g(z) :=1

2πi

∫Γ

ψ(ζ)ζ − z

dζ, z ∈ C \ Γ.

Then, by Theorem 7.8 and Cauchy’s integral theorem we find that

〈Aϕ, ψ〉 + 〈ϕ, Aψ〉 = 2〈 f−, g−〉 − 2〈 f+, g+〉 = −2∫|z|=R

f (z)g(z) dz→ 0, R→ ∞,

since f (z), g(z) = O(1/|z|), z→ ∞. �


Note that Theorem 7.13 justifies interchanging the order of integration in

∫Γ

(∫Γ

ϕ(ζ)ζ − z

dζ

)ψ(z) dz =

∫Γ

ϕ(ζ)

(∫Γ

ψ(z)ζ − z

dz

)dζ.

Example 7.14. Let Γ be the unit circle. Then, substituting z = eit, ζ = eiτ, we find

dζζ − z

=12

(cot

τ − t2+ i

)dτ,

i.e., the operator A can be expressed in the form

(Aϕ)(eit

)=

12πi

∫ 2π

0

{cot

τ − t2+ i

}ϕ(eiτ

)dτ, t ∈ [0, 2π],

with the integral to be understood as a Cauchy principal value. Consider the integralequation of the first kind

12π

∫ 2π

0cot

τ − t2

ϕ(τ) dτ = ψ(t), t ∈ [0, 2π], (7.16)

in the space C0,α[0, 2π]. Since

∫ 2π

0cot

t2

dt = 0,

by integrating (7.16) with respect to t and interchanging the order of integration (useTheorem 7.13 for ψ(z) = 1/z and transform the integrals), we find that

∫ 2π

0ψ(t) dt = 0 (7.17)

is a necessary condition for the solvability of (7.16). It is also sufficient, because bymaking use of A2 = I we see that when (7.17) is satisfied the general solution of(7.16) is given by

ϕ(t) = − 12π

∫ 2π

0cot

τ − t2

ψ(τ) dτ + c, t ∈ [0, 2π], (7.18)

where c is an arbitrary constant. �

The formulas (7.16) and (7.18) represent the inversion formulas of Hilbert. Thekernel in (7.16) and (7.18) is a called Hilbert kernel, since singular equations withthis kernel were first encountered by Hilbert [95]. Splitting the integral equationof Theorem 7.11 into its real and imaginary part, we readily find that the Hilbertinversion formulas relate the real and imaginary part of holomorphic functions inthe unit disk.


7.3 The Riemann Problem

The following boundary value problem was first formulated by Riemann in hisinaugural dissertation. Because a first attempt toward a solution was made byHilbert [95] in 1904 through the use of integral equations, what we will denote asthe Riemann problem in the literature is sometimes also called the Hilbert problem.

Riemann Problem. Find a function f that is holomorphic in D and in C \ D, whichcan be extended uniformly Holder continuously from D into D and from C \ D intoC \ D satisfying the boundary condition

f− = g f+ + h on Γ (7.19)

andf (z) → 0, z→ ∞, (7.20)

uniformly for all directions. Here, g and h are given uniformly Holder continuousfunctions on Γ.

Before we give a solution to this Riemann problem for the case where g vanishesnowhere on Γ, we want to indicate its connection to integral equations.

Theorem 7.15. Let a, b, h ∈ C0,α(Γ). Then the Cauchy integral (7.13) maps solu-tions ϕ ∈ C0,α(Γ) of the integral equation

aϕ + bAϕ = h (7.21)

linearly and bijectively onto solutions f of the Riemann problem with the boundarycondition

(a + b) f− = (a − b) f+ + h on Γ. (7.22)

Proof. By Theorems 7.8 and 7.10, the Cauchy integral maps C0,α(Γ) linearly andbijectively onto the linear space of sectionally holomorphic functions that vanish atinfinity and can be extended uniformly Holder continuously with Holder exponentα from D into D and from C \ D into C \D. From (7.15) and Theorem 7.11 we havethe relation

aϕ + bAϕ = a( f− − f+) + b( f− + f+) = (a + b) f− − (a − b) f+

between the density ϕ and the function f defined by the Cauchy integral. This endsthe proof. �

Definition 7.16. Let g be a complex-valued and nowhere vanishing function definedon the contour Γ. The index of the function g is the integer ind g given by the incre-ment of its argument along the contour in a counterclockwise direction divided by2π, i.e.,

ind g :=1

2π

∫Γ

d arg g.

7.3 The Riemann Problem 117

The index can also be expressed by the logarithm of g through

ind g =1

2πi

∫Γ

d ln g.

Now, with the aid of the Sokhotski–Plemelj jump relations, we explicitly solvethe homogeneous Riemann problem.

Theorem 7.17. Let g ∈ C0,α(Γ) be a nowhere vanishing function with ind g = κ.Then there exists a unique sectionally holomorphic function f0 satisfying the homo-geneous boundary condition

f0− = g f0+ on Γ

andlimz→∞ zκ f0(z) = 1

uniformly for all directions. It is called the canonical solution to the homogeneousRiemann problem, has the property that f0(z) � 0 for all z ∈ C, and is uniformlyHolder continuous up to the boundary.

Proof. Choose a point a ∈ D and define G ∈ C0,α(Γ) by

G(z) := (z − a)−κg(z), z ∈ Γ.Because G has index ind G = 0, we can reduce the homogeneous boundary condi-tion F− = G F+ by taking logarithms to obtain ln F− − ln F+ = ln G, where we maytake any branch of the logarithm to arrive at a single-valued function ln G ∈ C0,α(Γ).According to Theorem 7.10 we set

ψ(z) :=1

2πi

∫Γ

ln G(ζ)ζ − z

dζ, z ∈ C \ Γ,

and F(z) := eψ(z), z ∈ C \ Γ. Then, from ψ− − ψ+ = ln G on Γ we derive F− = GF+on Γ, and since ψ(z)→ 0, z→ ∞, we have F(z)→ 1, z→ ∞. Now the function f0,defined by,

f0(z) :=

⎧⎪⎪⎪⎨⎪⎪⎪⎩F(z), z ∈ D,

(z − a)−κF(z), z ∈ C \ D,

has the required properties.Let f be any other sectionally holomorphic function satisfying the homogeneous

boundary condition f− = g f+ on Γ. Then the quotient q := f / f0 is sectionally holo-morphic (observe f0(z) � 0 for all z ∈ C) satisfying q− = q+ on Γ. Hence, byMorera’s theorem, q is an entire function. From zκ f0(z) → 1 and zκ f (z) → 1 forz → ∞, it follows that q(z) → 1 for z → ∞, and Liouville’s theorem implies q = 1on C. �


From the proof we note that the general sectionally holomorphic function fsatisfying the homogeneous boundary condition f− = g f+ on Γ is obtained fromthe canonical solution f0 by multiplying it by an arbitrary entire function.

Theorem 7.18. Under the assumptions of Theorem 7.17, the homogeneous Riemannproblem admits max(κ, 0) linearly independent solutions.

Proof. For p an entire function, the product f := p f0 vanishes at infinity if andonly if

limz→∞ z−κp(z) = 0.

Hence, p must be a polynomial of degree less than or equal to κ − 1 if κ > 0 and itmust be zero if κ ≤ 0. �

By the procedure used in Theorem 7.17 it is also possible to treat the inhomo-geneous Riemann problem. But here we will proceed differently and use Theorem5.6 to obtain results on the existence of solutions to the inhomogeneous integralequation (7.21), which then by the equivalence stated in Theorem 7.15 imply resultson the existence of solutions to the inhomogeneous Riemann problem (see Prob-lem 7.1).

7.4 Integral Equations with Cauchy Kernel

For 0 < α, β ≤ 1 by C0,β,α(Γ × Γ) we denote the set of all functions k defined onΓ × Γ satisfying

|k(z1, ζ1) − k(z2, ζ2)| ≤ M(|z1 − z2|β + |ζ1 − ζ2|α) (7.23)

for all z1, z2, ζ1, ζ2 ∈ Γ and some constant M depending on k. Let 0 < α < β ≤ 1and let a ∈ C0,α(Γ) and k ∈ C0,β,α(Γ ×Γ). Then the operator K : C0,α(Γ)→ C0,α(Γ),defined by

(Kϕ)(z) := a(z)ϕ(z) +1πi

∫Γ

k(z, ζ)ζ − z

ϕ(ζ) dζ, z ∈ Γ,

is called a singular integral operator with a Cauchy kernel. The operator K0 :C0,α(Γ)→ C0,α(Γ), given by

(K0ϕ)(z) := a(z)ϕ(z) +b(z)πi

∫Γ

ϕ(ζ)ζ − z

dζ, z ∈ Γ,

where b(z) := k(z, z), z ∈ Γ, is called the dominant part of the operator K. Forthe coefficients a and b of the dominant part we will assume that a2 − b2 vanishesnowhere on Γ. The dominant part can be written in the short form K0 = aI +bA andis bounded by Corollary 7.9. The splitting of the operator K into its dominant partK0 and the remainder K − K0 is justified by the following theorem.

7.4 Integral Equations with Cauchy Kernel 119

Theorem 7.19. The singular integral operator K is bounded and the difference be-tween K and its dominant part K0 is compact from C0,α(Γ) into C0,α(Γ).

Proof. We choose 0 < γ < α and show that the difference H := K − K0 is boundedfrom C0,γ(Γ) into C0,α(Γ). Then the assertion follows from Theorems 2.21 and 7.4.

We write

(Hϕ)(z) =1πi

∫Γ

h(z, ζ)ϕ(ζ) dζ, z ∈ Γ,

with kernel

h(z, ζ) :=k(z, ζ) − k(z, z)

ζ − z, z � ζ. (7.24)

Since k ∈ C0,β,α(Γ × Γ), there exists a constant M such that

|h(z, ζ)| ≤ M|z − ζ |α−1, z � ζ. (7.25)

Hence, the kernel h is weakly singular and therefore we have

‖Hϕ‖∞ ≤ C0‖ϕ‖∞ (7.26)

for some constant C0.To establish Holder continuity, we proceed analogously to the proof of Theorem

7.8. Let z1, z2 ∈ Γ with 0 < |z1 − z2| ≤ R/4 and set ρ := 4|z1− z2|. Then, using (7.25),we can estimate ∣∣∣∣∣∣

∫Γ(z1;ρ)

{h(z1, ζ) − h(z2, ζ)}ϕ(ζ) dζ

∣∣∣∣∣∣

≤ M‖ϕ‖∞∫Γ(z1;ρ)

{|z1 − ζ |α−1 + |z2 − ζ |α−1} ds

≤ C′1‖ϕ‖∞∫ 2ρ

0σα−1dσ ≤ C1‖ϕ‖∞|z1 − z2|α

(7.27)

for some constants C1 and C′1 depending on M and α. Using (7.23), we can estimate

∣∣∣∣∣∣∫Γ\Γ(z1;ρ)

k(z1, ζ) − k(z2, ζ)ζ − z1

ϕ(ζ) dζ

∣∣∣∣∣∣

≤ M‖ϕ‖∞|z1 − z2|β∫Γ\Γ(z1;ρ)

ds|ζ − z1|

≤ C′2‖ϕ‖∞|z1 − z2|β{∫ R

ρ/2

dσσ+|Γ|R

}

≤ C′′2 ‖ϕ‖∞|z1 − z2|β{

ln1ρ+ 1

}≤ C2‖ϕ‖∞|z1 − z2|α

(7.28)


for some constants C2,C′2, and C′′2 depending on M, α, β, and Γ. Because for ζ ∈ Γwith |ζ − z1| ≥ ρ we have that |ζ − z1| ≤ 2|ζ − z2|, we derive the further estimate

∣∣∣∣∣∣∫Γ\Γ(z1;ρ)

{k(z2, ζ) − k(z2, z2)}{

1ζ − z1

− 1ζ − z2

}ϕ(ζ) dζ

∣∣∣∣∣∣

≤ M‖ϕ‖∞|z1 − z2|∫Γ\Γ(z1;ρ)

ds|ζ − z1| |ζ − z2|1−α

≤ C′3‖ϕ‖∞|z1 − z2|{∫ R

ρ/2σα−2dσ +

|Γ|R2−α

}

≤ C3‖ϕ‖∞|z1 − z2|α

(7.29)

for some constants C3 and C′3 depending on M, α, and Γ. Finally, we write

∫Γ\Γ(z1;ρ)

ϕ(ζ)ζ − z1

dζ =∫Γ\Γ(z1;ρ)

ϕ(ζ) − ϕ(z1)ζ − z1

dζ + ϕ(z1)∫Γ\Γ(z1;ρ)

dζζ − z1

and note from the proof of Theorem 7.7 that the second integral on the right-handside is bounded by 2π. For the first integral we have

∣∣∣∣∣∣∫Γ\Γ(z1;ρ)

ϕ(ζ) − ϕ(z1)ζ − z1

dζ

∣∣∣∣∣∣ ≤ |ϕ|γ∫Γ

ds|ζ − z1|1−γ ≤ C′4|ϕ|γ

for some constant C′4 depending on Γ and γ. Hence, using again (7.23), we obtain

∣∣∣∣∣∣{k(z1, z1) − k(z2, z2)}∫Γ\Γ(z1;ρ)

ϕ(ζ)ζ − z1

dζ

∣∣∣∣∣∣ ≤ C4‖ϕ‖γ|z1 − z2|α (7.30)

for some constant C4 depending on M, α and Γ. Combining (7.27)–(7.30), we find

π |(Hϕ)(z1) − (Hϕ)(z2)| ≤ (C1 +C2 +C3 + C4)‖ϕ‖γ |z1 − z2|α (7.31)

for all z1, z2 ∈ Γ with |z1 − z2| ≤ R/4. Now the desired result

‖Hϕ‖α ≤ C‖ϕ‖γfollows from (7.26) and (7.31) with the aid of Lemma 7.3. �

Theorem 7.20. The operators K0 = aI + bA and K0′ := aI − Ab are adjoint withrespect to the dual system 〈C0,α(Γ),C0,α(Γ)〉. They both have finite-dimensionalnullspaces and the index of the operator K0 is given by

ind K0 = inda − ba + b

.


Proof. That K0 and K0′ are adjoint is easily derived as a consequence of Theorem7.13. From Theorems 7.15 and 7.18 we obtain

dim N(K0) = max(κ, 0), (7.32)

where

κ := inda − ba + b

.

Analogous to Theorem 7.15, the homogeneous adjoint equation

aψ − Abψ = 0 (7.33)

is equivalent to the Riemann problem with homogeneous boundary condition

(a − b) f− = (a + b) f+ on Γ. (7.34)

Letψ solve the integral equation (7.33). Then ϕ := bψ solves aϕ−bAϕ = 0 and there-fore, by Theorem 7.15, the sectionally holomorphic function f defined by (7.13) sat-isfies the boundary condition (7.34). Conversely, let f be a solution to the Riemannproblem with boundary condition (7.34). Then for the function ψ := 2(a + b)−1 f−we have f− + f+ = aψ and f− − f+ = bψ. Hence, by Theorem 7.11 and its analoguefor C \ D, we find aψ − Abψ = f− + f+ − A( f− − f+) = 0.

With the aid of ind[(a+b)−1(a−b)] = − ind[(a−b)−1(a+b)], Theorem 7.18 nowyields

dim N(K0′) = max(−κ, 0) (7.35)

and ind K0 = dim N(K0) − dim N(K0′) = κ follows by combining (7.32) and(7.35). �

From now on, for symmetry, we assume that k ∈ C0,β,β(Γ×Γ) with 0 < α < β ≤ 1.The singular integral operators with Cauchy kernel

(Kϕ)(z) := a(z)ϕ(z) +1πi

∫Γ

k(z, ζ)ζ − z

ϕ(ζ) dζ, z ∈ Γ,

and

(K′ψ)(z) := a(z)ψ(z) − 1πi

∫Γ

k(ζ, z)ζ − z

ψ(ζ) dζ, z ∈ Γ,

are adjoint. This follows by writing K = K0 + H and K′ = K0′ + H′, where H andH′ have weakly singular kernels h(z, ζ) and h(ζ, z), respectively, with h given by(7.24). For weakly singular kernels, as in Theorem 4.7, the order of integration maybe interchanged.

Theorem 7.21. The operator (a2 − b2)−1K′ is a regularizer of K.

Proof. First, we observe that for each c ∈ C0,α(Γ) the difference Ac − cA is compactfrom C0,α(Γ) into C0,α(Γ). This follows from Theorem 7.19, applied to the kernel


c(ζ) contained in C0,β,α(Γ × Γ). Actually, the need for the compactness of the com-mutator Ac−cA was our reason for allowing different exponentsα and β in Theorem7.19. Now, using A2 = I, we derive

K0′K0 = (aI − Ab)(aI + bA) = (a2 − b2)I + M

where M := abA−Aba+ (b2A−Ab2)A is compact, since A is bounded by Corollary7.9. Hence, (a2 − b2)−1K0′ is a left regularizer of K0. Analogously,

K0K0′ = (aI + bA)(aI − Ab) = (a2 − b2)I + M

where M := bAa − aAb is compact, i.e., K0′(a2 − b2)−1 is a right regularizer ofK0. Again, the difference (a2 − b2)−1K0′ − K0′(a2 − b2)−1 is compact. Therefore,(a2 − b2)−1K0′ is a regularizer of K0. Now the assertion of the theorem follows fromthe fact that K = K0 + H and K = K0′ + H′ with compact H and H′. �

We now are in a position to derive the classical Noether theorems from our gen-eral theory on regularization in dual systems of Chapter 5.

Theorem 7.22 (First Noether Theorem). The singular integral operator withCauchy kernel has a finite-dimensional nullspace.

Proof. This follows from Theorems 5.6 and 7.21. �

Theorem 7.23 (Second Noether Theorem). The index of the singular integral op-erator K with Cauchy kernel is given by

ind K = inda − ba + b

.

Proof. This follows from Theorems 7.20 and 7.21 and Corollary 5.12. �

Theorem 7.24 (Third Noether Theorem). The inhomogeneous singular integralequation

Kϕ = h

is solvable if and only if〈h, ψ〉 = 0

is satisfied for all ψ ∈ N(K′).


Theorem 7.25. Let 0 < α < β ≤ 1, a ∈ C0,α(Γ), and k ∈ C0,β,β(Γ × Γ) and assumethat a2 − b2 vanishes nowhere on Γ, where b(z) := k(z, z), z ∈ Γ. Then, in the Holderspace C0,α(Γ), the number of linearly independent solutions of the homogeneoussingular integral equation

a(z)ϕ(z) +1πi

∫Γ

k(z, ζ)ζ − z

ϕ(ζ) dζ = 0, z ∈ Γ,


and of its adjoint equation

a(z)ψ(z) − 1πi

∫Γ

k(ζ, z)ζ − z

ψ(ζ) dζ = 0, z ∈ Γ,

are both finite, and their difference is given by the index

12πi

∫Γ

d lna − ba + b

.

The inhomogeneous singular integral equation

a(z)ϕ(z) +1πi

∫Γ

k(z, ζ)ζ − z

ϕ(ζ) dζ = h(z), z ∈ Γ,

is solvable if and only if ∫Γ

h(z)ψ(z) dz = 0

is satisfied for all solutions ψ of the homogeneous adjoint equation.

Proof. This is a reformulation of the three preceding theorems. �

For equations of the first kind, i.e., equations with a = 0, we note the followingcorollary.

Corollary 7.26. Singular integral operators of the first kind with Cauchy kernelhave index zero. In particular, injective singular integral operators of the first kindare bijective with bounded inverse.

Proof. The index zero follows from Theorem 7.23. The bounded inverse for an in-jective operator of the first kind is a consequence of the Riesz Theorem 3.4 togetherwith the fact that A provides an equivalent regularizer. �

With the aid of Example 7.14, choosing for Γ the unit circle, we obtain the fol-lowing corollary, which contains the equations for which Noether [185] proved thetheorems named after him.

Corollary 7.27. Let a and k be real-valued and 2π-periodic and assume that a2+b2

is strictly positive, where b(t) := k(t, t) for t ∈ [0, 2π]. Then for the singular integralequation with Hilbert kernel

a(t)ϕ(t) − 12π

∫ 2π

0k(t, τ) cot

τ − t2

ϕ(τ) dτ = h(t), t ∈ [0, 2π],

and its adjoint equation

a(t)ψ(t) +1

2π

∫ 2π

0k(τ, t) cot

τ − t2

ψ(τ) dτ = g(t), t ∈ [0, 2π],


the three Noether theorems are valid, i.e., the numbers of linearly independent solu-tions of the homogeneous equation and the homogeneous adjoint equation are bothfinite, and their difference is given by the index

1π

∫ 2π

0d arctan

ba.

The inhomogeneous equation is solvable if and only if

∫ 2π

0h(t)ψ(t) dt = 0

is satisfied for all solutions ψ of the homogeneous adjoint equation.

Proof. After transforming them onto the unit circle Γ by setting z = eit, ζ = eiτ, thetwo equations read

a(z)ϕ(z) − 1π

∫Γ

k(z, ζ)ζ − z

ϕ(ζ) dζ = h(z), z ∈ Γ,

and

a(z)ψ(z) +1π

∫Γ

k(ζ, z)ζ − z

ψ(ζ) dζ = g(z), z ∈ Γ,

for ϕ(eit) := ϕ(t) and ψ(eit) := ψ(t)/eit. We have set k(eit, eiτ) := (eit + eiτ)k(t, τ)/2eiτ

and a(eit) := a(t), h(eit) := h(t), and g(eit) := g(t)/eit. Hence, the corollary is estab-lished through transforming the Noether theorems for the pair of equations on theunit circle back into the equations on [0, 2π]. For the index we compute

lna + iba − ib

∣∣∣∣∣2π

0= 2 ln(a + ib)

∣∣∣∣∣2π

0= 2i arctan

ba

∣∣∣∣∣2π

0,

and the proof is finished. �

Concluding this section, we wish to mention that the above integral equations canalso be treated in various other normed function spaces. In this context we make useof Lax’s Theorem 4.13 to establish the boundedness of the Cauchy integral operatorA in L2(Γ).

Theorem 7.28. The Cauchy integral operator A is bounded from L2(Γ) into L2(Γ).

Proof. We want to apply Theorem 4.13 in the positive dual system (C0,α(Γ),C0,α(Γ))generated by the scalar product

(ϕ, ψ) :=∫Γ

ϕψ ds, ϕ, ψ ∈ C0,α(Γ).

Therefore, we need the adjoint A∗ of the Cauchy integral operator A with respect tothis scalar product. Complex integration and contour integration along Γ are related

7.5 Cauchy Integral and Logarithmic Potential 125

bydz = eiγ(z) ds(z) (7.36)

for all z ∈ Γ, where γ(z) denotes the angle between the tangent to Γ at the point zand the real axis. Using Theorem 7.13 and (7.36), we see that

∫Γ

Aϕ ψ ds = −∫Γ

ϕχ A(ψχ) ds,

where χ(z) := eiγ(z), z ∈ Γ. Hence, the adjoint A∗ is given by

A∗ψ = −χA(ψχ),

and A∗ : C0,α(Γ) → C0,α(Γ) is bounded, since A : C0,α(Γ) → C0,α(Γ) isbounded. Now the statement follows from Theorem 4.13 and the denseness ofC0,α(Γ) in L2(Γ). �

In particular, as a consequence of Theorem 7.28, the property A2 = I of Theorem7.12 remains valid in L2(Γ) and can be used to construct regularizers as in Theorem7.21.

7.5 Cauchy Integral and Logarithmic Potential

We now want to use the close connection between holomorphic and two-dimensionalharmonic functions to derive regularity results on the logarithmic single- and double-layer potential from the Sokhotski–Plemelj theorem.

In the Cauchy integral for fixed z ∈ C we substitute ζ − z = reiϑ, where r = |ζ − z|and ϑ = arg(ζ − z). Then

∫Γ

dζζ − z

=

∫Γ

d ln(ζ − z) =∫Γ

d(ln r + iϑ) =∫Γ

(∂ ln r∂s+ i

∂ϑ

∂s

)ds.

By the Cauchy–Riemann equations, applied to ln(ζ − z) = ln r + iϑ, we have

∂ϑ

∂s=∂ ln r∂ν= − ∂

∂νln

1r.

Consequently,

12πi

∫Γ

ϕ(ζ)ζ − z

dζ = − 12π

∫Γ

ϕ(ζ)∂

∂ν(ζ)ln

1|ζ − z| ds(ζ)

− 12πi

∫Γ

ϕ(ζ)∂

∂s(ζ)ln

1|ζ − z| ds(ζ).

(7.37)


This formula indicates that for real-valued densities the real part of the Cauchyintegral coincides with the logarithmic double-layer potential

v(z) :=1

2π

∫Γ

ϕ(ζ)∂

∂ν(ζ)ln

1|ζ − z| ds(ζ), z ∈ IR2 \ Γ.

Note that we use the same symbol z for the complex number z = x1 + ix2 and thevector z = (x1, x2) ∈ IR2. Hence, from Theorem 7.8 and the Sokhotski–Plemeljformula (7.14), we derive the following theorem on the regularity of the double-layer potential with a Holder continuous density. Recall that in Theorem 6.18 thedensity is only assumed to be continuous.

Theorem 7.29. The logarithmic double-layer potential v with Holder continuousdensity ϕ can be extended uniformly Holder continuously from D into D and fromIR2 \ D into IR2 \ D with limiting values

v±(z) =1

2π

∫Γ

ϕ(ζ)∂

∂ν(ζ)ln

1|ζ − z| ds(ζ) ± 1

2ϕ(z), z ∈ Γ. (7.38)

Furthermore, we have the inequalities

‖v‖α,D ≤ C‖ϕ‖α and ‖v‖α,IR2\D ≤ C‖ϕ‖αfor some constant C depending on α and Γ.

Recall the relation (7.36) between complex integration and contour integrationalong Γ. Since Γ is of class C2, the angle γ is continuously differentiable with re-spect to z. Therefore, from Theorem 7.8 we immediately deduce that

f (z) :=1

2π

∫Γ

ϕ(ζ)ζ − z

ds(ζ), z ∈ C \ Γ, (7.39)

is holomorphic in D and in C \ D and that it can be extended uniformly Holdercontinuously from D into D and from C \ D into C \ D with limiting values

f±(z) :=1

2π

∫Γ

ϕ(ζ)ζ − z

ds(ζ) ∓ i2

e−iγ(z)ϕ(z), z ∈ Γ. (7.40)

The gradient of the logarithmic single-layer potential

u(z) :=1

2π

∫Γ

ϕ(ζ) ln1|ζ − z| ds(ζ), z ∈ IR2 \ Γ,

is given by

grad u(z) =1

2π

∫Γ

ϕ(ζ)ζ − z|ζ − z|2 ds(ζ), z ∈ IR2 \ Γ.

We observe that the gradient of the single-layer potential u corresponds to the com-plex conjugate of the holomorphic function f given by the Cauchy integral (7.39).

7.5 Cauchy Integral and Logarithmic Potential 127

Therefore, observing the fact that the complex conjugate of ie−iγ(z) corresponds tothe normal ν(z) at z ∈ Γ, we can again use Theorem 7.8 to prove the followingtheorem.

Theorem 7.30. The first derivatives of the logarithmic single-layer potential u withHolder continuous density ϕ can be extended uniformly Holder continuously fromD into D and from IR2 \ D into IR2 \ D with limiting values

grad u±(z) =1

2π

∫Γ

ϕ(ζ) grad ln1|ζ − z| ds(ζ) ∓ 1

2ν(z)ϕ(z), z ∈ Γ. (7.41)


‖ grad u‖α,D ≤ C‖ϕ‖α and ‖ grad u‖α,IR2\D ≤ C‖ϕ‖αfor some constant C depending on α and Γ.

Recall that by C1,α(Γ) we denote the normed space of all functions defined on Γthat have a uniformly Holder continuous first derivative furnished with the norm

‖ϕ‖1,α := ‖ϕ‖∞ + ‖ϕ′‖0,αwith the prime indicating differentiation with respect to arc length.

Corollary 7.31. The single-layer operator S : C0,α(Γ)→ C1,α(Γ) given by

(Sϕ)(z) :=1π

∫Γ

ϕ(ζ) ln1|ζ − z| ds(ζ), z ∈ Γ, (7.42)

is bounded.

Consider once again the logarithmic double-layer potential v and assume thatϕ ∈ C1,α(Γ). For a = (a1, a2) ∈ IR2 we denote a⊥ = (a2,−a1). Then, for r = |ζ − z|,we write

∂

∂ν(ζ)ln

1r= ν(ζ) · gradζ ln

1r= −ν(ζ) · gradz ln

1r= − divz

(ln

1r

[τ(ζ)]⊥)

where we have made use of the relation ν = τ⊥ between the tangent vector τ and thenormal vector ν. From this, using the vector identity

grad divw = Δw + [grad divw⊥]⊥,

the identity [w⊥]⊥ = −w, and Δ ln 1/r = 0, we see that

gradz∂

∂ν(ζ)ln

1r=

[gradz

(τ(ζ) · gradz ln

1r

)]⊥.


Then, by partial integration, for the gradient of the double-layer potential we obtainthat

grad v(z) =−12π

[grad

∫Γ

τ(ζ) · gradζ ln1|ζ − z| ϕ(ζ) ds(ζ)

]⊥

=1

2π

[grad

∫Γ

dϕds

(ζ) ln1|ζ − z| ds(ζ)

]⊥, z ∈ IR2 \ Γ.

Therefore, from Theorem 7.30 we derive the following result which is due toMaue [164].

Theorem 7.32. The first derivatives of the logarithmic double-layer potential v withdensity ϕ ∈ C1,α(Γ) can be extended uniformly Holder continuously from D into Dand from IR2 \ D into IR2 \ D. The normal derivative is given by

∂v±∂ν

(z) =1

2πd

ds(z)

∫Γ

dϕds

(ζ) ln1|ζ − z| ds(ζ), z ∈ Γ, (7.43)

and the tangential derivative by

∂v±∂s

(z) = − 12π

∫Γ

dϕds

(ζ)∂

∂ν(z)ln

1|ζ − z| ds(ζ) ± 1

2dϕds

(z), z ∈ Γ. (7.44)


for some constant C depending on α and Γ.

Corollary 7.33. The operator T , defined by the normal derivative of the logarithmicdouble-layer potential

(Tϕ)(z) :=1π

∂

∂ν(z)

∫Γ

ϕ(ζ)∂

∂ν(ζ)ln

1|ζ − z| ds(ζ), z ∈ Γ, (7.45)

can be expressed by the single-layer integral operator S in the form

Tϕ =dds

Sdϕds

. (7.46)

It is a bounded operator from C1,α(Γ) into C0,α(Γ).

Since the singularity of the operator T is stronger than the Cauchy type singular-ity it is called a hypersingular operator.

We wish to mention that these results on the regularity of single- and double-layerpotentials with uniformly Holder continuous densities can be extended to the three-dimensional case by techniques similar to those used in the proofs of Theorems 7.5,7.8 and 7.19 (see [31]).

‖ grad v‖α,D ≤ C‖ϕ‖1,α and ‖ grad v‖α,IR2\D ≤ C‖ϕ‖1,α

7.6 Boundary Integral Equations in Holder Spaces 129

For the logarithmic single-layer potential with L2 densities we have the followingcontinuity property.

Theorem 7.34. The logarithmic single-layer potential with L2 density is continuousin all of IR2.

Proof. This can be shown analogously to Theorems 2.29 and 2.30 by cutting off thesingularity and estimating with the aid of the Cauchy–Schwarz inequality.

7.6 Boundary Integral Equations in Holder Spaces

Armed with Theorems 7.5, 7.6, 7.29, 7.30 and 7.32 together with their corollaries wenow revisit the boundary integral equations from potential theory in IRm, m = 2, 3,to make a few additions to the results of Section 6.4 in the classical Holder spacesetting. In the sequel we always assume for the exponent that 0 < α < 1 and thatD ⊂ IRm is a bounded domain with connected boundary ∂D ∈ C2. For conveniencewe recall the definition of the four main boundary integral operators given by thesingle- and double-layer operators

(Sϕ)(x) := 2∫∂DΦ(x, y)ϕ(y) ds(y)

and

(Kϕ)(x) := 2∫∂D

∂Φ(x, y)∂ν(y)

ϕ(y) ds(y)

and the normal derivative operators

(K′ϕ)(x) := 2∫∂D

∂Φ(x, y)∂ν(x)

ϕ(y) ds(y)

and

(Tϕ)(x) := 2∂

∂ν(x)

∫∂D

∂Φ(x, y)∂ν(y)

ϕ(y) ds(y)

for x ∈ ∂D. From Section 6.4 we know that K and K′ are adjoint with respectto the L2 bilinear form and, clearly, S is self-adjoint, i.e., 〈Sϕ, ψ〉 = 〈ϕ, Sψ〉 forall ϕ, ψ ∈ C(∂D). To derive further properties of the boundary integral operators,let u and v denote the double-layer potentials with densities ϕ and ψ in C1,α(∂D),respectively. Then by the jump relation of Theorem 6.18, Green’s Theorem 6.3 andthe asymptotic behavior (6.20) we find that

∫∂D

Tϕψ ds = 2∫∂D

∂u∂ν

(v+ − v−) ds = 2∫∂D

(u+ − u−)∂v

∂νds =

∫∂DϕTψ ds,

that is, T also is self-adjoint. Now, in addition, let w denote the single-layer potentialwith density ϕ ∈ C(∂D). Then, using the jump relations from Theorems 6.15, 6.18


and 6.19, we find∫∂D

SϕTψ ds = 4∫∂Dw∂v

∂νds = 4

∫∂Dv−

∂w−∂ν

ds =∫∂D

(K − I)ψ(K′ + I)ϕ ds,

whence ∫∂Dϕ STψ ds =

∫∂Dϕ(K2 − I)ψ ds

follows for all ϕ ∈ C(∂D) and ψ ∈ C1,α(∂D). Thus, we have proven the relation

S T = K2 − I (7.47)

and similarly it can be shown that the adjoint relation

TS = K′2 − I (7.48)

is also valid.

Theorem 7.35. The unique solution u to the interior Dirichlet problem with bound-ary value f ∈ C1,α(∂D) belongs to C1,α(D) and depends continuously on f , i.e.,

‖ grad u‖α,D ≤ C‖ f ‖1,α,∂D (7.49)

with some constant C depending on D and α. The Dirichlet-to-Neumann operatorA mapping f onto the normal derivative ∂u/∂ν on ∂D is a bounded operator A :C1,α(∂D)→ C0,α(∂D).

Proof. By Theorem 7.6 the operator K is compact from C1,α(∂D) into itself. Hence,for the unique solution ϕ of the integral equation ϕ − Kϕ = −2 f from Theorem6.22 we have that ϕ ∈ C1,α(∂D) and ‖ϕ‖1,α ≤ c‖ f ‖1,α with some constant c. Nowu ∈ C1,α(D) and the boundedness (7.49) follow from Theorem 7.32. The statementon the Dirichlet-to-Neumann operator is a consequence of A = −T (I −K)−1 and themapping properties of T and K. �

Theorem 7.36. For g ∈ C0,α(∂D) satisfying the solvability condition∫∂Dg ds = 0,

the unique solution u to the interior Neumann problem subject to the normalizationcondition

∫∂D

u ds = 0 belongs to C1,α(D) and depends continuously on g, i.e.,

‖ grad u‖α,D ≤ C‖g‖α,∂D (7.50)

with some constant C depending on D and α. The Neumann-to-Dirichlet operatorB mapping g onto the trace u|∂D is a bounded operator B : C1,α

0 (∂D) → C0,α0 (∂D)

where the subscripts indicate the subspaces of functions with mean value zero.

Proof. We refer back to the proof of Theorem 6.30. Since K′ is compact fromCα

0 (∂D) into itself and I + K′ has a trivial nullspace in C0(∂D), for the unique solu-tion ψ ∈ C0(∂D) of the integral equation ψ + K′ψ = 2g from Theorem 6.26 we havethat ψ ∈ Cα

0 (∂D) and ‖ψ‖α ≤ c‖g‖α with some constant c. Now u ∈ C1,α(D) and the

7.6 Boundary Integral Equations in Holder Spaces 131

boundedness (7.50) follow from Theorem 7.30. The statement on the Neumann-to-Dirichlet operator is a consequence of B = S (I + K′)−1 and the mapping propertiesof S and K′. �

Note that, by construction, the operators A : C0,α0 (∂D) → C1,α

0 (∂D) and B :C1,α

0 (∂D)→ C0,α0 (∂D) are inverse to each other.

Theorem 7.37. For a harmonic function u ∈ C2(D) ∩ C1,α(D) the boundary valueand the normal derivative satisfy

⎛⎜⎜⎜⎜⎜⎜⎝u

∂u/∂ν

⎞⎟⎟⎟⎟⎟⎟⎠ =⎛⎜⎜⎜⎜⎜⎜⎝−K S

−T K′

⎞⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎝

u

∂u/∂ν

⎞⎟⎟⎟⎟⎟⎟⎠ , (7.51)

i.e., the operator in (7.51) is a projection operator in the product space of the bound-ary values and the normal derivatives of harmonic functions. This projection oper-ator is known as the Calderon projection.

Proof. This follows from Green’s representation formula (6.4) using the jump rela-tions. �

Note that by inserting a single- or a double-layer potential into (7.51) the relations(7.47) and (7.48) can again be obtained.

Obviously, given the Dirichlet data f , the second equation in (7.51) leads to

g − K′g = −T f (7.52)

as an integral equation of the second kind for the unknown Neumann data g. Theoperator in (7.52) is the adjoint of the operator in the equation ϕ − Kϕ = −2 ffrom the double-layer potential approach (see Theorem 6.22). Therefore, (7.52) isuniquely solvable.

Conversely, given the Neumann data g, the first equation in (7.51) leads to

f + K f = S g (7.53)

as an integral equation of the second kind for the unknown Dirichlet data. Theoperator in this equation is the adjoint of the operator in the equation ϕ + K′ϕ = 2gfrom the single-layer potential approach (see Theorem 6.26). The solvability con-dition from the Fredholm alternative is satisfied for (7.53). For the solution of thehomogeneous adjoint equation ψ0 from Theorem 6.21 we have that Sψ0 = constantand therefore 〈ψ0, S g〉 = 〈Sψ0, g〉 = 0 provided g satisfies the solvability conditionfor the interior Neumann problem.

The integral equations (7.52) and (7.53) obtained from Green’s representationtheorem are also known as the direct approach to boundary integral equations forthe Dirichlet and Neumann problems. As an advantage of this approach we note that(7.52) and (7.53) are equations for the physically interesting unknown Neumann orDirichlet data, respectively, which in the potential approach that we have considered


so far need an additional evaluation of a potential. However, there is a conservationof computational costs since the equations (7.52) and (7.53) require evaluations ofpotentials for their right-hand sides. We also note that, if these equations are usedwithout the parallel discussion of the integral equations from the potential approach,for a complete existence analysis one still has to show that solutions to the integralequations (7.52) and (7.53) indeed lead to solutions of the boundary value problems.

7.7 Boundary Integral Equations of the First Kind

We will conclude this chapter by considering integral equations of the first kindfor the solution of the Dirichlet and Neumann problems. Given f ∈ C1,α(∂D), thesingle-layer potential

u(x) :=∫∂DΦ(x, y)ϕ(y) ds(y), x ∈ IRm, (7.54)

solves the interior Dirichlet problem in D provided the density ϕ ∈ C0,α(∂D) is asolution of the integral equation

Sϕ = 2 f . (7.55)

In two dimensions, where (7.55) is known as Symm’s [228] integral equation, dueto the logarithmic fundamental solution the operator S , in general, is not injective.For example, from Lemma 8.23 we can conclude that S 1 = 0 when ∂D is the unitcircle. The following theorem that we adopt from [126] ensures injectivity of Sunder a geometric assumption.

Theorem 7.38. In two dimensions, assume there exist z0 ∈ D such that |x − z0| � 1for all x ∈ Γ. Then the single-layer operator S : C(∂D)→ C(∂D) is injective.

Proof. By continuity, from |x− z0| � 1 for all x ∈ ∂D we have that either |x − z0| < 1for all x ∈ ∂D or |x−z0| > 1 for all x ∈ ∂D. We consider the case where |x−z0| < 1 forall x ∈ ∂D and choose a neighborhood U of z0 such that U ⊂ D and |x−z| < 1 for allx ∈ ∂D and all z ∈ U. For a solution ϕ ∈ C(∂D) of Sϕ = 0 the single-layer potentialu defined by (7.54) satisfies u = 0 on ∂D and by Theorem 6.15 is continuous in IR2.The maximum-minimum principle (Theorem 6.9) implies that u = 0 in D.

We will show that u = 0 also in IR2 \ D. To this end we prove that∫∂Dϕ ds = 0

and assume without loss of generality that∫∂Dϕ ds ≥ 0. We consider the harmonic

function

v(x) := u(x) +1

2πln |x − z|

∫∂Dϕ ds, x ∈ IR2 \ D,

for some z ∈ U. Then

v(x) =1

2πln |x − z|

∫∂Dϕ ds ≤ 0, x ∈ ∂D,

7.7 Boundary Integral Equations of the First Kind 133

since |x− z| < 1 for x ∈ ∂D and z ∈ U. Furthermore, elementary estimates show that

v(x) =1

2π |x|2 x ·∫∂D

(y − z) ϕ(y) ds(y) + O

(1|x|2

), x → ∞. (7.56)

Consequently v(x)→ 0, x→ ∞, and the maximum-minimum principle implies thatv(x) ≤ 0 for all x ∈ IR2 \ D. Thus we can conclude from (7.56) that

e ·∫∂D

(y − z) ϕ(y) ds(y) ≤ 0

for all e ∈ IR2 with |e| = 1 which in turn yields∫∂Dy ϕ(y) ds(y) = z

∫∂Dϕ(y) ds(y).

Since this holds for all z ∈ U we finally conclude that∫∂Dϕ ds = 0.

Now we have u = v and consequently u(x) → 0, x → ∞ . Therefore, in view ofu = 0 on ∂D, by the uniqueness for the exterior Dirichlet problem (Theorem 6.12)we obtain that that u = 0 in IR2 \ D and consequently u = 0 in IR2. Hence ϕ = 0follows by the jump relation of Theorem 6.19.

The case where |x − z0| > 1 for all x ∈ ∂D is treated analogously. �

Theorem 7.39. In three dimensions the single-layer operator S : C(∂D) → C(∂D)is injective.

Proof. For a solution ϕ ∈ C(∂D) to Sϕ = 0 the single-layer potential (7.54) solvesboth the homogeneous interior Dirichlet problem in D and the homogeneous exte-rior Dirichlet problem in IR3 \ D. Therefore u = 0 in IR3 by Theorem 6.12 whenceϕ = 0 follows by the jump relation of Theorem 6.19. �

Theorem 7.40. For f ∈ C1,α(∂D) there exists a unique solution of the integral equa-tion of the first kind (7.55) if m = 3 or if m = 2 and the geometric condition ofTheorem 7.38 is satisfied.

Proof. By Theorems 7.5 and 7.6 the operators K′ : C0,α(∂D) → C0,α(∂D) andK : C1,α(∂D) → C1,α(∂D) are compact and by Corollaries 7.31 and 7.33 theoperators S : C0,α(∂D) → C1,α(∂D) and T : C1,α(∂D) → C0,α(∂D) are bounded.Therefore by (7.47) and (7.48) the self-adjoint operator T is a regularizer of theself-adjoint operator S (and vice versa). Now Theorem 5.6 implies that S is a Fred-holm operator with index zero and the statement of the theorem follows from thetwo uniqueness results of Theorems 7.38 and 7.39. �

To avoid the geometric condition in the uniqueness result of Theorem 7.38for Symm’s integral equation we describe a modified single-layer approach forthe Dirichlet problem in IR2. For notational brevity we introduce the mean valueoperator M by

M : ϕ → 1|∂D|

∫∂Dϕ ds. (7.57)


Then the modified single-layer potential

u(x) :=∫∂DΦ(x, y)[ϕ(y) − Mϕ] ds(y) + Mϕ, x ∈ IR2 \ ∂D, (7.58)

solves the interior (and the exterior) Dirichlet problem with boundary conditionu = f on ∂D provided the density ϕ ∈ C0,α(∂D) solves the integral equation

S0ϕ = 2 f (7.59)

with the operatorS0 := S − SM + M. (7.60)

Note that our modification of the logarithmic single-layer potential in (7.58) ensuresboundedness of u at infinity for the exterior problem. As we will see it is also crucialfor the interior problem since it ensures injectivity of the modified operator S0.

Theorem 7.41. The modified single-layer operator S0 : C0,α(Γ)→ C1,α(Γ) is bijec-tive.

Proof. Clearly, S0 is a compact perturbation of S since the difference S0 − S isbounded and has finite-dimensional range. Therefore the operator T remains a reg-ularizer of S0. Hence by Theorem 5.6 is suffices to show that S0 is injective. Letϕ ∈ C(∂D) be a solution to S0ϕ = 0. Then the single-layer potential u given by(7.58) solves the homogeneous interior and exterior Dirichlet problem. Hence, bythe uniqueness Theorem 6.12, we have u = 0 in D and in IR2 \ D. From the behav-ior at infinity we deduce

∫∂Dϕ ds = 0, and then the jump relation of Theorem 6.19

implies ϕ = 0 on ∂D. �

The single-layer potential (7.54) with the solution ϕ of the integral equation ofthe first kind (7.55) also provides the solution to the exterior Dirichlet problem inthree dimensions. In the two-dimensional case the solution to the exterior Dirichletproblem is obtained by the modified single-layer potential and the solution of (7.59).

We now turn to the Neumann problem. Given g ∈ C0,α(∂D), the double-layerpotential

v(x) :=∫∂D

∂Φ(x, y)∂ν(y)

ϕ(y) ds(y), x ∈ IRm \ ∂D, (7.61)

solves the interior Neumann problem in D provided the density ϕ ∈ C1,α(∂D) is asolution of the integral equation

Tϕ = 2g. (7.62)

Theorem 7.42. The hypersingular operator T : C1,α(∂D)→ C0,α(∂D) has nullspaceN(T ) = span{1}.Proof. For a solution ϕ ∈ C1,α(∂D) to Tϕ = 0 the double-layer potential (7.61)solves both the homogeneous interior Neumann problem in D and the homogeneousexterior Neumann problem in IRm \ D. Therefore v = 0 in IRm \ D and v = constant

7.8 Logarithmic Single-Layer Potential on an Arc 135

in D by Theorem 6.13 whence N(T ) ⊂ span{1} follows by the jump relation ofTheorem 6.18. The proof is completed by the observation that T1 = 0 by Example6.17. �

Theorem 7.43. For each g ∈ C0,α(∂D) satisfying the solvability condition∫∂Dg ds=0 for the interior Neumann problem there exists a solution of the integral

equation of the first kind (7.62). Two solutions can differ only by a constant.

Proof. In the proof of Theorem 7.40 we already noted that the self-adjoint operatorS is a regularizer of the self-adjoint operator T . Thus the statement of the theoremagain follows from Theorem 5.6. �

We note that in two dimensions, with the aid of either Theorem 7.40 or Theorem7.41, the hypersingular integral equation (7.62) can also be solved using the relation(7.46) (see also Problem 7.4).

The double-layer potential (7.61) with the solution ϕ of (7.62) also provides thesolution to the exterior Neumann problem in the two-dimensional case. Recall fromTheorem 6.29 that the condition

∫∂Dg ds = 0 is also necessary for the exterior Neu-

mann problem. In three dimensions, in order to obtain the correct asymptotics at in-finity, we add the single-layer potential u with density given by the natural chargeψ0

from Theorem 6.21 to the double-layer potential (7.61), noting that ∂u+/∂ν = −ψ0

on ∂D, see (6.38). The modified double-layer potential

v(x) :=∫∂D

∂Φ(x, y)∂ν(y)

ϕ(y) ds(y) −∫∂Dg ds

∫∂DΦ(x, y)ψ0(y) ds(y), x ∈ IR3 \ ∂D,

solves the exterior Neumann problem in three dimensions provided the densityϕ ∈ C1,α(∂D) solves the integral equation

Tϕ = 2g − 2ψ0

∫∂Dg ds. (7.63)

Since∫∂Dψ0 ds = 1 the right-hand side of (7.63) is in the range of T .

We conclude this section by noting that we can also obtain boundary integralequations of the first kind in terms of the operator S and T from the Calderon pro-jection in the direct approach. These two equations coincide with (7.52) and (7.53)with the roles of the known and unknown functions reversed. We leave their discus-sion as an exercise to the reader.

7.8 Logarithmic Single-Layer Potential on an Arc

In the final section of this chapter we wish to indicate how the modifications ofthe integral equations of the first kind of Section 7.7 arising from the Dirichlet orNeumann problem for the exterior of an arc can be reduced to the case of a closedboundary contour by using a cosine substitution as suggested by Multhopp [178].


For brevity we confine ourselves to the case of the single-layer potential for theDirichlet problem. For the Neumann problem we refer to Monch [172].

Assume that Γ ⊂ IR2 is an arc of class C3, i.e.,

Γ = {x(s) : s ∈ [−1, 1]},where x : [−1, 1] → IR2 is an injective and three times continuously differentiablefunction with x′(s) � 0 for all s ∈ [−1, 1]. We consider the Dirichlet problem for theLaplace equation in the exterior of the arc Γ: Given a function f ∈ C1,α(Γ), find abounded solution u ∈ C2(IR2 \ Γ) ∩ C(IR2) to the Laplace equation

Δu = 0 in IR2 \ Γ (7.64)

satisfying the Dirichlet boundary condition

u = f on Γ. (7.65)

Note that besides continuity we do not explicitly assume any condition for thebehavior of the solution at the two endpoints z1 := x(1) and z−1 := x(−1) of Γ.

Theorem 7.44. The Dirichlet problem for the exterior of an arc has at most onesolution.

Proof. The statement follows analogously to the proof of Theorem 6.12 from themaximum-minimum principle and the boundedness condition at infinity. �

As in the case of a closed boundary curve (compare (7.58)) we will establishexistence by seeking the solution in the form

u(x) :=1

2π

∫Γ

[ψ(y) − Mψ] ln1|y − x| ds(y) + Mψ, x ∈ IR2 \ Γ, (7.66)

of a single-layer potential. Here, analogous to (7.57) we have abbreviated

Mψ :=1|Γ|

∫Γ

ψ ds.

For the case of the arc Γ the density ψ is assumed to be of the form

ψ(x) =ψ(x)√|x − z1| |x − z−1|

, x ∈ Γ \ {z1, z−1}, (7.67)

where ψ ∈ C0,α(Γ). Note that the integrals in (7.66) exist as improper integrals withrespect to the two endpoints of Γ.

Recall the cut-off function h from the proof of Theorem 2.29 and, for n ∈ IN,denote by un the single-layer potential (7.66) with the density ψ − Mψ replaced byχn, where χn(x) := h(n|x − z1|)h(n|x − z−1|)[ψ(x) − Mψ]. Then, by interpreting Γas part of a closed contour Γ and setting χn = 0 on Γ \ Γ, from Theorem 6.15 we

7.8 Logarithmic Single-Layer Potential on an Arc 137

conclude that un is continuous in all of IR2. For sufficiently large n we have∫Γ(z1;1/n)

1|y − z1|3/4 ds(y) ≤ C1

n1/4

for some constant C1. Therefore, using Holder’s inequality for integrals and (7.67),we can estimate

∫Γ(z1;1/n)

∣∣∣∣∣[ψ(y) − Mψ] ln1|y − x|

∣∣∣∣∣ ds(y) ≤ C

n1/6

(∫Γ

∣∣∣∣∣ln 1|y − x|

∣∣∣∣∣3

ds(y)

)1/3

for sufficiently large n and some constant C. Analogous to the logarithmic single-layer potential the integral on the right-hand side is continuous and therefore isbounded on compact subsets of IR2. Therefore we can conclude that

∫Γ(z1;1/n)

∣∣∣∣∣[ψ(y) − Mψ] ln1|y − x|

∣∣∣∣∣ ds(y)→ 0, n→ ∞,

uniformly on compact subsets of IR2. From this and the corresponding property forthe integral over a neighborhood of the other endpoint z−1, it follows that u is theuniform limit of a sequence of continuous functions and therefore is continuous inall of IR2 (see also [91], p. 276). Therefore, as in the case of a closed boundarycurve (compare (7.59)), the single-layer potential (7.66) solves the Dirichlet prob-lem (7.64)–(7.65) provided the density satisfies the integral equation

12π

∫Γ

[ψ(y) − Mψ] ln1|y − x| ds(y) + Mψ = f (x), x ∈ Γ. (7.68)

From the uniqueness Theorem 7.44 and the jump relations of Theorem 6.19 we canconclude that the integral equation (7.68) has at most one solution.

Following Multhopp [178] we substitute s = cos t into the parameterization Γ ={x(s) : s ∈ [−1, 1]} to transform the integral equation (7.68) into the parameterizedform

14π

∫ π

0

{− ln(4[cos t − cos τ]2) + p(t, τ)

}ϕ(τ) dτ = g(t), t ∈ [0, π]. (7.69)

Here we have set

ϕ(t) := | sin t x′(cos t)|ψ(x(cos t)), g(t) := f (x(cos t)), t ∈ [0, π],

and the kernel p is given by

p(t, τ) := ln4[cos t − cos τ]2

|x(cos t) − x(cos τ)|2 +4π|Γ|

− 1|Γ|

∫ π

0ln

1|x(cos t) − x(cosσ)|2 |x

′(cosσ)| sinσ dσ


for t � τ. (Compare Problem 7.2 and use (7.70).) Solving the integral equation(7.68) for the function ψ on Γ is equivalent to solving the integral equation (7.69)for the even 2π-periodic function ϕ.

Writingx(s) − x(σ)

s − σ =

∫ 1

0x′[s + λ(σ − s)] dλ,

it can be seen that the kernel p can be extended as a twice continuously differentiablefunction on IR2 that is even and 2π-periodic with respect to both variables. Hence,in view of Theorem 7.4 the integral operator

(Lϕ)(t) :=1

4π

∫ π

0{p(t, τ) − 4}ϕ(τ) dτ, t ∈ [0, π],

is a compact operator from the Holder space C0,α2π,e ⊂ C0,α(IR) of even 2π-periodic

Holder continuous functions into the Holder space of even 2π-periodic Holder con-tinuously differentiable functions C1,α

2π,e ⊂ C1,α(IR). From the identity

ln(4[cos t − cos τ]2) = ln(4 sin2 t − τ

2

)+ ln

(4 sin2 t + τ

2

)(7.70)

it follows that for even and 2π-periodic functions ϕ and g the integral equation

14π

∫ π

0

{− ln(4[cos t − cos τ]2) + 4

}ϕ(τ) dτ = g(t), t ∈ [0, π],

is equivalent to

14π

∫ 2π

0

{− ln

(4 sin2 t − τ

2

)+ 2

}ϕ(τ) dτ = g(t), t ∈ [0, 2π].

From Problem 7.2, by choosing x(t) = (cos t, sin t) as the parameterization of theunit circle, it can be seen that the latter equation corresponds to the parameterizedform of the single-layer integral equation (7.59) for the case of the unit disk D.Hence, as a consequence of Theorem 7.41, the integral operator

(L0ϕ)(t) :=1

4π

∫ π

0

{− ln(4[cos t − cos τ]2) + 4

}ϕ(τ) dτ, t ∈ [0, π],

is a bounded and bijective operator from C0,α2π,e into C1,α

2π,e with a bounded inverse.Therefore, writing (7.69) in the form L0ϕ + Lϕ = g, from Corollary 3.6 and theinjectivity for (7.68) we have the following theorem.

Theorem 7.45. For each f ∈ C1,α(Γ) the single-layer integral equation (7.68) hasa unique solution ψ of the form (7.67) with ψ ∈ C0,α(Γ).

From this we finally conclude the following result.

Problems 139

Theorem 7.46. The Dirichlet problem (7.64)–(7.65) for the exterior of an arc isuniquely solvable.

From the form (7.67) of the density of the single-layer potential we expect thatthe gradient of the solution develops singularities at the endpoints of the arc. For acloser study of the nature of these singularities we refer to Grisvard [73].

Problems

7.1. Let g ∈ C0,α(Γ) be a nowhere vanishing function. Show that the inhomogeneous Riemannproblem with boundary conditions f− = g f+ + h on Γ is solvable if and only if the condition∫Γ

h f− dz = 0 is satisfied for all solutions f to the homogeneous adjoint Riemann problem with

boundary condition f+ = g f−.

7.2. Use a regular 2π-periodic parameterization Γ = {x(t) : 0 ≤ t ≤ 2π} with counterclockwiseorientation for the boundary curve to transform the integral equation of the first kind (7.59) for theDirichlet problem into the form

14π

∫ 2π

0

{− ln

(4 sin2 t − τ

2

)+ p(t, τ)

}ψ(τ) dτ = f (t), 0 ≤ t ≤ 2π,

where ψ(t) := |x′(t)|ψ(x(t)), f (t) := f (x(t)), and where the kernel is given by

p(t, τ) := ln4 sin2 t − τ

2|x(t) − x(τ)|2 −

1|Γ|

∫ 2π

0ln

1|x(t) − x(σ)|2 |x

′(σ)| dσ + 4π|Γ| , t � τ.

Show that the kernel p is continuously differentiable provided Γ is of class C2. Transform theintegral equation of the first kind (7.62) for the Neumann problem into the form

14π

∫ 2π

0

{cot

τ − t2+ q(t, τ)

}ϕ′(τ) dτ = g(t), 0 ≤ t ≤ 2π,

where ϕ(t) := ϕ(x(t)), g(t) := |x′(t)| g(x(t)), and where the kernel is given by

q(t, τ) := 2x′(t) · [x(τ) − x(t)]|x(t) − x(τ)|2 − cot

τ − t2

, t � τ.

Show that the kernel q is continuous provided Γ is of class C2.Hint: Use the integrals of Lemma 8.23 from the next chapter for m = 0.

7.3. Formulate and prove Noether’s theorems for the singular integral equation with Cauchy kernelwhere the complex integration is replaced by contour integration.Hint: Use (7.36).

7.4. Show that in two dimensions the bounded operator R : C1,α(∂D) → C0,α(∂D) given by R f :=f ′ +M f in terms of the derivative f ′ with respect to arc length and the mean value operator (7.57)is injective. Therefore Symm’s integral equation (7.55) is equivalent to RS f = 2R f . Use this toconclude that RS differs from the singular operator with Cauchy kernel

k(z, ζ) = (ζ − z)∂

∂s(z)ln

1|ζ − z| , z � ζ,


only by a compact operator. Show that

k(z, ζ) ==Re{h(z) (ζ − z)}|ζ − z|2 (ζ − z) =

12

h(z) +12

h(z)(ζ − z)2

|ζ − z|2 , z � ζ,

where τ = (τ1 , τ2) is the unit tangent vector to ∂D and h(z) := τ1(z) + iτ2(z). Use a parametricrepresentation of ∂D to conclude that

(ζ − z)2

|ζ − z|2 , z � ζ,

can be extended as a continuously differentiable function onto ∂D × ∂D and therefore k belongsto C0,1,1(∂D × ∂D). Consequently one can apply Noether’s theorems in the form of Problem 7.3to obtain an alternative proof for the existence result of Theorem 7.40. The same approach can beused for the modified single-layer approach of Theorem 7.41 which is equivalent to RS0 f = 2R f .The idea to reduce the logarithmic single-layer potential integral equation by differentiation toa Cauchy type integral equation is due to Fichera [53] and in a modified version to Hsiao andMacCamy [102].

7.5. Show that for each real-valued function ϕ ∈ C0,α(Γ) there exists a function f that is holomor-phic in D, which can be extended continuously from D into D and that has real part Re f = ϕ onthe boundary Γ. Show that two functions with this property can differ only by a purely imaginaryconstant.Hint: Solve a Dirichlet problem for the real part of f and use (7.37) and Problem 7.4.

Chapter 8Sobolev Spaces

In this chapter we study the concept of weak solutions to boundary value problemsfor harmonic functions. We shall extend the classical theory of boundary integralequations as described in the two previous chapters from the spaces of continuous orHolder continuous functions to appropriate Sobolev spaces. For the sake of brevitywe will confine ourselves to interior boundary value problems in two dimensions.

We wish to mention that our introduction of Sobolev spaces of periodic functionsand Sobolev spaces on a closed contour differs from the usual approach in tworegards. Firstly, instead of using the Fourier transform we only need Fourier series,and secondly, the Sobolev spaces in the contour case are defined via a global ratherthan a local procedure. The motivation for our approach is the hope that this analysismight make the basic ideas on these Sobolev spaces more easily accessible. Wealso want to emphasize that we do not rely heavily on Lebesgue integration as anecessary prerequisite. For our purpose, it will be sufficient to understand the Hilbertspace L2[0, 2π] as the completion of the space C[0, 2π] of continuous functions withrespect to the mean square norm.

8.1 The Sobolev Space Hp[0, 2π]

As the basis of our presentation of Sobolev spaces we begin with a brief review ofthe classical Fourier series expansion. For a function ϕ ∈ L2[0, 2π] the series

∞∑m=−∞

ϕmeimt, (8.1)

where

ϕm :=1

2π

∫ 2π

0ϕ(t)e−imtdt,

is called the Fourier series of ϕ, its coefficients ϕm are called the Fourier coeffi-cients of ϕ. On L2[0, 2π], as usual, the mean square norm is introduced by the scalar


141

142 8 Sobolev Spaces

product

(ϕ, ψ) :=∫ 2π

0ϕ(t)ψ(t) dt.

We denote by fm the trigonometric monomials

fm(t) := eimt

for t ∈ IR and m ∈ ZZ. Then the set { fm : m ∈ ZZ} is an orthogonal system. By theWeierstrass approximation theorem (see [40]), the trigonometric polynomials aredense with respect to the maximum norm in the space of 2π-periodic continuousfunctions, and C[0, 2π] is dense in L2[0, 2π] in the mean square norm. Therefore,by Theorem 1.28, the orthogonal system is complete and the Fourier series (8.1)converges in the mean square norm. Because of the orthonormality factor ‖ fm‖22 =2π, Parseval’s equality assumes the form

∞∑m=−∞

|ϕm|2 = 12π

∫ 2π

0|ϕ(t)|2dt =

12π‖ϕ‖22. (8.2)

We will now define subspaces Hp[0, 2π] of L2[0, 2π] by requiring for their ele-ments ϕ a certain decay of the Fourier coefficients ϕm as |m| → ∞.

Definition 8.1. Let 0 ≤ p < ∞. By Hp[0, 2π] we denote the space of all functionsϕ ∈ L2[0, 2π] with the property

∞∑m=−∞

(1 + m2)p|ϕm|2 < ∞

for the Fourier coefficients ϕm of ϕ. The space Hp[0, 2π] is called a Sobolev space.Frequently we will abbreviate Hp = Hp[0, 2π]. Note that H0[0, 2π] coincides withL2[0, 2π].

Theorem 8.2. The Sobolev space Hp[0, 2π] is a Hilbert space with the scalar prod-uct defined by

(ϕ, ψ)p :=∞∑

m=−∞(1 + m2)pϕmψm

for ϕ, ψ ∈ Hp[0, 2π] with Fourier coefficients ϕm and ψm, respectively. Note that thenorm on Hp[0, 2π] is given by

‖ϕ‖p =⎧⎪⎪⎨⎪⎪⎩∞∑

m=−∞(1 + m2)p|ϕm|2

⎫⎪⎪⎬⎪⎪⎭1/2

.

The trigonometric polynomials are dense in Hp[0, 2π].

Proof. We leave it as an exercise to verify that Hp is a linear space and that(· , ·)p is a scalar product. That (· , ·)p is well defined can be concluded from the

8.1 The Sobolev Space Hp[0, 2π] 143

Cauchy–Schwarz inequality

∣∣∣∣∣∣∣∞∑

m=−∞(1 + m2)pϕmψm

∣∣∣∣∣∣∣2

≤∞∑

m=−∞(1 + m2)p|ϕm|2

∞∑m=−∞

(1 + m2)p|ψm|2.

To prove that Hp is complete, let (ϕn) be a Cauchy sequence, i.e., given ε > 0, thereexists N(ε) ∈ IN such that ‖ϕn − ϕk‖p < ε for all n, k ≥ N(ε), or in terms of theFourier coefficients ϕm,n of ϕn,

∞∑m=−∞

(1 + m2)p|ϕm,n − ϕm,k |2 < ε2

for all n, k ≥ N(ε). From this we observe that

M2∑m=−M1

(1 + m2)p|ϕm,n − ϕm,k|2 < ε2 (8.3)

for all M1, M2 ∈ IN and all n, k ≥ N(ε). Therefore, since C is complete, there existsa sequence (ϕm) in C such that ϕm,n → ϕm, n → ∞, for each m ∈ ZZ. Passing to thelimit k → ∞ in (8.3) now yields

M2∑m=−M1

(1 + m2)p|ϕm,n − ϕm |2 ≤ ε2

for all M1, M2 ∈ IN and all n ≥ N(ε). Hence

∞∑m=−∞

(1 + m2)p|ϕm,n − ϕm|2 ≤ ε2

for all n ≥ N(ε). From this we conclude that

ϕ :=∞∑

m=−∞ϕm fm

defines a function ϕ ∈ Hp with ‖ϕ − ϕn‖p → 0, n→ ∞. Hence, Hp is complete.Let ϕ ∈ Hp with Fourier coefficients ϕm. Then for the partial sums

ϕn :=n∑

m=−n

ϕm fm

of the Fourier series we have that

‖ϕ − ϕn‖2p =∞∑

|m|=n+1

(1 + m2)p|ϕm|2 → 0, n→ ∞,


since the series∑∞

m=−∞(1 +m2)p|ϕm|2 converges. Therefore, the trigonometric poly-nomials are dense in Hp. �

Theorem 8.3. If q > p then Hq[0, 2π] is dense in Hp[0, 2π] with compact imbeddingfrom Hq[0, 2π] into Hp[0, 2π].

Proof. From (1+m2)p ≤ (1+m2)q for m ∈ ZZ it follows that Hq ⊂ Hp with boundedimbedding

‖ϕ‖p ≤ ‖ϕ‖q (8.4)

for all ϕ ∈ Hq. Then the denseness of Hq in Hp is a consequence of the densenessof the trigonometric polynomials in Hp.

We denote the imbedding operator by I : Hq → Hp. For n ∈ IN we definefinite-dimensional bounded operators In : Hq → Hp by setting

Inϕ :=n∑

m=−n

ϕm fm

for ϕ ∈ Hq with Fourier coefficients ϕm. Then

‖(In − I)ϕ‖2p =∞∑

|m|=n+1

(1 + m2)p|ϕm|2

≤ 1(1 + n2)q−p

∞∑|m|=n+1

(1 + m2)q|ϕm|2

≤ 1(1 + n2)q−p

‖ϕ‖2q.

Hence, compactness of the imbedding follows from Theorem 2.22. �

Theorem 8.4. Let p > 1/2. Then the Fourier series for ϕ ∈ Hp[0, 2π] convergesabsolutely and uniformly. Its limit is continuous and 2π-periodic and coincides withϕ almost everywhere. This imbedding of the Sobolev space Hp[0, 2π] in the spaceC2π of 2π-periodic continuous functions is compact.

Proof. For the Fourier series of ϕ ∈ Hp[0, 2π], by the Cauchy–Schwarz inequality,we conclude that

⎧⎪⎪⎨⎪⎪⎩∞∑

m=−∞

∣∣∣ϕmeimt∣∣∣⎫⎪⎪⎬⎪⎪⎭

2

≤∞∑

m=−∞

1(1 + m2)p

∞∑m=−∞

(1 + m2)p|ϕm|2

for all t ∈ IR. Since the series

∞∑m=−∞

1(1 + m2)p


converges for p > 1/2, this estimate establishes the absolute and uniform conver-gence of the Fourier series. Its limit is continuous and 2π-periodic because it is theuniform limit of a sequence of continuous and 2π-periodic functions. It must coin-cide with ϕ almost everywhere, because we already have convergence of the Fourierseries to ϕ in the mean square norm, which is weaker than the maximum norm.

The above estimate also shows that

‖ϕ‖∞ ≤ C‖ϕ‖pfor all ϕ ∈ Hp[0, 2π] and some constant C depending on p. Therefore, the imbed-ding from Hp[0, 2π] into C2π is bounded, and consequently, by Theorems 2.21 and8.3, is compact for p > 1/2. �

For the introduction of the Sobolev spaces Hp(Γ) for a closed contour Γ in IR2

we shall need additional norms that are equivalent to ‖ · ‖p. We will denote them by‖ · ‖p,[p], where [p] is the largest integer less than or equal to p. For their definitionwe have to distinguish three cases. For k ∈ IN, by Ck

2π we will denote the space of ktimes continuously differentiable 2π-periodic functions from IR into C.

Theorem 8.5. For k ∈ IN we have Ck2π ⊂ Hk[0, 2π] and on Ck

2π the norm ‖ · ‖k isequivalent to

‖ϕ‖k,k :=

(∫ 2π

0

{|ϕ(t)|2 + |ϕ(k)(t)|2

}dt

)1/2

. (8.5)

Proof. Using ∫ 2π

0ϕ(k)(t)e−imtdt = (im)k

∫ 2π

0ϕ(t)e−imtdt (8.6)

and Parseval’s equality (8.2) for the Fourier coefficients of ϕ and ϕ(k) we obtain

‖ϕ‖2k,k = 2π∞∑

m=−∞(1 + m2k)|ϕm|2.

Now the inequalities

(1 + m2k) ≤ (1 + m2)k ≤ (2m2)k ≤ 2k(1 + m2k), m ∈ ZZ,

yield the equivalence of the norms ‖ · ‖k and ‖ · ‖k,k. �

Theorem 8.5 shows that for k ∈ IN the Sobolev spaces Hk[0, 2π] are isomorphicto the classical Sobolev spaces given by the completion of the space of k timescontinuously differentiable 2π-periodic functions with respect to the norm ‖ · ‖k,k.

The norms considered in the following theorem and its corollaries are known asSobolev–Slobodeckij norms. To some extent they resemble Holder norms.


Theorem 8.6. For 0 < p < 1 on C12π the norm ‖ · ‖p is equivalent to

‖ϕ‖p,0 :=

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

∫ 2π

0|ϕ(t)|2dt +

∫ 2π

0

∫ 2π

0

|ϕ(t) − ϕ(τ)|2∣∣∣∣∣sint − τ

2

∣∣∣∣∣2p+1

dτdt

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

1/2

. (8.7)

Proof. We first observe that the norm ‖ · ‖p,0 corresponds to the scalar product

(ϕ, ψ) :=∫ 2π

0ϕ(t)ψ(t) dt +

∫ 2π

0

∫ 2π

0

{ϕ(t) − ϕ(τ)}{ψ(t) − ψ(τ)}∣∣∣∣∣sint − τ

2

∣∣∣∣∣2p+1

dτdt

on C12π. Note that the second term on the right-hand side is well defined, since by the

mean value theorem continuously differentiable functions are uniformly Holder con-tinuous with Holder exponent 1. Straightforward integration shows that the trigono-metric monomials fm are orthogonal with respect to this scalar product, i.e.,

( fm, fk) = (2π + γm)δmk,

where

γm := 16π∫ π

0

sin2 m2

t

sin2p+1 t2

dt,

δmk = 1 for k = m, and δmk = 0 for k � m. Using 2t/π < sin t < t for 0 < t < π/2,we can estimate

22p+5π

∫ π

0

sin2 m2

t

t2p+1dt < γm < 16π2p+2

∫ π

0

sin2 m2

t

t2p+1dt.

For m > 0, we substitute mt = τ and use

0 <∫ π

0

sin2 τ

2τ2p+1

dτ ≤∫ mπ

0

sin2 τ

2τ2p+1

dτ <∫ ∞

0

sin2 τ

2τ2p+1

dτ < ∞

to obtainc0|m|2p < γm < c1|m|2p (8.8)

for all m � 0 with some constants c0 and c1 depending on p. Using the estimate|eimt − eimτ| ≤ |m(t − τ)|, from Parseval equality (8.2) for ϕ ∈ C1

2π and the Cauchy–Schwarz inequality, proceeding as in the proof of Theorem 8.4 we see that the series

ϕ(t) − ϕ(τ) =∞∑

m=−∞ϕm

{eimt − eimτ

}


is absolutely and uniformly convergent for all t, τ ∈ [0, 2π]. Therefore we can inte-grate termwise to derive

∫ 2π

0

∫ 2π

0

|ϕ(t) − ϕ(τ)|2∣∣∣∣∣sint − τ

2

∣∣∣∣∣2p+1

dτdt =∞∑

m=−∞γm|ϕm|2.

Now the statement of the theorem follows from the inequalities (8.8) and

(1 + |m|2p) ≤ 2|m|2p ≤ 2(1 + m2)p ≤ 2p+1(1 + |m|2p), m ∈ ZZ,

together with Parseval’s equality (8.2). �

Corollary 8.7. When p = k + q, where k ∈ IN and 0 < q < 1, then on Ck+12π the norm

‖ · ‖p is equivalent to

‖ϕ‖p,k :={‖ϕ‖20 + ‖ϕ(k)‖2q,0

}1/2. (8.9)

Proof. Combine the proofs of Theorems 8.5 and 8.6 (see Problem 8.1). �

Corollary 8.8. For a nonnegative integer k let f ∈ Ck2π and assume that 0 ≤ p ≤ k.

Then for all ϕ ∈ Hp[0, 2π] the product fϕ belongs to Hp[0, 2π] and

‖ fϕ‖p ≤ C{‖ f ‖∞ + ‖ f (k)‖∞

}‖ϕ‖p

for some constant C depending on p.

Proof. Use the equivalent norms of Theorems 8.5 and 8.6 and Corollary 8.7 (seeProblem 8.1). �

Definition 8.9. For 0 ≤ p < ∞ by H−p[0, 2π] we denote the dual space of Hp[0, 2π],i.e., the space of bounded linear functionals on Hp[0, 2π].

The space H−p is characterized by the following theorem.

Theorem 8.10. For F ∈ H−p[0, 2π] the norm is given by

‖F‖p =⎧⎪⎪⎨⎪⎪⎩∞∑

m=−∞(1 + m2)−p|Fm|2

⎫⎪⎪⎬⎪⎪⎭1/2

,

where Fm = F( fm). Conversely, to each sequence (Fm) in C satisfying

∞∑m=−∞

(1 + m2)−p|Fm|2 < ∞

there exists a bounded linear functional F ∈ H−p[0, 2π] with F( fm) = Fm.


Proof. Assume the sequence (Fm) satisfies the required inequality and define a func-tional F : Hp[0, 2π]→ C by

F(ϕ) :=∞∑

m=−∞ϕmFm

for ϕ ∈ Hp[0, 2π] with Fourier coefficients ϕm. Then from the Cauchy–Schwarzinequality we have

|F(ϕ)|2 ≤∞∑

m=−∞(1 + m2)−p|Fm|2

∞∑m=−∞

(1 + m2)p|ϕm|2.

Hence, F is well defined and bounded with

‖F‖p ≤⎧⎪⎪⎨⎪⎪⎩∞∑

m=−∞(1 + m2)−p|Fm|2

⎫⎪⎪⎬⎪⎪⎭1/2

.

On the other hand, for n ∈ IN the function ϕn :=∑n

m=−n(1 + m2)−pFm fm has norm

‖ϕn‖p =⎧⎪⎪⎨⎪⎪⎩

n∑m=−n

(1 + m2)−p|Fm|2⎫⎪⎪⎬⎪⎪⎭

1/2

.

Therefore

‖F‖p ≥ |F(ϕn)|‖ϕn‖p =

⎧⎪⎪⎨⎪⎪⎩n∑

m=−n

(1 + m2)−p|Fm|2⎫⎪⎪⎬⎪⎪⎭

1/2

.

Hence, ⎧⎪⎪⎨⎪⎪⎩∞∑

m=−∞(1 + m2)−p|Fm|2

⎫⎪⎪⎬⎪⎪⎭1/2

≤ ‖F‖p,

and this completes the proof. �

Theorem 8.11. For each function g ∈ L2[0, 2π] the sesquilinear duality pairing

G(ϕ) :=1

2π

∫ 2π

0ϕ(t)g(t) dt, ϕ ∈ Hp[0, 2π], (8.10)

defines a linear functional G ∈ H−p[0, 2π]. In this sense, L2[0, 2π] is a subspaceof each dual space H−p[0, 2π], and the trigonometric polynomials are dense inH−p[0, 2π].

Proof. Denote by gm the Fourier coefficients of g. Then, since G( fm) = gm, by thesecond part of Theorem 8.10 we have G ∈ H−p. Let F ∈ H−p with F( fm) = Fm and


define a sequence (Fn) in H−p by

Fn(ϕ) :=1

2π

∫ 2π

0ϕ(t)gn(t) dt,

where gn is the trigonometric polynomial gn :=∑n

m=−n Fm fm. Then we have

‖F − Fn‖2p =∞∑

|m|=n+1

(1 + m2)−p|Fm|2 → 0, n→ ∞,

analogous to Theorem 8.2. �

Obviously, H−p becomes a Hilbert space by appropriately extending the defini-tion of the scalar product from Theorem 8.2 to negative p. For p = 0 the dualitymap described in Theorem 8.11 is bijective with ‖G‖0 = ‖g‖0. Therefore, we canidentify H−0 and H0 and obtain a Hilbert scale of Hilbert spaces Hp for all real pwith compact imbedding from Hq into Hp for q > p.

The Sobolev spaces Hp frequently are called interpolation spaces because of theinterpolating properties indicated by the following two theorems.

Theorem 8.12. Let p < q and r = λp + (1 − λ)q with 0 < λ < 1. Then

‖ϕ‖r ≤ ‖ϕ‖λp ‖ϕ‖1−λq

for all ϕ ∈ Hq[0, 2π].

Proof. With the aid of Holder’s inequality we estimate

‖ϕ‖2r =∞∑

m=−∞{(1 + m2)p|ϕm|2}λ{(1 + m2)q|ϕm|2}1−λ

≤⎧⎪⎪⎨⎪⎪⎩∞∑

m=−∞(1 + m2)p|ϕm|2

⎫⎪⎪⎬⎪⎪⎭λ ⎧⎪⎪⎨⎪⎪⎩

∞∑m=−∞

(1 + m2)q|ϕm|2⎫⎪⎪⎬⎪⎪⎭

1−λ= ‖ϕ‖2λp ‖ϕ‖2−2λ

q ,


For real s consider the operator Ds transferring the trigonometric monomialsfm(t) = eimt to

Ds fm := (1 + m2)s fm.

For each p ∈ IR it clearly defines a bounded linear operator from Hp+2s onto Hp

with‖Dsϕ‖p = ‖ϕ‖p+2s (8.11)

for all ϕ ∈ Hp+2s. Obviously, Ds+t = DsDt for all s, t ∈ IR. In particular, D−s is theinverse of Ds. For s = 1 the operator D = D1 corresponds to the second derivativeDϕ = ϕ − ϕ′′ if ϕ is twice continuously differentiable. Therefore, we may interpret


D as a generalization of this derivative for elements of the Sobolev spaces and theoperators Ds as intermediate derivatives. With the aid of Ds we can transform thescalar products by

(ϕ, ψ)p+s = (Ds/2ϕ,Ds/2ψ)p (8.12)

for all ϕ, ψ ∈ Hp+s and(ϕ, ψ)p+s = (ϕ,Dsψ)p (8.13)

for all ϕ ∈ Hp+s and ψ ∈ Hp+2s.Noting that D fm = (1 + m2) fm, i.e., the fm are eigenfunctions of the operator D

with eigenvalues 1 + m2, we indicate obvious possibilities for extending this proce-dure to introduce interpolation spaces generated by other operators.

Theorem 8.13. Let p, q ∈ IR, r > 0, and let A be a linear operator mapping Hp

boundedly into Hq and Hp+r boundedly into Hq+r, i.e.,

‖Aϕ‖q ≤ C1‖ϕ‖pfor all ϕ ∈ Hp and

‖Aϕ‖q+r ≤ C2‖ϕ‖p+r

for all ϕ ∈ Hp+r and some constants C1 and C2. Then

‖Aϕ‖q+λr ≤ C1−λ1 Cλ

2‖ϕ‖p+λr

for all 0 < λ < 1 and all ϕ ∈ Hp+λr, i.e., A maps Hp+λr boundedly into Hq+λr for0 < λ < 1.

Proof. By A∗ : Hq → Hp we denote the adjoint of A : Hp → Hq, i.e.,

(Aϕ, ψ)q = (ϕ, A∗ψ)p (8.14)

for all ϕ ∈ Hp and ψ ∈ Hq. By Theorem 4.11 we have

‖A∗ψ‖p ≤ C1‖ψ‖qfor all ψ ∈ Hq. From this, making use of (8.11), we observe that the operator

A := D−r/2A∗Dr/2

is bounded from Hq+r into Hp+r with

‖Aψ‖p+r ≤ C1‖ψ‖q+r

for all ψ ∈ Hq+r.For all ϕ ∈ Hp+r and ψ ∈ Hq+r, with the aid of (8.13) and (8.14), we deduce

(Aϕ, ψ)q+r/2= (Aϕ,Dr/2ψ)q= (ϕ, A∗Dr/2ψ)p= (ϕ,D−r/2A∗Dr/2ψ)p+r/2.

8.2 The Sobolev Space Hp(Γ) 151

Hence, the bounded operators A : Hp+r → Hq+r and A : Hq+r → Hp+r are adjointwith respect to the positive dual systems (Hp+r,Hp+r) and (Hq+r,Hq+r) generatedby the (p+ r/2)– and (q+ r/2)–scalar products. Therefore, applying Lax’s Theorem4.13 and using the denseness of Hp+r in Hp+r/2, we conclude that A is bounded fromHp+r/2 into Hq+r/2 with

‖Aϕ‖q+r/2 ≤ C1/21 C1/2

2 ‖ϕ‖p+r/2

for all ϕ ∈ Hp+r/2.Repeating this argument, we see that


2‖ϕ‖p+λr

for all ϕ ∈ Hp+λr and all rational binary numbers 0 < λ < 1. For arbitrary 0 < λ < 1we can choose a monotonic decreasing sequence (λn) of binary rationals such thatλn → λ, n→ ∞. Then, using (8.4), we have

‖Aϕ‖q+λr ≤ ‖Aϕ‖q+λnr ≤ C1−λn1 Cλn

2 ‖ϕ‖p+λnr

for all ϕ ∈ Hp+r and n ∈ IN. Passing to the limit n→ ∞ we see that


2‖ϕ‖p+λr

for all ϕ ∈ Hp+r, and because Hp+r is dense in Hp+λr, for all ϕ ∈ Hp+λr. The idea touse Lax’s theorem to prove this interpolation theorem is due to Kirsch [122]. �

8.2 The Sobolev Space Hp(Γ)

Let Γ be the boundary of a simply connected bounded domain D ⊂ IR2 of class Ck,k ∈ IN. With the aid of a regular and k-times continuously differentiable 2π-periodicparametric representation

Γ = {z(t) : t ∈ [0, 2π)}for 0 ≤ p ≤ k we can define the Sobolev space Hp(Γ) as the space of all functionsϕ ∈ L2(Γ) with the property that ϕ ◦ z ∈ Hp[0, 2π]. By ϕ ◦ z, as usual, we denotethe 2π-periodic function given by (ϕ ◦ z)(t) := ϕ(z(t)), t ∈ IR. The scalar product andnorm on Hp(Γ) are defined through the scalar product on Hp[0, 2π] by

(ϕ, ψ)Hp(Γ) := (ϕ ◦ z, ψ ◦ z)Hp[0,2π].

Without loss of generality we have restricted the parameter domain to be the inter-val [0, 2π). However, we must allow the possibility of different regular parametricrepresentations for the boundary curve Γ. Therefore, we have to convince ourselvesthat our definition is invariant with respect to the parameterization.


Theorem 8.14. Assume that z and z are two different regular 2π-periodic para-metric representations for the boundary Γ of a simply connected bounded domainD ⊂ IR2 of class Ck, k ∈ IN, i.e., Γ = {z(t) : t ∈ [0, 2π)} and Γ = {z(t) : t ∈ [0, 2π)}.Then, for 0 ≤ p ≤ k the Sobolev spaces

Hp(Γ) :={ϕ ∈ L2(Γ) : ϕ ◦ z ∈ Hp[0, 2π]

}

with scalar product(ϕ, ψ)Hp(Γ) := (ϕ ◦ z, ψ ◦ z)Hp[0,2π]

andHp(Γ) :=

{ϕ ∈ L2(Γ) : ϕ ◦ z ∈ Hp[0, 2π]

}with scalar product

(ϕ, ψ)Hp(Γ) := (ϕ ◦ z, ψ ◦ z)Hp[0,2π]

are isomorphic.

Proof. Assume that both parameterizations have the same orientation. (The caseof two opposite parameterizations is treated analogously.) Because of the 2π-periodicity, without loss of generality, we may assume that z(0) = z(0). Denoteby z−1 : Γ → [0, 2π) the inverse of z : [0, 2π) → Γ. Then the regularity ofthe parameterization implies that the mapping f : [0, 2π) → [0, 2π) given byf := z−1 ◦ z is bijective and k-times continuously differentiable with f (0) = 0.By setting f (t + 2π) := f (t) + 2π it can be extended as a k-times continuously dif-ferentiable function on IR.

To establish the statement of the theorem, obviously, it suffices to show that forall ϕ ∈ Hp[0, 2π] we have ϕ ◦ f ∈ Hp[0, 2π] with

‖ϕ ◦ f ‖p ≤ C‖ϕ‖p, (8.15)

where C is some constant depending on k and p. For this we make use of the equiv-alent norms of Theorems 8.5 and 8.6 and Corollary 8.7. If p ∈ IN, we use the normgiven by (8.5). Because of Theorem 8.5 there exist positive constants c1 and c2 suchthat

c1‖ϕ‖2j, j ≤ ‖ϕ‖2j ≤ c2‖ϕ‖2j, jfor all ϕ ∈ Cp

2π and all j = 1, . . . , p. Then, using the chain rule and the Cauchy–Schwarz inequality, we can estimate

‖ϕ ◦ f ‖2p ≤ c2

∫ 2π

0

{|ϕ( f (t))|2 +

∣∣∣∣∣ dp

dtpϕ( f (t))

∣∣∣∣∣2}

dt ≤ c2Mp∑

j=0

∫ 2π

0

∣∣∣ϕ( j)( f (t))∣∣∣2 dt,

where M is some constant depending on p that contains bounds on the derivativesof f up to order p. Substituting s = f (t) on the right-hand side and using (8.4), we


get the further estimates

‖ϕ ◦ f ‖2p ≤ c2Mγ

p∑j=0

∫ 2π

0

∣∣∣ϕ( j)(s)∣∣∣2 ds ≤ c2Mγ

p∑j=1

‖ϕ‖2j, j ≤c2

c1Mγp ‖ϕ‖2p.

Here, we have set γ := ‖1/ f ′‖∞. This completes the proof of (8.15) for p ∈ IN,since Cp

2π is dense in Hp[0, π].If 0 < p < 1, we use the norm given by (8.7). From l’Hopital’s rule and the

property f (t + 2π) := f (t) + 2π we can conclude that the function

b(t, τ) :=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∣∣∣∣∣∣∣∣∣∣sin

f (t) − f (τ)2

sint − τ

2

∣∣∣∣∣∣∣∣∣∣,

t − τ2π� ZZ,

| f ′(t)|, t − τ2π∈ ZZ,

is continuous and 2π-periodic with respect to both variables. Hence, b is boundedand we can estimate

|ϕ( f (t)) − ϕ( f (τ))|2∣∣∣∣∣sint − τ

2

∣∣∣∣∣2p+1

≤ B2p+1 |ϕ( f (t)) − ϕ( f (τ))|2∣∣∣∣∣sinf (t) − f (τ)

2

∣∣∣∣∣2p+1

,

where B is a bound on b. From this, substituting s = f (t) and σ = f (τ), we obtain

∫ 2π

0

∫ 2π

0

|ϕ( f (t)) − ϕ( f (τ))|2∣∣∣∣∣sint − τ

2

∣∣∣∣∣2p+1

dτdt ≤ γ2B2p+1∫ 2π

0

∫ 2π

0

|ϕ(s) − ϕ(σ)|2∣∣∣∣∣sins − σ

2

∣∣∣∣∣2p+1

dσds.

With this inequality, (8.15) follows from the equivalence of the norms ‖ · ‖p and‖ · ‖p,0 on C1

2π and the denseness of C12π in Hp[0, 2π].

Finally, the case of an arbitrary noninteger positive p is settled by combining thetwo previous cases with the aid of the norm given by (8.9). �

The classical Sobolev space H1(D) for a bounded domain D ⊂ IR2 with boundary∂D of class C1 is defined as the completion of the space C1(D) of continuouslydifferentiable functions with respect to the norm

‖u‖H1(D) :=

(∫D

{|u(x)|2 + | grad u(x)|2

}dx

)1/2

.

For a detailed description of this space, in particular, its equivalent definition throughthe concept of weak derivatives, and its extension to the Sobolev spaces Hk(D) of ar-bitrary order k ∈ IN we refer to Adams and Fournier [2], Gilbarg and Trudinger [63],McLean [165], and Treves [236]. Since each Cauchy sequence with respect to


‖ · ‖H1(D) is also a Cauchy sequence with respect to ‖ · ‖L2 (D), we may interpret H1(D)as a subspace of L2(D). Obviously the gradient can be extended from C1(D) as abounded linear operator from H1(D) into L2(D) (see Problem 2.1).

We want to illustrate the connection between Sobolev spaces on the domain Dand Sobolev spaces on its boundaryΓ through the simplest case of the trace theorem.For functions defined on the closure D their values on the boundary are clearlydefined and the restriction of the function to the boundary Γ is called the trace. Theoperator mapping a function onto its trace is called the trace operator.

As a first step, we consider continuously differentiable functions u in the stripIR×[0, 1] that are 2π-periodic with respect to the first variable, i.e., u(t+2π, ·) = u(t, ·)for all t ∈ IR. By Q we denote the rectangle Q := [0, 2π) × [0, 1]. For 0 ≤ η ≤ 1 weconsider the Fourier coefficients

um(η) :=1

2π

∫ 2π

0u(t, η)e−imtdt.

By Parseval’s equality we have

∞∑m=−∞

|um(η)|2 = 12π

∫ 2π

0|u(t, η)|2dt, 0 ≤ η ≤ 1.

Because the um and u are continuous, by Dini’s theorem (see Problem 8.2), the seriesis uniformly convergent. Hence, we can integrate term by term to obtain

∞∑m=−∞

∫ 1

0|um(η)|2dη =

12π‖u‖2L2(Q). (8.16)

Similarly, from

u′m(η) =1

2π

∫ 2π

0

∂u∂η

(t, η)e−imtdt

and

imum(η) =1

2π

∫ 2π

0

∂u∂t

(t, η)e−imtdt

we see that ∞∑m=−∞

∫ 1

0|u′m(η)|2dη =

12π

∥∥∥∥∥∂u∂η

∥∥∥∥∥2

L2(Q)(8.17)

and ∞∑m=−∞

∫ 1

0m2|um(η)|2dη =

12π

∥∥∥∥∥∂u∂t

∥∥∥∥∥2

L2(Q). (8.18)

In the second step we shall show that

‖u(· , 0)‖2H1/2[0,2π] ≤1π‖u‖2H1(Q) (8.19)


for all u with u(· , 1) = 0. The latter property implies that um(1) = 0 for all m ∈ ZZ.From this, by the Cauchy–Schwarz inequality, we obtain that

|um(0)|2 = 2 Re∫ 0

1u′m(η)um(η) dη ≤ 2

{∫ 1

0|u′m(η)|2dη

}1/2 {∫ 1

0|um(η)|2dη

}1/2

for all m ∈ ZZ and consequently, using again the Cauchy–Schwarz inequality, we canestimate

‖u(· , 0)‖2H1/2[0,2π]

=

∞∑m=−∞

(1 + m2)1/2|um(0)|2

≤ 2

⎧⎪⎪⎨⎪⎪⎩∞∑

m=−∞

∫ 1

0|u′m(η)|2dη

⎫⎪⎪⎬⎪⎪⎭1/2 ⎧⎪⎪⎨⎪⎪⎩

∞∑m=−∞

(1 + m2)∫ 1

0|um(η)|2dη

⎫⎪⎪⎬⎪⎪⎭1/2

=1π

∥∥∥∥∥∂u∂η

∥∥∥∥∥L2(Q)

{‖u‖2L2(Q) +

∥∥∥∥∥∂u∂t

∥∥∥∥∥2

L2(Q)

}1/2

≤ 1π‖u‖2

H1(Q).

We are now ready to establish the following theorem.

Theorem 8.15. Let D be a simply connected bounded domain in IR2 with boundaryΓ of class C2. Then there exists a positive constant C such that

‖u‖H1/2(Γ) ≤ C‖u‖H1(D)

for all u ∈ C1(D).

Proof. As in the proof of Theorem 7.8, we choose a parallel strip

Dh := {x + ηhν(x) : x ∈ Γ, η ∈ [0, 1]}with some h > 0 such that each y ∈ Dh is uniquely representable through projectiononto Γ in the form y = x + ηhν(x) with x ∈ Γ and η ∈ [0, 1]. Here, for convenience,deviating from our general rule, we assume the normal ν is directed into D. By Γh

we denote the interior boundary Γh := {y = x + hν(x) : x ∈ Γ} of Dh. Then, througha regular 2π-periodic parametric representation

Γ = {z(t) : 0 ≤ t < 2π}of the contour Γ we have a parameterization

ζ(t, η) = z(t) + ηhν(z(t)), 0 ≤ t < 2π, 0 ≤ η ≤ 1, (8.20)

of the strip Dh. In particular, ζ provides a bijective and continuously differentiablemapping from Q onto Dh with a continuously differentiable inverse. Thus, applying


the chain rule, for all u ∈ C1(Dh) with u = 0 on Γh from (8.19) we conclude that

‖u‖H1/2(Γ) = ‖u ◦ z‖H1/2[0,2π] ≤ 1√π‖u ◦ ζ‖H1(Q) ≤ c‖u‖H1(Dh),

where c is some constant containing a bound on the first derivatives of the mappingζ and its inverse.

Finally, to extend the estimate to arbitrary u ∈ C1(D), we choose a cut-off func-tion g ∈ C1(D) given by g(y) = 0 for y � Dh and g(y) = f (η) for y ∈ Dh withy = x + ηhν(x). Here, f is defined by

f (η) := (1 − η)2(1 + 3η)

and satisfies f (0) = f ′(0) = 1 and f (1) = f ′(1) = 0. The property f ′(0) = 1 will notbe used in this proof but will be needed later, in the proof of Theorem 8.18. Then,applying the product rule, we obtain

‖u‖H1/2(Γ) = ‖gu‖H1/2(Γ) ≤ c‖gu‖H1(D) ≤ C‖u‖H1(D)

for all u ∈ C1(D) and some constant C containing a bound on g and its first deriva-tives. �

Corollary 8.16. The trace operator can be uniquely extended as a continuous oper-ator γ : H1(D)→ H1/2(Γ).

Proof. This follows from Theorem 8.15 and Problem 2.1. �

We proceed by showing that the trace operator γ of Corollary 8.16 has a boundedright inverse.

Theorem 8.17. There exists a bounded linear operator β : H1/2(Γ) → H1(D) suchthat γβ = I.

Proof. We use the notations introduced in the proof of Theorem 8.15 and first con-sider the strip IR × [0, 1]. For m ∈ ZZ we introduce functions wm : [0, 1]→ IR by

wm(η) := (1 − η)2|m|, 0 ≤ η ≤ 1.

Elementary calculations yield

∫ 1

0[wm(t)]2dt =

14|m| + 1

(8.21)

and ∫ 1

0[w′m(t)]2dt =

4m2

4|m| − 1. (8.22)


Let ψ be a trigonometric polynomial with Fourier coefficients ψm (of which onlyfinitely many are different from zero). We define a function v on IR × [0, 1] by

v(t, η) :=∞∑

m=−∞ψmwm(η)eimt

with the properties v(· , 0) = ψ and v(· , 1) = ∂η v(· , 1) = 0. With the aid of (8.21),(8.22) and the orthogonality of the trigonometric monomials, we obtain

‖v‖2H1(Q) = 2π∞∑

m=−∞

{1 + m2

4|m| + 1+

4m2

4|m| − 1

}|ψm|2

whence the estimate‖v‖H1(Q) ≤ c ‖ψ‖H1/2[0,2π] (8.23)

follows for some constant c independent of ψ.In terms of the bijective mapping ζ : Q → Dh given by (8.20) we define a

function u ∈ C1(D) by u = v ◦ ζ−1 in Dh and u = 0 in D \ Dh. Since γu = ψ ◦ z−1,on the subspace U ⊂ H1/2(Γ) given by the images ψ ◦ z−1 of all trigonometricpolynomials ψ we now define a linear operator β : U → H1(D) by setting

β(ψ ◦ z−1) := u.

By (8.23) it is bounded and clearly it satisfies γβ = I. Now the statement followsfrom the denseness of the trigonometric polynomials in H1/2[0, 2π] andProblem 2.1. �

In the following theorem, by | · |L1(Γ) we will denote the semi-norm

|u|L1(Γ) :=∣∣∣∣∣∫Γ

u ds∣∣∣∣∣ .

Theorem 8.18. Let D be as in Theorem 8.15. Then there exists a positive constantC such that

‖u‖2L2(D) ≤ C{|u|2L1(Γ) + ‖ grad u‖2L2(D)

}and

‖u‖2L2(D) ≤ C{‖u‖2L2(Γ) + ‖ grad u‖2L2(D)

}

for all u ∈ H1(D).

Proof. Since C1(D) is dense in H1(D), it suffices to establish the inequalities for allu ∈ C1(D). In addition, obviously, we only need to verify the first inequality.

We use the notations introduced in the proof of Theorem 8.15 and consider afunction u in the strip IR × [0, 1]. From

∫ 2π

0u(t, η) dt −

∫ 2π

0u(t, 0) dt =

∫ 2π

0

∫ η

0

∂u∂ξ

(t, ξ) dξdt,


using the Cauchy–Schwarz inequality, we derive

∣∣∣∣∣∣∫ 2π

0u(t, η) dt

∣∣∣∣∣∣2

≤ 2

⎧⎪⎪⎨⎪⎪⎩∣∣∣∣∣∣∫ 2π

0u(t, 0) dt

∣∣∣∣∣∣2

+ 2π∥∥∥∥∥∂u∂η

∥∥∥∥∥2

L2(Q)

⎫⎪⎪⎬⎪⎪⎭ .

Integrating this inequality with respect to η we obtain

∫ 1

0

∣∣∣∣∣∣∫ 2π

0u(t, η) dt

∣∣∣∣∣∣2

dη ≤ C1

⎧⎪⎪⎨⎪⎪⎩∣∣∣∣∣∣∫ 2π

0u(t, 0) dt

∣∣∣∣∣∣2

+ ‖ grad u‖2L2(Q)

⎫⎪⎪⎬⎪⎪⎭for some constant C1. Combining this with (8.16) and (8.18), we see that

‖u‖2L2(Q)

≤ 2π∫ 1

0|u0(η)|2dη +

∥∥∥∥∥∂u∂t

∥∥∥∥∥2

L2(Q)

≤ C2

⎧⎪⎪⎨⎪⎪⎩∣∣∣∣∣∣∫ 2π

0u(t, 0) dt

∣∣∣∣∣∣2

+ ‖ grad u‖2L2(Q)

⎫⎪⎪⎬⎪⎪⎭for some constant C2. Substituting back into the domain D as in the proof ofTheorem 8.15, we see that there exists a constant C3 such that

‖u‖2L2(Dh) ≤ C3

{|u|2L1(Γ) + ‖ grad u‖2L2(D)

}. (8.24)

For this transformation, without loss of generality we have assumed that the pa-rameter t in the representation of the boundary Γ is given through the arc lengthmultiplied by 2π/|Γ|.

Now let u ∈ C1(D) satisfy u = 0 and grad u = 0 on the boundary Γ. Then wecan extend u to a continuously differentiable function on IR2 by setting u = 0 inthe exterior of D. Choose R large enough such that D is contained in the disk withradius R and center at the origin. Then from

u(x) =∫ x1

−R

∂u∂x1

dx1,

with the aid of the Cauchy–Schwarz inequality, we see that

|u(x)|2 ≤ 2R∫ R

−R

∣∣∣∣∣ ∂u∂x1

∣∣∣∣∣2

dx1.

Integrating this inequality and using u = 0 outside of D yields

‖u‖2L2(D) ≤ 4R2‖ grad u‖2L2(D) (8.25)

for all u ∈ C1(D) with u = 0 and grad u = 0 on Γ.

8.3 Weak Solutions to Boundary Value Problems 159

Finally, for arbitrary u ∈ C1(D) we use the function g introduced in the proof ofTheorem 8.15 to decompose

u = gu + (1 − g)u

and then apply (8.24) to the first term and (8.25) to the second term on the right-handside. Observing that

‖ grad(gu)‖L2(D) ≤ C4

{‖ grad u‖L2(D) + ‖u‖L2(Dh)

}

for some constant C4 depending on g ends the proof. �

Corollary 8.19. On H1(D) the norm ‖ · ‖H1(D) is equivalent to each of the two norms

‖u‖ :=(|u|2L1(Γ) + ‖ grad u‖2L2(D)

)1/2

and‖u‖ :=

(‖u‖2L2(Γ) + ‖ grad u‖2L2(D)

)1/2.

Proof. This follows from Theorems 8.3, 8.15, and 8.18.

8.3 Weak Solutions to Boundary Value Problems

We will demonstrate how the Sobolev spaces H1/2(Γ) and H−1/2(Γ) occur in a nat-ural way through the solution of boundary value problems in a weak formulation.For a function u ∈ H1(D), the integral

∫D| grad u|2dx

is called the Dirichlet integral. For a harmonic function u the Dirichlet integral rep-resents the energy of the potential u. Therefore, it is natural to attempt to developan approach for the solution of boundary value problems for harmonic functionsin which it is required that the solutions have finite energy, i.e., they belong to theSobolev space H1(D). Since the functions contained in H1(D), in general, are nottwice continuously differentiable in D and since they do not attain boundary val-ues or normal derivatives on the boundary in the classical sense, we have to extendthe classical formulation of the boundary value problems by generalized versions.Our procedure will be quite typical in the sense that we use specific properties ofclassical solutions to formulate what we mean by a weak solution.

For each solution u ∈ C2(D) ∩ C1(D) to the Laplace equation Δu = 0 in D, bythe first Green’s Theorem 6.3 we have

∫D

grad u · grad v dx = 0 (8.26)


for all v ∈ C1(D) with v = 0 on Γ. Since the integral in this equation is well definedfor all u, v ∈ H1(D), we call a function u ∈ H1(D) a weak solution to the Laplaceequation in D if (8.26) is satisfied for all v ∈ H1

0(D), where H10(D) denotes the

subspace of all functions v ∈ H1(D) satisfying v = 0 on Γ in the sense of the traceoperator.

Weak Interior Dirichlet Problem. For a given f ∈ H1/2(Γ), find a weak solutionu ∈ H1(D) to the Laplace equation in D such that

u = f on Γ (8.27)

in the sense of the trace operator γ of Corollary 8.16.

For each solution u ∈ C2(D) ∩C1(D) to the Neumann problem Δu = 0 in D with∂u/∂ν = g on Γ by the first Green’s theorem we have

∫D

grad u · grad v dx =∫Γ

gv ds

for all v ∈ C1(D). The following lemma extends this property to weak solutions ofthe Laplace equation by introducing a weak formulation of the normal derivative.

Lemma 8.20. Let u ∈ H1(D) be a weak solution to the Laplace equation in D. Thenthere exists a unique g ∈ H−1/2(Γ) such that

∫D

grad u · grad v dx =∫Γ

g γv ds (8.28)

for all v ∈ H1(D). The integral in (8.28) has to be understood in the sense of thebilinear duality pairing ∫

Γ

gϕ ds := g(ϕ) (8.29)

for ϕ ∈ H1/2(Γ) and g satisfies

‖g‖H−1/2(Γ) ≤ C ‖u‖H1(D) (8.30)

for some constant C depending only on D.

Proof. In terms of the right inverse β of Theorem 8.17, we define g ∈ H−1/2(Γ) by

g(ϕ) :=∫

Dgrad u · grad βϕ dx (8.31)

for ϕ ∈ H1/2(Γ). Indeed g is bounded because of

|g(ϕ)| ≤ ‖u‖H1(D)‖βϕ‖H1(D) ≤ ‖β‖ ‖u‖H1(D)‖ϕ‖H1/2(Γ).

From this inequality the estimate (8.30) follows.


In order to show that g satisfies (8.28), for v ∈ H1(D) we consider v0 = v − βγv.Then v0 ∈ H1

0(D) and consequently

∫D

grad u · grad v0 dx = 0,

since u is a weak solution to the Laplace equation. Therefore, in view of the defini-tion (8.31) we obtain

∫D

grad u · grad v dx =∫

Dgrad u · grad(v0 + βγv) dx = g(γv)

as required. To establish uniqueness, we observe that g(γv) = 0 for all v ∈ H1(D), byTheorem 8.17, implies that g(ϕ) = 0 for all ϕ ∈ H1/2(Γ) and consequently g = 0. �

We particularly note that if the weak normal derivative of a weak solution tothe Laplace equation in the sense of Lemma 8.20 is used, Green’s theorem remainsvalid.

We are now in the position to introduce the following weak formulation of theinterior Neumann problem.

Weak Interior Neumann Problem. For a given g ∈ H−1/2(Γ), find a weak solutionu ∈ H1(D) to the Laplace equation in D such that

∂u∂ν= g on Γ (8.32)

with the weak normal derivative in the sense of Lemma 8.20.

Theorem 8.21. The weak interior Dirichlet problem has at most one solution. Twosolutions to the weak interior Neumann problem can differ only by a constant.

Proof. The difference u := u1 − u2 between two solutions to the Dirichlet problem isa weak solution with homogeneous boundary condition u = 0 on Γ. Then we mayinsert v = u in (8.26) and obtain ‖ grad u‖L2(D) = 0. Since u = 0 on Γ, from Corollary8.19 we see that u = 0 in D. As a consequence of (8.28) for two solutions of theNeumann problem the difference u := u1 − u2 − c again satisfies ‖ grad u‖L2(D) = 0for all constants c. We choose c such that

∫Γ

u ds = 0. Then from Corollary 8.19 weobtain u = 0 in D. �

To establish the existence of weak solutions we will proceed analogously to theclassical case and try to find the solution in the form of a logarithmic single- ordouble-layer potential with densities in H−1/2(Γ) or H1/2(Γ). Therefore, we firstneed to investigate the properties of these potentials in our Sobolev space setting.This will be achieved through making use of corresponding properties in the spacesof uniformly Holder continuous functions. For the remainder of this section, by αwe will always denote a Holder exponent with 0 < α < 1.


Theorem 8.22. The operators K : H1/2(Γ) → H1/2(Γ) defined by the logarithmicdouble-layer potential

(Kϕ)(x) :=1π

∫Γ

ϕ(y)∂

∂ν(y)ln

1|x − y| ds(y), x ∈ Γ,

and K′ : H−1/2(Γ) → H−1/2(Γ) defined by the formal normal derivative of thelogarithmic single-layer potential

(K′ψ)(x) :=1π

∫Γ

ψ(y)∂

∂ν(x)ln

1|x − y| ds(y), x ∈ Γ,

are compact and adjoint in the dual system (H1/2(Γ),H−1/2(Γ))L2(Γ), i.e.,

(Kϕ, ψ)L2(Γ) = (ϕ,K′ψ)L2(Γ) (8.33)

for all ϕ ∈ H1/2(Γ) and ψ ∈ H−1/2(Γ). Here, again, we use the notation ( f , g)L2(Γ) :=g( f ) for f ∈ H1/2(Γ) and g ∈ H−1/2(Γ).

Proof. We shall show that K is bounded from H1/2(Γ) into H1(Γ). Then the assertionof the compactness of K follows from Theorems 2.21 and 8.3.

For functions ϕ ∈ C1,α(Γ) from Theorem 7.32, we know that the gradient of thedouble-layer potential vwith density ϕ can be expressed in terms of the derivatives ofthe single-layer potentialwwith density dϕ/ds. In particular, by (7.44) the tangentialderivative is given by

∂v−∂s

(x) = −12

dϕds

(x) − 12π

∫Γ

dϕds

(y)∂

∂ν(x)ln

1|x − y| ds(y), x ∈ Γ.

From this, with the aid of the jump relations of Theorem 7.29 for double-layer po-tentials with uniformly Holder continuous densities, we conclude that

dKϕds

(x) = −1π

∫Γ

dϕds

(y)∂

∂ν(x)ln

1|x − y| ds(y), x ∈ Γ.

Performing a partial integration yields

dKϕds

(x) =1π

∫Γ

{ϕ(y) − ϕ(x)} ∂

∂s(y)∂

∂ν(x)ln

1|x − y| ds(y), x ∈ Γ,

where the integral has to be understood in the sense of a Cauchy principal value.Now we use a 2π-periodic parameterizationΓ = {x(t) : t ∈ [0, 2π)} of Γ to transformthe integral into

dKϕds

(x(t)) =∫ 2π

0

k(t, τ)

sint − τ

2

{ϕ(x(τ)) − ϕ(x(t))} dτ, t ∈ [0, 2π],


where k is a bounded weakly singular kernel. Thus, by the Cauchy–Schwarz in-equality, we can estimate

∫ 2π

0

∣∣∣∣∣dKϕds

(x(t))∣∣∣∣∣2

dt ≤ M∫ 2π

0

∫ 2π

0

|ϕ(x(τ)) − ϕ(x(t))|2sin2 t − τ

2

dτdt ≤ M‖ϕ ◦ x‖21/2,0

for some constant M. Hence, by Theorem 8.6,∥∥∥∥∥dKϕ

ds

∥∥∥∥∥L2(Γ)≤ c‖ϕ‖H1/2(Γ)

for some constant c. Since K has continuous kernel (see Problem 6.1), it is boundedfrom L2(Γ) into L2(Γ). Therefore we conclude

‖Kϕ‖H1(Γ) ≤ C‖ϕ‖H1/2(Γ)

for all ϕ ∈ C1,α(Γ) and some constant C. This completes the proof of the bounded-ness of K from H1/2(Γ) into H1(Γ), since C1,α(Γ) is dense in H1/2(Γ) byTheorem 8.2.

For ϕ, ψ ∈ C1,α(Γ), clearly (8.33) holds and we can estimate

|(ϕ,K′ψ)L2(Γ)| = |(Kϕ, ψ)L2(Γ)| ≤ ‖Kϕ‖H1(Γ) ‖ψ‖H−1(Γ) ≤ C‖ϕ‖H1/2(Γ) ‖ψ‖H−1(Γ).

From this it follows that

‖K′ψ‖H−1/2(Γ) ≤ C‖ψ‖H−1(Γ)

for all ψ ∈ C1,α(Γ) and, because C1,α(Γ) is dense in H−1(Γ), also for all ψ ∈ H−1(Γ).Therefore K′ is bounded from H−1(Γ) into H−1/2(Γ) and the statement on the com-pactness of K′ again follows from Theorems 2.21 and 8.3.

Finally, the validity of (8.33) for all ϕ ∈ H1/2(Γ) and ψ ∈ H−1/2(Γ) followsfrom the boundedness of K and K′ by the denseness of C1,α(Γ) in H1/2(Γ) and inH−1/2(Γ). �

Lemma 8.23. For the trigonometric monomials we have the integrals

12π

∫ 2π

0ln

(4 sin2 t

2

)eimtdt =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

0, m = 0,

− 1|m| , m = ±1,±2, . . . .

Proof. Integrating the geometric sum

1 + 2m−1∑k=1

eikt + eimt = i (1 − eimt) cott2, 0 < t < 2π, (8.34)


we obtain ∫ 2π

0eimt cot

t2

dt = 2πi, m = 1, 2, . . . , (8.35)

in the sense of Cauchy principal values. Integrating the identity

ddt

{[eimt − 1

]ln

(4 sin2 t

2

)}= im eimt ln

(4 sin2 t

2

)+

[eimt − 1

]cot

t2

yields ∫ 2π

0eimt ln

(4 sin2 t

2

)dt = − 1

im

∫ 2π

0eimt cot

t2

dt = −2πm

for m = 1, 2, . . . , and the statement is proven for m � 0. To evaluate the integral form = 0 we set

I :=∫ 2π

0ln

(4 sin2 t

2

)dt.

Then

2I =∫ 2π

0ln

(4 sin2 t

2

)dt +

∫ 2π

0ln

(4 cos2 t

2

)dt

=

∫ 2π

0ln(4 sin2 t) dt =

12

∫ 4π

0ln

(4 sin2 t

2

)dt = I,

and therefore I = 0. �

By Problem 7.2 the operator S0 in the following theorem corresponds to the mod-ified single-layer operator of Theorem 7.41 for the case of the unit circle.

Theorem 8.24. The operator S0 : Hp[0, 2π]→ Hp+1[0, 2π], defined by

(S0ϕ)(t) :=1

2π

∫ 2π

0

{ln

(4 sin2 t − τ

2

)− 2

}ϕ(τ) dτ, t ∈ [0, 2π], (8.36)

is bounded and has a bounded inverse for all p ∈ IR. (For negative p the integral in(8.36) has to be understood as the duality pairing.)

Proof. From Lemma 8.23, for fm(t) = eimt, we have

S0 fm = βm fm, m ∈ ZZ,

where βm = −1/|m| for m � 0 and β0 = −2. This implies the boundedness ofS0 : Hp[0, 2π] → Hp+1[0, 2π] and its invertibility with the inverse operator S −1

0 :Hp+1[0, 2π]→ Hp[0, 2π] given by

S −10 fm =

1βm

fm, m ∈ ZZ,

which again is bounded. �


Theorem 8.25. The operators S : H−1/2(Γ) → H1/2(Γ) defined by the single-layerpotential

(Sϕ)(x) :=1π

∫Γ

ϕ(y) ln1|x − y| ds(y), x ∈ Γ,

and T : H1/2(Γ) → H−1/2(Γ) defined by the normal derivative of the double-layerpotential

(Tψ)(x) :=1π

∂

∂ν(x)

∫Γ

ψ(y)∂

∂ν(y)ln

1|x − y| ds(y), x ∈ Γ,

are bounded. The operator S is self-adjoint with respect to the dual systems(H−1/2(Γ),H1/2(Γ))L2(Γ) and (H1/2(Γ),H−1/2(Γ))L2(Γ), i.e.,

(Sϕ, ψ)L2(Γ) = (ϕ, Sψ)L2(Γ) (8.37)

for all ϕ, ψ ∈ H−1/2(Γ). The operator T is self-adjoint with respect to the dualsystems (H1/2(Γ),H−1/2(Γ))L2(Γ) and (H−1/2(Γ),H1/2(Γ))L2(Γ), i.e.,

(Tϕ, ψ)L2(Γ) = (ϕ, Tψ)L2(Γ) (8.38)

for all ϕ, ψ ∈ H1/2(Γ).

Proof. We use a 2π-periodic parameterization Γ = {x(t) : t ∈ [0, 2π)} of Γ. ByTheorem 8.14, we may assume that the parameter t is given by the arc length on Γmultiplied by 2π/|Γ|. Then, for ϕ ∈ C0,α(Γ), we can write (see Problem 7.2)

(Sϕ)(x(t)) =|Γ|4π2

∫ 2π

0

{− ln

(4 sin2 t − τ

2

)+ p(t, τ)

}ϕ(x(τ)) dτ

for t ∈ [0, 2π], where p is 2π-periodic and continuously differentiable. By the pre-vious Theorem 8.24, the leading term with the logarithmic singularity representsa bounded operator from H−1/2(Γ) into H1/2(Γ). Because the kernel function p iscontinuously differentiable, using the Cauchy–Schwarz inequality and an integra-tion by parts, the second term on the right is seen to be bounded from H0(Γ) intoH1(Γ) and from H−1(Γ) into H0(Γ). Hence, by the interpolation Theorem 8.13, it isalso bounded from H−1/2(Γ) into H1/2(Γ).

For ϕ, ψ ∈ C1,α(Γ) we use the relation (7.46) between S and T and a partialintegration to write

(Tϕ, ψ)L2(Γ) =

(dds

Sdϕds

, ψ

)L2(Γ)

= −(S

dϕds

,dψds

)L2(Γ)

. (8.39)


Then we can estimate

|(Tϕ, ψ)|L2(Γ) ≤∥∥∥∥∥S

dϕds

∥∥∥∥∥H1/2(Γ)

∥∥∥∥∥dψds

∥∥∥∥∥H−1/2(Γ)

≤ C∥∥∥∥∥dϕ

ds

∥∥∥∥∥H−1/2(Γ)

∥∥∥∥∥dψds

∥∥∥∥∥H−1/2(Γ)

≤ C‖ϕ‖H1/2(Γ) ‖ψ‖H1/2(Γ)

for some constant C. Since C1,α(Γ) is dense in H1/2(Γ), this implies that T isbounded from H1/2(Γ) into H−1/2(Γ).

The self-adjointness property (8.37) follows as in the proof of Theorem 8.22 byextending it from the case of smooth functions. Finally, combining (8.37) and (8.39)yields (8.38). �

Theorem 8.26. The logarithmic single-layer potential defines a bounded linear op-erator from H−1/2(Γ) into H1(D). The logarithmic double-layer potential defines abounded linear operator from H1/2(Γ) into H1(D).

Proof. Let u be the single-layer potential with density ϕ ∈ C0,α(Γ). Then, by Green’stheorem and the jump relations of Theorem 7.30, we have

∫D| grad u|2dx =

∫Γ

u∂u∂ν

ds =14

∫Γ

S ϕ (ϕ + K′ϕ) ds.

Therefore, by Theorems 8.22 and 8.25, with a constant c we can estimate

‖ grad u‖2L2(D) ≤14‖S ϕ‖H1/2(Γ) ‖ϕ + K′ϕ‖H−1/2(Γ) ≤ c ‖ϕ‖2H−1/2(Γ).

Since S is bounded from L2(Γ) into L2(Γ) (see Problem 5.5), using Theorem 8.18,we see that

‖u‖H1(D) ≤ C‖ϕ‖H−1/2(Γ)

for some constant C. Now the statement on the single-layer potential follows fromthe denseness of C0,α(Γ) in H−1/2(Γ).

The case of the double-layer potential v with density ψ is treated analogouslythrough the relation

∫D| grad v|2dx =

14

∫Γ

T ϕ (Kϕ − ϕ) ds,

which again follows from Green’s theorem and the jump relations. Note that K isbounded from L2(Γ) into L2(Γ) since it has a continuous kernel. �

We are now prepared for the solution of the weak Dirichlet and Neumann prob-lem via boundary integral equations.


Theorem 8.27. The weak Dirichlet problem has a unique solution. The mappingtaking the given boundary data f ∈ H1/2(Γ) into the solution u ∈ H1(D) is bounded.

Proof. Analogous to the classical treatment we try to find the solution in the form ofa double-layer potential

u(x) =1

2π

∫Γ

ϕ(y)∂

∂ν(y)ln

1|x − y| ds(y), x ∈ D. (8.40)

If ϕ ∈ C1,α(Γ), by Theorem 7.32 and Green’s theorem, we have u ∈ C1(D) and∫

Dgrad u · grad v dx =

12

∫Γ

Tϕ v ds (8.41)

for all v ∈ C1(D) and2u = Kϕ − ϕ on Γ.

In the first equation, by Theorems 8.25 and 8.26 and the trace theorem, both sidesare continuous bilinear mappings from H1/2(Γ) × H1(D) → C. Hence, the double-layer potential with density ϕ ∈ H1/2(Γ) satisfies (8.26) for all v ∈ H1

0(D), that is,it is a weak solution to the Laplace equation. In the second equation both sides areagain bounded from H1/2(Γ) into H1/2(Γ) by Theorems 8.22 and 8.26 and Corollary8.16. Therefore u is a weak solution to the Dirichlet problem provided the density ϕsolves the integral equation

ϕ − Kϕ = −2 f .

Let ϕ ∈ H1/2(Γ) be a solution to the homogeneous equation ϕ − Kϕ = 0. Then,since K has continuous kernel (see Problem 6.1), we have ϕ ∈ C(Γ) and from theclassical Theorem 6.21 we deduce that N(I −K) = {0} in H1/2(Γ). Now existence ofa solution in H1/2(Γ) to the inhomogeneous equation follows from the Riesz theoryby Theorems 3.4 and 8.22.

The statement on the continuous dependence of the solution on the given bound-ary data is a consequence of the boundedness of the inverse of I − K (Theorem 3.4)and Theorem 8.26. �

Corollary 8.28. The Dirichlet-to-Neumann operator A taking the boundary valuesof a harmonic function in H1(D) into its normal derivative is bounded from H1/2(Γ)into H−1/2(Γ).

Proof. From Theorem 8.27 and Lemma 8.20. �

Since each weak solution u ∈ H1(D) to the Laplace equation can be viewed asthe solution of a weak Dirichlet problem, from the representation of this solution asthe double-layer potential (8.40) we can conclude that weak solutions to the Laplaceequation are twice continuously differentiable in D, i.e., they are automatically clas-sical solutions. The weak formulation only affects the behavior of the solutions atthe boundary Γ.


Theorem 8.29. The weak interior Neumann problem is solvable if and only if∫Γ

g ds = 0.

The operator mapping the given boundary data g ∈ H−1/2(Γ) into the unique solu-tion u ∈ H1(D) satisfying the additional condition

∫Γ

u ds = 0 is bounded.

Proof. By Theorem 7.30, the single-layer potential

u(x) =1

2π

∫Γ

ψ(y) ln1|x − y| ds(y), x ∈ D,

with density ψ ∈ C0,α(Γ) belongs to C1(D). It satisfies∫

Dgrad u · grad v dx =

12

∫Γ

(ψ + K′ψ) v ds

for all v ∈ C1(D). As in the previous proof, both sides of this equation are continuousfrom H−1/2(Γ) × H1(D) → C by Theorems 8.22 and 8.26 and the trace theorem.Hence, the single-layer potential u with density ψ ∈ H−1/2(Γ) is a weak solution tothe Neumann problem provided ψ solves the integral equation

ψ + K′ψ = 2g.

As in the proof of Theorem 8.27 from Theorem 6.21 we conclude that N(I+K) =span{1} in H1/2(Γ). Therefore, by Theorem 8.22 and the Fredholm alternative, theinhomogeneous equation ψ + K′ψ = 2g is solvable in H−1/2(Γ) provided g satisfiesthe solvability condition

∫Γg ds = 0. The necessity of the latter condition for the

existence of a weak solution to the Neumann problem follows from (8.28) for v = 1.The continuous dependence of the solution on the given boundary data is a con-

sequence of Theorem 8.26 analogous to the classical case of Theorem 6.30. �

Corollary 8.30. The Neumann-to-Dirichlet operator B taking the normal derivativeof a harmonic function in H1(D) into its boundary values is bounded from H−1/2

0 (Γ)into H1/2

0 (Γ). Here, the subscripts indicate the subspaces with mean value zero.

Proof. From Theorem 8.29 and Corollary 8.16. �

As in Section 7.7, it is also possible to obtain the weak solvability to the Dirichletand Neumann problem through integral equations of the first kind

S0ψ = 2 f

andTϕ = 2g

respectively. Here, S0 denotes the modified single-layer operator as introduced by(7.60). By denseness and continuity arguments it can be seen that the relations (7.47)


and (7.48) carry over into the Sobolev space setting. Therefore, analogues of The-orems 7.41 and 7.43 can be proven with the aid of Theorem 5.6. Alternatively, bi-jectivity of S0 : H−1/2(Γ) → H1/2(Γ) can also be obtained by Corollary 3.5 andTheorem 8.24 (see also Problem 8.3) and the hypersingular integral equation withthe operator T can also be solved using the relation (7.46).

For the solution of the single-layer potential integral equation of Section 7.8 foran arc in a Sobolev space setting we refer to [143].

Since weak solutions to the Laplace equation in D are classical solutions (seep. 167) and Green’s theorem remains valid for them (see p. 161), from the proof ofGreen’s representation formula Theorem 6.5 it can be seen that this formula remainsvalid for harmonic functions in H1(D). The jump-relations are also preserved inthe Sobolev space setting (we have used them already in the proofs of Theorems8.27 and 8.29). Therefore the Calderon projection of Theorem 7.37 is also validfor harmonic functions in H1(D) with the projection operator acting on H1/2(Γ) ×H−1/2(Γ). This opens up the option of also considering the direct approach for theboundary integral equations in the Sobolev space setting as in Section 7.6. To avoidrepetitions, we leave this as an exercise to the reader.

Our presentation clearly demonstrates that the results on weak solutions heavilyrely on the classical analysis on single- and double-layer potentials with uniformlyHolder continuous densities. The mapping properties of the integral operators in theSobolev spaces essentially follow from the corresponding properties in the Holderspaces through applications of functional analytic tools, in particular, denseness andcontinuation arguments. For a proof of the mapping property of the single-layeroperator from Theorem 8.25 with the aid of Lax’s Theorem 4.13, we refer to [32].

The major advantage of the weak approach to the boundary integral equations,besides the fact that it allows less regular boundary data, stems from the fact thatSobolev spaces are Hilbert spaces rather than just normed spaces. This, as we haveseen, adds some elegance to the analysis.

Another advantage of the Sobolev space setting is the possibility to extend theanalysis to less regular boundaries such as Lipschitz domains. These domains are de-fined analogously to domains of class C1 on p. 30 with the continuous differentiabil-ity replaced by Lipschitz continuity, i.e., Holder continuity with Holder exponent 1.In particular, Lipschitz domains can have corners. For extensions of the mappingproperties of single- and double layer potentials of this section to Lipschitz domainsand its implications for the existence analysis of the boundary integral equations werefer to McLean [165].

For a more detailed study of boundary integral equations in Sobolev space set-tings in the framework of pseudodifferential operators (see Taylor [230, 231] andTreves [237]) including the extensions of the analysis to exterior problems and tothree-dimensional problems we refer to Hsiao and Wendland [104] and the litera-ture therein. For extensions of the approach based on Lax’s Theorem 4.13 we referto Kirsch [121, 123].


Problems

8.1. Work out the proofs of Corollaries 8.7 and 8.8.

8.2. Prove Dini’s theorem: Let D ⊂ IRm be compact and let (ϕn) be a nondecreasing sequence ofcontinuous real-valued functions on D converging pointwise to a continuous function ϕ. Then (ϕn)converges uniformly on D to ϕ.

8.3. Let K : [0, 2π] × [0, 2π] → IR be n-times continuously differentiable. Show that the integraloperator A with kernel K is compact from Hp[0, 2π] into Hq[0, 2π] for all |p|, |q| ≤ n.

8.4. Formulate and prove the analogue of the Noether theorems in the form of Corollary 7.26 inthe Sobolev spaces Hp[0, 2π].Hint: Use the integrals (8.35).

8.5. Use the separation of variables solutions of the Laplace equation in polar coordinates to verifythe regularity and mapping properties of the weak solution to the Dirichlet and Neumann problemfor the case where D is the unit disk.

Chapter 9The Heat Equation

The temperature distribution u in a homogeneous and isotropic heat conductingmedium with conductivity k, heat capacity c, and mass density ρ satisfies the partialdifferential equation

∂u∂t= κΔu

where κ = k/cρ. This is called the equation of heat conduction or, shortly, the heatequation; it was first derived by Fourier. Simultaneously, the heat equation alsooccurs in the description of diffusion processes. The heat equation is the standardexample for a parabolic differential equation. In this chapter we want to indicate theapplication of Volterra-type integral equations of the second kind for the solution ofinitial boundary value problems for the heat equation. Without loss of generality weassume the constant κ = 1. For a more comprehensive study of integral equations ofthe second kind for the heat equation we refer to Cannon [25], Friedman [58], andPogorzelski [193].

9.1 Initial Boundary Value Problem: Uniqueness

The mathematical modeling of the heat conduction in a medium with a given tem-perature distribution at some initial time and a given temperature distribution on theboundary of the medium for all times leads to the following initial boundary valueproblem for the heat equation.

Initial Boundary Value Problem. Let D ⊂ IRm, m = 1, 2, 3, be a bounded domainwith boundary Γ := ∂D and let T denote a positive real number. Find a func-tion u ∈ C(D × [0, T ]) that is twice continuously differentiable with respect to thespace variable x and continuously differentiable with respect to the time variable tin D × (0, T ] and that satisfies the heat equation

∂u∂t= Δu in D × (0, T ], (9.1)


171

172 9 The Heat Equation

the initial conditionu(· , 0) = w in D, (9.2)

and the Dirichlet boundary condition

u = f on Γ × [0, T ]. (9.3)

Here, w ∈ C(D) and f ∈ C(Γ × [0, T ]) are given continuous functions subject to thecompatibility condition

w = f (· , 0) on Γ. (9.4)

Analogous to the case of the Laplace equation as studied in the previous threechapters, instead of the Dirichlet boundary condition also a Neumann boundarycondition

∂u∂ν= g on Γ × [0, T ]

can be considered. Here, as in Chapters 6–8, we denote by ν the unit normal to Γdirected into the exterior of D and the given function g ∈ C(Γ × [0, T ]) must satisfythe compatibility condition

∂w

∂ν= g(· , 0) on Γ.

For our short introduction to the use of Volterra type integral equations for initialboundary value problems for the heat equation for brevity we choose to only con-sider the Dirichlet boundary condition. However, for the Neumann boundary condi-tion in the one-dimensional case see Problem 9.5.

Our aim is to establish that this initial boundary value problem (9.1)–(9.3) has aunique solution that depends continuously on the given initial and boundary data.As in the case of the Dirichlet problem for the Laplace equation, uniqueness andcontinuous dependence follow from a maximum-minimum principle.

Theorem 9.1 (Weak Maximum-Minimum Principle). Assume that the functionu ∈ C(D × [0, T ]) is twice continuously differentiable with respect to the spacevariable and continuously differentiable with respect to the time variable and solvesthe heat equation in D × (0, T ]. Then u attains both its maximum and its minimumon the parabolic boundary

B := {(x, 0) : x ∈ D} ∪ {(x, t) : x ∈ Γ, t ∈ [0, T ]}.Proof. We confine ourselves to the proof for the maximum. Define a continuousfunction v on D × [0, T ] by

v(x, t) := u(x, t) + ε(T − t)

9.1 Initial Boundary Value Problem: Uniqueness 173

with ε > 0. Assume that v attains its maximum value in the parabolic interiorD × (0, T ], i.e., there exists a point (x, t) ∈ D × (0, T ] such that

v(x, t) = max(y,τ)∈D×[0,T ]

v(y, τ).

Then the necessary conditions for a maximum

∂2v

∂x2i

(x, t) ≤ 0, i = 1, . . . ,m,

and∂v

∂t(x, t) ≥ 0

must be fulfilled. Hence, we have

Δv(x, t) − ∂v∂t

(x, t) ≤ 0,

which is a contradiction to

Δv − ∂v∂t= ε > 0

throughout D × (0, T ]. Therefore, v attains its maximum on the parabolic boundaryB. Then we can conclude that

u(x, t) ≤ v(x, t) ≤ max(y,τ)∈B

v(y, τ) ≤ max(y,τ)∈B

u(y, τ) + εT

for all (x, t) ∈ D × [0, T ]. Because ε can be chosen arbitrarily, it follows that

u(x, t) ≤ max(y,τ)∈B

u(y, τ),

which ends the proof. �

We wish to mention that analogous to the maximum-minimum principle Theo-rem 6.9 for harmonic functions, there is also a strong maximum-minimum principle,stating that, under the conditions of Theorem 9.1, a solution to the heat equation can-not assume its maximum or minimum in the parabolic interior unless it is constant.For a proof in the case m = 1 we refer to Cannon [25].

The maximum-minimum principle implies uniqueness and continuous depen-dence for the initial boundary value problem as formulated in the following twotheorems.

Theorem 9.2. The initial boundary value problem (9.1)–(9.3) for the heat equationhas at most one solution.

Theorem 9.3. The solution to the initial boundary value problem (9.1)–(9.3) for theheat equation depends continuously in the maximum norm on the given initial andboundary data.


In the statement of Theorem 9.3 we have tacitly assumed that we will be able toestablish existence of the solution.

9.2 Heat Potentials

The function

G(x, t; y, τ) :=1√

4π(t − τ)m exp

{− |x − y|

2

4(t − τ)

}, t > τ, (9.5)

is called fundamental solution of the heat equation. Straightforward differentiationshows that G satisfies the heat equation with respect to the variables x and t. With theaid of this fundamental solution, as in the analysis of Laplace’s equation in Chap-ter 6, we will define so-called heat potentials. Then we shall use these potentials tosolve the initial boundary value problem. For the remainder of this chapter, in thecase m ≥ 2, we will always assume that the bounded domain D ⊂ IRm is of class C2.We will take care of the initial condition through the Poisson integral introduced inthe following theorem.

Theorem 9.4. Let w be a continuous function on IRm with compact support, i.e., wvanishes outside some compact set. Then the Poisson integral

u(x, t) :=1√

4πtm

∫IRm

exp

{−|x − y|

2

4t

}w(y) dy (9.6)

defines an infinitely differentiable solution of the heat equation in IRm × (0,∞). Itcan be continuously extended into IRm × [0,∞) with initial values

u(· , 0) = w. (9.7)

Proof. Obviously, for all x ∈ IRm and t ∈ (0,∞) differentiation and integration can beinterchanged to show that the Poisson integral is infinitely differentiable and satisfiesthe heat equation. Substituting y = x + 2

√t z, we obtain

1√4πt

m

∫IRm

exp

{−|x − y|

2

4t

}w(y) dy =

1√π

m

∫IRmw(x + 2

√t z) e−|z|

2dz.

Since w has compact support, it is bounded and uniformly continuous. Let M be abound for w. Then for ε > 0 we choose R such that

1√π

m

∫|z|≥R

e−|z|2

dz <ε

4M.

Because w is uniformly continuous there exists δ > 0 such that |w(x) − w(y)| < ε/2for all x and y with |x − y| < δ. Setting η = δ2/4R2 and making use of the integral

9.2 Heat Potentials 175

(see Problem 9.1) ∫IRm

e−|z|2dz =

√π

m(9.8)

and the fact that for |z| ≤ R and t < η we have 2√

t |z| < 2√ηR = δ, we deduce that

∣∣∣∣∣∣1√

4πtm

∫IRm

exp

{−|x − y|

2

4t

}w(y) dy − w(x)

∣∣∣∣∣∣

=

∣∣∣∣∣∣1√π

m

∫IRm

{w(x + 2

√t z) − w(x)

}e−|z|

2dz

∣∣∣∣∣∣

<ε

2√π

m

∫|z|≤R

e−|z|2dz +

2M√π m

∫|z|≥R

e−|z|2dz < ε

for all x ∈ IRm and all t < η. This implies continuity of u at t = 0 and (9.7). �

In order to deal with the boundary condition we will need surface heat poten-tials. Analogous to the terminology used for harmonic functions we define single-and double-layer potentials. For a function ϕ ∈ C(Γ × [0, T ]) the single-layer heatpotential is given by

u(x, t) :=∫ t

0

∫Γ

G(x, t; y, τ)ϕ(y, τ) ds(y)dτ (9.9)

and the double-layer heat potential by

u(x, t) :=∫ t

0

∫Γ

∂G(x, t; y, τ)∂ν(y)

ϕ(y, τ) ds(y)dτ. (9.10)

These potentials are well defined for x ∈ D and t ∈ (0, T ] with the time integralto be understood as an improper integral with respect to the upper limit. For thedouble-layer potential we assume the unit normal vector ν to the boundary surfaceΓ to be directed into the exterior of D.

In the case of one space dimension for an interval D := (a, b) the two potentialshave the form

u(x, t) :=∫ t

0

1√4π(t − τ)

[exp

{− (x − a)2

4(t − τ)

}+ exp

{− (x − b)2

4(t − τ)

}]ϕ(b, τ) dτ (9.11)

and

u(x, t) :=∫ t

0

a − x

4√π√

(t − τ)3

exp

{− (x − a)2

4(t − τ)

}ϕ(a, τ) dτ

+

∫ t

0

x − b

4√π√

(t − τ)3

exp

{− (x − b)2

4(t − τ)

}ϕ(b, τ) dτ.

(9.12)


Standard analysis again shows that both the single- and double-layer heat poten-tials are infinitely differentiable solutions to the heat equation in D× (0, T ]. In addi-tion they can be continuously extended into D× [0, T ] with initial values u(· , 0) = 0in D. For the discussion of the boundary values we will confine our analysis to thedouble-layer potential.

Theorem 9.5. The double-layer heat potential with continuous density ϕ can becontinuously extended from D × (0, T ] into D × (0, T ] with limiting values

u(x, t) =∫ t

0

∫Γ

∂G(x, t; y, τ)∂ν(y)

ϕ(y, τ) ds(y)dτ − 12ϕ(x, t) (9.13)

for x ∈ Γ and t ∈ (0, T ]. Here the time integral exists as improper integral.

Proof. We carry out the proof for the cases m = 2, 3, and leave it to the reader towork out the details for the simpler case m = 1 (see Problem 9.3). For x ∈ D wemay interchange the integrations over Γ and [0, t], since the integrand is continuouson Γ × [0, t] with value zero at the upper limit τ = t. Then we substitute for the timeintegral

σ =|x − y|

2√

t − τto obtain

u(x, t) =∫ t

0

1√4π(t − τ)

m

∫Γ

ν(y) · (x − y)2(t − τ)

exp

{− |x − y|

2

4(t − τ)

}ϕ(y, τ) ds(y)dτ

=1√π

m

∫Γ

ν(y) · (x − y)|x − y|m

∫ ∞

|x−y|/2√tσm−1e−σ

2ϕ

(y, t − |x − y|

2

4σ2

)dσds(y).

Therefore, we can view the double-layer heat potential as a harmonic double-layerpotential with the density

ψ(x, y, t) :=1√π

m

∫ ∞

|x−y|/2√tσm−1e−σ

2ϕ

(y, t − |x − y|

2

4σ2

)dσ

= |x − y|m∫ t

0

1√4π(t − τ)

m1

2(t − τ)exp

{− |x − y|

2

4(t − τ)

}ϕ(y, τ) dτ

depending on the time t as a parameter. First, we will show that ψ is continuous onIRm × Γ × (0, T ] with

limx→y ψ(x, y, t) = γmϕ(y, t) (9.14)

for all y ∈ Γ and t ∈ (0, T ], where

γm :=1√π

m

∫ ∞

0sm−1e−s2

ds

9.2 Heat Potentials 177

and the limit holds uniformly on Γ and compact subintervals of (0, T ]. Clearly thefunction ψ is continuous for all x � y and all t ∈ (0, T ]. We establish the limit (9.14)by splitting

ψ(x, y, t) =1√π

m

∫ √|x−y||x−y|/2√t

σm−1e−σ2ϕ

(y, t − |x − y|

2

4σ2

)dσ

+1√π

m

∫ ∞√|x−y|

σm−1e−σ2

{ϕ

(y, t − |x − y|

2

4σ2

)− ϕ(y, t)

}dσ

+ϕ(y, t)1√π

m

∫ ∞√|x−y|

σm−1e−σ2dσ =: I1 + I2 + I3,

with obvious definitions of I1, I2 and I3. Clearly we have

limx→y I1(x, y, t) = 0

uniformly on Γ and on compact subintervals of (0, T ] and

limx→y I3(x, y, t) = γmϕ(y, t)

uniformly on Γ × [0, T ]. Since ϕ is uniformly continuous, for any ε > 0 there existsδ > 0 such that

|ϕ(y, t1) − ϕ(y, t2)| < εfor all y ∈ Γ and all t1, t2 ∈ [0, T ] with |t1 − t2| < δ. Then for all |x − y| < 4δ and allσ ≥ √|x − y| we have

|x − y|24σ2

≤ |x − y|4

< δ

and therefore ∣∣∣∣∣∣ϕ(y, t − |x − y|

2

4σ2

)− ϕ(y, t)

∣∣∣∣∣∣ < ε.Hence,

|I2(x, y, t)| < ε√π

m

∫ ∞√|x−y|

σm−1e−σ2dσ ≤ γmε

and thuslimx→y I2(x, y, t) = 0

uniformly on Γ × [0, T ]. Now, using γ2 = 1/2π and γ3 = 1/4π, from Theorem 6.18we can deduce that

limh→0

u(x − hν(x), t) =∫ t

0

∫Γ

∂G(x, t; y, τ)∂ν(y)

ϕ(y, τ) ds(y)dτ−12ϕ(x, t) (9.15)


with uniform convergence on Γ and on compact subintervals of (0, T ]. Note that weneed a slightly generalized version of Theorem 6.18 where the density is allowed toalso depend on x and a parameter t. It is left to the reader to go over the proof ofTheorem 6.18 to see that it can be extended to this more general case.

Finally, the statement on the continuity of the double-layer heat potential onD × (0, T ] follows from the fact that the right-hand side of (9.15) is continuouson Γ × [0, T ] by the following Theorem 9.6. �

Consider the integral operator H : C(Γ × [0, T ])→ C(Γ × [0, T ]) defined by

(Hϕ)(x, t) := 2∫ t

0

∫Γ

∂G(x, t; y, τ)∂ν(y)

ϕ(y, τ) ds(y)dτ (9.16)

for x ∈ Γ and t ∈ (0, T ] with improper integral over [0, t]. For its kernel

2∂G(x, t; y, τ)

∂ν(y)=

1√4π(t − τ)

mν(y) · (x − y)

t − τ exp

{− |x − y|

2

4(t − τ)

},

using Lemma 6.16, we can estimate

2∣∣∣∣∣∂G(x, t; y, τ)

∂ν(y)

∣∣∣∣∣ ≤ L√t − τ m

|x − y|2t − τ exp

{− |x − y|

2

4(t − τ)

}, t > τ,

with some constant L. Applying the inequality (see Problem 9.2)

sβe−s ≤ ββe−β, (9.17)

which is valid for all 0 < s, β < ∞, for the special case

s =|x − y|24(t − τ)

and β = 1 + m/2 − α, we derive the further estimate

2∣∣∣∣∣∂G(x, t; y, τ)

∂ν(y)

∣∣∣∣∣ ≤ M(t − τ)α|x − y|m−2α

, t > τ, x � y, (9.18)

for all 0 < α < 1 + m/2 and some constant M depending on L and α. Hence,choosing 1/2 < α < 1, the kernel of H is seen to be weakly singular with respectto both the integrals over Γ and over time. Therefore, proceeding as in the proofs ofTheorems 2.29 and 2.30, we can establish the following result.

Theorem 9.6. The double-layer heat potential operator H is a compact operatorfrom C(Γ × [0, T ]) into C(Γ × [0, T ]).

Summarizing, the double-layer heat potential with continuous density is contin-uous in D × [0, T ] with the exception of possible discontinuities on the boundaryΓ for t = 0. From (9.18) we observe that (Hϕ)(· , 0) = 0. Therefore we can expect

9.3 Initial Boundary Value Problem: Existence 179

continuity in D × [0, T ] only if ϕ(· , 0) = 0 on Γ. And indeed, in this case we canextend the density ϕ continuously onto Γ × (−∞, T ] by setting ϕ(· , t) = 0 for t < 0.Then, as a consequence of Theorem 9.5, we can state the following corollary.

Corollary 9.7. The double-layer heat potential is continuous on D× [0, T ] providedthe continuous density satisfies ϕ(· , 0) = 0.

9.3 Initial Boundary Value Problem: Existence

We now return to the initial boundary value problem. First we reduce it to the specialcase of a homogeneous initial condition. To this end, consider

u(x, t) := v(x) +1√

4πtm

∫IRm

exp

{−|x − y|

2

4t

}{w(y) − v(y)} dy. (9.19)

Here, v denotes the unique solution to the Dirichlet problem for Laplace’s equationin D with boundary condition v = f (· , 0) on Γ (see Theorem 6.23). Because of thecompatibility condition (9.4), the function w − v can be continuously extended intoIRm by setting it to zero outside D. Then, from Theorems 6.6 and 9.4, we deduce that(9.19) defines an infinitely differentiable solution to the heat equation in D × (0, T ]that is continuous in D × [0, T ] and satisfies the initial condition u(· , 0) = w in D.Hence, by superposition, it suffices to treat the special case of the initial boundaryvalue problem with initial condition

u(· , 0) = 0 in D (9.20)

and boundary conditionu = f on Γ × [0, T ], (9.21)

where f satisfies the compatibility condition

f (· , 0) = 0 on Γ. (9.22)

In order to deal with the boundary condition (9.21) we seek the solution in the formof a double-layer heat potential. Again we will leave the case where m = 1 as anexercise for the reader (see Problem 9.4).

Theorem 9.8. The double-layer heat potential (9.10) solves the initial boundaryvalue problem (9.20)–(9.22) provided that the continuous density ϕ for all x ∈ Γand t ∈ (0, T ] solves the integral equation

ϕ(x, t) − 2∫ t

0

∫Γ

∂G(x, t; y, τ)∂ν(y)

ϕ(y, τ) ds(y)dτ = −2 f (x, t). (9.23)


Proof. This follows from Theorem 9.5 and Corollary 9.7. The compatibility condi-tion (9.22) for f ensures that ϕ(· , 0) = 0 for solutions to (9.23). �

By Theorem 9.6 and the Riesz theory Theorem 3.4, the inhomogeneous integralequation ϕ−Hϕ = −2 f is solvable if the corresponding homogeneous integral equa-tion ϕ − Hϕ = 0 has only the trivial solution. From the weak singularity (9.18) wesee that for the integral operator H the integration over Γ corresponds to a com-pact operator. Therefore we can estimate for the maximum norm with respect to thespace variable by

‖(Hϕ)(· , t)‖∞,Γ ≤ C∫ t

0

1(t − τ)α

‖ϕ(· , τ)‖∞,Γ dτ (9.24)

for all 0 < t ≤ T and some constant C depending on Γ and α. Note that (see Problem9.1) ∫ t

σ

dτ(t − τ)α(τ − σ)β

=1

(t − σ)α+β−1

∫ 1

0

dssβ(1 − s)α

≤ 1(t − σ)α+β−1

∫ 1

0

ds[s(1 − s)]α

(9.25)

for β ≤ α < 1. Then, from (9.24), by induction and by interchanging the integrals in

∫ t

0

∫ τ

0

dσ(t − τ)α(τ − σ)β

dτ =∫ t

0

∫ t

σ

dτ(t − τ)α(τ − σ)β

dσ,

we find that

‖(Hkϕ)(· , t)‖∞,Γ ≤ CkIk−1∫ t

0

1(t − τ)k(α−1)+1

‖ϕ(· , τ)‖∞,Γ dτ

for all k ∈ IN and all 0 < t ≤ T , where

I :=∫ 1

0

ds[s(1 − s)]α

.

Hence there exists an integer k such that for H := Hk we have an estimate of theform

‖(Hϕ)(· , t)‖∞,Γ ≤ M∫ t

0‖ϕ(· , τ)‖∞,Γ dτ (9.26)

for all 0 ≤ t ≤ T and some constant M.Now, let ϕ be a solution to the homogeneous equation ϕ − Hϕ = 0. Then, by

iteration, ϕ also solves ϕ− Hϕ = 0. Proceeding as in the proof of Theorem 3.10, i.e.,as for Volterra equations with a continuous kernel, from (9.26) we can derive that

‖ϕ(· , t)‖∞,Γ ≤ ‖ϕ‖∞ Mntn

n!, 0 ≤ t ≤ T,

Problems 181

for all n ∈ IN. Hence ‖ϕ(· , t)‖∞,Γ = 0 for all t ∈ [0, T ], i.e., ϕ = 0. Thus we haveestablished the following existence theorem.

Theorem 9.9. The initial boundary value problem for the heat equation has aunique solution.

We wish to mention that, by using a single-layer heat potential, the analysis ofthis chapter can be carried over to an initial boundary value problem where theDirichlet boundary condition is replaced by a Neumann boundary condition. Forthe one-dimensional case, see Problem 9.5.

The idea to proceed along the same line in the case of the heat equation as forthe Laplace equation is due to Holmgren [100, 101] and Gevrey [62]. The first rig-orous existence proof was given by Muntz [179] using successive approximationsfor the integral equation in two dimensions. Integral equations of the first kind forthe heat equation have been investigated by Arnold and Noon [7], Costabel [34],Costabel and Saranen [35] and Hsiao and Saranen [103] in Sobolev spaces, and byBaderko [13] in Holder spaces.

Problems

9.1. Prove the integrals (9.8) and (9.25).

9.2. Prove the inequality (9.17).

9.3. Carry out the proof of Theorem 9.5 in the one-dimensional case.

9.4. Show that the double-layer heat potential (9.12) solves the initial boundary value problem forthe heat equation in (a, b) × (0, T ] with homogeneous initial condition u(· , 0) = 0 and Dirichletboundary conditions

u(a, ·) = f (a, ·) and u(b, ·) = f (b, ·)(with compatibility condition f (a, 0) = f (b, 0) = 0) provided the densities satisfy the system ofVolterra integral equations

ϕ(a, t) −∫ t

0h(t, τ)ϕ(b, τ) dτ = −2 f (a, t),

ϕ(b, t) −∫ t

0h(t, τ)ϕ(a, τ) dτ = −2 f (b, t)

for 0 ≤ t ≤ T with kernel

h(t, τ) :=a − b

2√π

1√

(t − τ)3

exp

{− (a − b)2

4(t − τ)

}, 0 ≤ τ < t.

Establish existence and uniqueness for this system.

9.5. Show that the single-layer heat potential (9.11) solves the initial boundary value problem forthe heat equation in (a, b) × (0, T ] with homogeneous initial condition u(· , 0) = 0 and Neumannboundary conditions

−∂u(a, ·)∂x

= g(a, ·) and∂u(b, ·)∂x

= g(b, ·)


(with compatibility condition g(a, 0) = g(b, 0) = 0) provided the densities satisfy the system ofVolterra integral equations

ϕ(a, t) +∫ t

0h(t, τ)ϕ(b, τ) dτ = 2g(a, t),

ϕ(b, t) +∫ t

0h(t, τ)ϕ(a, τ) dτ = 2g(b, t)

for 0 ≤ t ≤ T with the kernel h given as in Problem 9.4. Establish existence and uniqueness forthis system.

Chapter 10Operator Approximations

In subsequent chapters we will study the numerical solution of integral equations.It is our intention to provide the basic tools for the investigation of approximatesolution methods and their error analysis. We do not aim at a complete review of allthe various numerical methods that have been developed in the literature. However,we will develop some of the principal ideas and illustrate them with a few instructiveexamples.

A fundamental concept for approximately solving an operator equation

Aϕ = f

with a bounded linear operator A : X → Y mapping a Banach space X into a Banachspace Y is to replace it by an equation

Anϕn = fn

with an approximating sequence of bounded linear operators An : X → Y and anapproximating sequence fn → f , n→ ∞. For practical purposes, the approximatingequations will be chosen so that they can be reduced to solving a finite-dimensionallinear system. In this chapter we will provide an error analysis for such generalapproximation schemes. In particular, we will derive convergence results and errorestimates for the cases where we have either norm or pointwise convergence of thesequence An → A, n → ∞. For the latter case, we will present the concept ofcollectively compact operators.

10.1 Approximations via Norm Convergence

Theorem 10.1. Let X and Y be Banach spaces and let A : X → Y be a boundedlinear operator with a bounded inverse A−1 : Y → X, i.e., an isomorphism. Assumethe sequence An : X → Y of bounded linear operators to be norm convergent


183

184 10 Operator Approximations

‖An − A‖ → 0, n→ ∞. Then for sufficiently large n, more precisely for all n with

‖A−1(An − A)‖ < 1,

the inverse operators A−1n : Y → X exist and are bounded by

‖A−1n ‖ ≤

‖A−1‖1 − ‖A−1(An − A)‖ . (10.1)

For the solutions of the equations

Aϕ = f and Anϕn = fn

we have the error estimate

‖ϕn − ϕ‖ ≤ ‖A−1‖1 − ‖A−1(An − A)‖ {‖(An − A)ϕ‖ + ‖ fn − f ‖} .

Proof. If ‖A−1(An − A)‖ < 1, then by the Neumann series Theorem 2.14, the inverse[I − A−1(A − An)]−1 of I − A−1(A − An) = A−1An exists and is bounded by

‖ [I − A−1(A − An)]−1‖ ≤ 11 − ‖A−1(An − A)‖ .

But then [I − A−1(A − An)]−1A−1 is the inverse of An and bounded by (10.1). Theerror estimate follows from

An(ϕn − ϕ) = fn − f + (A − An)ϕ,


In Theorem 10.1 from the unique solvability of the original equation we con-clude unique solvability of the approximating equation provided the approximationis sufficiently close in the operator norm. The converse situation is described in thefollowing theorem.

Theorem 10.2. Assume there exists some N ∈ IN such that for all n ≥ N the inverseoperators A−1

n : Y → X exist and are uniformly bounded. Then the inverse operatorA−1 : Y → X exists and is bounded by

‖A−1‖ ≤ ‖A−1n ‖

1 − ‖A−1n (An − A)‖ (10.2)

for all n with ‖A−1n (An − A)‖ < 1. For the solutions of the equations

Aϕ = f and Anϕn = fn

10.2 Uniform Boundedness Principle 185


‖ϕn − ϕ‖ ≤ ‖A−1n ‖

1 − ‖A−1n (An − A)‖ {‖(An − A)ϕn‖ + ‖ fn − f ‖}.

Proof. This follows from Theorem 10.1 by interchanging the roles of A and An. �

Note that Theorem 10.2 provides an error bound that, in principle, can be eval-uated, because it involves A−1

n and ϕn but neither A−1 nor ϕ. The error bound ofTheorem 10.1 shows that the accuracy of the approximate solution depends on howwell Anϕ approximates Aϕ for the exact solution as expressed through the followingcorollary.

Corollary 10.3. Under the assumptions of Theorem 10.1 we have the error estimate

‖ϕn − ϕ‖ ≤ C {‖(An − A)ϕ‖ + ‖ fn − f ‖} (10.3)

for all sufficiently large n and some constant C.

Proof. This is an immediate consequence of Theorem 10.1. �

In Chapter 11 we will apply the error analysis of Theorems 10.1 and 10.2 tothe approximation of integral equations of the second kind by replacing the kernelsthrough so-called degenerate kernels.

10.2 Uniform Boundedness Principle

To develop a similar analysis for the case where the sequence (An) is merely point-wise convergent, i.e., Anϕ → Aϕ, n → ∞, for all ϕ, we will have to bridge the gapbetween norm and pointwise convergence. This goal will be achieved through com-pactness properties and one of the necessary tools will be provided by the principleof uniform boundedness that we will develop in this section.

Analogous to IRm, in Banach spaces we have the following properties of nestedballs.

Lemma 10.4. Let X be a Banach space and let (Bn) be a sequence of closed ballswith radii rn and the properties Bn+1 ⊂ Bn for n = 1, 2, . . . and limn→∞ rn = 0. Thenthere exists a unique ϕ ∈ X with ϕ ∈ ⋂∞

n=1 Bn.

Proof. We denote the centers of the balls Bn by ϕn. Then for m ≥ n we have

‖ϕm − ϕn‖ ≤ rn → 0, n→ ∞, (10.4)

since ϕm ∈ Bm ⊂ Bn. Hence (ϕn) is a Cauchy sequence and consequently ϕn → ϕ,n→ ∞, for some ϕ ∈ X. Passing to the limit m→ ∞ in (10.4) yields ‖ϕ − ϕn‖ ≤ rn,that is, ϕ ∈ Bn for all n ∈ IN.


Assume that ϕ, ψ are two elements satisfying ϕ, ψ ∈ ⋂∞n=1 Bn. Then

‖ϕ − ψ‖ ≤ ‖ϕ − ϕn‖ + ‖ϕn − ψ‖ ≤ 2rn → 0, n→ ∞,that is, ϕ = ψ. �

Theorem 10.5 (Baire). Let X be a Banach space and let (Un) be a sequence ofclosed subsets of X such that

⋃∞n=1 Un contains an open ball. Then for some n ∈ IN

the set Un contains an open ball.

Proof. By assumption there exists an open ball V ⊂ ⋃∞n=1 Un. We assume that none

of the sets Un contains an open ball. Then by induction we construct sequences (rn)in IR and (ϕn) in X with the properties

0 < rn ≤ 1n, Bn := B[ϕn; rn] ⊂ V, Bn ⊂ X \ Un, Bn ⊂ Bn−1

for n = 1, 2, . . . .We start the sequence by choosing B0 = B[ϕ0; r0] ⊂ V arbitrary andassume that for some n ≥ 1 the partial sequences r0, . . . , rn and ϕ0, . . . , ϕn with therequired properties are constructed. Then the set (X\Un+1)∩B(ϕn; rn) is open and notempty, since otherwise B(ϕn; rn) ⊂ Un+1 would contradict our assumption that Un+1

does not contain an open ball. Therefore there exist rn+1 with 0 < rn+1 ≤ 1/(n + 1)and ϕn+1 ∈ X such that

Bn+1 = B[ϕn+1; rn+1] ⊂ (X \ Un+1) ∩ Bn.

The sequence (Bn) satisfies the assumptions of Lemma 10.4. Hence there existssome ϕ ∈ X such that ϕ ∈ ⋂∞

n=1 Bn. Then on one hand we have that

ϕ ∈∞⋂

n=1

Bn ⊂∞⋂

n=1

(X \ Un) = X \∞⋃

n=1

Un,

that is, ϕ �⋃∞

n=1 Un. One the other hand we also have that

ϕ ∈∞⋂

n=1

Bn ⊂ V ⊂∞⋃

n=1

Un,

i.e., our assumption that none of the sets Un contains an open ball leads to a contra-diction and the theorem is proven. �

Theorem 10.6 (Uniform boundedness principle). Let the set A := {A : X → Y}of bounded linear operators A mapping a Banach space X into a normed space Y bepointwise bounded, i.e., for each ϕ ∈ X there exists a positive number Cϕ dependingon ϕ such that ‖Aϕ‖ ≤ Cϕ for all A ∈ A. Then the set A is uniformly bounded, i.e.,there exists some constant C such that ‖A‖ ≤ C for all A ∈ A.

Proof. We defineUn := {ϕ ∈ X : ‖Aϕ‖ ≤ n, A ∈ A} .


Because of the continuity of the operators A the sets Un are closed. Clearly, for eachϕ ∈ X we have ϕ ∈ Un for all n ≥ Cϕ and therefore X =

⋃∞n=1 Un. By Baire’s

Theorem 10.5 there exist n ∈ IN and an open ball B(ϕ0; r) with radius r > 0 centeredat some ϕ0 ∈ X such that B(ϕ0; r) ⊂ Un, that is, ‖Aϕ‖ ≤ n for all ϕ ∈ B(ϕ0; r) and allA ∈ A.

Finally, for all ψ ∈ X with ‖ψ‖ ≤ 1 and all A ∈ A, this implies that

‖Aψ‖ = 1r‖A(rψ + ϕ0) − Aϕ0‖ ≤ 2n

r,

since rψ + ϕ0 ∈ B(ϕ0; r). Therefore ‖A‖ ≤ 2n/r for all A ∈ A. �

As a simple consequence of the uniform boundedness principle we note that thelimit operator of a pointwise convergent sequence of bounded linear operators againis bounded.

Corollary 10.7. Let X be a Banach space, Y be a normed space, and assume thatthe bounded linear operators An : X → Y converge pointwise with limit operatorA : X → Y. Then the convergence is uniform on compact subsets U of X, i.e.,

supϕ∈U‖Anϕ − Aϕ‖ → 0, n→ ∞.

Proof. For ε > 0 consider the open balls B(ϕ; r) = {ψ ∈ X : ‖ψ − ϕ‖ < r} with centerϕ ∈ X and radius r = ε/3C, where C is a bound on the operators An. Clearly

U ⊂⋃ϕ∈U

B(ϕ; r)

forms an open covering of U. Since U is compact, there exists a finite subcovering

U ⊂m⋃

j=1

B(ϕ j; r).

Pointwise convergence of (An) guarantees the existence of an integer N(ε) such that

‖Anϕ j − Aϕ j‖ < ε

3

for all n ≥ N(ε) and all j = 1, . . . ,m. Now let ϕ ∈ U be arbitrary. Then ϕ is containedin some ball B(ϕ j; r) with center ϕ j. Hence for all n ≥ N(ε) we have

‖Anϕ − Aϕ‖ ≤ ‖Anϕ − Anϕ j‖ + ‖Anϕ j − Aϕ j‖ + ‖Aϕ j − Aϕ‖

≤ ‖An‖ ‖ϕ − ϕ j‖ + ε3 + ‖A‖ ‖ϕ j − ϕ‖ ≤ 2Cr +ε

3= ε.

Therefore the convergence is uniform on U. �


Although the Banach open mapping theorem does not play a central role in thisbook, for completeness of our presentation of the functional analysis for boundedlinear operators we include the following proof since with Baire’s theorem its mainingredient is already available to us.

Theorem 10.8 (Banach open mapping theorem). Let X and Y be Banach spacesand let A : X → Y be a bijective bounded linear operator. Then its inverse A−1 :Y → X is also bounded, i.e., A is an isomorphism.

Proof. 1. We begin by showing that there exists a closed ball centered at some ψ ∈ Ywith radius ρ such that

B[ψ; ρ] ⊂ A(B[0; 1]). (10.5)

To this end, for f ∈ Y we choose m ∈ IN satisfying m ≥ ‖A−1 f ‖. Then we have that

f = AA−1 f ∈ A(B[0; m]) ⊂ A(B[0; m]),

whence

Y =∞⋃

m=1

A(B[0; m])

follows. Baire’s Theorem 10.5 now implies that there exist m ∈ IN and an openball with radius 2ρm centered at some ψm ∈ Y with B(ψm; 2ρm) ⊂ A(B[0; m]) andconsequently

B[ψm; ρm] ⊂ A(B[0; m]).

Setting ψ := ψm/m and ρ := ρm/m for f ∈ B[ψ; ρ] we have that m f ∈ B[ψm; ρm].Therefore there exists a sequence (ϕn) in X with ‖ϕn‖ ≤ m and Aϕn → m f as n→ ∞,that is, (10.5) is proven.2. In the next step we shift the center of the ball and show that

B[0; rρ] ⊂ A(B[0; r]) (10.6)

for all r > 0. Let f ∈ B[0; rρ] and set

f1 = ψ +fr, f2 = ψ − f

r.

Then f1, f2 ∈ B[ψ; ρ] ⊂ A(B[0; 1]) as a consequence of (10.5). Hence, for j = 1, 2,there exist sequences (ϕn, j) in X satisfying ‖ϕn, j‖ ≤ 1 and f j = limn→∞ Aϕn, j. Thisimplies

f =r2

( f1 − f2) = limn→∞ A

r2

(ϕn,1 − ϕn,2)

and consequently f ∈ A(B[0; r]).3. Now we complete the proof of the theorem by showing that

‖A−1 f ‖ ≤ 2 (10.7)


for all f ∈ Y with ‖ f ‖ ≤ ρ. To this end by induction we construct a sequence ( fn) inY with the properties

fn ∈ A(B[0; 1/2n−1]

)and ∥∥∥∥∥∥∥ f −

n∑k=1

fk

∥∥∥∥∥∥∥ ≤ρ

2n

for all n ∈ IN. By (10.6) we have that f ∈ A(B[0; 1]) and therefore there exists anf1 ∈ A(B[0; 1]) with ‖ f − f1‖ ≤ ρ/2. Assume we have constructed the sequence upto the n-th term. Then again by (10.6) we have

f −n∑

k=1

fk ∈ A (B[0; 1/2n])

and consequently there exists an fn+1 ∈ A(B[0; 1/2n]) with∥∥∥∥∥∥∥ f −

n+1∑k=1

fk

∥∥∥∥∥∥∥ ≤ρ

2n+1

as the next element of the sequence. Now clearly the partial sums gn :=∑n

k=1 fkconverge to f . Because of fk ∈ A

(B[0; 1/2k−1]

)we have ‖A−1 fk‖ ≤ 1/2k−1, and

consequently absolute convergence

∞∑k=1

‖A−1 fk‖ ≤∞∑

k=1

12k−1

= 2.

By Problem 1.5 the latter implies the existence of ϕ ∈ X with the properties

ϕ =

∞∑k=1

A−1 fk = limn→∞ A−1gn

and ‖ϕ‖ ≤ 2. The continuity of A finally yields

Aϕ = A limn→∞ A−1gn = lim

n→∞ AA−1gn = limn→∞ gn = f

and ‖A−1 f ‖ = ‖ϕ‖ ≤ 2, that is, (10.7) is proven. �

We leave it as an exercise for the reader to show that a function A : U ⊂ X → Ymapping a subset U of a normed space into a normed space Y is continuous if andonly if for each open set V ⊂ A(U) the pre-image

A−1(V) := {ϕ ∈ U : Aϕ ∈ V}


is open. Analogous to the concept of a continuous function an open function, or anopen mapping, is defined as a function for which the images of open sets are open.Consequently, the previous theorem on the bounded inverse implies that a bijectivebounded linear operator between Banach spaces is an open map and this explainswhy the theorem is known as the open mapping theorem.

10.3 Collectively Compact Operators

Motivated through Corollary 10.7 and following Anselone [5], we introduce theconcept of collectively compact operators.

Definition 10.9. A set A = {A : X → Y} of linear operators mapping a normedspace X into a normed space Y is called collectively compact if for each boundedset U ⊂ X the image setA(U) = {Aϕ : ϕ ∈ U, A ∈ A} is relatively compact.

Clearly, every operator in a collectively compact set is compact. Each finite setof compact operators is collectively compact. A sequence (An) is called collectivelycompact when the corresponding set is. Pointwise convergence An → A, n→ ∞, ofa collectively compact sequence implies compactness of the limit operator, since

A(U) ⊂ {Anϕ : ϕ ∈ U, n ∈ IN}.Theorem 10.10. Let X, Z be normed spaces and let Y be a Banach space. LetA bea collectively compact set of operators mapping X into Y and let Ln : Y → Z bea pointwise convergent sequence of bounded linear operators with limit operatorL : Y → Z. Then

‖(Ln − L)A‖ → 0, n→ ∞,uniformly for all A ∈ A, that is, supA∈A ‖(Ln − L)A‖ → 0, n→ ∞.Proof. The set U := {Aϕ : ‖ϕ‖ ≤ 1, A ∈ A} is relatively compact. By Corollary 10.7the convergence Lnψ → Lψ, n → ∞, is uniform for all ψ ∈ U. Hence, for everyε > 0 there exists an integer N(ε) such that

‖(Ln − L)Aϕ‖ < εfor all n ≥ N(ε), all ϕ ∈ X with ‖ϕ‖ ≤ 1, and all A ∈ A. Therefore

‖(Ln − L)A‖ ≤ εfor all n ≥ N(ε) and all A ∈ A. �

Corollary 10.11. Let X be a Banach space and let An : X → X be a collectivelycompact and pointwise convergent sequence with limit operator A : X → X. Then

‖(An − A)A‖ → 0 and ‖(An − A)An‖ → 0, n→ ∞.

10.4 Approximations via Pointwise Convergence 191

10.4 Approximations via Pointwise Convergence

The following error analysis for equations of the second kind takes advantage ofCorollary 10.11 and is due to Brakhage [19] and Anselone and Moore [6].

Theorem 10.12. Let A : X → X be a compact linear operator in a Banach spaceX and let I − A be injective. Assume the sequence An : X → X to be collectivelycompact and pointwise convergent Anϕ → Aϕ, n → ∞, for all ϕ ∈ X. Then forsufficiently large n, more precisely for all n with

‖(I − A)−1(An − A)An‖ < 1,

the inverse operators (I − An)−1 : X → X exist and are bounded by

‖(I − An)−1‖ ≤ 1 + ‖(I − A)−1An‖1 − ‖(I − A)−1(An − A)An‖ . (10.8)


ϕ − Aϕ = f and ϕn − Anϕn = fn


‖ϕn − ϕ‖ ≤ 1 + ‖(I − A)−1An‖1 − ‖(I − A)−1(An − A)An‖ {‖(An − A)ϕ‖ + ‖ fn − f ‖}.

Proof. By the Riesz Theorem 3.4 the inverse operator (I − A)−1 : X → X exists andis bounded. The identity

(I − A)−1 = I + (I − A)−1A

suggestsBn := I + (I − A)−1An

as an approximate inverse for I − An. Elementary calculations yield

Bn(I − An) = I − Sn, (10.9)

whereSn := (I − A)−1(An − A)An.

From Corollary 10.11 we conclude that ‖Sn‖ → 0, n → ∞. For ‖Sn‖ < 1 theNeumann series Theorem 2.14 implies that (I − Sn)−1 exists and is bounded by

‖(I − Sn)−1‖ ≤ 11 − ‖Sn‖ .

Now (10.9) implies first that I − An is injective, and therefore, since An is compact,by Theorem 3.4 the inverse (I − An)−1 exists. Then (10.9) also yields (I − An)−1 =


(I − Sn)−1Bn, whence the estimate (10.8) follows. The error estimate follows from

(I − An)(ϕn − ϕ) = fn − f + (An − A)ϕ,


Analogous to Theorem 10.1, in Theorem 10.12 from the unique solvability ofthe original equation we conclude unique solvability of the approximating equationprovided the approximation is sufficiently close. The converse situation is describedthrough the following theorem.

Theorem 10.13. Assume there exists some N ∈ IN such that for all n ≥ N the inverseoperators (I−An)−1 exist and are uniformly bounded. Then the inverse (I−A)−1 existsand is bounded by

‖(I − A)−1‖ ≤ 1 + ‖(I − An)−1A‖1 − ‖(I − An)−1(An − A)A‖ (10.10)

for all n with‖(I − An)−1(An − A)A‖ < 1.


ϕ − Aϕ = f and ϕn − Anϕn = fn


‖ϕn − ϕ‖ ≤ 1 + ‖(I − An)−1A‖1 − ‖(I − An)−1(An − A)A‖ {‖(An − A)ϕn‖ + ‖ fn − f ‖}.

Proof. This follows from Theorem 10.12 by interchanging the roles of A and An. �

We note that Theorem 10.13 provides an error estimate that, in principle, can beevaluated, since it involves (I−An)−1 and ϕn but neither (I−A)−1 nor ϕ. From Theo-rem 10.12 we can conclude that the accuracy of the approximate solution essentiallydepends on how well Anϕ approximates Aϕ for the exact solution as expressed inthe following corollary.

Corollary 10.14. Under the assumptions of Theorem 10.12 we have the errorestimate

‖ϕn − ϕ‖ ≤ C {‖(An − A)ϕ‖ + ‖ fn − f ‖} (10.11)


Proof. This is an immediate consequence of Theorem 10.12. �

In Chapter 12 we will apply the error analysis of Theorems 10.12 and 10.13 tothe approximate solution of integral equations of the second kind by using numericalquadratures.

10.5 Successive Approximations 193

10.5 Successive Approximations

We now briefly revisit the investigation of the convergence of successive approxi-mations

ϕn+1 := Aϕn + f

to solve the equation of the second kind ϕ − Aϕ = f , where A : X → X is abounded linear operator in a Banach space X. So far we have used ‖A‖ < 1 asa sufficient condition for convergence (see Theorem 2.15). It is our aim now toshow that, similar to the case of finite-dimensional linear equations, convergenceof successive approximations can be characterized through the spectral radius ofthe operator A. For the notion of the resolvent set ρ(A), the resolvent R(λ; A) =(λI − A)−1, the spectrum σ(A), and the spectral radius r(A) of a bounded linearoperator A recall Definition 3.8.

Theorem 10.15. Let A : X → X be a bounded linear operator mapping a Banachspace X into itself. Then the Neumann series

(λI − A)−1 =

∞∑k=0

λ−k−1Ak (10.12)

converges in the operator norm for all |λ| > r(A) and diverges for all |λ| < r(A).

Proof. Provided the series (10.12) converges it defines a bounded linear operatorS (λ). As in the proof of Theorem 2.14, this operator S (λ) can be seen to be theinverse of λI − A, since |λ|−n−1‖An‖ → 0, n → ∞, is necessary for the convergenceof the series (10.12).

Let λ0 belong to the resolvent set ρ(A) of A. Then, by Theorem 2.14, for all λwith

|λ − λ0| ‖R(λ0; A)‖ < 1

the series

T (λ) :=∞∑

k=0

(λ0 − λ)kR(λ0; A)k+1 (10.13)

converges in the operator norm and defines a bounded linear operator with

T (λ) = R(λ0; A) [I − (λ0 − λ)R(λ0; A)]−1= [I − (λ0 − λ)R(λ0; A)]−1R(λ0; A).

Hence,(λI − A)T (λ) = [λ0I − A − (λ0 − λ)I] T (λ)

= [I − (λ0 − λ)R(λ0; A)] (λ0I − A)T (λ) = I

and similarly T (λ)(λI − A) = I. Therefore λ ∈ ρ(A) and T (λ) = R(λ; A) for all λwith |λ − λ0| ‖R(λ0; A)‖ < 1. In particular, this implies that the resolvent set ρ(A) isan open subset of C.


For a bounded linear functional F on the space L(X, X) of bounded linear opera-tors in X by the continuity of F from (10.13) we observe that the Taylor series

F((λI − A)−1

)=

∞∑k=0

(λ0 − λ)kF(R(λ0; A)k+1

)

converges for all λ with |λ − λ0| ‖R(λ0; A)‖ < 1, i.e., the function

λ → F((λI − A)−1

)

is an analytic function from ρ(A) into C. Therefore the integral

Jn(F) :=1

2πi

∫|λ|=r

λnF((λI − A)−1

)dλ (10.14)

is well defined for all n ∈ IN and r > r(A), and by Cauchy’s integral theorem doesnot depend on r. If we choose r > ‖A‖, by Theorem 2.14 the series (10.12) convergesand we can insert it into (10.14) to obtain

Jn(F) =∞∑

k=0

12πi

∫|λ|=r

λn−k−1F(Ak) dλ = F(An).

By the Corollary 2.11 of the Hahn–Banach theorem, for each n ∈ IN there ex-ists a bounded linear functional Fn ∈ (L(X, X))∗ with the properties ‖Fn‖ = 1 andFn(An) = ‖An‖. Therefore, estimating in (10.14), we obtain

‖An‖ = Fn(An) = Jn(Fn) ≤ rn

2π

∫|λ|=r‖(λI − A)−1‖ |dλ| ≤ γrr

n (10.15)

with a bound γr on the continuous function λ → λ ‖(λI − A)−1‖ on the circle ofradius r centered at the origin.

Now let |λ| > r(A) and choose r such that |λ| > r > r(A). Then from (10.15) itfollows that ∣∣∣λ−n

∣∣∣ ‖An‖ ≤ γr

(r|λ|

)n

and therefore we arrived at a convergent geometric series as majorant for (10.12).Thus by Problem 1.5 the series (10.12) converges in the Banach space L(X, X).

On the other hand, assume that the Neumann series converges for |λ0| < r(A).Then there exists a constant M such that |λ0|−k−1 ‖Ak‖ ≤ M for all k ∈ IN. Hence,for all λ with |λ| > |λ0| the Neumann series has a convergent geometric series as amajorant, and thus it converges in the Banach space X (see Problem 1.5). Its limitrepresents the inverse of λI − A. Hence all λ with |λ| > |λ0| belong to the resolventset ρ(A), which is a contradiction to |λ0| < r(A). �


Theorem 10.16. Let A : X → X be a bounded linear operator in a Banach space Xwith spectral radius r(A) < 1. Then the successive approximations

ϕn+1 := Aϕn + f , n = 0, 1, 2, . . . ,

converge for each f ∈ X and each ϕ0 ∈ X to the unique solution of ϕ − Aϕ = f .

Proof. This follows, as in the proof of Theorem 2.15, from the convergence of theNeumann series for λ = 1 > r(A). �

That the condition r(A) < 1 for convergence cannot be weakened is demonstratedby the following remark.

Remark 10.17. Let A have spectral radius r(A) > 1. Then the successive approxi-mations with ϕ0 = 0 cannot converge for all f ∈ X.

Proof. Assume the statement is false. Convergence of the successive approximationsfor all f ∈ X is equivalent to pointwise convergence of the Neumann series for λ = 1.Hence, by the uniform boundedness principle Theorem 10.6, there exists a positiveconstant C such that ∥∥∥∥∥∥∥

n∑k=0

Ak

∥∥∥∥∥∥∥ ≤ C

for all n ∈ IN. Then, by the triangle inequality, it follows that ‖An‖ ≤ 2C for alln ∈ IN. Since r(A) > 1, there exists λ0 ∈ σ(A) with |λ0| > 1. For the partial sumsSn :=

∑nk=0 λ

−k−10 Ak of the Neumann series we have

‖Sm − Sn‖ ≤ 2C|λ0|−n−1

|λ0| − 1

for all m > n. Therefore (Sn) is a Cauchy sequence in the Banach space L(X, X)(see Theorem 2.6). This implies convergence of the Neumann series for λ0, whichcontradicts λ0 ∈ σ(A). �

If A is compact then the successive approximations, in general, cannot convergeif r(A) = 1. By Theorem 3.9 all spectral values of a compact operator different fromzero are eigenvalues. Hence, for r(A) = 1, there exists f ∈ X with f � 0 and λ0 with|λ0| = 1 such that A f = λ0 f . But then the successive approximations with ϕ0 = 0satisfy

ϕn =

n−1∑k=0

λk0 f , n = 1, 2, . . . ,

and therefore they diverge.Usually, in practical calculations, some approximation of the operator A will be

used. Therefore we have to establish that our results on convergence remain stableunder small perturbations. This is done through the following two theorems.


Theorem 10.18. Let A : X → X be a bounded linear operator in a Banach space Xwith spectral radius r(A) < 1 and let the sequence Am : X → X of bounded linearoperators be norm convergent ‖Am −A‖ → 0, m→ ∞. Then for all sufficiently largem the equation ϕ − Amϕ = f can be solved by successive approximations.

Proof. Choose λ0 ∈ (r(A), 1). Then R(λ; A) = (λI − A)−1 exists and is bounded forall λ ∈ C with |λ| ≥ λ0. Since R(λ; A) is analytic and ‖R(λ; A)‖ → 0, |λ| → ∞ (thisfollows from the Neumann series expansion of the resolvent), we have

C := sup|λ|≥λ0

‖(λI − A)−1‖ < ∞.

Therefore, because ‖Am − A‖ → 0, m → ∞, there exists an integer N such that‖(λI − A)−1(Am − A)‖ < 1 for all m ≥ N and all |λ| ≥ λ0. But then from Theorem10.1, applied to λI−A and λI−Am, we deduce that (λI−Am)−1 exists and is boundedfor all m ≥ N and all |λ| ≥ λ0. Hence r(Am) < 1 for all m ≥ N and the statementfollows from Theorem 10.16. �

Similarly, based on Theorem 10.12, we can prove the following theorem.

Theorem 10.19. Let A : X → X be a compact linear operator in a Banach spaceX with spectral radius r(A) < 1 and let the sequence Am : X → X of collectivelycompact linear operators be pointwise convergent Amϕ → Aϕ, m → ∞, for allϕ ∈ X. Then for all sufficiently large m the equation ϕ − Amϕ = f can be solved bysuccessive approximations.

We conclude this section with two examples for the application of Theorem10.16.

Theorem 10.20. Volterra integral equations of the second kind with continuous orweakly singular kernels can be solved by successive approximations.

Proof. Volterra integral operators have spectral radius zero. For continuous kernelsthis is a consequence of Theorems 3.9 and 3.10. The case of a weakly singular ker-nel can be dealt with by reducing it to a continuous kernel through iteration as in theproof of Theorem 9.9. �

Historically, the first existence proofs for Volterra integral equations of thesecond kind actually were obtained by directly establishing the convergence ofsuccessive approximations.

The following classical potential theoretic result goes back to Plemelj [192].

Theorem 10.21. The integral operators K,K′ : C(∂D) → C(∂D), defined by (6.35)and (6.36), have spectrum

−1 ∈ σ(K) = σ(K′) ⊂ [−1, 1).


Proof. By Theorem 3.9, the spectrumσ(K′)\{0} of the compact operator K′ consistsonly of eigenvalues. Let λ � −1 be an eigenvalue of K′ with eigenfunction ϕ anddefine the single-layer potential u with density ϕ. Then, from the jump relations ofTheorems 6.15 and 6.19, we obtain u+ = u− and

∂u±∂ν=

12

K′ϕ ∓ 12ϕ =

12

(λ ∓ 1)ϕ

on ∂D. From this we see that

(λ + 1)∫∂D

u+∂u+∂ν

ds = (λ − 1)∫∂D

u−∂u−∂ν

ds.

In addition, by the Gauss theorem Corollary 6.4, we deduce∫∂Dϕ ds = 0. Therefore,

in view of (6.18) the single-layer potential u has the asymptotic behavior

u(x) = O

(1|x|m−1

), grad u(x) = O

(1|x|m

), |x| → ∞,

uniformly for all directions. Now we can apply Green’s Theorem 6.3 to obtain

(1 + λ)∫

IRm\D| grad u|2dx = (1 − λ)

∫D| grad u|2dx. (10.16)

Assume that∫

IRm\D | grad u|2dx =∫

D| grad u|2dx = 0. Then grad u = 0 in IRm and

from the jump relations we have ϕ = 0, which is a contradiction to the fact that ϕis an eigenfunction. Hence, from (10.16) we conclude that λ is contained in [−1, 1].From Theorem 6.21 we already know that −1 is an eigenvalue of K′ whereas 1 is notan eigenvalue. Finally, using the fact that σ(K′) is real, the equality σ(K) = σ(K′)is a consequence of the Fredholm alternative for the adjoint operators K and K′. �

Corollary 10.22. The successive approximations

ϕn+1 :=12ϕn +

12

Kϕn − f , n = 0, 1, 2, . . . , (10.17)

converge uniformly for all ϕ0 and all f in C(∂D) to the unique solution ϕ of the inte-gral equation ϕ − Kϕ = −2 f from Theorem 6.22 for the interior Dirichlet problem.

Proof. We apply Theorem 10.16 to the equation ϕ − Aϕ = − f with the operatorA := 2−1(I + K). Then we have

σ(A) = {1/2 + λ/2 : λ ∈ σ(K)}and therefore r(A) < 1 by Theorem 10.21. �

Remark 10.23. The successive approximations (10.17) also converge in L2(∂D)and in the Sobolev space H1/2(∂D).


Proof. Applying Theorem 4.20 it can be seen that the spectrum of the operator K isthe same for all three spaces C(∂D), L2(∂D) and H1/2(∂D). �

We note that the proof of Corollary 10.22 relies on the compactness of K sincewe made use of the spectral Theorem 3.9. As pointed out in Section 6.5, the double-layer potential operator K no longer remains compact for domains with corners or,more generally, for Lipschitz domains. For an alternative approach to establish con-vergence of Neumann’s iteration method in Lipschitz domains we refer to Steinbachand Wendland [226] and further references therein.

We already mentioned in Problem 6.5 that Neumann used the successive approx-imation scheme of Corollary 10.22 to give the first rigorous existence proof for theDirichlet problem in two-dimensional convex domains.

Problems

10.1. Prove the Banach–Steinhaus theorem: Let A : X → Y be a bounded linear operator and letAn : X → Y be a sequence of bounded linear operators from a Banach space X into a normed spaceY. For pointwise convergence Anϕ → Aϕ, n → ∞, for all ϕ ∈ X it is necessary and sufficient that‖An‖ ≤ C for all n ∈ IN and some constant C and that Anϕ → Aϕ, n → ∞, for all ϕ ∈ U, where Uis some dense subset of X.

10.2. Show that a sequence An : X → Y of compact linear operators mapping a normed space Xinto a normed space Y is collectively compact if and only if for each bounded sequence (ϕn) in Xthe sequence (Anϕn) is relatively compact in Y.

10.3. Let X and Y be Banach spaces, S : X → Y be a bounded linear operator with bounded inverseS −1 : Y → X, and A : X → Y be compact. Formulate and prove extensions of Theorems 10.12 and10.13 for the approximation of the equation

Sϕ − Aϕ = f

by equationsSnϕn − Anϕn = fn,

where the An : X → Y are collectively compact with pointwise convergence An → A, n → ∞,the Sn : X → Y are bounded with pointwise convergence Sn → S , n → ∞, and with uniformlybounded inverses S −1

n : Y → X, and fn → f , n→∞.

10.4. Solve the Volterra integral equation

ϕ(x) −∫ x

0ex−yϕ(y) dy = f (x)

by successive approximations.

10.5. How can the integral equation from Theorem 6.26 for the interior Neumann problem besolved by successive approximations?

Chapter 11Degenerate Kernel Approximation

In this chapter we will consider the approximate solution of integral equations of thesecond kind by replacing the kernels by degenerate kernels, i.e., by approximatinga given kernel K(x, y) through a sum of a finite number of products of functions ofx alone by functions of y alone. In particular, we will describe the construction ofappropriate degenerate kernels by interpolation of the given kernel and by orthonor-mal expansions. The corresponding error analysis will be settled by our results inSection 10.1. We also include a discussion of some basic facts on piecewise linearinterpolation and trigonometric interpolation, which will be used in this and subse-quent chapters.

The degenerate kernel approximation is the simplest method for numericallysolving integral equations of the second kind as far as its understanding and er-ror analysis are concerned. However, because the actual implementation requiresthe numerical evaluation of either a single or double integral for each matrix coef-ficient and for each right-hand side of the approximating linear system, in general,it cannot compete in efficiency with other methods, for example, with the Nystrommethod of Chapter 12 or the projection methods of Chapter 13.

For further studies on degenerate kernel methods, we refer to Atkinson [11],Baker [14], Fenyo and Stolle [52], Hackbusch [76], Kantorovic and Krylov [117]and Kanwal [118].

11.1 Degenerate Operators and Kernels

Let 〈X, X〉 be a dual system, generated by a bounded non-degenerate bilinear form,and let An : X → X be a bounded linear operator of the form

Anϕ =

n∑j=1

〈ϕ, b j〉a j, (11.1)


199

200 11 Degenerate Kernel Approximation

where a1, . . . , an and b1, . . . , bn are elements of X such that the a1, . . . , an are linearlyindependent. For the sake of notational clarity we restrict ourselves to the case wherethe index n for the operator An coincides with the number of elements a1, . . . , an andb1, . . . , bn. The solution of the equation of the second kind ϕn − Anϕn = f with sucha degenerate operator reduces to solving a finite-dimensional linear system.

Theorem 11.1. Each solution of the equation

ϕn −n∑

j=1

〈ϕn, b j〉a j = f (11.2)

has the form

ϕn = f +n∑

k=1

γkak, (11.3)

where the coefficients γ1, . . . , γn satisfy the linear system

γ j −n∑

k=1

〈ak, b j〉γk = 〈 f , b j〉, j = 1, . . . , n. (11.4)

Conversely, to each solution γ1, . . . , γn of the linear system (11.4) there correspondsa solution ϕn of (11.2) defined by (11.3).

Proof. Let ϕn be a solution of (11.2). Writing γk := 〈ϕn, bk〉 for k = 1, . . . , n, andsolving for ϕn we obtain the form (11.3). Taking the bilinear form of (11.3) with b j

we find that the coefficients γ1, . . . , γn must satisfy the linear system (11.4). Con-versely, let γ1, . . . , γn be a solution to the linear system (11.4) and define ϕn by(11.3). Then

ϕn −n∑

j=1

〈ϕn, b j〉a j = f +n∑

j=1

γ ja j −n∑

j=1

⟨f +

n∑k=1

γkak, b j

⟩a j

= f +n∑

j=1

⎧⎪⎪⎨⎪⎪⎩γ j −n∑

k=1

〈ak, b j〉γk − 〈 f , b j〉⎫⎪⎪⎬⎪⎪⎭ a j = f ,


For a given operator A we will use finite-dimensional approximations An of theform (11.1) to obtain approximate solutions of equations of the second kind by usingTheorems 10.1 and 10.2 for the operators I − A and I − An. We wish to mention thatthis approach via finite-dimensional approximations can also be used to establish theFredholm alternative for the case of linear operators in Banach spaces, which canbe split into the sum of an operator of the form (11.1) and an operator with normless than one (see Problems 11.1 and 11.2). For integral operators with continuousor weakly singular kernels such an approximation can be achieved by Problem 2.3

11.2 Interpolation 201

and by using an approximation of weakly singular kernels by continuous kernels, asin the proof of Theorem 2.29.

In the case of the dual system 〈C(G),C(G)〉 introduced in Theorem 4.4 the finite-dimensional operator An has the form

(Anϕ)(x) =∫

GKn(x, y)ϕ(y) dy, x ∈ G,

of an integral operator with a degenerate kernel

Kn(x, y) =n∑

j=1

a j(x)b j(y).

The solution of the integral equation of the second kind

ϕn(x) −∫

G

n∑j=1

a j(x)b j(y)ϕn(y) dy = f (x), x ∈ G, (11.5)

with such a degenerate kernel is described through Theorem 11.1. We will use inte-gral operators An with degenerate kernels Kn as approximations for integral opera-tors A with continuous kernels K in the sense of Section 10.1. In view of Theorems2.13, 10.1, and 10.2 we need approximations with the property that

‖An − A‖∞ = maxx∈G

∫G|Kn(x, y) − K(x, y)| dy

becomes small.Here, we do not attempt to give a complete account of the various possibilities

for degenerate kernel approximations. Instead, we will introduce the reader to thebasic ideas by considering a few simple examples.

11.2 Interpolation

One important method to construct degenerate kernels approximating a given con-tinuous kernel is by interpolation. For its investigation, and for use in subsequentchapters, we collect some basic facts on interpolation.

Theorem 11.2. Let Un ⊂ C(G) be an n-dimensional subspace and let x1, . . . , xn ben points in G such that Un is unisolvent with respect to x1, . . . , xn, i.e., each func-tion from Un that vanishes in x1, . . . , xn vanishes identically. Then, given n valuesg1, . . . , gn, there exists a uniquely determined function u ∈ Un with the interpolationproperty

u(x j) = g j, j = 1, . . . , n.


With the interpolation data given by the values g j = g(x j), j = 1, . . . , n, of a functiong ∈ C(G) the mapping g → u defines a bounded linear operator Pn : C(G) → Un

which is called interpolation operator.

Proof. Let Un = span{u1, . . . , un}. Then the solution to the interpolation problem isgiven by

u =n∑

k=1

γkuk,

where the coefficients γ1, . . . , γn are determined by the uniquely solvable linearsystem

n∑k=1

γkuk(x j) = g j, j = 1, . . . , n.

Let L1, . . . , Ln denote the Lagrange basis for Un, i.e., we have the interpolationproperty

Lk(x j) = δ jk, j, k = 1, . . . , n,

where δ jk is the Kronecker symbol with δ jk = 1 for k = j, and δ jk = 0 for k � j.Then, from

Png =

n∑k=1

g(xk)Lk (11.6)

we conclude the linearity and boundedness of Pn. �

From (11.6) we have that

‖Pn‖∞ ≤ maxx∈G

n∑k=1

|Lk(x)|.

Now choose z ∈ G such that

n∑k=1

|Lk(z)| = maxx∈G

n∑k=1

|Lk(x)|

and a function f ∈ C(G) with ‖ f ‖∞ = 1 and

n∑k=1

f (xk)Lk(z) =n∑

k=1

|Lk(z)|.

Then

‖Pn‖∞ ≥ ‖Pn f ‖∞ ≥ |(Pn f )(z)| = maxx∈G

n∑k=1

|Lk(x)|,

11.2 Interpolation 203

and therefore, in terms of the Lagrange basis, we have the norm

‖Pn‖∞ = maxx∈G

n∑k=1

|Lk(x)|. (11.7)

For examples we confine ourselves to interpolation on a finite interval [a, b] inIR with a < b. First we consider linear splines, i.e., continuous piecewise linearfunctions. Let x j = a + jh, j = 0, . . . , n, denote an equidistant subdivision with stepsize h = (b − a)/n and let Un be the space of continuous functions on [a, b] whoserestrictions on each of the subintervals [x j−1, x j], j = 1, . . . , n, are linear. Existenceand uniqueness for the corresponding linear spline interpolation are evident. Here,the Lagrange basis is given by the so-called hat functions

L j(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1h

(x − x j−1), x ∈ [x j−1, x j], j ≥ 1,

1h

(x j+1 − x), x ∈ [x j, x j+1], j ≤ n − 1,

0, otherwise.

(11.8)

Because the L j are nonnegative, we have

n∑j=0

|L j(x)| =n∑

j=0

L j(x) = 1

for all x ∈ [a, b]. Therefore, (11.7) implies that

‖Pn‖∞ = 1 (11.9)

for piecewise linear interpolation. For the interpolation of twice continuously differ-entiable functions g we have the following error estimate.

Theorem 11.3. Let g ∈ C2[a, b]. Then, for the error in piecewise linear interpola-tion we have the estimate

‖Png − g‖∞ ≤ 18

h2‖g′′‖∞. (11.10)

Proof. Obviously, the maximum of |Png − g| in [x j, x j+1] is attained at an interiorpoint ξ where

g′(ξ) = (Png)′(ξ) =1h{g(x j+1) − g(x j)}.


Without loss of generality we may assume that ξ − x j ≤ h/2. Then, using Taylor’sformula, we derive

(Png)(ξ) − g(ξ) = g(x j) + (Png)′(ξ)(ξ − x j) − g(ξ)

= g(x j) − g(ξ) − (x j − ξ)g′(ξ)

=12

(x j − ξ)2g′′(η),

with some η ∈ (x j, ξ). Hence,

maxx j≤x≤x j+1

|(Png)(x) − g(x)| ≤ 18

h2‖g′′‖∞

and the estimate (11.10) follows. �

11.3 Trigonometric Interpolation

As we have seen in Chapters 6–8 the parameterization of boundary integral equa-tions for two-dimensional boundary value problems leads to integral equations forperiodic functions. For these, instead of a local approximation by piecewise linearinterpolation (or higher-order spline interpolation), quite often it will be preferableto use a global approximation in the form of trigonometric interpolation, since thisleads to a more rapid convergence. To support this preference for using trigono-metric polynomial approximations we quote from Atkinson [11]: . . . the most effi-cient numerical methods for solving boundary integral equations on smooth planarboundaries are those based on trigonometric polynomial approximations, and suchmethods are sometimes called spectral methods. When calculations using piecewisepolynomial approximations are compared with those using trigonometric polyno-mial approximations, the latter are almost always the more efficient.

Let t j = jπ/n, j = 0, . . . , 2n − 1, be an equidistant subdivision of the interval[0, 2π] with an even number of grid points. Then, given the values g0, . . . , g2n−1,there exists a unique trigonometric polynomial of the form

u(t) =α0

2+

n−1∑k=1

[αk cos kt + βk sin kt

]+αn

2cos nt (11.11)

11.3 Trigonometric Interpolation 205

with the interpolation property u(t j) = g j, j = 0, . . . , 2n − 1. Its coefficients aregiven by

αk =1n

2n−1∑j=0

g j cos kt j, k = 0, . . . , n,

βk =1n

2n−1∑j=0

g j sin kt j, k = 1, . . . , n − 1.

From this we deduce that the Lagrange basis for the trigonometric interpolation hasthe form

L j(t) =1

2n

⎧⎪⎪⎨⎪⎪⎩1 + 2n−1∑k=1

cos k(t − t j) + cos n(t − t j)

⎫⎪⎪⎬⎪⎪⎭ (11.12)

for t ∈ [0, 2π] and j = 0, . . . , 2n−1. Using the real part of the geometric sum (8.34),we can transform (11.12) into

L j(t) =1

2nsin n(t − t j) cot

t − t j

2, t � t j. (11.13)

Theorem 11.4. For n ≥ 2, the trigonometric interpolation operator with 2n equidis-tant interpolation points has norm

‖Pn‖∞ < 3 +2π

ln(2n). (11.14)

Proof. The function λ : [0, π]→ IR defined by

λ(t) :=2n−1∑j=0

|L j(t)|

is even and has period π/n. Therefore, in view of (11.7), we have

‖Pn‖∞ = max0≤t≤π/2n

λ(t).

For 0 ≤ t ≤ π/2n, using (11.12) and (11.13), we find

λ(t) ≤ 3 +1

2n

2n−2∑j=2

∣∣∣∣∣cott − t j

2

∣∣∣∣∣ .

Since the cotangent function s → cot(s/2) is positive and monotonically decreasingon the interval (0, π], for 0 ≤ t ≤ π/2n in view of t1 − t ≥ π/2n we can estimate

12n

n∑j=2


2

∣∣∣∣∣ < 12π

∫ tn

t1

cots − t

2ds < −1

πln sin

π

4n.


Analogously we have

12n

2n−2∑j=n+1


2

∣∣∣∣∣ < 12π

∫ t2n−1

tn+1

cott − s

2ds < −1

πln sin

π

4n.

Now, using the inequality

sinπ

4n>

12n

the statement of the lemma follows by piecing the above inequalities together. �

The estimate (11.14) gives the correct behavior for large n, because proceedingas in the proof of Theorem 11.4 we can show that

‖Pn‖∞ ≥ λ(π

2n

)>

12n

cotπ

4n− 2π

ln sinπ

4n(11.15)

for all n ≥ 2. This estimate implies that ‖Pn‖∞ → ∞ as n → ∞. Therefore, fromthe uniform boundedness principle Theorem 10.6 it follows that there exists a con-tinuous 2π-periodic function g for which the sequence (Png) does not converge tog in C[0, 2π]. However, by the Banach–Steinhaus theorem (see Problem 10.1) andthe denseness of the trigonometric polynomials in C[0, 2π], the following lemmaimplies mean square convergence

‖Png − g‖2 → 0, n→ ∞,continuous 2π-periodic functions g.

Lemma 11.5. The trigonometric interpolation operator satisfies

‖Png‖2 ≤√

3π ‖g‖∞ (11.16)

for all n ∈ IN and all continuous 2π-periodic functions g.

Proof. Using (11.12), elementary integrations yield

∫ 2π

0L j(t)Lk(t) dt =

π

nδ jk − (−1) j−k π

4n2(11.17)

for j, k = 0, . . . , 2n − 1. Then, by the triangle inequality we can estimate

‖Png‖22 ≤ ‖g‖2∞2n−1∑j,k=0

∣∣∣∣∣∣∫ 2π

0L j(t)Lk(t) dt

∣∣∣∣∣∣ ≤ 3π ‖g‖2∞,

and (11.16) is proven. �

Provided the interpolated function g has additional regularity, we also have uni-form convergence of the trigonometric interpolation polynomials as expressed by


the following three theorems. For 0 < α ≤ 1, by C0,α2π ⊂ C0,α(IR) we denote the

space of 2π-periodic Holder continuous functions. Furthermore, for m ∈ IN byCm,α

2π ⊂ Cm,α(IR) we denote the space of 2π-periodic and m-times Holder contin-uously differentiable functions g equipped with the norm

‖g‖m,α = ‖g‖∞ + ‖g(m)‖0,α.The following theorem provides a uniform error estimate and implies uniform con-vergence for the trigonometric interpolation of Holder continuous functions.

Theorem 11.6. Let m ∈ IN∪ {0} and 0 < α ≤ 1. Then for the trigonometric interpo-lation we have

‖Png − g‖∞ ≤ Cln nnm+α

‖g‖m,α (11.18)

for all g ∈ Cm,α2π and some constant C depending on m and α.

Proof. For g ∈ Cm,α2π let pn be a best approximation of g with respect to trigonometric

polynomials of the form (11.11) and the maximum norm (see Theorem 1.24). Then,by Jackson’s theorem (see [40, 166]) there exists a constant c that does not dependon g, such that

‖pn − g‖∞ ≤ cnm+α

‖g‖m,α. (11.19)

Now, writingPng − g = Pn(g − pn) − (g − pn),

the estimate (11.18) follows from (11.14) and (11.19). �

Theorem 11.6 can be extended to

‖Png − g‖�,β ≤ Cln n

nm−�+α−β ‖g‖m,α (11.20)

for g ∈ Cm,α2π , m, � ∈ IN ∪ {0} with � ≤ m, 0 < β ≤ α ≤ 1 and some constant

C depending on m, �, α, and β (see Prossdorf and Silbermann [199, 200]). For thetrigonometric interpolation of 2π-periodic analytic functions we have the followingerror estimate (see [133]).

Theorem 11.7. Let g : IR → IR be analytic and 2π-periodic. Then there exists astrip D = IR × (−s, s) ⊂ C with s > 0 such that g can be extended to a holomorphicand 2π-periodic bounded function g : D → C. The error for the trigonometricinterpolation can be estimated by

‖Png − g‖∞ ≤ Mcoth

s2

sinh ns, (11.21)

where M denotes a bound for the holomorphic function g on D.

Proof. Because g : IR → IR is analytic, at each point t ∈ IR the Taylor expansionprovides a holomorphic extension of g into some open disk in the complex plane


with radius r(t) > 0 and center t. The extended function again has period 2π, sincethe coefficients of the Taylor series at t and at t + 2π coincide for the 2π-periodicfunction g : IR → IR. The disks corresponding to all points of the interval [0, 2π]provide an open covering of [0, 2π]. Because [0, 2π] is compact, a finite number ofthese disks suffices to cover [0, 2π]. Then we have an extension into a strip D withfinite width 2s contained in the union of the finite number of disks. Without loss ofgenerality we may assume that g is bounded on D.

Let 0 < σ < s be arbitrary. By Γ we denote the boundary of the rectangle[−π/2n, 2π− π/2n] × [−σ, σ] with counterclockwise orientation. A straightforwardapplication of the residue theorem yields

12πi

∫Γ

cotτ − t

2sin nτ

g(τ) dτ =2g(t)sin nt

− 1n

2n−1∑j=0

(−1) jg(t j) cott − t j

2

for −π/2n ≤ t < 2π − π/2n, t � tk, k = 0, . . . , 2n − 1. Hence, in view of (11.13) weobtain

g(t) − (Png)(t) =sin nt4πi

∫Γ

cotτ − t

2sin nτ

g(τ) dτ,

where we can obviously drop the restriction that t does not coincide with an in-terpolation point. From the periodicity of the integrand and since, by the Schwarzreflection principle, g enjoys the symmetry property g(τ) = g(τ), we find the repre-sentation

g(t) − (Png)(t) =1

2πsin nt Re

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩∫ iσ+2π

iσ

i cotτ − t

2sin nτ

g(τ) dτ

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭ (11.22)

with 0 < σ < s. Because of

| sin nτ| ≥ sinh nσ and | cot τ/2| ≤ cothσ/2

for Im τ = σ, the bound (11.21) now follows by estimating in (11.22) and then pass-ing to the limit σ→ s. �

We can summarize Theorem 11.7 by the estimate

‖Png − g‖∞ ≤ Ce−ns (11.23)

for the trigonometric interpolation of periodic analytic functions, where C and s aresome positive constants depending on g, i.e., the interpolation error decays at leastexponentially.

We conclude this section on trigonometric interpolation with an error estimate inthe Sobolev spaces Hp[0, 2π]. Note that by Theorem 8.4 each function g ∈ Hp[0, 2π]is continuous if p > 1/2.


Theorem 11.8. For the trigonometric interpolation we have

‖Png − g‖q ≤ Cnp−q

‖g‖p, 0 ≤ q ≤ p,12< p, (11.24)

for all g ∈ Hp[0, 2π] and some constant C depending on p and q.

Proof. Consider the monomials fm(t) = eimt and write m = 2kn + μ, where k is aninteger and −n < μ ≤ n. Since fm(t j) = fμ(t j) for j = 0, . . . , 2n − 1, we have

Pn fm = Pn fμ =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

fμ, μ � n,

12

( fn + f−n), μ = n.

Therefore the trigonometric interpolation polynomial is given by

Png =

∞∑k=−∞

⎧⎪⎪⎪⎨⎪⎪⎪⎩12g2kn+n( fn + f−n) +

n−1∑μ=−n+1

g2kn+μ fμ

⎫⎪⎪⎪⎬⎪⎪⎪⎭in terms of the Fourier coefficients gk of g ∈ Hp[0, 2π]. Hence,

‖g − Png‖2q = S1 + S2 + S3,

where

S1 :=∑|m|≥n+1

(1 + m2)q|gm|2, S2 :=n−1∑

μ=−n+1

(1 + μ2)q

∣∣∣∣∣∣∣∑k�0

g2kn+μ

∣∣∣∣∣∣∣2

,

S3 := (1 + n2)q

⎧⎪⎪⎪⎨⎪⎪⎪⎩∣∣∣∣∣∣∣gn − 1

2

∞∑k=−∞

g2kn+n

∣∣∣∣∣∣∣2

+

∣∣∣∣∣∣∣g−n − 12

∞∑k=−∞

g2kn+n

∣∣∣∣∣∣∣2⎫⎪⎪⎪⎬⎪⎪⎪⎭ .

Since p ≥ q, we can estimate

S1 ≤ (1 + n2)q−p∑|m|≥n+1

(1 + m2)p|gm|2 ≤ c1n2q−2p‖g‖2p

with some constant c1 depending on p and q. For the second term, with the aid ofthe Cauchy–Schwarz inequality we find

S2 ≤ c2

n−1∑μ=−n+1

(1 + μ2)q

n2p

∑k�0

|2kn + μ|2p|g2kn+μ|2


with some constant c2 depending on p and q. Similarly we can estimate

S3 ≤ c3(1 + n2)q

n2p

∞∑k=−∞|2kn + n|2p|g2kn+n|2

with some constant c3 depending on p and q. Since q ≥ 0, we have

(1 + μ2)q

n2p≤ 22qn2q−2p, |μ| ≤ n,

and thus we obtain

S2 + S3 ≤ c4n2q−2pn∑

μ=−n+1

∞∑k=−∞|2kn + μ|2p|g2kn+μ|2 ≤ c5n2q−2p‖g‖2p

with some constants c4 and c5 depending on p and q. Summarizing the two estimateson S1 and S2 + S3 concludes the proof. �

11.4 Degenerate Kernels via Interpolation

Now we return to integral equations of the second kind of the form

ϕ(x) −∫ b

aK(x, y)ϕ(y) dy = f (x), a ≤ x ≤ b.

Recalling the interpolation Theorem 11.2, we approximate a given continuouskernel K by the kernel Kn interpolating K with respect to x, i.e., Kn(· , y) ∈ Un

andKn(x j, y) = K(x j, y), j = 1, . . . , n,

for each y in [a, b]. Then we can write

Kn(x, y) =n∑

j=1

L j(x)K(x j, y).

Hence, Kn is a degenerate kernel with a j = L j and b j = K(x j, ·). In this case thelinear system (11.4) reads

γ j −n∑

k=1

γk

∫ b

aK(x j, y)Lk(y) dy =

∫ b

aK(x j, y) f (y) dy (11.25)

11.4 Degenerate Kernels via Interpolation 211

for j = 1, . . . , n, and the solution of the integral equation (11.5) is given by

ϕn = f +n∑

k=1

γkLk.

We illustrate the method by two examples, where we use piecewise linear interpo-lation and trigonometric interpolation as discussed in the two previous sections.

From Theorems 2.13 and 11.3, for the approximation of an integral operator Awith a twice continuously differentiable kernel K by a degenerate kernel operatorAn via linear spline interpolation we conclude the estimate

‖An − A‖∞ ≤ 18

h2(b − a)

∥∥∥∥∥∥∂2K∂x2

∥∥∥∥∥∥∞. (11.26)

Therefore, by Theorem 10.1, the corresponding approximation of the solution to theintegral equation is of order O(h2). In principle, using Theorem 10.2, it is possibleto derive computable error bounds based on the estimate (11.26). In general, thesebounds will be difficult to evaluate in applications. However, in most practical prob-lems it will be sufficient to judge the accuracy of the computed solution by refiningthe subdivision and then comparing the results for the fine and the coarse grid withthe aid of the convergence order, i.e., in the case of linear splines with the aid of theapproximation order O(h2).

Only in special cases the integrals for the coefficients and the right-hand side ofthe linear system (11.25) can be evaluated in closed form. Therefore the degeneratekernel method, in principle, is only a semi-discrete method. For a fully discretemethod we have to incorporate a numerical evaluation of the integrals occurring in(11.25). As a general rule, these numerical quadratures should be performed so thatthe approximation order for the solution to the integral equation is maintained.

To be consistent with our approximations, we replace K(x j, ·) by its linear splineinterpolation, i.e., we approximate

∫ b

aK(x j, y)Lk(y) dy ≈

n∑m=0

K(x j, xm)∫ b

aLm(y)Lk(y) dy, (11.27)

for j, k = 0, . . . , n. Straightforward calculations yield the tridiagonal matrix

W =h6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

2 11 4 1

1 4 1. . . . . .

1 4 11 2

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

for the weights wmk =∫ b

aLm(y)Lk(y) dy. For the right-hand side of (11.25) we

simultaneously replace f and K(x j, ·) by its spline interpolations. This leads to the


approximations

∫ b

aK(x j, y) f (y) dy ≈

n∑k,m=0

K(x j, xm) f (xk)∫ b

aLm(y)Lk(y) dy (11.28)

for j = 0, . . . , n. We now investigate the influence of these approximations on theerror analysis. For this we interpret the solution of the system (11.25) with theapproximate values for the coefficients and the right-hand sides as the solution ϕn

of a further approximate equation

ϕn − Anϕn = fn,

namely, of the degenerate kernel equation

ϕn(x) −∫ b

aKn(x, y)ϕn(y) dy = fn(x), a ≤ x ≤ b,

with

Kn(x, y) :=n∑

j,m=0

K(x j, xm)L j(x)Lm(y) and fn(x) :=n∑

m=0

f (xm)Lm(x).

Provided that the kernel K is twice continuously differentiable, writing

K(x, y) − Kn(x, y) = K(x, y) − [PnK(· , y)](x)

+[Pn[K(· , y) − [PnK(· , ·)](y)](x),(11.29)

by Theorem 11.3 and (11.9), we can estimate

|K(x, y) − Kn(x, y)| ≤ 18

h2

{∥∥∥∥∥∥∂2K∂x2

∥∥∥∥∥∥∞ +∥∥∥∥∥∥∂2K∂y2

∥∥∥∥∥∥∞}

for all a ≤ x, y ≤ b. Hence, for the integral operator An with kernel Kn we have‖An − A‖∞ = O(h2). When f is twice continuously differentiable, we also have‖ fn − f ‖∞ = O(h2). Now, Theorem 10.1 yields the error estimate ‖ϕn − ϕ‖ = O(h2).Therefore we have the following theorem.

Theorem 11.9. The degenerate kernel approximation via equidistant piecewise lin-ear interpolation with the matrix entries and the right-hand sides of the approximat-ing linear system (11.25) replaced by the quadratures (11.27) and (11.28) approxi-mates the solution to the integral equation with order O(h2) if the kernel K and theright-hand side f are twice continuously differentiable.


ϕ(x) − 12

∫ 1

0(x + 1)e−xyϕ(y) dy = e−x − 1

2+

12

e−(x+1), 0 ≤ x ≤ 1. (11.30)

11.4 Degenerate Kernels via Interpolation 213

Obviously it has the solution ϕ(x) = e−x. For its kernel we have

max0≤x≤1

∫ 1

0|K(x, y)|dy = sup

0<x≤1

x + 12x

(1 − e−x) < 1.

Therefore, by Corollary 2.16, the solution is unique.Table 11.1 shows the error between the approximate and the exact solution at the

points x = 0, 0.25, 0.5, 0.75, and 1 for various n. It clearly exhibits the behaviorO(h2) as predicted by Theorem 11.9. �

Table 11.1 Numerical results for Example 11.10

n x = 0 x = 0.25 x = 0.5 x = 0.75 x = 1

4 0.004808 0.005430 0.006178 0.007128 0.0083318 0.001199 0.001354 0.001541 0.001778 0.002078

16 0.000300 0.000338 0.000385 0.000444 0.00051932 0.000075 0.000085 0.000096 0.000111 0.000130

From Theorem 11.6, for the approximation of an integral operator A with a 2π-periodic and m-times continuously differentiable kernel K by degenerate kernels viatrigonometric interpolation we conclude the estimate

‖An − A‖∞ ≤ O

(1

nm−ε

)(11.31)

for all ε > 0. If the kernel is analytic, then from Theorem 11.7 we have

‖An − A‖∞ ≤ O(e−ns) (11.32)

for some s > 0. For the derivation of (11.32) we have to use the fact that for thekernel function, which is assumed to be analytic and periodic with respect to bothvariables, there exists a strip D with width 2s such that K(· , τ) can be continuedholomorphicly and uniformly bounded into D for all τ ∈ [0, 2π]. We leave it to thereader to go over the argument given in the proof of Theorem 11.7 to verify thisproperty. Now, by Theorem 10.1, the corresponding approximation of the solutionto the integral equation is also of order O (nε−m) or O(e−ns), respectively.

Again we have to describe the numerical evaluation of the coefficients and theright-hand side of the linear system (11.25). We proceed analogously to the preced-ing example and approximate the matrix entries by replacing K(t j, ·) by its trigono-metric interpolation polynomial, i.e.,

∫ 2π

0K(t j, τ)Lk(τ) dτ ≈

2n−1∑m=0

K(t j, tm)∫ 2π

0Lm(τ)Lk(τ) dτ (11.33)


for j, k = 0, . . . , 2n − 1 with the integrals on the right-hand side given by (11.17).For the right-hand side of (11.25) we simultaneously replace f and K(t j, ·) by itstrigonometric interpolation. This leads to

∫ 2π

0K(t j, τ) f (τ) dτ ≈

2n−1∑k,m=0

K(t j, tm) f (tk)∫ 2π

0Lm(τ)Lk(τ) dτ (11.34)

for j = 0, . . . , 2n − 1. Note that despite the global nature of the trigonometric inter-polation and its Lagrange basis, due to the simple structure of the weights (11.17) inthe quadrature rule, the approximate computation of the matrix elements and right-hand side is not too costly. In view of (11.18) and splitting analogous to (11.29), weremain with a total error O (nε−m) or O(e−ns), respectively, for the approximate solu-tion of the integral equation. Here we use that ln n e−ns = O(e−nσ) for all 0 < σ < s.Hence we have the following theorem.

Theorem 11.11. The degenerate kernel approximation via trigonometric interpola-tion with the matrix entries and the right-hand sides of the approximating linearsystem (11.25) replaced by the quadratures (11.33) and (11.34) approximates thesolution to the integral equation of order O (nε−m) or O(e−ns), respectively, if the 2π-periodic kernel K and the 2π-periodic right-hand side f are m-times continuouslydifferentiable or analytic, respectively.


ϕ(t) +abπ

∫ 2π

0

ϕ(τ) dτa2 + b2 − (a2 − b2) cos(t + τ)

= f (t), 0 ≤ t ≤ 2π, (11.35)

corresponding to the Dirichlet problem for the Laplace equation in the interior of anellipse with semi-axis a ≥ b > 0 (see Problem 6.2). We numerically want to solvethe case where the unique solution is given by

ϕ(t) = ecos t cos(sin t) = Re∞∑

m=0

1m!

eimt, 0 ≤ t ≤ 2π.

Then, the right-hand side becomes

f (t) = ϕ(t) + ec cos t cos(c sin t), 0 ≤ t ≤ 2π,

where c = (a − b)/(a + b) (compare also Problem 11.4).Table 11.2 shows the error between the approximate and the exact solution at

the three points t = 0, π/2, and π for various n and depending on the ratio b/a. Itclearly shows the exponential decay of the error: doubling the number of grid pointsdoubles the number of correct digits in the approximate solution. The rate of theexponential decay depends on the parameters a and b, which describe the locationof the singularities of the integrands in the complex plane, i.e., they determine thevalue for the strip parameter s. �

11.5 Degenerate Kernels via Expansions 215


2n t = 0 t = π/2 t = π

4 −0.10752855 −0.03243176 0.03961310a=1 8 −0.00231537 0.00059809 0.00045961

b=0.5 16 −0.00000044 0.00000002 −0.0000000032 0.00000000 0.00000000 0.00000000

4 −0.56984945 −0.18357135 0.06022598a=1 8 −0.14414257 −0.00368787 −0.00571394

b=0.2 16 −0.00602543 −0.00035953 −0.0004540832 −0.00000919 −0.00000055 −0.00000069

11.5 Degenerate Kernels via Expansions

A second possibility for constructing approximate degenerate kernels is by expan-sions, in particular, by orthonormal expansions. Let (· , ·) denote a scalar product onC(G) and let {u1, u2, . . .} be a complete orthonormal system (recall Theorem 1.28).Then a given continuous kernel K is expanded with respect to x for each fixed y ∈ G,i.e., K(x, y) is approximated by the partial sum

Kn(x, y) :=n∑

j=1

u j(x)(K(· , y), u j)

of the Fourier series. For this degenerate kernel the linear system (11.4) reads

γ j −n∑

k=1

γk

∫G

uk(y)(K(· , y), u j) dy =∫

Gf (y)(K(· , y), u j) dy (11.36)

for j = 1, . . . , n. Usually, the scalar product will be given in terms of an integral.Therefore the system (11.36) requires a double integration for each coefficient andfor each right-hand side. Since it will turn out that the degenerate kernel method viaorthonormal expansion is closely related to the Galerkin method (see Section 13.6),we omit further discussion and a description of the numerical implementation.

For the orthonormal expansion degenerate kernel approximations, in principle,we can derive error estimates from Theorems 10.1 and 10.2 only with respect tothe scalar product norm. In the subsequent analysis, for later theoretical use, wewill describe how estimates in the maximum norm can be obtained under additionalregularity assumptions on the kernel.

The Chebyshev polynomials of the first kind are defined by

Tn(z) := cos(n arccos z), −1 ≤ z ≤ 1, n = 0, 1, . . . . (11.37)


From T0(z) = 1 and T1(z) = z and from the recursion formula

Tn+1(z) + Tn−1(z) = 2zTn(z),

which follows from the cosine addition theorem, we observe that the Tn indeedare polynomials of degree n and therefore, in particular, well defined on the wholecomplex plane. A substitution x = cos t readily shows that

∫ 1

−1

Tn(x)Tm(x)√1 − x2

dx =π

2[δnm + δn0],

i.e., the Tn form an orthogonal system with respect to the scalar product

(ϕ, ψ) :=∫ 1

−1

ϕ(x)ψ(x)√1 − x2

dx (11.38)

on C[−1, 1]. By the same substitution, the denseness of the trigonometric polyno-mials in C[0, 2π] implies the denseness of span{T0, T1, . . .} in C[−1, 1] with respectto the scalar product (11.38), i.e., the Tn form a complete orthogonal system.

Theorem 11.13. Let g : [−1, 1] → IR be analytic. Then there exists an ellipse Ewith foci at −1 and 1 such that g can be extended to a holomorphic and boundedfunction g : D → C, where D denotes the open interior of E. The orthonormalexpansion with respect to the Chebyshev polynomials

g =a0

2T0 +

∞∑n=1

anTn (11.39)

with coefficients

an =2π

∫ 1

−1

g(x)Tn(x)√1 − x2

dx

is uniformly convergent with the estimate∥∥∥∥∥∥∥g −

a0

2T0 −

n∑m=1

amTm

∥∥∥∥∥∥∥∞≤ 2M

R − 1R−n.

Here, R is given through the semi-axis a and b of E by R = a + b and M is a boundon g in D.

Proof. The existence of the holomorphic extension of the analytic function g :[−1, 1] → IR into the open interior D of some ellipse with foci at −1 and 1 isshown analogously to the proof of Theorem 11.7.

The function

z =w

2+

12w

11.5 Degenerate Kernels via Expansions 217

maps the annulus 1/R < |w| < R of the w–plane onto the interior D of the ellipse Eof the z–plane. For each holomorphic function g on D the function g, defined by

g(w) := 2g

(w

2+

12w

),

is holomorpic in the annulus. Therefore it can be expanded into a Laurent series

g(w) =∞∑

n=−∞anw

n

with coefficients

an =1πi

∫|w|=r

g

(w

2+

12w

)dwwn+1

,

where 1/R < r < R. By substituting w = 1/w and using Cauchy’s integral theoremwe find that an = a−n for all n ∈ IN. Hence,

g(w) = a0 +

∞∑n=1

an

(wn +

1wn

).

In particular, this Laurent expansion converges uniformly on the circle |w| = 1.Writing w = eit, we derive

Tn

(w

2+

12w

)=

12

(wn +

1wn

)

first on the unit circle, and then, since both sides of the equation represent ratio-nal functions, for all w � 0. Inserting this into the previous series, we obtain theexpansion (11.39) and its uniform convergence on [−1, 1].

Estimating the coefficients of the Laurent expansion and then passing to the limitr → R yields

|an| ≤ 2MRn

, n = 0, 1, 2 . . . ,

with a bound M for g on D. Hence, using |Tn(x)| ≤ 1 for all −1 ≤ x ≤ 1 and alln ∈ IN, the remainder can be estimated by

∣∣∣∣∣∣∣∞∑

m=n+1

amTm(x)

∣∣∣∣∣∣∣ ≤∞∑

m=n+1

|am| ≤ 2MR − 1

R−n,

which finishes the proof. �

From Theorem 11.13, analogous to the derivation of the estimate (11.32), forthe approximation of an integral operator A with an analytic kernel by orthonormalexpansion with respect to Chebyshev polynomials we find the estimate

‖An − A‖∞ = O(R−n) (11.40)

with some constant R > 1 depending on the kernel of A.


In closing this chapter, we wish to mention the use of Taylor expansions as an-other method to construct degenerate kernels that may be useful in special cases (seeProblem 11.3).

Problems

11.1. Let 〈X, Y〉 be a dual system with two normed spaces X and Y and let An : X → X andBn : Y → Y be adjoint finite-dimensional operators of the form

Anϕ =n∑

j=1

〈ϕ, bj〉aj, Bnψ =n∑

j=1

〈aj, ψ〉bj

with linearly independent elements a1, . . . , an ∈ X and b1, . . . , bn ∈ Y. By reduction to linearsystems as in Theorem 11.1, establish the Fredholm alternative for the operators I − An and I − Bn.

11.2. Let 〈X, Y〉 be a dual system with two Banach spaces X and Y, let An : X → X and Bn : Y → Ybe adjoint finite-dimensional operators as in Problem 11.1, and let S : X → X and T : Y → Y beadjoint operators with norm less than one. With the aid of Theorem 2.14 and Problem 11.1 establishthe Fredholm alternative for the operators I − A and I − B where A = An + S and B = Bn + T .Hint: Transform the equations with the operators I − A and I − B equivalently into equations withI − (I − S )−1An and I − Bn(I − T )−1.

11.3. Solve the integral equation (11.30) approximately through degenerate kernels by using theTaylor series for exy .

11.4. Show that the eigenvalues of the integral equation (11.35) are described by

abπ

∫ 2π

0

einτdτ(a2 + b2) − (a2 − b2) cos(t + τ)

=

(a − ba + b

)n

e−int, n = 0, 1, 2, . . . .

11.5. Show that, analogous to (11.40), we have ‖An − A‖∞ = o (1/√

n) for the approximation ofcontinuously differentiable kernels through degenerate kernels via Chebyshev polynomials.Hint: Use Dini’s theorem from Problem 8.2.

Chapter 12Quadrature Methods

In this chapter we shall describe the quadrature or Nystrom method for the approx-imate solution of integral equations of the second kind with continuous or weaklysingular kernels. As we have pointed out in Chapter 11, the implementation of thedegenerate kernel method, in general, requires some use of numerical quadrature.Therefore it is natural to try the application of numerical integration in a more di-rect approach to approximate integral operators by numerical integration operators.This will lead to a straightforward but widely applicable method for approximatelysolving equations of the second kind. The reason we placed the description of thequadrature method after the degenerate kernel method is only because its error anal-ysis is more involved.

Through numerical examples we will illustrate that the Nystrom method pro-vides a highly efficient method for the approximate solution of the boundary in-tegral equations of the second kind for two-dimensional boundary value problemsas considered in Chapters 6–8. We also wish to point out that, despite the fact thatour numerical examples for the application of the Nystrom method are all for one-dimensional integral equations, this method can be applied to multi-dimensionalintegral equations provided appropriate numerical quadratures are available. Thelatter requirement puts some limits on the application to the weakly singular integralequations arising for boundary value problems in more than two space dimensions.

12.1 Numerical Integration

We start with a brief account of the basics of numerical quadrature. In general, aquadrature formula is a numerical method for approximating an integral of the form

Q(g) :=∫

Gw(x)g(x) dx, (12.1)


219

220 12 Quadrature Methods

where w is some weight function and g ∈ C(G). Throughout this chapter G ⊂ IRm

will be compact and Jordan measurable. We consider only quadrature rules of theform

Qn(g) :=n∑

j=1

α(n)j g(x(n)

j )

with quadrature points x(n)1 , . . . , x(n)

n contained in G and real quadrature weights

α(n)1 , . . . , α(n)

n . For the sake of notational clarity we restrict ourselves to the casewhere the index n of the quadrature rule coincides with the number of quadra-ture points. Occasionally, we also will write x1, . . . , xn instead of x(n)

1 , . . . , x(n)n and

α1, . . . , αn instead of α(n)1 , . . . , α(n)

n . The basic numerical integrations are interpola-tory quadratures. They are constructed by replacing the integrand g by an inter-polation with respect to the quadrature points x1, . . . , xn, usually a polynomial, atrigonometric polynomial, or a spline, and then integrating the interpolating func-tion instead of g. Clearly, polynomial interpolation quadratures of degree n − 1 onan interval [a, b] with n quadrature points integrate polynomials of degree less thann exactly. The classical Newton–Cotes rules are a special case of these interpolatoryquadratures with polynomial interpolation on an equidistantly spaced subdivision ofthe interval [a, b]. Since the Newton–Cotes rules have unsatisfactory convergencebehavior as the degree of the interpolation increases, it is more practical to use so-called composite rules. These are obtained by subdividing the interval of integrationand then applying a fixed rule with low interpolation order to each subinterval. Themost frequently used quadrature rules of this type are the composite trapezoidal ruleand the composite Simpson’s rule. For convenience we will discuss their error anal-ysis, which we base on the representation of the remainder terms by means of Peanokernels. Let x j = a + jh, j = 0, . . . , n, be an equidistant subdivision with step sizeh = (b − a)/n.

Theorem 12.1. Let g ∈ C2[a, b]. Then the remainder

RT (g) :=∫ b

ag(x) dx − h

[12g(x0) + g(x1) + · · · + g(xn−1) +

12g(xn)

]

for the composite trapezoidal rule can be estimated by

|RT (g)| ≤ 112

h2(b − a) ‖g′′‖∞.

Proof. Define the Peano kernel for the trapezoidal rule by

KT (x) :=12

(x − x j−1)(x j − x), x j−1 ≤ x ≤ x j,

for j = 1, . . . , n. Then, straightforward partial integrations yield

∫ b

aKT (x)g′′(x) dx = −RT (g).

12.1 Numerical Integration 221

Now the estimate follows from∫ b

aKT (x) dx =

h2

12(b − a)

and the observation that KT is nonnegative on [a, b]. �

Theorem 12.2. Let g ∈ C4[a, b] and let n be even. Then the remainder

RS (g) :=∫ b

ag(x) dx − h

3[g(x0) + 4g(x1) + 2g(x2) + 4g(x3) + 2g(x4)

+ · · · + 2g(xn−2) + 4g(xn−1) + g(xn)]

for the composite Simpson’s rule can be estimated by

|RS (g)| ≤ 1180

h4(b − a) ‖g(4)‖∞.

Proof. With the Peano kernel of Simpson’s rule given by

KS (x) :=

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

h18

(x − x2 j−2)3 − 124

(x − x2 j−2)4, x2 j−2 ≤ x ≤ x2 j−1,

h18

(x2 j − x)3 − 124

(x2 j − x)4, x2 j−1 ≤ x ≤ x2 j,

for j = 1, . . . , n/2, via

∫ b

aKS (x)g(4)(x) dx = −RS (g),

the proof is analogous to the proof of Theorem 12.1. �

More satisfying than Newton–Cotes quadratures are Gauss quadratures. Theseare polynomial interpolation quadratures with n quadrature points that are chosensuch that polynomials of degree less than 2n are integrated exactly. For the integral(12.1) over an interval [a, b] with integrable positive weight function w the quadra-ture points of the Gauss quadrature are given by the n distinct zeros in (a, b) of thepolynomial of order n that is orthogonal to all polynomials of degree less than n

with respect to the scalar product ( f , g) :=∫ b

aw fg dx. The Legendre polynomial Ln

of degree n is defined by

Ln(x) :=1

2nn!dn

dxn(x2 − 1)n.


If m < n, by repeated partial integration we see that

∫ 1

−1xm dn

dxn(x2 − 1)ndx = 0.

Therefore Ln is orthogonal to all polynomials of degree less than n with respectto the L2 scalar product on [−1, 1]. Hence, the quadrature points of the Gauss–Legendre quadrature are given by the n zeros −1 < ξ1 < ξ2 < · · · < ξn < 1 ofLn and the corresponding quadrature weights can be computed as

α j :=2(1 − ξ2

j )

[nLn−1(ξ j)]2, j = 1, . . . , n.

We note that these weights are positive. For a more detailed analysis of Gaussquadratures we refer to Davis and Rabinowitz [41] and Engels [47].

Definition 12.3. A sequence (Qn) of quadrature formulas is called convergent ifQn(g) → Q(g), n → ∞, for all g ∈ C(G), i.e., if the sequence of linear functionals(Qn) converges pointwise to the integral Q.

With the aid of the uniform boundedness principle we can state the followingnecessary and sufficient conditions for convergence.

Theorem 12.4 (Szego). The quadrature formulas (Qn) converge if and only ifQn(g)→ Q(g), n→ ∞, for all g in some dense subset U ⊂ C(G) and

supn∈IN

n∑j=1

|α(n)j | < ∞.

Proof. It is left as an exercise to show that

‖Qn‖∞ =n∑

j=1

|α(n)j |.

Now the statement follows by the Banach–Steinhaus theorem (see Problem 10.1). �

Corollary 12.5 (Steklov). Assume that Qn(1) → Q(1), n → ∞, and that thequadrature weights are all nonnegative. Then the quadrature formulas (Qn) con-verge if and only if Qn(g)→ Q(g), n→ ∞, for all g in some dense subset U ⊂ C(G).

Proof. This follows from

n∑j=1

|α(n)j | =

n∑j=1

α(n)j = Qn(1)→ Q(1), n→ ∞,

and the preceding theorem. �

12.1 Numerical Integration 223

In particular, from Theorems 12.1 and 12.2 and Corollary 12.5, we observe thatthe composite trapezoidal and the composite Simpson’s rule are convergent. Alsothe sequence of Gauss–Legendre quadratures is convergent because of their positiveweights.

We conclude our remarks on numerical quadrature by describing an error esti-mate for the integration of periodic analytic functions due to Davis [39] (see alsoProblem 12.2). Note that for periodic functions the composite trapezoidal rule coin-cides with the composite rectangular rule.

Theorem 12.6. Let g be as in Theorem 11.7. Then the error

RT (g) :=1

2π

∫ 2π

0g(t) dt − 1

2n

2n−1∑j=0

g( jπ

n

)

for the composite trapezoidal rule can be estimated by

|RT (g)| ≤ M (coth ns − 1).

Proof. Using the integrals (8.35), we integrate the remainder term (11.22) for thetrigonometric interpolation to obtain

RT (g) =1

2πRe

{∫ iσ+2π

iσ(1 − i cot nτ) g(τ) dτ

}

for all 0 < σ < s. This can also be shown directly through the residue theorem. Nowthe estimate follows from |1 − i cot nτ| ≤ coth nσ − 1 for Im τ = σ and passing tothe limit σ→ s. �

We can summarize Theorem 12.6 by the estimate

|RT (g)| ≤ Ce−2ns (12.2)

for the composite trapezoidal rule for periodic analytic functions, where C and sare some positive constants depending on g. The improvement by the factor 2 inthe exponent as compared with (11.23) reflects the fact that the trapezoidal ruleintegrates trigonometric polynomials not only of degree less than or equal to n butalso of degree less than or equal to 2n − 1 exactly.

For m-times continuously differentiable 2π-periodic functions g the Euler–Mac-Laurin expansion yields the estimate

|RT (g)| ≤ Cnm‖g(m)‖∞ (12.3)

for some constant C depending on m (see [142]).Obviously, numerical quadrature of 2π-periodic functions is equivalent to nu-

merical quadrature of contour integrals over the unit circle, or more generally overclosed curves in IR2 that are homeomorphic to the unit circle. As an analogue in IR3


we briefly describe a numerical quadrature scheme for the integration of analyticfunctions over closed surfaces Γ which are homeomorphic to the unit sphere S2.Clearly, it suffices to describe the quadrature for integrating over the unit sphere.For this we recall the quadrature points ξ j and the weights α j of the Gauss–Legendrequadrature and choose a set of points x jk on the unit sphere given in spherical coor-dinates by

x jk := (sin θ j cosϕk, sin θ j sin ϕk, cos θ j)

for j = 1, . . . , n and k = 0, . . . , 2n − 1 where θ j := arccos ξ j and ϕk = πk/n. Then,writing the integral over the unit sphere as a double integral

∫S2g ds =

∫ 2π

0

∫ π

0g(sin θ cosϕ, sin θ sin ϕ, cos θ) sin θ dθdϕ

=

∫ 2π

0

∫ 1

−1g

(√1 − ξ2 cosϕ,

√1 − ξ2 sin ϕ, ξ

)dξdϕ,

we apply the Gauss–Legendre quadrature to the integration over ξ and the trape-zoidal rule to the integration over ϕ to obtain the so-called Gauss trapezoidalproduct rule ∫

S2g ds ≈ π

n

n∑j=1

2n−1∑k=0

α jg(x jk) (12.4)

for the numerical integration over the unit sphere. Wienert [249] has shown that foranalytic functions g there exist positive constants C and σ depending on g such thatthe quadrature error in (12.4) can be estimated by Ce−nσ, i.e, as for the compositetrapezoidal rule we have exponential convergence as n→ ∞.

For a comprehensive study of numerical integration, we refer the reader to Davisand Rabinowitz [41] and Engels [47].

12.2 Nystrom’s Method

We choose a convergent sequence (Qn) of quadrature formulas for the integralQ(g) =

∫Gg(x) dx and approximate the integral operator

(Aϕ)(x) :=∫

GK(x, y)ϕ(y) dy, x ∈ G, (12.5)

with continuous kernel K, as introduced in Theorem 2.13, by a sequence of numer-ical integration operators

(Anϕ)(x) :=n∑

k=1

α(n)k K(x, x(n)

k )ϕ(x(n)k ), x ∈ G. (12.6)

12.2 Nystrom’s Method 225

Then the solution to the integral equation of the second kind

ϕ − Aϕ = f

is approximated by the solution of

ϕn − Anϕn = f ,

which reduces to solving a finite-dimensional linear system.

Theorem 12.7. Let ϕn be a solution of

ϕn(x) −n∑

k=1

αkK(x, xk)ϕn(xk) = f (x), x ∈ G. (12.7)

Then the values ϕ(n)j = ϕn(x j), j = 1, . . . , n, at the quadrature points satisfy the

linear system

ϕ(n)j −

n∑k=1

αkK(x j, xk)ϕ(n)k = f (x j), j = 1, . . . , n. (12.8)

Conversely, let ϕ(n)j , j = 1, . . . , n, be a solution of the system (12.8). Then the function

ϕn defined by

ϕn(x) := f (x) +n∑

k=1

αkK(x, xk)ϕ(n)k , x ∈ G, (12.9)

solves equation (12.7).

Proof. The first statement is trivial. For a solution ϕ(n)j , j = 1, . . . , n, of the system

(12.8), the function ϕn defined by (12.9) has values

ϕn(x j) = f (x j) +n∑

k=1

αkK(x j, xk)ϕ(n)k = ϕ

(n)j , j = 1, . . . , n.

Inserting this into (12.9), we see that ϕn satisfies (12.7). �

The formula (12.9) may be viewed as a natural interpolation of the values ϕ(n)j ,

j = 1, . . . , n, at the quadrature points to obtain the approximating function ϕn andgoes back to Nystrom [187].

The error analysis will be based on the following result.

Theorem 12.8. Assume the quadrature formulas (Qn) are convergent. Then the se-quence (An) is collectively compact and pointwise convergent (i.e., Anϕ → Aϕ,n→ ∞, for all ϕ ∈ C(G)), but not norm convergent.


Proof. Because the quadrature formulas (Qn) are assumed to be convergent, by The-orem 12.4, there exists a constant C such that the weights satisfy

n∑k=1

|α(n)k | ≤ C

for all n ∈ IN. Then we can estimate

‖Anϕ‖∞ ≤ C maxx,y∈G |K(x, y)| ‖ϕ‖∞ (12.10)

and|(Anϕ)(x1) − (Anϕ)(x2)| ≤ C max

y∈G|K(x1, y) − K(x2, y)| ‖ϕ‖∞ (12.11)

for all x1, x2 ∈ G. Now let U ⊂ C(G) be bounded. Then from (12.10) and (12.11)we see that {Anϕ : ϕ ∈ U, n ∈ IN} is bounded and equicontinuous because K isuniformly continuous on G ×G. Therefore, by the Arzela–Ascoli Theorem 1.18 thesequence (An) is collectively compact.

Since the quadrature is convergent, for fixed ϕ ∈ C(G) the sequence (Anϕ) ispointwise convergent, i.e., (Anϕ)(x) → (Aϕ)(x), n → ∞, for all x ∈ G. As a conse-quence of (12.11), the sequence (Anϕ) is equicontinuous. Hence it is uniformly con-vergent ‖Anϕ−Aϕ‖∞ → 0, n→ ∞, i.e., we have pointwise convergence Anϕ→ Aϕ,n→ ∞, for all ϕ ∈ C(G) (see Problem 12.1).

For ε > 0 choose a function ψε ∈ C(G) with 0 ≤ ψε(x) ≤ 1 for all x ∈ G suchthat ψε(x) = 1 for all x ∈ G with min j=1,...,n |x − x j| ≥ ε and ψε(x j) = 0, j = 1, . . . , n.Then

‖Aϕψε − Aϕ‖∞ ≤ maxx,y∈G|K(x, y)|

∫G{1 − ψε(y)} dy→ 0, ε→ 0,

for all ϕ ∈ C(G) with ‖ϕ‖∞ = 1. Using this result, we derive

‖A − An‖∞ = sup‖ϕ‖∞=1

‖(A − An)ϕ‖∞ ≥ sup‖ϕ‖∞=1

supε>0‖(A − An)ϕψε‖∞

= sup‖ϕ‖∞=1

supε>0‖Aϕψε‖∞ ≥ sup

‖ϕ‖∞=1‖Aϕ‖∞ = ‖A‖∞,

whence we see that the sequence (An) cannot be norm convergent. �

Corollary 12.9. For a uniquely solvable integral equation of the second kind witha continuous kernel and a continuous right-hand side, the Nystrom method with aconvergent sequence of quadrature formulas is uniformly convergent.


In principle, using Theorem 10.13, it is possible to derive computable errorbounds (see Problem 12.3). Because these, in general, will be too complicated toevaluate, as already mentioned in Chapter 11, in applications it will usually be

12.2 Nystrom’s Method 227

sufficient to estimate the error by extrapolation from the convergence order. For thediscussion of the error based on the estimate (10.11) of Corollary 10.14 we need thenorm ‖(A − An)ϕ‖∞. It can be expressed in terms of the error for the correspondingnumerical quadrature by

‖(A − An)ϕ‖∞ = maxx∈G

∣∣∣∣∣∣∣∫

GK(x, y)ϕ(y) dy −

n∑k=1

αkK(x, xk)ϕ(xk)

∣∣∣∣∣∣∣and requires a uniform estimate for the error of the quadrature applied to the inte-gration of K(x, ·)ϕ. Therefore, from the error estimate (10.11), it follows that un-der suitable regularity assumptions on the kernel K and the exact solution ϕ, theconvergence order of the underlying quadrature formulas carries over to the con-vergence order of the approximate solutions to the integral equation. We illustratethis through the case of the trapezoidal rule. Under the assumption ϕ ∈ C2[a, b] andK ∈ C2([a, b] × [a, b]), by Theorem 12.1, we can estimate

‖(A − An)ϕ‖∞ ≤ 112

h2(b − a) maxx,y∈G

∣∣∣∣∣∣∂2

∂y2K(x, y)ϕ(y)

∣∣∣∣∣∣ .

Table 12.1 Numerical solution of (11.30) by the trapezoidal rule

n x = 0 x = 0.25 x = 0.5 x = 0.75 x = 1

4 0.007146 0.008878 0.010816 0.013007 0.0154798 0.001788 0.002224 0.002711 0.003261 0.003882

16 0.000447 0.000556 0.000678 0.000816 0.00097132 0.000112 0.000139 0.000170 0.000204 0.000243

Example 12.10. We use the composite trapezoidal rule for approximately solvingthe integral equation (11.30) of Example 11.10. By the numerical results for thedifference between the approximate and exact solutions given in Table 12.1 the ex-pected convergence rate O(h2) is clearly demonstrated.

We now use the composite Simpson’s rule for the integral equation (11.30). Thenumerical results in Table 12.2 show the convergence order O(h4), which we expectfrom the error estimate (10.11) and Theorem 12.2. �

After comparing the last two examples, we wish to emphasize the major ad-vantage of Nystrom’s method compared with the degenerate kernel method ofChapter 11. The matrix and the right-hand side of the linear system (12.8) are ob-tained by just evaluating the kernel K and the given function f at the quadraturepoints. Therefore, without any further computational effort we can improve consid-erably on the approximations by choosing a more accurate numerical quadrature.For example, in the degenerate kernel method, it is much more involved to design


Table 12.2 Numerical solution of (11.30) by Simpson’s rule

n x = 0 x = 0.25 x = 0.5 x = 0.75 x = 1

4 0.00006652 0.00008311 0.00010905 0.00015046 0.000214168 0.00000422 0.00000527 0.00000692 0.00000956 0.00001366

16 0.00000026 0.00000033 0.00000043 0.00000060 0.00000086

a scheme with convergence order O(h4) rather than O(h2) from the linear splineinterpolation.

Example 12.11. For the integral equation (11.35) of Example 11.12 we use thetrapezoidal rule. Since we are dealing with periodic analytic functions, from Theo-rem 12.6 and the error estimate (12.2), we expect an exponentially decreasing errorbehavior that is exhibited by the numerical results in Table 12.3. Note that for peri-odic analytic functions the trapezoidal rule, in general, yields better approximationsthan Simpson’s rule. �

Table 12.3 Numerical solution of (11.35) by the trapezoidal rule

2n t = 0 t = π/2 t = π

4 −0.15350443 0.01354412 −0.00636277a = 1 8 −0.00281745 0.00009601 −0.00004247

b = 0.5 16 −0.00000044 0.00000001 −0.0000000132 0.00000000 0.00000000 0.00000000

4 −0.69224130 −0.06117951 −0.06216587a = 1 8 −0.15017166 −0.00971695 −0.01174302

b = 0.2 16 −0.00602633 −0.00036043 −0.0004549832 −0.00000919 −0.00000055 −0.00000069

Example 12.11 is quite typical for the efficiency of the Nystrom method withthe composite trapezoidal rule for solving two-dimensional boundary value prob-lems for the Laplace equation by using parameterized versions of the boundary in-tegral equations analogous to the integral equation of Problem 6.1 for the interiorDirichlet problem. For analytic boundary curves and boundary data or for boundarycurves of class Cm+2, in general, the trapezoidal rule yields an approximation orderof the form (12.2) or (12.3), respectively. Hence, this method is extremely efficient,with the main advantage being its simplicity and its high approximation order. Asa consequence of the latter, in general, the linear systems required for a sufficientlyaccurate approximation will be rather small, i.e., the number of quadrature pointswill be small. Therefore, despite the fact that they have full matrices, the linear

12.3 Weakly Singular Kernels 229

systems may be solved conveniently by elimination methods and do not requireadditional sophisticated efforts for its efficient solution.

For variants of the Nystrom method for the two-dimensional double-layer poten-tial integral equation for the Dirichlet problem in domains with corners, includingan error analysis that mimics the existence approach as described in Section 6.5, werefer to Kress [138] and Kress, Sloan, and Stenger [152] (see also Atkinson [11]).

We confine ourselves to the above examples for Nystrom’s method for equationsof the second kind with a continuous kernel. In particular, we refrain from presentingnumerical examples for the use of the Gauss trapezoidal product rule (12.4) for inte-gral equations of the second kind with continuous kernels on spheres. For a greatervariety, the reader is referred to Anderssen et al. [4], Atkinson [11], Baker [14],Delves and Mohamed [42], Golberg and Chen [65], Hackbusch [76], and Prossdorfand Silbermann [200].

12.3 Weakly Singular Kernels

We will now describe the application of Nystrom’s method for the approximatesolution of integral equations of the second kind with weakly singular kernels of theform

(Aϕ)(x) :=∫

Gw(|x − y|)K(x, y)ϕ(y) dy, x ∈ G ⊂ IRm. (12.12)

Here, we assume the weight function w : (0,∞) → IR to represent the weak sin-gularity, i.e., w is continuous and satisfies |w(t)| ≤ Mtα−m for all t > 0 and somepositive constants M and α. The remaining part K of the kernel is required to becontinuous. We choose a sequence (Qn) of quadrature rules

(Qng)(x) :=n∑

j=1

α(n)j (x)g(x(n)

j ), x ∈ G,

for the weighted integral

(Qg)(x) :=∫

Gw(|x − y|)g(y) dy, x ∈ G,

with quadrature weights depending continuously on x. Then we approximate theweakly singular integral operator by a sequence of numerical integration operators

(Anϕ)(x) :=n∑

k=1

α(n)k (x)K(x, x(n)

k )ϕ(x(n)k ), x ∈ G. (12.13)

Appropriately modified, Theorem 12.7 for the solution of the approximating equa-tion of the second kind, of course, remains valid. The linear system corresponding


to (12.8) now assumes the form

ϕ(n)j −

n∑k=1

α(n)k (x j)K(x j, xk)ϕ(n)

k = f (x j), j = 1, . . . , n. (12.14)

Due to the appearance of the weight function w in the quadrature Q, this form ofNystrom’s method for weakly singular kernels is also known as product integrationmethod.

For the error analysis we will assume that the sequence (Qn) of quadrature for-mulas converges, i.e., Qng → Qg, n → ∞, uniformly on G for all g ∈ C(G).Then applying the Banach–Steinhaus theorem to the sequence of linear operatorsQn : C(G) → C(G), we observe that for convergence of the sequence (Qn) it isnecessary and sufficient that Qng → Qg, n → ∞, uniformly on G for all g in somedense subset of C(G), and that there exists a constant C such that the weights satisfy

n∑k=1

|α(n)k (x)| ≤ C (12.15)

for all n ∈ IN and all x ∈ G. Since the weights are not constant, this bound alonewill not ensure collective compactness of the operator sequence (An). Therefore, wewill also assume that the weights satisfy

limy→x

supn∈IN

n∑k=1

|α(n)k (y) − α(n)

k (x)| = 0 (12.16)

uniformly for all x ∈ G. Then, writing

(Anϕ)(x1) − (Anϕ)(x2) =n∑

k=1

α(n)k (x1){K(x1, x(n)

k ) − K(x2, x(n)k )}ϕ(x(n)

k )

+

n∑k=1

{α(n)k (x1) − α(n)

k (x2)}K(x2, x(n)k )ϕ(x(n)

k )

we can estimate

|(Anϕ)(x1) − (Anϕ)(x2)| ≤ C maxy∈G|K(x1, y) − K(x2, y)| ‖ϕ‖∞

+maxx,y∈G

|K(x, y)| supn∈IN

n∑k=1

|α(n)k (x1) − α(n)

k (x2)| ‖ϕ‖∞.

Hence, analogous to Theorem 12.8, we can prove the following result.

Theorem 12.12. Assume that the quadrature formulas (Qn) converge and satisfy thecondition (12.16). Then the sequence (An), given by (12.13), is collectively compactand pointwise convergent Anϕ → Aϕ, n → ∞, for all ϕ ∈ C(G), but not normconvergent.


For a systematic study of approximations satisfying condition (12.16) we refer toSloan [222]. We confine ourselves to a special case by considering a weakly singularoperator with a logarithmic singularity

(Aϕ)(t) :=1

2π

∫ 2π

0ln

(4 sin2 t − τ

2

)K(t, τ)ϕ(τ) dτ, 0 ≤ t ≤ 2π, (12.17)

in the space C2π ⊂ C(IR) of 2π-periodic continuous functions. The kernel func-tion K is assumed to be continuous and 2π-periodic with respect to both variables.The fundamental solutions to elliptic partial differential equations, for example theLaplace or the Helmholtz equation, in two space dimensions contain a logarithmicsingularity. Therefore, the boundary integral equation approach to two-dimensionalboundary value problems via a periodic parameterization of the boundary curve, ingeneral, leads to such logarithmic singularities (see also Problem 7.2).

According to the general ideas outlined in Section 12.1 and following Kuss-maul [154] and Martensen [161], we construct numerical quadratures for the im-proper integral

(Qg)(t) :=1

2π

∫ 2π

0ln

(4 sin2 t − τ

2

)g(τ) dτ, t ∈ [0, 2π],

by replacing the continuous periodic function g by its trigonometric interpolationpolynomial described in Section 11.3. Using the Lagrange basis we obtain

(Qng)(t) =2n−1∑j=0

R(n)j (t)g(t j), t ∈ [0, 2π], (12.18)

with the equidistant quadrature points t j = jπ/n and the quadrature weights

R(n)j (t) =

12π

∫ 2π

0ln

(4 sin2 t − τ

2

)L j(τ) dτ, j = 0, . . . , 2n − 1. (12.19)

Using Lemma 8.23, from the form (11.12) of the Lagrange basis we derive

R(n)j (t) = −1

n

⎧⎪⎪⎨⎪⎪⎩n−1∑m=1

1m

cos m(t − t j) +1

2ncos n(t − t j)

⎫⎪⎪⎬⎪⎪⎭ (12.20)

for j = 0, . . . , 2n − 1.This quadrature is uniformly convergent for all trigonometric polynomials, since

by construction Qn integrates trigonometric polynomials of degree less than or equalto n exactly. We will now establish convergence for all 2π-periodic continuous func-tions and the validity of (12.16).


Using the integrals in Lemma 8.23 and Parseval’s equality it can be seen that foreach t ∈ [0, 2π] the function

ft(τ) :=1

2πln

(4 sin2 t − τ

2

), τ ∈ IR,

t − τ2π� ZZ,

belongs to L2[0, 2π] with

‖ ft‖22 =1π

∞∑m=1

1m2=π

6.

For fixed t ∈ [0, 2π] we choose a 2π-periodic continuous function g with ‖g‖∞ = 1satisfying g(t j) = 1 if R(n)

j (t) ≥ 0 and g(t j) = −1 if R(n)j (t) < 0. Then, in view of

(12.19), we can write

2n−1∑j=0

|R(n)j (t)| =

2n−1∑j=0

R(n)j (t)g(t j) =

12π

∫ 2π

0ln

(4 sin2 t − τ

2

)(Png)(τ) dτ.

From this, using Lemma 11.5 and the Cauchy–Schwarz inequality, we can estimate

2n−1∑j=0

|R(n)j (t)| ≤ ‖ ft‖2 ‖Png‖2 ≤

√3π ‖ ft‖2 = π√

2,

and this establishes the uniform boundedness (12.15) for the weights (12.20).Similarly,

2n−1∑j=0

|R(n)j (t1) − R(n)

j (t2)| ≤ √3π ‖ ft1 − ft2‖2,

and again from (8.23) and Parseval’s equality we see that

‖ ft1 − ft2‖22 =1

2π

∞∑m=−∞m�0

1m2

∣∣∣eimt1 − eimt2∣∣∣2 ,

whence the validity of condition (12.16) for the weights (12.20) follows. Hence, forthe sequence

(Anϕ)(t) :=2n−1∑j=0

R(n)j (t)K(t, t j)ϕ(t j) (12.21)

generated by the quadratures (12.18) we can state the following theorem.

Theorem 12.13. The sequence (An) given by (12.21) is collectively compact andpointwise convergent to the integral operator A with logarithmic singularity givenby (12.17).

Therefore our general convergence and error analysis based on Theorems 10.12and 10.13 is applicable. By Theorem 11.7, in the case of analytic functions the


quadrature error is exponentially decreasing. By the estimate (10.11), this behav-ior is inherited by the approximate solution to the integral equation provided thatthe kernel K and the exact solution ϕ are analytic (see Problem 12.4). The matrixentering into (12.14) is a circulant matrix, i.e.,

R(n)k (t j) = R(n)

k− j, j, k = 0, . . . , 2n − 1,

with the weights

R(n)j := −1

n

⎧⎪⎪⎨⎪⎪⎩n−1∑m=1

1m

cosm jπ

n+

(−1) j

2n

⎫⎪⎪⎬⎪⎪⎭ , j = 0,±1 . . . ,±(2n − 1). (12.22)

Example 12.14. We consider the Neumann boundary value problem for the reducedwave equation or Helmholtz equation

�u + κ2u = 0

with a positive wave number κ in the open unit disk D with given normal derivative

∂u∂ν= g on ∂D.

We assume the unit normal ν to be directed into the exterior of D. The fundamentalsolution to the two-dimensional Helmholtz equation is given by

Φ(x, y) :=i4

H(1)0 (κ|x − y|), x � y,

where H(1)0 denotes the Hankel function of the first kind and of order zero.

We decomposeH(1)

0 = J0 + i Y0,

where J0 and Y0 are the Bessel and Neumann functions of order zero, and note thepower series

J0(z) =∞∑

k=0

(−1)k

(k!)2

( z2

)2k

(12.23)

and

Y0(z) =2π

(ln

z2+C

)J0(z) − 2

π

∞∑k=1

ak(−1)k

(k!)2

( z2

)2k(12.24)

with ak =∑k

m=1 1/m and Euler’s constant C = 0.57721 . . . . From these expansionswe deduce the asymptotics

Φ(x, y) =1

2πln

1|x − y| +

i4− 1

2π

(ln

κ

2+ C

)+ O

(|x − y|2 ln

1|x − y|

)


for |x − y| → 0. Therefore, the fundamental solution to the Helmholtz equation hasthe same singular behavior as the fundamental solution of Laplace’s equation. As aconsequence, Green’s representation Theorem 6.5 and the potential theoretic jumprelations of Theorems 6.15 and 6.18 can be carried over to the Helmholtz equation.For details we refer to [31]. In particular, it can be shown that the unknown boundaryvalues ϕ = u on ∂D of the solution u satisfy the integral equation

ϕ(x) + 2∫∂Dϕ(y)

∂Φ(x, y)∂ν(y)

ds(y) = 2∫∂Dg(y)Φ(x, y) ds(y) (12.25)

for x ∈ ∂D. With the exception of a countable set of wave numbers κ accumulatingonly at infinity, for which the homogeneous Neumann problem admits nontrivialsolutions, this integral equation is uniquely solvable.

We use the representation z(t) = (cos t, sin t), 0 ≤ t ≤ 2π, for the unit circleand, by straightforward calculations, transform the integral equation (12.25) intothe parametric form

ϕ(t) − 12π

∫ 2π

0K(t, τ)ϕ(τ) dτ =

12π

∫ 2π

0L(t, τ)g(τ) dτ (12.26)

for 0 ≤ t ≤ 2π. Here we have set ϕ(t) := ϕ(z(t)) and g(t) := g(z(t)), and the kernelsare given by

K(t, τ) := iπκ∣∣∣∣∣sin

t − τ2

∣∣∣∣∣ H(1)1

(2κ

∣∣∣∣∣sint − τ

2

∣∣∣∣∣)

and

L(t, τ) := iπH(1)0

(2κ

∣∣∣∣∣sint − τ

2

∣∣∣∣∣)

for t � τ. For computing the normal derivative of the fundamental solution we usedthe relation

H(1) ′0 = −H(1)

1 = −J1 − i Y1,

where H(1)1 denotes the Hankel function of the first kind and of order one represented

in terms of the Bessel and Neumann functions J1 and Y1 of order one.Observing the expansions (12.23) and (12.24) and their term-by-term derivatives,

we can split the kernels into

K(t, τ) = K1(t, τ) ln(4 sin2 t − τ

2

)+ K2(t, τ) (12.27)

andL(t, τ) = L1(t, τ) ln

(4 sin2 t − τ

2

)+ L2(t, τ), (12.28)


whereK1(t, τ) := −κ sin

t − τ2

J1

(2κ sin

t − τ2

),

K2(t, τ) := K(t, τ) − K1(t, τ) ln(4 sin2 t − τ

2

),

L1(t, τ) := −J0

(2κ sin

t − τ2

),

L2(t, τ) := L(t, τ) − L1(t, τ) ln(4 sin2 t − τ

2

).

The kernels K1,K2, L1, and L2 turn out to be analytic. In particular, from

limz→0

z H(1)1 (z) =

2πi

and

limz→0

(H(1)

0 (z) − 2iπ

ln z J0(z)

)=

2iπ

(C − ln 2) + 1,

we deduce the diagonal terms

K2(t, t) = K(t, t) = 1

andL2(t, t) = −2 ln

κ

2− 2C + iπ

for 0 ≤ t ≤ 2π. Note that despite the continuity of the kernel K, for numericalaccuracy it is advantageous to incorporate the quadrature (12.18) because of thelogarithmic singularities of the derivatives of the kernel K. Now, in the spirit ofTheorem 12.7, we approximate the integral equation (12.26) by the linear system

ϕ j −2n−1∑k=0

{R(n)

k− jK1(t j, tk) +12n

K2(t j, tk)

}ϕk

=

2n−1∑k=0

{R(n)

k− jL1(t j, tk) +12n

L2(t j, tk)

}g(tk), j = 0, . . . , 2n − 1,

for approximating values ϕ j for ϕ(t j).For a numerical example we consider the case in which the exact solution to the

Neumann problem is given by u(x) = Y0(κ|x − x0|), where x0 = (q, 0) with q > 1.Then the Neumann boundary data are given by

g(t) = −κ 1 − q cos tv(t)

Y1(κv(t))


and the exact solution of the integral equation by ϕ(t) = Y0(κv(t)), where

v2(t) = 1 + q2 − 2q cos t.

The numerical results contained in Table 12.4 confirm the exponentially decreasingbehavior of the error we expect from our general error analysis. �


2n t = 0 t = π/2 t = π

4 −0.07907728 0.12761991 0.24102137κ = 1 8 −0.01306333 0.00867450 0.01155067q = 2 16 −0.00023494 0.00003924 0.00004517

32 −0.00000019 0.00000001 0.00000000

4 −0.13910590 0.39499045 0.68563472κ = 1 8 −0.07111636 0.07511294 0.10540051

q = 1.5 16 −0.00659502 0.00277753 0.0038607432 −0.00005924 0.00000558 0.00000616

8 0.35729406 0.21301358 −0.16596385κ = 5 16 0.01138634 0.00181974 0.00377600q = 2 32 −0.00000558 −0.00000006 −0.00000010

64 0.00000003 0.00000002 0.00000000

8 −0.54146680 −0.35298932 0.09706015κ = 5 16 −0.05669554 −0.02916764 −0.02480085

q = 1.5 32 −0.00021246 −0.00000877 −0.0000133264 −0.00000006 0.00000000 0.00000003

As at the end of Section 12.2, we wish to point out that Example 12.14 is typ-ical for the efficiency of the Nystrom method based on trigonometric interpolatoryquadratures for logarithmic singularities applied to boundary value problems forthe Helmholtz equation in two dimensions. For details we refer to [32] (see alsoProblem 12.5). There is no immediate extension of this approach to three dimen-sions since as opposed to the logarithmic singularity of the fundamental solutionsto the Laplace and Helmholtz equation in IR3 the square root singularities in thethree-dimensional case cannot be split off in the same elegant manner. However, itis possible to use the above technique in three dimensions for rotationally symmetricboundaries, see [137, 161].

12.4 Nystrom’s Method in Sobolev Spaces 237

12.4 Nystrom’s Method in Sobolev Spaces

In this final section we wish to illustrate that the Nystrom method can also be ana-lyzed in the Sobolev spaces Hp[0, 2π] and in Holder spaces. We confine our presen-tation to the weakly singular integral operator with logarithmic singularity definedby (12.17) and begin by investigating its mapping properties in Sobolev spaces.

Theorem 12.15. Assume that K is infinitely differentiable and 2π-periodic with re-spect to both variables. Then the operator A given by (12.17) is bounded fromHp[0, 2π] into Hp+1[0, 2π] for all p ≥ 0.

Proof. We write

K(t, τ) =∞∑

m=−∞Km(t)eimτ, (12.29)

where

Km(t) :=1

2π

∫ 2π

0K(t, τ)e−imτdτ

denote the Fourier coefficients of K(t, ·). The infinite differentiability of Kimplies that

supm∈ZZ|m|p‖K(k)

m ‖∞ < ∞ (12.30)

for all k ∈ IN∪{0} and all p ≥ 0 (see (8.6)). Therefore, the series (12.29) and its term-by-term derivatives of arbitrary order converge uniformly. Recall the trigonometricmonomials fm(t) = eimt. Then, for trigonometric polynomials ϕ we have

(Aϕ)(t) =∞∑

m=−∞Km(t) (A0(ϕ fm))(t),

where

(A0ϕ)(t) :=1

2π

∫ 2π

0ln

(4 sin2 t − τ

2

)ϕ(τ) dτ, 0 ≤ t ≤ 2π. (12.31)

By Theorem 8.24 the operator A0 is bounded from Hp[0, 2π] into Hp+1[0, 2π] for allp ∈ IR. Using Corollary 8.8 we can estimate

‖Aϕ‖p+1 ≤ C1

∞∑m=−∞

[‖K(k)

m ‖∞ + ‖Km‖∞]‖A0(ϕ fm)‖p+1,

where k is the smallest integer larger than or equal to p + 1 and C1 is some constantdepending on p. The boundedness of A0 and again Corollary 8.8 yield that

‖A0(ϕ fm)‖p+1 ≤ C2‖ϕ fm‖p ≤ C3|m|k‖ϕ‖p


for all m ∈ ZZ and some constants C2 and C3 depending on p. The last two inequali-ties, together with (12.30), imply that

‖Aϕ‖p+1 ≤ C‖ϕ‖pfor all trigonometric polynomials ϕ and some constant C depending on p. Now, inview of Theorem 8.2, the proof is complete. �

Corollary 12.16. Under the assumptions of Theorem 12.15, the integral operatorA : Hp[0, 2π]→ Hp[0, 2π] is compact for all p ≥ 0.

Proof. This follows from the imbedding Theorem 8.3. �

Corollary 12.16 allows the application of the Riesz Theorem 3.4. In view ofTheorem 8.4 and the smoothing property of A from Theorem 12.15, the inverseoperator (I − A)−1 : Hp[0, 2π]→ Hp[0, 2π] exists and is bounded if and only if thehomogeneous equation ϕ − Aϕ = 0 has only the trivial continuous solution ϕ = 0.

The operator A0 corresponds to the single-layer potential operator for the unitdisk. Therefore, by Theorem 7.41 the operator A0 is bounded from the Holder spaceC0,α

2π into C1,α2π , and proceeding as in the proof of Theorem 12.15 we can establish

the following result.

Theorem 12.17. Under the assumptions of Theorem 12.15 the operator A isbounded from C0,α

2π into C1,α2π for all 0 < α < 1.

Theorem 12.18. Assume that 0 ≤ q ≤ p and p > 1/2. Then under the assump-tions of Theorem 12.15 for the operator sequence (An) given by (12.21) we have theestimate

‖Anϕ − Aϕ‖q+1 ≤ Cnq−p‖ϕ‖p (12.32)

for all ϕ ∈ Hp[0, 2π] and some constant C depending on p and q.

Proof. As in the proof of Theorem 12.15 we can write

(Anϕ)(t) − (Aϕ)(t) =∞∑

m=−∞Km(t) (A0[Pn(ϕ fm) − ϕ fm])(t)

and estimate analogously. Using the boundedness of A0, Theorem 11.8, and Corol-lary 8.8, we obtain

‖A0[Pn(ϕ fm) − ϕ fm]‖q+1 ≤ c1‖Pn(ϕ fm) − ϕ fm‖q≤ c2nq−p‖ϕ fm‖p ≤ c3|m|p+1nq−p‖ϕ‖p

for all m ∈ ZZ and some constants c1, c2, and c3 depending on p and q. Now the proofcan be completed as in Theorem 12.15. �

Corollary 12.19. Under the assumptions of Theorem 12.15, for the operators An :Hp[0, 2π]→ Hp[0, 2π], we have norm convergence ‖An − A‖p → 0, n → ∞, for allp ≥ 1.

Problems 239

Proof. Setting q = p − 1 in (12.32) yields

‖Anϕ − Aϕ‖p ≤ Cn‖ϕ‖p

for all ϕ ∈ Hp[0, 2π] and all n ∈ IN. This establishes the assertion. �

Now we are in the position to apply Theorem 10.1 and the error estimate (10.3)for the approximate solution of ϕ−Aϕ = f by ϕn −Anϕn = fn in the Sobolev spacesHp[0, 2π] for p ≥ 1. We note that because of the imbedding Theorem 8.4, conver-gence of the approximate solutions in the Sobolev space Hp[0, 2π] for p > 1/2 alsoimplies uniform convergence. As opposed to the case of the convergence and erroranalysis in the space of continuous functions, due to the regularity properties in theSobolev spaces, here we do not rely on collective compactness. We leave it as anexercise to the reader to extend this analysis to the integral operator

(Aϕ)(t) =∫ 2π

0K(t, τ)ϕ(τ) dτ

with infinitely differentiable 2π-periodic kernel K and its approximations

(Anϕ)(t) =π

n

2n−1∑k=0

K(t, tk)ϕ(tk)

by the composite trapezoidal rule. We also leave it as an exercise to carry Theorem12.18 and its Corollary 12.19 over to Holder spaces as in Theorem 12.17 by usingthe error estimate (11.20).

For an error analysis of a Nystrom method for Cauchy-type singular integralequations in Sobolev spaces, which imitates the regularization of singular equationsconsidered in Chapter 7, we refer to Kirsch and Ritter [130].

Problems

12.1. Let X and Y be normed spaces, let U ⊂ X be compact, and let (ϕn) be an equicontinuoussequence of functions ϕn : U → Y converging pointwise on U to some function ϕ : U → Y. Thenthe sequence (ϕn) converges uniformly on U.

12.2. Let D = IR × (−s, s) denote a strip in the complex plane. Show that the space H(D) of2π-periodic holomorphic functions defined on D with the property

limσ→s

∫ 2π

0{| f (t + iσ)|2 + | f (t − iσ)|2} dt < ∞

is a Hilbert space with the scalar product

( f , g) := limσ→s

∫ 2π

0{ f (t + iσ)g(t + iσ) + f (t − iσ)g(t − iσ)} dt.


Show that the functions

fn(t) =1

(4π cosh 2ns)1/2eint, n ∈ ZZ,

form a complete orthonormal system in H(D). Use Cauchy’s integral formula to verify that theremainder for the trapezoidal rule is a bounded linear functional on H(D). Apply the Riesz repre-sentation Theorem 4.10 and Parseval’s equality from Theorem 1.28 to derive the error bound

|Rn(g)| ≤ 2M(e4ns − 1)1/2

which is slightly better than the estimate given in Theorem 12.6 (see [47] and [134]).

12.3. Derive bounds on ‖(An − A)A‖∞ and ‖(An − A)An‖∞ for the numerical integration operatorsusing the trapezoidal rule.

12.4. Let D = IR × (−s, s) denote a strip in the complex plane. Consider the Banach space B(D)of 2π-periodic continuous functions defined on D that are holomorphic in D, furnished with themaximum norm

‖ f ‖∞,D := maxt∈D| f (t)|.

Prove that under suitable assumptions on K the integral operator given by (12.17) is compactin B(D). Use this result to deduce that solutions to integral equations of the second kind withthe logarithmic kernel of (12.17) are analytic provided the kernel K and the right-hand side areanalytic.Hint: By the Arzela–Ascoli theorem, show that integral operators with analytic kernels are compactin B(D). Use

(Aϕ)(t) :=1

2π

∫ 2π

0ln

(4 sin2 s

2

)K(t, t + s)ϕ(t + s) ds

for the definition of A on B(D). Approximate ln sin2(s/2) by its Fourier series and use Theorem2.22 and the Cauchy–Schwarz inequality.

12.5. Use a regular 2π-periodic parameterization ∂D = {x(t) : 0 ≤ t ≤ 2π} with counterclock-wise orientation for an arbitrary boundary curve ∂D to derive a parametric form of the inte-gral equation (12.25) for the Neumann problem for the reduced wave equation corresponding to(12.26). Describe an appropriate treatment of the logarithmic singularity as in Example 12.14 (see[32, 135, 140, 154] for related integral equations).

Chapter 13Projection Methods

The application of the quadrature method, in principle, is confined to equations ofthe second kind. To develop numerical methods that can also be used for equationsof the first kind we will describe projection methods as a general tool for approxi-mately solving linear operator equations. After introducing into the principal ideasof projection methods and their convergence and error analysis we shall considercollocation and Galerkin methods as special cases. We do not intend to give a com-plete account of the numerous implementations of collocation and Galerkin methodsfor integral equations that have been developed in the literature. Our presentation ismeant as an introduction to these methods by studying their basic concepts anddescribing their numerical performance through a few typical examples.

For a more exhaustive study of projection methods in general and of collo-cation and Galerkin methods, we refer to Atkinson [11], Baker [14], Fenyo andStolle [52], Hackbusch [76], Krasnoselski et al. [132], Mikhlin and Prossdorf [170],Prossdorf and Silbermann [199, 200], Rjasanow and Steinbach [207], Saranen andVainikko [213], Sauter and Schwab [214] and Steinbach [225]. For an introductionto projection methods for boundary integral equations with an extensive bibliogra-phy, we refer to the review paper by Sloan [223].

13.1 The Projection Method

We describe the approximate solution of linear operator equations by projectingthem onto subspaces, which for practical calculations we assume to be finite-dimensional.

Definition 13.1. Let X be a normed space and U ⊂ X a nontrivial subspace.A bounded linear operator P : X → U with the property Pϕ = ϕ for all ϕ ∈ Uis called a projection operator from X onto U.


241

242 13 Projection Methods

Theorem 13.2. A nontrivial bounded linear operator P mapping a normed space Xinto itself is a projection operator if and only if P2 = P. Projection operators satisfy‖P‖ ≥ 1.

Proof. Let P : X → U be a projection operator. Then, from Pϕ ∈ U it follows thatP2ϕ = P(Pϕ) = Pϕ for all ϕ ∈ X. Conversely, let P2 = P and set U := P(X). Then,for all ϕ ∈ U we may write ϕ = Pψ for some ψ ∈ X and obtain Pϕ = ϕ. FinallyP = P2, by Remark 2.5, implies ‖P‖ ≤ ‖P‖2, whence ‖P‖ ≥ 1. �

An important example for projection operators is given by the so-called orthog-onal projection, i.e., by the best approximation in pre-Hilbert spaces in the sense ofTheorem 1.26.

Theorem 13.3. Let U be a nontrivial complete subspace of a pre-Hilbert space X.Then the operator P mapping each element ϕ ∈ X into its unique best approximationwith respect to U is a projection operator. It is called orthogonal projection onto Uand satisfies ‖P‖ = 1.

Proof. Trivially, we have Pϕ = ϕ for all ϕ ∈ U. From the orthogonality conditionof Theorem 1.25 for the best approximation in pre-Hilbert spaces we readily verifythat P is linear and

‖ϕ‖2 = ‖Pϕ + (ϕ − Pϕ)‖2 = ‖Pϕ‖2 + ‖ϕ − Pϕ‖2 ≥ ‖Pϕ‖2

for all ϕ ∈ X. Hence ‖P‖ ≤ 1, and Theorem 13.2 yields ‖P‖ = 1. �

A second important example for projection operators is given by interpolationoperators.

Theorem 13.4. The interpolation operator introduced in Theorem 11.2 is a projec-tion operator.

Note that for polynomial or trigonometric polynomial interpolation, because ofthe general non-convergence results due to Faber (see [182, 217]) combined withthe uniform boundedness principle Theorem 10.6, the interpolation operators arenot uniformly bounded with respect to the maximum norm (see also the estimate(11.15)).

Definition 13.5. Let X and Y be Banach spaces and let A : X → Y be an injectivebounded linear operator. Let Xn ⊂ X and Yn ⊂ Y be two sequences of subspaces withdim Xn = dim Yn = n and let Pn : Y → Yn be projection operators. Given f ∈ Y, theprojection method, generated by Xn and Pn, approximates the equation

Aϕ = f (13.1)

for ϕ ∈ X by the projected equation

PnAϕn = Pn f (13.2)

13.1 The Projection Method 243

for ϕn ∈ Xn. This projection method is called convergent for the operator A if thereexists n0 ∈ IN such that for each f ∈ A(X) the approximating equation (13.2) has aunique solution ϕn ∈ Xn for all n ≥ n0 and these solutions convergeϕn → ϕ, n→ ∞,to the unique solution ϕ of Aϕ = f .

In terms of operators, convergence of the projection method means that for alln ≥ n0 the finite-dimensional operators PnA : Xn → Yn are invertible and we havepointwise convergence (PnA)−1PnAϕ → ϕ, n → ∞, for all ϕ ∈ X. In general, wecan expect convergence only if the subspaces Xn possess the denseness property

infψ∈Xn

‖ψ − ϕ‖ → 0, n→ ∞, (13.3)

for all ϕ ∈ X. Therefore, in the subsequent analysis of this chapter we will alwaysassume that this condition is fulfilled.

Since PnA : Xn → Yn is a linear operator between two finite-dimensional spaces,carrying out the projection method reduces to solving a finite-dimensional linearsystem. In the following sections we shall describe the collocation and the Galerkinmethod as projection methods in the sense of this definition obtained via interpola-tion and orthogonal projection operators, respectively. Here, we first proceed with ageneral convergence and error analysis.

Theorem 13.6. A projection method with projection operators Pn : Y → Yn con-verges for an injective linear operator A : X → Y from a Banach space X into aBanach space Y if and only if there exist n0 ∈ IN and a positive constant M suchthat for all n ≥ n0 the finite-dimensional operators

PnA : Xn → Yn

are invertible and the operators (PnA)−1PnA : X → Xn are uniformly bounded, i.e.,

‖(PnA)−1PnA‖ ≤ M (13.4)

for all n ≥ n0 and some constant M. In case of convergence we have the errorestimate

‖ϕn − ϕ‖ ≤ (1 + M) infψ∈Xn

‖ψ − ϕ‖. (13.5)

Proof. Provided the projection method converges for the operator A, the uniformboundedness (13.4) is a consequence of Theorem 10.6. Conversely, if the assump-tions of the theorem are satisfied we can write

ϕn − ϕ = [(PnA)−1PnA − I]ϕ.

Since for all ψ ∈ Xn, trivially, we have (PnA)−1PnAψ = ψ, it follows that

ϕn − ϕ = [(PnA)−1PnA − I](ϕ − ψ).


Hence, we have the error estimate (13.5) and, with the aid of the denseness (13.3),the convergence follows. �

The uniform boundedness condition (13.4) is also known as the stability condi-tion for the projection method. The error estimate (13.5) of Theorem 13.6 is usuallyreferred to as Cea’s lemma (see [26]). It indicates that the error in the projectionmethod is determined by how well the exact solution can be approximated by ele-ments of the subspace Xn.

We now state the main stability property of the projection method with respectto perturbations of the operator.

Theorem 13.7. Assume that A : X → Y is a bijective bounded linear operator froma Banach space X into a Banach space Y and that the projection method generatedby the projection operators Pn : Y → Yn is convergent for A. Let B : X → Y be abounded linear operator such that either(a) supϕ∈Xn, ‖ϕ‖=1 ‖PnBϕ‖ is sufficiently small for all sufficiently large n, or(b) B is compact and A + B is injective.Then the projection method with the projection operators Pn also converges forA + B.

Proof. The operator A satisfies the conditions of Theorem 13.6, i.e., there existsn0 ∈ IN such that for all n ≥ n0 the operators PnA : Xn → Yn are invertible andsatisfy ‖(PnA)−1PnA‖ ≤ M for some constant M. By the open mapping Theorem10.8 the inverse operator A−1 : Y → X is bounded. We will show that for sufficientlylarge n the inverse operators of I + (PnA)−1PnB : Xn → Xn exist and are uniformlybounded if (a) or (b) is satisfied.

In case (a), assume that M ‖A−1‖ supϕ∈Xn, ‖ϕ‖=1 ‖PnBϕ‖ ≤ q for all n ≥ n0 withsome n0 ≥ n0 and some q < 1. Then writing

(PnA)−1PnB = (PnA)−1P2nB = (PnA)−1PnAA−1PnB

we can estimate ‖(PnA)−1PnB‖ ≤ q for all n ≥ n0. Therefore by Theorem 2.14, theinverse operators [I + (PnA)−1PnB]−1 : Xn → Xn exist and are uniformly boundedfor all n ≥ n0.

In case (b), by the Riesz theory I + A−1B : X → X has a bounded inverse, sinceA−1B is compact. Since A−1 is bounded, the pointwise convergence of the sequence(PnA)−1PnA → I, n → ∞, on X implies pointwise convergence (PnA)−1Pn → A−1,n → ∞, on Y. From the pointwise convergence of (PnA)−1Pn and the compactnessof B, by Theorem 10.10, we derive norm convergence

‖[I + A−1B] − [I + (PnA)−1PnB]‖ → 0, n→ ∞.Therefore, by Theorem 10.1, the inverse [I + (PnA)−1PnB]−1 : Xn → Xn exists andis uniformly bounded for all sufficiently large n.

We now set S := A + B. Then from

PnS = PnA[I + (PnA)−1PnB]

13.1 The Projection Method 245

it follows that PnS : Xn → Yn is invertible for sufficiently large n with the inversegiven by

(PnS )−1 = [I + (PnA)−1PnB]−1(PnA)−1.

From (PnS )−1PnS = [I + (PnA)−1PnB]−1(PnA)−1PnA(I + A−1B) we can estimate

‖(PnS )−1PnS ‖ ≤ ‖[I + (PnA)−1PnB]−1‖M ‖I + A−1B‖and observe that the condition (13.4) is satisfied for S . This completes the proof. �

Because, in general, projection methods are only semi-discrete methods, in actualnumerical calculations instead of (13.2) an approximate version of the form

PnAnϕn = Pn fn (13.6)

will usually be solved, where An is some approximation to A and fn approximatesf . For this we can state the following result.

Theorem 13.8. Assume that A : X → Y is a bijective bounded linear operatorfrom a Banach space X into a Banach space Y and that for the projection operatorsPn : Y → Yn and the approximating bounded linear operators An : X → Y point-wise convergence

PnAn − PnA→ 0, n→ ∞,and

supϕ∈Xn, ‖ϕ‖=1

‖PnAn − PnA‖ → 0, n→ ∞,

is satisfied. Then for sufficiently large n the approximate equation (13.6) is uniquelysolvable and we have an error estimate

‖ϕn − ϕ‖ ≤ C{ infψ∈Xn

‖ψ − ϕ‖ + ‖(PnAn − PnA)ϕ‖ + ‖Pn( fn − f )‖} (13.7)

for the solution ϕ of (13.1) and some constant C.

Proof. By setting B = An − A in Theorem 13.7 for the case (a) it follows that forsufficiently large n the inverse operators of PnAn : Xn → Yn exist and are uniformlybounded. With the aid of

ϕn − ϕn = [(PnAn)−1 − (PnA)−1]PnAϕ + (PnAn)−1Pn( fn − f )

= (PnAn)−1[PnA − PnAn](ϕn − ϕ)

+(PnAn)−1[PnA − PnAn]ϕ + (PnAn)−1Pn( fn − f )

the error estimate follows from Theorem 13.6 and the uniform boundedness princi-ple Theorem 10.6. �


13.2 Projection Methods for Equations of the Second Kind

For an equation of the second kind

ϕ − Aϕ = f (13.8)

with a bounded linear operator A : X → X we need only a sequence of subspacesXn ⊂ X and projection operators Pn : X → Xn. Then the projection method assumesthe form

ϕn − PnAϕn = Pn f . (13.9)

Note that each solution ϕn ∈ X of (13.9) automatically belongs to Xn. When A iscompact, from Theorem 13.7 we have the following convergence property.

Corollary 13.9. Let A : X → X be a compact linear operator in a Banach space X,I − A be injective, and the projection operators Pn : X → Xn converge pointwisePnϕ→ ϕ, n→ ∞, for all ϕ ∈ X. Then the projection method converges for I − A.

Proof. We apply the second case of Theorem 13.7. �

We wish to give a second proof of the latter convergence result based directlyon Theorem 10.1. This will also give us the opportunity to point out that the pro-jection method for equations of the second kind may converge without pointwiseconvergence of the projection operators on all of X.

Theorem 13.10. Let A : X → X be a compact linear operator in a Banach spaceX and I − A be injective. Assume that the projection operators Pn : X → Xn satisfy‖PnA − A‖ → 0, n → ∞. Then, for sufficiently large n, the approximate equation(13.9) is uniquely solvable for all f ∈ X and we have an error estimate

‖ϕn − ϕ‖ ≤ M‖Pnϕ − ϕ‖ (13.10)

for some positive constant M depending on A.

Proof. Theorems 3.4 and 10.1, applied to I − A and I − PnA, imply that for allsufficiently large n the inverse operators (I−PnA)−1 exist and are uniformly bounded.To verify the error bound, we apply the projection operator Pn to (13.8) and obtain

ϕ − PnAϕ = Pn f + ϕ − Pnϕ.

Subtracting this from (13.9) we find

(I − PnA)(ϕn − ϕ) = Pnϕ − ϕ,whence the estimate (13.10) follows. �

Note that for a compact operator A pointwise convergence Pnϕ → ϕ, n → ∞,for all ϕ ∈ X, by Theorem 10.10 implies that ‖PnA − A‖ → 0, n → ∞. But norm

13.2 Projection Methods for Equations of the Second Kind 247

convergence PnA−A→ 0, n→ ∞, may be satisfied without pointwise convergenceof the projection operator sequence (Pn) as we will see in our discussion of thecollocation method (see Theorems 13.15 and 13.16). Then, of course, from (13.10)we can assure convergence only if we have convergence Pnϕ → ϕ, n → ∞, for theexact solution ϕ.

Since projection methods are only semi-discrete, in actual numerical calculationsinstead of (13.9) an approximate version of the form

ϕn − PnAnϕn = Pn fn (13.11)

needs to be solved, where An is an approximation of A and fn approximates f . Thenwe can state the following result.

Corollary 13.11. Under the assumptions of Theorem 13.10 on the operator A andthe projection operators Pn assume that pointwise convergence



‖PnAn − PnA‖ → 0, n→ ∞,

is satisfied. Then, for sufficiently large n, the approximate equation (13.11) isuniquely solvable and we have an error estimate

‖ϕn − ϕ‖ ≤ M{‖Pnϕ − ϕ‖ + ‖Pn(An − A)ϕ‖ + ‖Pn( fn − f )‖} (13.12)

for some positive constant M.

Proof. We apply Theorem 10.1 to the operators I − PnA : Xn → Xn and I − PnAn :Xn → Xn to establish the existence and uniform boundedness of the inverse opera-tors (I − PnAn)−1 : Xn → Xn for sufficiently large n. From Theorem 10.1 we alsoobtain the error estimate

‖ϕn − ϕn‖ ≤ C {‖(PnAn − PnA)ϕn‖ + ‖Pn( fn − f )‖}for some constant C. Now (13.12) follows from (13.10) and the uniform bounded-ness principle Theorem 10.6. �

We conclude this section with a theorem that will enable us to derive convergenceresults for collocation and Galerkin methods for equations of the first kind. Considerthe equation

Sϕ − Aϕ = f , (13.13)

where S : X → Y is assumed to be a bijective bounded linear operator mapping aBanach space X into a Banach space Y and where A : X → Y is compact and S − Ais injective. Then, for the projection method

Pn(S − A)ϕn = Pn f (13.14)


with subspaces Xn and Yn and projection operators Pn : Y → Yn chosen as inDefinition 13.5 we have the following theorem.

Theorem 13.12. Assume that Yn = S (Xn) and ‖PnA − A‖ → 0, n → ∞. Then, forsufficiently large n, the approximate equation (13.14) is uniquely solvable and wehave an error estimate

‖ϕn − ϕ‖ ≤ M‖PnSϕ − Sϕ‖ (13.15)

for some positive constant M depending on S and A.

Proof. By Theorem 10.8 the inverse S −1 : Y → X is bounded. The equation (13.14)is equivalent to

S −1PnS (I − S −1A)ϕn = S −1PnS S −1 f ,

i.e.,Qn(I − S −1A)ϕn = QnS −1 f ,

where Qn := S −1PnS : X → Xn obviously is a projection operator. Using‖QnS −1A − S −1A‖ = ‖S −1(PnA − A)‖ and Qnϕ − ϕ = S −1(PnSϕ − Sϕ), theassertion follows from Theorem 13.10. �

The essence of Theorem 13.12 lies in the fact that the projection method isapplied directly to the given equation whereas the convergence result is based onthe regularized equation of the second kind.

In actual numerical calculations, instead of (13.14) an approximate version of theform

Pn(S − An)ϕn = Pn fn (13.16)

will usually be solved, where An and fn are approximations of A and f , respectively.Then, analogous to Corollary 13.11, we have the following result.

Corollary 13.13. Under the assumptions of Theorem 13.12 on the operators S andA assume that



‖PnAn − PnA‖ → 0, n→ ∞.

Then, for sufficiently large n, the approximate equation (13.16) is uniquely solvableand we have an error estimate

‖ϕn − ϕ‖ ≤ M{‖PnSϕ − Sϕ‖ + ‖Pn(An − A)ϕ‖ + ‖Pn( fn − f )‖} (13.17)

for some positive constant M.

13.3 The Collocation Method 249

13.3 The Collocation Method

The collocation method for approximately solving the equation

Aϕ = f , (13.18)

roughly speaking, consists of seeking an approximate solution from a finite-dimen-sional subspace by requiring that (13.18) is satisfied only at a finite number of so-called collocation points. To be more precise, let Y = C(G) and A : X → Y be abounded linear operator. Let Xn ⊂ X and Yn ⊂ Y denote a sequence of subspaceswith dim Xn = dim Yn = n. Choose n points x(n)

1 , . . . , x(n)n in G (we also will write

x1, . . . , xn instead of x(n)1 , . . . , x(n)

n ) such that the subspace Yn is unisolvent (see The-orem 11.2) with respect to these points. Then the collocation method approximatesthe solution of (13.18) by an element ϕn ∈ Xn satisfying

(Aϕn)(x j) = f (x j), j = 1, . . . , n. (13.19)

Let Xn = span{u1, . . . , un}. Then we express ϕn as a linear combination

ϕn =

n∑k=1

γkuk

and immediately see that (13.19) is equivalent to the linear system

n∑k=1

γk(Auk)(x j) = f (x j), j = 1, . . . , n, (13.20)

for the coefficients γ1, . . . , γn. The collocation method can be interpreted as a pro-jection method with the interpolation operator Pn : Y → Yn described in Theorem13.4. Indeed, because the interpolating function is uniquely determined by its valuesat the interpolation points, equation (13.19) is equivalent to

PnAϕn = Pn f .

Hence, our general error and convergence results for projection methods apply tothe collocation method.

Note that the collocation method can also be applied in function spaces otherthan the space C(G), for example in Holder spaces or Sobolev spaces.

For an equation of the second kind ϕ−Aϕ = f in X = C(G) with a bounded linearoperator A : X → X we need only one sequence of subspaces Xn ⊂ X and assume Xn

to be unisolvent with respect to the collocation points. Here, the equations (13.20)assume the form

n∑k=1

γk{uk(x j) − (Auk)(x j)} = f (x j), j = 1, . . . , n. (13.21)


We shall first consider the collocation method for integral equations of the secondkind

ϕ(x) −∫

GK(x, y)ϕ(y) dy = f (x), x ∈ G,

with a continuous or weakly singular kernel K. If we use the Lagrange basis for Xn

and write

ϕn =

n∑k=1

γkLk

then γ j = ϕn(x j), j = 1, . . . , n, and the system (13.21) becomes

γ j −n∑

k=1

γk

∫G

K(x j, y)Lk(y) dy = f (x j), j = 1, . . . , n. (13.22)

From the last equation we observe that the collocation method for equations of thesecond kind is closely related to the degenerate kernel method via interpolation.Obviously, the matrix of the system (13.22) coincides with the matrix (11.25) fromthe degenerate kernel approach. But the right-hand sides of both systems are differ-ent and the solutions γ1, . . . , γn have different meanings. However, if the right-handside f happens to belong to Xn, then f = Pn f and we may express the collocationsolution ϕn also in the form

ϕn = f +n∑

k=1

γkLk.

Now, for the coefficients γ1, . . . , γn we obtain the same system as (11.25), and inthis case the collocation method and the degenerate kernel method via interpolationwill yield the same numerical results.

Analogous to the degenerate kernel method the collocation method is only asemi-discrete method. In principle, the simplest approach to make the collocationmethod fully discrete is to use a quadrature operator An of the form (12.6) or (12.13)from the previous chapter with the collocation points as quadrature points to approx-imate A in the sense of Corollary 13.11. In this case, however, the solution ϕn of

ϕn − PnAnϕn = Pn f

will coincide at the collocation points with the solution ψn of the correspondingNystrom equation

ψn − Anψn = f ,

since both equations lead to the same linear system if we use the Lagrange basis forthe collocation method. In particular, we have the relation

ψn = f + Anϕn.


This follows from the fact that χn := f + Anϕn satisfies

χn − AnPnχn = f + Anϕn − AnPn( f + Anϕn) = f

and the observation that AnPnχ = Anχ for all χ, since the collocation and quadraturepoints coincide. Hence, this type of approximation can be analyzed and interpretedmore conveniently as a Nystrom method. Therefore, in the sequel for integral equa-tions of the second kind we will consider only fully discrete collocation methodswhich are based on numerical integration of the matrix elements.

In the literature, a broad variety of collocation methods exists corresponding tovarious choices for the subspaces Xn, for the basis functions u1, . . . , un, and for thecollocation points x1, . . . , xn. We briefly discuss two possibilities based on linearsplines and trigonometric interpolation. When we use linear splines (or more gen-eral spline interpolations) we have pointwise convergence of the corresponding in-terpolation operator as can be seen from the Banach–Steinhaus theorem by usingTheorem 11.3 and the norm (11.9) of the linear spline interpolation operator. There-fore, in this case, Corollary 13.9 applies, and we can state the following theorem.

Theorem 13.14. The collocation method with linear splines converges for uniquelysolvable integral equations of the second kind with continuous or weakly singularkernels.

Provided the exact solution of the integral equation is twice continuously differ-entiable, from Theorems 11.3 and 13.6 we derive an error estimate of the form

‖ϕn − ϕ‖∞ ≤ M‖ϕ′′‖∞h2

for the linear spline collocation approximate solution ϕn. Here, M denotes someconstant depending on the kernel K.

In most practical problems the evaluation of the matrix entries in (13.22) willrequire numerical integration. For twice continuously differentiable kernels, as in thecase of the degenerate kernel method, we suggest the quadrature rule (11.27). Then,according to our remarks concerning the connection between the equations (11.25)and (13.22), the collocation method will produce the same numerical results as thedegenerate kernel method. So we do not have to repeat the numerical calculationsfor equation (11.30) of Example 11.10 using the quadrature (11.27). However, to

Table 13.1 Collocation method for Example 11.10

n x = 0 x = 0.25 x = 0.5 x = 0.75 x = 1

4 0.003984 0.004428 0.004750 0.004978 0.0051358 0.000100 0.001112 0.001193 0.001250 0.001291

16 0.000250 0.000278 0.000298 0.000313 0.00032332 0.000063 0.000070 0.000075 0.000078 0.000081


illustrate that it does not pay off to evaluate the matrix elements more accurately,in Table 13.1 we give the numerical results for the error between the approximateand exact solution for the spline collocation for equation (11.30) using the exactmatrix entries. Comparing Tables 11.1 and 13.1, we observe that the improvementin the accuracy for the solution of the integral equation is only marginal. In general,instead of investing computing efforts in a highly accurate evaluation of the matrixelements it is more efficient to increase the number of collocation points, i.e., toincrease the size of the linear system.

We proceed discussing the collocation method based on trigonometric interpola-tion with equidistant knots t j = jπ/n, j = 0, .., 2n − 1.

Theorem 13.15. The collocation method with trigonometric polynomials convergesfor uniquely solvable integral equations of the second kind with 2π-periodic contin-uously differentiable kernels and right-hand sides.

Proof. The integral operator A with 2π-periodic continuously differentiable kernelK maps C2π into C1

2π, and we have

‖Aϕ‖∞ + ‖(Aϕ)′‖∞ ≤ 2π

{‖K‖∞ +

∥∥∥∥∥∂K∂t

∥∥∥∥∥∞}‖ϕ‖∞

for all ϕ ∈ C2π. From Theorems 8.4 and 11.8 we can conclude that

‖PnAϕ − Aϕ‖∞ ≤ c1‖PnAϕ − Aϕ‖Hq ≤ c2nq−1‖Aϕ‖H1

for all ϕ ∈ C2π, all 1/2 < q < 1, and some constants c1 and c2 depending on q.Combining these two inequalities and using Theorem 8.5 yields

‖PnA − A‖∞ ≤ C nq−1

for all 1/2 < q < 1 and some constant C depending on q. Now the assertion followsfrom Theorem 13.10. �

One possibility for the implementation of the collocation method is to use the

monomials fm(t) = eimt as basis functions. Then the integrals∫ 2π

0K(t j, τ)eikτdτ have

to be integrated numerically. Using fast Fourier transform techniques (see Elliott andRao [46], Nussbaumer [186] and Rao-et-al [202]) these quadratures can be carriedout very rapidly. A second even more efficient possibility is to use the Lagrange basisas already described in connection with the degenerate kernel method. This leads tothe quadrature rule (11.33) for the evaluation of the matrix elements in (13.22).Because of the simple structure of (11.17), the only computational effort besidesthe kernel evaluation is the computation of the row sums

∑2n−1m=0 (−1)mK(t j, tm) for

j = 0, . . .2n − 1. Of course, with this quadrature the collocation method again willcoincide with the degenerate kernel method via trigonometric interpolation.

We now proceed to illustrate the application of the collocation method for equa-tions of the second kind with weakly singular kernels. As in Sections 12.3 and 12.4


we consider the operator

(Aϕ)(t) :=1

2π

∫ 2π

0ln

(4 sin2 t − τ

2

)K(t, τ)ϕ(τ) dτ, 0 ≤ t ≤ 2π, (13.23)

where the kernel function K is assumed to be 2π-periodic and infinitely differen-tiable.

Theorem 13.16. The collocation method with trigonometric polynomials convergesfor uniquely solvable integral equations of the second kind with a logarithmic singu-larity of the form (13.23) provided the kernel function K is 2π-periodic and infinitelydifferentiable and the right-hand side is 2π-periodic and continuously differentiable.

Proof. From Theorems 8.4, 11.8, and 12.15 it follows that

‖PnAϕ − Aϕ‖∞ ≤ c1‖PnAϕ − Aϕ‖Hq ≤ c2nq−1‖Aϕ‖H1 ≤ c3nq−1‖ϕ‖H0 ≤ c4nq−1‖ϕ‖∞for all ϕ ∈ C[0, 2π], all 1/2 < q < 1, and some constants c1, c2, c3, and c4 dependingon q. Hence ‖PnA − A‖∞ ≤ c4 nq−1 for all 1/2 < q < 1 and the proof is completedusing Theorem 13.10. �

To evaluate the matrix elements, as usual, we replace K(t j, ·) by its trigonometricinterpolation. This yields the approximations

12π

∫ 2π

0ln

(4 sin2 t j − τ

2

)K(t j, τ)Lk(τ) dτ ≈

2n−1∑m=0

Sk− j,m− jK(t j, tm), (13.24)

where

Skm :=1

2π

∫ 2π

0ln

(4 sin2 τ

2

)Lk(τ)Lm(τ) dτ, k,m = 0, . . . , 2n − 1,

(and Skm = S−k,m = Sk,−m). A practical numerical evaluation of the weights in (13.24)rests on the observation that the quadrature rule (12.18) with the number of gridpoints 2n doubled to 4n integrates trigonometric polynomials of degree less than orequal to 2n exactly, provided the coefficient of the term sin 2nt vanishes. Thereforewe have

Skm =

4n−1∑j=0

R(2n)j Lk(t(2n)

j )Lm(t(2n)j ), k,m = 0, . . . , 2n − 1.

With the quadrature rule (13.24) for the evaluation of the matrix elements, thesystem (13.22) actually corresponds to solving the equation (13.11) with

(Anϕ)(t) :=1

2π

∫ 2π

0ln

(4 sin2 t − τ

2

) 2n−1∑m=0

K(t, tm)Lm(τ)ϕ(τ) dτ. 0 ≤ t ≤ 2π.


Since the kernel K is assumed infinitely differentiable, from Theorems 8.4 and 11.8,for the kernel function

Kn(t) :=2n−1∑m=0

K(t, tm)Lm(τ)

of An, we have ‖Kn − K‖∞ = O (n−p) for p ∈ IN, and consequently ‖An − A‖∞ =O (n−p). Therefore Theorem 11.4 implies convergence ‖Pn(An − A)‖∞ → 0, n→ ∞.Hence Corollary 13.11 and its error estimate (13.12) applies. In the case when theexact solution ϕ is infinitely differentiable, then from Theorems 8.4 and 11.8 and(13.12) we conclude that ‖ϕn −ϕ‖ = O(n−p) for each p ∈ IN. If both the kernel func-tion K and the exact solution ϕ are analytic, then from Theorem 11.7 and (13.12)we conclude that the convergence is exponential.


2n t = 0 t = π/2 t = π

4 −0.10828581 0.15071347 0.22404278κ = 1 8 −0.01368742 0.00827850 0.01129843q = 2 16 −0.00023817 0.00003702 0.00004366

32 −0.00000019 0.00000001 0.00000000

4 −0.19642139 0.43455120 0.66382871κ = 1 8 −0.07271273 0.07400151 0.10462955

q = 1.5 16 −0.00662395 0.00275639 0.0038456232 −0.00005937 0.00000548 0.00000609

8 0.18032618 0.02704260 −0.36058686κ = 5 16 0.00617287 −0.00852966 −0.00357971q = 2 32 −0.00000686 −0.00000015 −0.00000009

64 0.00000003 0.00000002 −0.00000000

8 −0.17339263 0.01532368 0.46622097κ = 5 16 −0.04130869 −0.01351387 −0.01226315

q = 1.5 32 −0.00022670 −0.00000972 −0.0000133564 −0.00000006 0.00000000 0.00000003

Example 13.17. We apply the collocation method based on trigonometric poly-nomials for the integral equation (12.26) of Example 12.14. We use the splitting(12.27) of the kernel in a logarithmic and a smooth part, and then evaluate the cor-responding matrix elements with the quadrature rules (13.24) and (11.33), respec-tively. The right-hand side of the system (13.22) is evaluated by numerical integra-tion, as in Example 12.14. The numerical results for the error between the approxi-mate and the exact solution in Table 13.2 show the exponential convergence, whichwe expect from our error analysis. �

13.4 Collocation Methods for Equations of the First Kind 255

In general, the implementation of the collocation method as described by our twoexamples can be used in all situations where the required numerical quadratures forthe matrix elements can be carried out in closed form for the chosen approximat-ing subspace and collocation points. In all these cases, of course, the quadratureformulas that are required for the related Nystrom method will also be available.Because the approximation order for both methods will usually be the same,Nystrom’s method is preferable; it requires the least computational effort for evalu-ating the matrix elements.

However, the situation changes in cases where no straightforward quadraturerules for the application of Nystrom’s method are available. This, in particular,occurs for the boundary integral equations described in Chapter 6 in the case ofthree space dimensions. Here, the collocation method is the most important numer-ical approximation method. Usually, the boundary surface is subdivided into a fi-nite number of segments, like curved triangles and squares. Then the approximationspace is chosen to consist of some kind of low-order polynomial splines with respectto these surface elements, which possess appropriate smoothness properties acrossthe boundary curves of the segments. Within each segment, depending on the degreeof freedom in the chosen splines, a number of collocation points are selected. Thenthe integrals for the matrix elements in the collocation system (13.22) are evalu-ated using numerical integration. Since the integral equations usually have a weaklysingular kernel, calculation of the improper integrals for the diagonal elements ofthe matrix, where the collocation points and the surface elements coincide, needsspecial attention. With these few remarks we have given a somewhat vague out-line of a very effective method known as boundary element method. For a detaileddescription and a review of the corresponding literature we refer the reader to Atkin-son [11], Brebbia, Telles, and Wrobel [20], Chen and Zhou [28], Hackbusch [76],Rjasanow and Steinbach [207], Sauter and Schwab [214] and Steinbach [225]. Forglobal numerical approximation methods for three-dimensional boundary integralequations using spherical harmonics for the approximating subspaces and quadra-ture formulas for single- and double-layer potentials analogous to the Gauss trape-zoidal product rule (12.4) for smooth integrands, we refer to Atkinson [11], Coltonand Kress [32] and Ganesh, Graham and Sloan [61, 68]

13.4 Collocation Methods for Equations of the First Kind

We continue our study of collocation methods by indicating its applications to inte-gral equations of the first kind, basing the convergence and error analysis on Theo-rem 13.12 and its Corollary 13.13.

Consider a singular integral equation of the first kind with Hilbert kernel

12π

∫ 2π

0

{cot

τ − t2+ K(t, τ)

}ϕ(τ) dτ = f (t), 0 ≤ t ≤ 2π, (13.25)


as discussed in Corollary 7.27. Because of the Cauchy-type singularity, this equa-tion, and correspondingly its error analysis, has to be treated in the Holder spaceC0,α

2π or in a Sobolev space Hp[0, 2π]. For convenience, the 2π-periodic kernel func-tion K is assumed to be infinitely differentiable with respect to both variables. Weexpress the leading singular part of (13.25) in the form

(Rϕ)(t) :=1

2π

∫ 2π

0

{cot

τ − t2+ 2

}ϕ(τ) dτ, 0 ≤ t ≤ 2π.

For the constant kernel function K = 2, equation (13.25) corresponds to a modifiedversion of the operator A in Example 7.14 from which we can conclude that thesingular integral operator R : C0,α

2π → C0,α2π is bounded and has a bounded inverse.

Analogously, from the integrals (8.35) it can be seen that R : Hp[0, 2π]→ Hp[0, 2π]is bounded and has a bounded inverse for all p ≥ 0.

Choose Xn to be the subspace of trigonometric polynomials of the form

ϕ(t) =n∑

j=0

α j cos jt +n−1∑j=1

β j sin jt, (13.26)

and note that Xn is unisolvent with respect to the equidistantly spaced grid t j = jπ/n,j = 0, . . . , 2n − 1 (see Section 11.3). From the integrals (8.35) it follows that Yn :=R(Xn) is given by the trigonometric polynomials of the form

ϕ(t) =n−1∑j=0

α j cos jt +n∑

j=1

β j sin jt. (13.27)

Observe that Yn is not unisolvent with respect to the t j, j = 0, . . . , 2n − 1, that,however, it is unisolvent with respect to the intersparsed set of points t j = t j + π/2n,j = 0, . . . , 2n − 1. Hence, we use the latter points as collocation points.

Denote by Pn : C0,α2π → Yn the trigonometric interpolation operator with respect

to the points t j, j = 0, . . . , 2n − 1. Then, from the error estimate (11.20), for theintegral operator A given by

(Aϕ)(t) :=1

2π

∫ 2π

0K(t, τ)ϕ(τ) dτ, 0 ≤ t ≤ 2π, (13.28)

we can conclude that

‖PnA − A‖0,α ≤ c1 max0≤τ≤2π

‖PnK(·, τ) − K(·, τ)‖0,α ≤ c2ln nnm

max0≤τ≤2π

‖K(·, τ)‖m,α

for all m ∈ IN and some constants c1 and c2 depending on m. This implies that‖PnA − A‖0,α → 0, n → ∞. Analogously, for the numerical quadrature operator An

given by

(Anϕ)(t) :=1

2n

2n−1∑k=0

K(t, tk)ϕ(tk), 0 ≤ t ≤ 2π, (13.29)


it can be seen that

‖PnAn − PnA‖0,α ≤ ‖Pn‖0,α ‖An − A‖0,α → 0, n→ ∞.Hence we can apply Theorem 13.12 and its Corollary 13.13 to obtain the followingresult.

Theorem 13.18. Both the semi-discrete collocation method (with trigonometricpolynomials of the form (13.26) and collocation points jπ/n+π/2n, j = 0, . . . , 2n−1)and the corresponding fully discrete method using the numerical quadrature oper-ator An converge in C0,α

2π for uniquely solvable integral equations of the first kind ofthe form (13.25) with Hilbert kernel provided K and f are infinitely differentiable.

From the error estimates (13.15) and (13.17) and Theorems 8.4 and 11.8 wecan deduce convergence of order O(n−m) for all m ∈ IN if the exact solution ϕ isinfinitely differentiable. Analogously, from Theorem 11.7 we conclude exponentialconvergence when the kernel function K and the exact solution ϕ are both analytic.

Note that here, as opposed to equations of the second kind, it makes sense toconsider a fully discrete version via quadrature operators, since for equations of thefirst kind there is no Nystrom method available. Of course, a fully discrete versionof the collocation method for (13.25) can also be based on numerical evaluation ofthe matrix elements as described in the previous section.

We leave it as an exercise for the reader to prove a variant of Theorem 13.18 inthe Sobolev spaces Hp[0, 2π] for p > 1/2.

For the numerical implementation we need the integrals

I jk :=1

2π

∫ 2π

0cot

τ − t j

2Lk(τ) dτ, j, k = 0, . . . , 2n − 1,

for the Lagrange basis (11.12). With the aid of the integrals (8.35) it can be seen that

I jk =12n{1 − cos n(tk − t j)} cot

tk − t j

2, j, k = 0, . . . , 2n − 1.

Then the fully discrete collocation method for the integral equation of the first kind(13.25) using the numerical quadrature operator An leads to the linear system

2n−1∑k=0

{I jk +

12n

K (t j, tk)

}γk = f (t j), j = 0, . . . , 2n − 1,

for the coefficients γ0, . . . , γ2n−1 in the representation ϕn =∑2n−1

k=0 γkLk of the ap-proximate solution.

Example 13.19. The integral equation (13.25) with kernel

K(t, τ) = 2 − (a2 − b2) sin(t + τ)a2 + b2 − (a2 − b2) cos(t + τ)


corresponds to the Cauchy singular integral equation RS0 = 2R f of Problem 7.4 foran ellipse with semi-axis a ≥ b > 0. Therefore, in particular, we have uniqueness.Table 13.3 gives some numerical results (for the difference between the approximateand the exact solution) in the case where the exact solution is given by

ϕ(t) = 1 − ecos t cos(sin t), 0 ≤ t ≤ 2π.

Then, setting c = (a − b)/(a + b), the right-hand side becomes

f (t) = ec cos t sin(c sin t) + ecos t sin(sin t), 0 ≤ t ≤ 2π,

as can be derived from (8.35) and Problem 11.4 �


2n t = 0 t = π/2 t = π

4 0.16740241 −0.00528236 0.00992820a = 1 8 0.00258801 −0.00022139 −0.00003859

b = 0.5 16 0.00000040 −0.00000004 −0.0000000132 0.00000000 0.00000000 0.00000000

4 0.26639845 −0.04603478 −0.00756302a = 1 8 0.06215996 −0.01726862 −0.00546249

b = 0.2 16 0.00267315 −0.00071327 −0.0002201032 0.00000409 −0.00000109 −0.00000034

Finally, we consider an integral equation of the first kind

S0ϕ − Aϕ = f (13.30)

where S0 and A have logarithmic singularities and are given by

(S0ϕ)(t) :=1

2π

∫ 2π

0

{ln

(4 sin2 t − τ

2

)− 2

}ϕ(τ) dτ (13.31)

and

(Aϕ)(t) :=1

2π

∫ 2π

0

{K(t, τ) ln

(4 sin2 t − τ

2

)+ L(t, τ)

}ϕ(τ) dτ (13.32)

for 0 ≤ t ≤ 2π. Here, we assume K and L to be infinitely differentiable and 2π-periodic with respect to both variables such that K(t, t) = 0 for all 0 ≤ t ≤ 2π. Forapplications of integral equations with such a logarithmic singularity we recall theremarks in Section 12.3 on two-dimensional boundary value problems.


By Theorem 8.24 the operator S0 : Hp[0, 2π]→ Hp+1[0, 2π] is bounded and hasa bounded inverse for all p ≥ 0. From the integrals in Lemma 8.23 we concludethat S0 maps the space of trigonometric polynomials of the form (13.26) into itself.Hence, here we can use the set t j = jπ/n, j = 0, . . . , 2n − 1, as collocation points.

Theorem 13.20. The semi-discrete collocation method (with trigonometric polyno-mials of the form (13.26) and collocation points jπ/n, j = 0, . . . , 2n − 1,) convergesfor uniquely solvable integral equations of the first kind of the form (13.30) withlogarithmic kernel in the Sobolev space Hp[0, 2π] for each p > 1/2 provided K, L,and f are infinitely differentiable (and K(t, t) = 0 for all t ∈ [0, 2π]).

Proof. For the derivative of Aϕ we can write

ddt

(Aϕ)(t) =1

2π

∫ 2π

0

{K(t, τ) ln

(4 sin2 t − τ

2

)+ L(t, τ)

}ϕ(τ) dτ,

where

K(t, τ) =∂K(t, τ)∂t

and

L(t, τ) =∂L(t, τ)∂t

+ K(t, τ) cott − τ

2

are infinitely differentiable as a consequence of the assumption K(t, t) = 0. There-fore, for integers p ≥ 0, from Theorem 12.15 we have that

∥∥∥∥∥ ddt

Aϕ∥∥∥∥∥

p+1≤ c1‖ϕ‖p

for all ϕ ∈ Hp[0, 2π] and some constant c1 depending on p. This implies that Ais bounded from Hp[0, 2π] into Hp+2[0, 2π] for all integers p ≥ 0. Then, by theinterpolation Theorem 8.13, it follows that A : Hp[0, 2π]→ Hp+2[0, 2π] is boundedfor arbitrary p ≥ 0.

Now, for p > 1/2, from Theorem 11.8 we conclude that

‖PnAϕ − Aϕ‖p+1 ≤ c2

n‖Aϕ‖p+2 ≤ c3

n‖ϕ‖p (13.33)

for all ϕ ∈ Hp[0, 2π] and some constants c2 and c3 depending on p. This impliesnorm convergence ‖PnA − A‖ → 0, n→ ∞, for the operator PnA − A : Hp[0, 2π]→Hp+1[0, 2π], and the assertion follows from Theorem 13.12. (Note that instead ofTheorem 13.12, we could have used the stability Theorem 13.7.) �

Finally, we need to describe a fully discrete variant. For this, as in Chapter 12(see 12.21), we use the quadrature operators defined by

(Anϕ)(t) :=2n−1∑k=0

{R(n)

k (t)K(t, tk) +1

2nL(t, tk)

}ϕ(tk), 0 ≤ t ≤ 2π. (13.34)


We also recall their construction via interpolatory quadratures

(Anϕ)(t) =1

2π

∫ 2π

0

{[Pn(K(t, ·)ϕ](τ) ln

(4 sin2 t − τ

2

)+ [Pn(L(t, ·)ϕ](τ)

}dτ.

In order to apply Corollary 13.13 we prove the following lemma, which improveson the estimates of Theorem 12.18.

Lemma 13.21. Assume that 0 ≤ q ≤ p and p > 1/2. Then, for the quadratureoperators An, we have the estimate

‖(PnAn − PnA)ϕ‖q+1 ≤ C nq−p−1‖ϕ‖p (13.35)

for all trigonometric polynomials ϕ of degree less than or equal to n and someconstant C depending on p and q.

Proof. For simplicity we only treat the case where L = 0. We denote by Tn the spaceof all trigonometric polynomials of degree less than or equal to n. Recalling theFourier series (12.29) from the proof of Theorem 12.15, we write

(PnAn − PnA)ϕ = Pn

∞∑m=−∞

KmA0[Pn( fmϕ) − fmϕ]

in terms of the bounded linear operator A0 : Hq[0, 2π] → Hq+1[0, 2π] given by(12.31) and the trigonometric monomials fm(t) = eimt. From this, using the propertyPn( fg) = Pn( f Png) of interpolation operators, we deduce that

(PnAn − PnA)ϕ = Pn

∞∑m=−∞

KmPnA0[Pn( fmϕ) − fmϕ].

By Theorem 11.8, the operators Pn : Hq+1[0, 2π] → Hq+1[0, 2π] are uniformlybounded. Therefore, using Corollary 8.8 and the property (12.30) of the Fouriercoefficients Km and denoting by k the smallest integer larger than or equal to q, wecan estimate

‖(PnAn − PnA)ϕ‖q+1 ≤ c1

∞∑m=−∞

1(1 + |m|)3+k+p−q

‖PnA0[Pn( fmϕ) − fmϕ]‖q+1

for all ϕ ∈ Tn and some constant c1.In the case where |m| ≥ n/2 we can estimate

‖PnA0[Pn( fmϕ) − fmϕ]‖q+1 ≤ c2‖ fmϕ‖q ≤ c3|m|k‖ϕ‖qfor all ϕ ∈ Tn and some constants c2 and c3. From this we conclude that

∑|m|≥n/2

1(1 + |m|)3+k+p−q

‖PnA0[Pn( fmϕ) − fmϕ]‖q+1 ≤ c4

np−q+1‖ϕ‖p (13.36)


for some constant c4. For 0 ≤ m ≤ n/2, writing

ϕ =2n∑j=0

ϕn− j fn− j

in view of Pn fm+n− j = Pn fm− j−n for j = 0, . . . ,m and Pn fm+n− j = fm+n− j for j =m + 1, . . . , 2n we deduce that

Pn( fmϕ) − fmϕ =m∑

j=0

ϕn− j[Pn fm− j−n − fm− j+n].

From this, with the aid of the integrals in Lemma 8.23, we find that

PnA0[Pn( fmϕ) − fmϕ] =m∑

j=0

ϕn− j

{1

n + m − j− 1

n − m + j

}Pn fm− j−n,

and, using the inequalities

1n − m + j

− 1n + m − j

≤ 4mn2

and (n + j − m)2 ≤ n2 ≤ 4(n − j)2

for 0 ≤ j ≤ m ≤ n/2, we can estimate

‖PnA0[Pn( fmϕ) − fmϕ]‖2q+1 ≤c5m2

n4

m∑j=0

[1 + (n − j)2]q+1|ϕn− j|2

≤ c6m2

n2p−2q+2

m∑j=0

[1 + (n − j)2]p|ϕn− j|2 ≤ c6m2

n2p−2q+2‖ϕ‖2p

for all ϕ ∈ Tn, all 0 ≤ m ≤ n/2, and some constants c5 and c6. From this and thecorresponding inequality for −n/2 ≤ m ≤ 0 we deduce that

∑|m|≤n/2

1(1 + |m|)3+k+p−q

‖PnA0[Pn( fmϕ) − fmϕ]‖q+1 ≤ c7

np−q+1‖ϕ‖p

for all ϕ ∈ Tn and some constant c7. From this, together with (13.36), the statementof the lemma follows. �

Theorem 13.22. The fully discrete collocation method using the quadrature oper-ators (13.34) converges for uniquely solvable integral equations of the first kind ofthe form (13.30) with logarithmic kernel in the Sobolev space Hp[0, 2π] for eachp > 1/2 provided K, L, and f are infinitely differentiable (and K(t, t) = 0 for allt ∈ [0, 2π]).

Proof. From Theorems 11.8 and 12.18 we conclude uniform boundedness of theoperators PnAn − PnA : Hp[0, 2π] → Hp+1[0, 2π] and convergence (PnAn −PnA)ϕ→ 0, n→ ∞, for all trigonometric polynomials ϕ. By the Banach–Steinhaus


theorem this implies pointwise convergence in Hp[0, 2π]. From Lemma 13.21, set-ting q = p, we have that

‖(PnAn − PnA)ϕ‖p+1 ≤ Cn‖ϕ‖p, (13.37)

for all trigonometric polynomials of the form (13.26). Now the statement followsfrom Corollary 13.13 in connection with Theorem 13.20. (We note that instead ofCorollary 13.13 we may use Theorem 13.8.) �

From the error estimate (13.17) and Theorems 8.4 and 11.8 we have convergenceof order O(n−m) for all m ∈ IN if the exact solution ϕ is infinitely differentiable. FromTheorem 11.7 we obtain exponential convergence when the kernel functions K andL and the exact solution ϕ are analytic.

The above convergence and error analysis for the fully discrete trigonometricpolynomial collocation for integral equations of the first kind with a logarithmicsingularity follows Saranen and Vainikko [212, 213]. It differs from an earlier ap-proach by Kress and Sloan [151] and Prossdorf and Saranen [198], because it doesnot require a further decomposition of the logarithmic singularity, which compli-cates the numerical implementation of the fully discrete scheme.

Since the quadrature operators (13.34) are exact for constant kernel functions Kand L and trigonometric polynomials ϕ ∈ Xn, the fully discrete collocation methodfor (13.30) leads to the linear system

2n−1∑k=0

{R(n)

k− j[1 − K(t j, tk)] − 12n

[L(t j, tk) + 2]

}γk = f (t j), j = 0, . . . , 2n − 1,

for the coefficients in ϕn =∑2n−1

k=0 γkLk (see (12.22)).

Example 13.23. The integral equation (13.30) with kernel K(t, τ) = 0 and

L(t, τ) = − ln{a2 + b2 − (a2 − b2) cos(t + τ)} − 3 (13.38)

essentially corresponds to the integral equation (7.59) for an ellipse with semi-axis0 < b ≤ a ≤ 1. Therefore, in particular, it is uniquely solvable. Table 13.4 givessome numerical results for the error between the approximate and the true solutionfor the case in which the exact solution is given by

ϕ(t) = ecos t cos(t + sin t), 0 ≤ t ≤ 2π.

With the aid of the integrals in Lemma 8.23 and a partial integration for the integralsin Problem 11.4 it can be seen that the right-hand side becomes

f (t) = 2 − ecos t cos(sin t) − ec cos t cos(c sin t), 0 ≤ t ≤ 2π,

with c = (a − b)/(a + b). �

13.5 A Collocation Method for Hypersingular Equations 263


2n t = 0 t = π/2 t = π

4 -0.62927694 -0.19858702 0.20711238a = 1 8 -0.02649085 0.00602690 0.01355414

b = 0.5 16 -0.00000724 0.00000110 0.0000045732 0.00000000 0.00000000 0.00000000

4 -0.77913195 -0.22694775 0.20990539a = 1 8 -0.08896065 0.02823001 0.03008962

b = 0.2 16 -0.00167596 0.00010607 0.0000022632 -0.00000114 0.00000009 -0.00000001

13.5 A Collocation Method for Hypersingular Equations

In this section we consider a collocation method for solving a certain type of hy-persingular integral equations such as equation (7.62) arising from the solutionof the Neumann boundary value problem for the Laplace equation via a double-layer potential approach. This method exploits the relation (7.46) between the nor-mal derivative of the double-layer potential and the single-layer potential and usestrigonometric differentiation. To this end we begin with a short description of thelatter. We denote by

D : g → g′

the differentiation operator and note that D : Hp[0, 2π] → Hp−1[0, 2π] is boundedfor all p with the nullspace given by the constant functions. From Section 11.3 werecall the trigonometric interpolation operator Pn and define Dn := DPn, i.e., weapproximate the derivative Dg of a 2π-periodic function g by the derivative Dngof the unique trigonometric polynomial Png of the form (13.26) that interpolates(Png)(t j) = g(t j) at the interpolation points t j = π j/n for j = 0, . . . , 2n− 1. From theLagrange basis (11.12), by straightforward differentiation we obtain that

(Dng)(t j) =2n−1∑k=0

d(n)k− jg(tk), j = 0, . . . , 2n − 1,

where

d(n)j =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

(−1) j

2cot

jπ2n

, j = ±1, . . . ,±(2n − 1),

0, j = 0.

(13.39)

From Theorem 11.8, we immediately have the error estimate

‖Dng − Dg‖q−1 ≤ Cnp−q

‖g‖p, 0 ≤ q ≤ p,12< p, (13.40)


for all g ∈ Hp[0, 2π] and some constant C depending on p and q as consequence ofthe boundedness of D : Hp[0, 2π]→ Hp−1[0, 2π].

We defineT0 := DS0D + M

where S0 is given by (13.31) and M is the mean value operator given by

M : g → 12π

∫ 2π

0g(t) dt.

In view of Theorem 8.24, the hypersingular operator T0 : Hp[0, 2π]→ Hp−1[0, 2π]is bounded with a bounded inverse for all p ∈ IR. Now we consider a hypersingularequation of the form

T0ϕ − DADϕ − Bϕ − λϕ = f (13.41)

where λ is a constant and A and B are of the form (13.32) with infinitely differen-tiable 2π-periodic kernels KA, LA and KB, LB respectively such that KA(t, t) = 0 forall 0 ≤ t ≤ 2π. From the integrals in Lemma 8.23 we see that T0 maps the spaceof trigonometric polynomials of the form (13.26) into itself. Hence, as for equation(13.30), we can use the set t j = jπ/n, j = 0, . . . , 2n − 1, as collocation points.

Analogous to Theorem 13.20 we now can state the following result.

Theorem 13.24. The semi-discrete collocation method (with trigonometric polyno-mials of the form (13.26) and collocation points jπ/n, j = 0, . . . , 2n − 1,) convergesfor uniquely solvable hypersingular integral equations of the form (13.41) in theSobolev space Hp[0, 2π] for p ≥ 1 provided KA, LA,KB, LB and f are infinitely dif-ferentiable (and KA(t, t) = 0 for all t ∈ [0, 2π]).

Proof. All constants occurring in this proof depend on p. For p ≥ 1, by Theorem11.8 and the boundedness of A : Hp−1[0, 2π] → Hp+1[0, 2π] as established in theproof of Theorem 13.20 we obtain that

‖DPnADϕ − DADϕ‖p−1 ≤ c1

n‖ϕ‖p (13.42)

for all ϕ ∈ Hp[0, 2π] and some constant c1. With the triangle inequality, for p ≥ 1again by Theorem 11.8 we can estimate

‖PnDψ − DPnψ‖p−1 ≤ ‖PnDψ − Dψ‖p−1 + ‖D(ψ − Pnψ)‖p−1 ≤ c2

n‖ψ‖p+1

for all ψ ∈ Hp+1[0, 2π] and some constant c2. From this, setting ψ = ADϕ, weobtain that

‖PnDADϕ − DPnADϕ‖p−1 ≤ c3

n‖ϕ‖p (13.43)

for all ϕ ∈ Hp[0, 2π] and some constant c3. Now we combine (13.42) and (13.43) toobtain

‖PnDADϕ − DADϕ‖p−1 ≤ c4

n‖ϕ‖p (13.44)

for all ϕ ∈ Hp[0, 2π] and some constant c4.

13.5 A Collocation Method for Hypersingular Equations 265

For the second part of the operator in (13.41), using Theorem 11.8 and the bound-edness of B : Hp[0, 2π]→ Hp+1[0, 2π], for p ≥ 1 we can estimate

‖PnBϕ − Bϕ + λ(Pnϕ − ϕ)‖p−1 ≤ c5

n‖ϕ‖p

for all ϕ ∈ Hp[0, 2π] and some constant c5. With this estimate and (13.44) the proofcan be completed as in Theorem 13.20. �

We conclude this section by describing a fully discrete variant via trigonometricdifferentiation. As in the previous Section 13.4, for the approximation of A and Bwe use the quadrature operators An and Bn given analogously to (13.34) and approx-imate DAD by DnAnDn. For all trigonometric polynomials ϕ of the form (13.26), inview of Dnϕ = Dϕ, we can transform

Pn(DnAnDn−DAD)ϕ = PnD(PnAn−A)Dϕ = PnDPn(An−A)Dϕ+PnD(PnA−A)Dϕ.

Note that the interpolation operators Pn : Hp[0, 2π] → Hp[0, 2π] are bounded forp > 1/2 as a consequence of Theorem 11.8. Now assume that p > 3/2. Then withthe aid of Lemma 13.21, replacing p by p − 1 and setting q = p − 1, as in the proofof Theorem 13.22, for the first term on the right-hand side we can estimate

‖PnDPn(An − A)Dϕ‖p−1 ≤ c1

n‖ϕ‖p

for all trigonometric polynomials ϕ of the form (13.26) and some constant c1.For the second term, from Theorem 11.8 and the boundedness of the operatorA : Hp−1[0, 2π]→ Hp+1[0, 2π], we conclude that

‖PnD(PnA − A)Dϕ‖p−1 ≤ c2

n‖ϕ‖p

for all ϕ ∈ Hp[0, 2π] and some constant c2. Combining both estimates we find that

‖Pn(DnAnDn − DAD)ϕ‖p−1 ≤ cn‖ϕ‖p (13.45)

for all trigonometric polynomials ϕ of the form (13.26) and some constant c. Using(13.45) and a corresponding estimate for the operator B that can readily be obtainedfrom Theorem 12.18 and proceeding as in the proof of Theorem 13.22 the followingresult on the convergence of the fully discrete method can be proven.

Theorem 13.25. The fully discrete collocation method using the approximate oper-ators DnAnDn and Bn converges for uniquely solvable hypersingular integral equa-tions of the form (13.41) in the Sobolev space Hp[0, 2π] for p > 3/2 provided thekernels and the right-hand side are infinitely differentiable (and KA(t, t) = 0 for allt ∈ [0, 2π]).

As in the case of Theorem 13.22 the convergence is of order O(n−m) for all m ∈ INif the exact solution ϕ is infinitely differentiable.


From the Lagrange basis (11.12) and the integrals in Lemma 8.23 we obtain that

(PnT0ϕ)(t j) =2n−1∑k=0

b(n)k− jϕ(tk), j = 0, . . . , 2n − 1,

for all trigonometric polynomials ϕ of degree n the form (13.26) where

b(n)j =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

12n+

14n

[(−1) j − 1] sin−2 jπ2n

, j = ±1, . . . ,±(2n − 1),

n2 + 12n

, j = 0.

(13.46)

With this the fully discrete collocation method for (13.41) leads to the linear systemfor the coefficients in ϕn =

∑2n−1k=0 γkLk of the form

2n−1∑k=0

(U jk − V jk

)γk = f (t j), j = 0, . . . , 2n − 1.

Here the matrix

U jk := b(n)k− j −

2n−1∑�=0

d(n)�− j

2n−1∑m=0

{R(n)

m−�KA(t�, tm) +1

2nLA(t�, tm)

}d(n)

k−m

corresponds to the operator T0 − A and the matrix

V jk := R(n)k− jKB(t j, tk)] +

12n

LB(t j, tk) + λδ jk

corresponds to the operator B + λI. Note that the two summations in the expressionfor U jk represent multiplications by Toeplitz matrices. i.e., convolutions that can beefficiently computed via fast Fourier transform techniques (see [46, 186, 202]).

Example 13.26. The hypersingular equation (13.41) with λ = 0 and kernels KA =

KB = LB = 0, and LA given by the right-hand side in (13.38) corresponds to thehypersingular integral equation (7.62) for the ellipse with semi-axis 0 < b ≤ a ≤ 1.The addition of the mean value term M ensures that it is uniquely solvable. Table13.5 gives some numerical results for the error between the approximate and exactsolution for the case in which the exact solution is given by

ϕ(t) = ecos t cos(sin t), 0 ≤ t ≤ 2π.

Then analogous to Example 13.23 the right-hand side becomes

f (t) = 1 + ecos t cos(t + sin t) − cec cos t cos(t + c sin t), 0 ≤ t ≤ 2π,

with c = (a − b)/(a + b). For further numerical examples we refer to [24, 146]. �

13.6 The Galerkin Method 267


2n t = 0 t = π/2 t = π

4 0.71791866 0.17589838 -0.57725146a = 1 8 0.01173558 -0.00389596 -0.00530693

b = 0.5 16 0.00000112 -0.00000018 -0.0000006332 0.00000000 0.00000000 0.00000000

4 2.39803796 0.31205324 -2.65316812a = 1 8 0.10282364 -0.04584838 -0.04338518

b = 0.2 16 0.00165318 -0.00038918 -0.0003304132 0.00000118 -0.00000028 -0.00000024

13.6 The Galerkin Method

For operator equations in Hilbert spaces the projection method via orthogonalprojection into finite-dimensional subspaces leads to the Galerkin method, namedafter the Russian engineer Galerkin [60]. Let X and Y be Hilbert spaces and letA : X → Y be an injective bounded linear operator. Let Xn ⊂ X and Yn ⊂ Y besubspaces with dim Xn = dim Yn = n, and let Pn : Y → Yn be the orthogonalprojection operator described in Theorem 13.3. Then ϕn ∈ Xn is a solution of theprojection method, generated by Xn and Pn, for the equation Aϕ = f if and only if

(Aϕn, g) = ( f , g), g ∈ Yn. (13.47)

This follows immediately from the fact that (13.47), by Theorem 1.25, is equivalentto Pn(Aϕn − f ) = 0. The equation (13.47) is called the Galerkin equation.

In the literature, the Galerkin method is also known as the Petrov–Galerkinmethod and the special case where X = Y and Xn = Yn is also called the Bubnov–Galerkin method. When the operator A is self-adjoint and positive definite, as wewill briefly explain, the Bubnov–Galerkin method coincides with the Rayleigh–Ritzmethod. A bounded linear operator A : X → X is called self-adjoint if A = A∗; it iscalled positive definite if it is self-adjoint and satisfies

(Aϕ, ϕ) > 0 (13.48)

for all ϕ ∈ X with ϕ � 0. A positive definite operator clearly is injective. We candefine an additional scalar product on X by

(ϕ, ψ)E := (Aϕ, ψ), (13.49)

with the corresponding norm ‖ · ‖E called the energy norm. Consider the so-calledenergy functional, defined on X by

E(ϕ) := (Aϕ, ϕ) − 2 Re( f , ϕ) (13.50)


for all ϕ ∈ X with f ∈ X a given element. For f ∈ A(X) we can transform

E(ϕ) − E(A−1 f ) = ‖ϕ − A−1 f ‖2E .Hence, solving the equation Aϕ = f is equivalent to minimizing the energy func-tional E over X. Based on the classical work of Rayleigh [203] from 1896 andRitz [206] from 1908, the Rayleigh–Ritz method consists of an approximation byminimizing E only over a finite-dimensional subspace Xn. This, obviously, is equiv-alent to finding a best approximation to the exact solution with respect to Xn inthe energy norm. By Theorems 1.25 and 1.26, the best approximation exists and isuniquely determined by the orthogonality condition

(ϕn, g)E = (A−1 f , g)E

for all g ∈ Xn. But this coincides with the Galerkin equation (13.47), i.e., theRayleigh–Ritz method is a special case of the Galerkin method. Note that we haveconvergence of the Rayleigh–Ritz method if the denseness condition (13.3) is satis-fied with respect to the energy norm.

First Bubnov in 1913 and then, in more details, Galerkin [60] in 1915 approachedand extended this approximation method without relying on a minimization formu-lation. Later, Petrov [188] first considered the general situation of the form (13.47)of the Galerkin method.

Assume that Xn = span{u1, . . . , un} and Yn = span{v1, . . . , vn}. Then we expressϕn as a linear combination ϕn =

∑nk=1 γkuk and find that solving the equation (13.47)

is equivalent to the linear system

n∑k=1

γk(Auk, v j) = ( f , v j), j = 1, . . . , n, (13.51)

for the coefficients γ1, . . . , γn. Usually, the scalar product will have the form of anintegral. Therefore, in the case of integral equations, the system (13.51) requires adouble integration for each of the matrix elements. This is a disadvantage of theGalerkin method, making it considerably inferior in numerical efficiency comparedwith the collocation method. Its major advantage is based on the simplicity of theorthogonal projection and the exploitation of the Hilbert space structure. In particu-lar, for integral equations of the second kind, Corollary 13.9 can always be applied,because the denseness assumption (13.3) implies pointwise convergence of the or-thogonal projection operators. Hence, we can state the following theorem.Theorem 13.27. Let A : X → X be a compact linear operator in a Hilbert space Xand assume that I − A is injective. Then the Bubnov–Galerkin method converges.

In contrast to our previous methods, the Galerkin method applies more easily toequations of the first kind. We proceed with a few general results.


Definition 13.28. A bounded linear operator A : X → X in a Hilbert space X iscalled strictly coercive if there exists a constant c > 0 such that

Re(Aϕ, ϕ) ≥ c‖ϕ‖2 (13.52)

for all ϕ ∈ X.

Theorem 13.29 ((Lax–Milgram). In a Hilbert space X a strictly coercive boundedlinear operator A : X → X has a bounded inverse A−1 : X → X.

Proof. Using the Cauchy–Schwarz inequality, we can estimate

‖Aϕ‖ ‖ϕ‖ ≥ Re(Aϕ, ϕ) ≥ c‖ϕ‖2.Hence

‖Aϕ‖ ≥ c‖ϕ‖ (13.53)

for all ϕ ∈ X. From (13.53) we observe that Aϕ = 0 implies ϕ = 0, i.e., A is injective.Next we show that the range A(X) is closed. Let ψ ∈ A(X) and let (ψn) be a

sequence from A(X) with ψn → ψ, n→ ∞. Then we can write ψn = Aϕn with someϕn ∈ X, and from (13.53) we find that

c‖ϕn − ϕm‖ ≤ ‖ψn − ψm‖for all n,m ∈ IN. Therefore, (ϕn) is a Cauchy sequence in X and consequently con-verges: ϕn → ϕ, n→ ∞, with some ϕ ∈ X. Then ψ = Aϕ, since A is continuous andA(X) = A(X) is proven.

Knowing now that A(X) is complete, denote by P : X → A(X) the orthogonalprojection operator. Let f ∈ X be arbitrary. Then by Theorem 1.25, we have thatP f − f ⊥ A(X). In particular, (P f − f , A(P f − f )) = 0. Hence, from (13.52) wesee that f = P f ∈ A(X). Therefore, A is surjective. Finally, the boundedness of theinverse ‖A−1‖ ≤ 1/c is a consequence of (13.53). �

Theorem 13.30. Let A : X → X be a strictly coercive operator. Then the Bubnov–Galerkin method converges.

Proof. As in the previous proof we can estimate

‖PnAϕ‖ ‖ϕ‖ ≥ Re(PnAϕ, ϕ) = Re(Aϕ, ϕ) ≥ c‖ϕ‖2

for all ϕ ∈ Xn, since orthogonal projections are self-adjoint (see Problem 13.1).Hence,

‖PnAϕ‖ ≥ c‖ϕ‖ (13.54)

for all ϕ ∈ Xn. This implies that An = PnA : Xn → Xn is injective and, consequently,surjective. For all ϕ ∈ X, using (13.54) and Theorem 13.3, we can estimate

c‖(PnA)−1PnAϕ‖ ≤ ‖PnA(PnA)−1PnAϕ‖ = ‖PnAϕ‖ ≤ ‖A‖ ‖ϕ‖.


Therefore, ‖A−1n PnA‖ ≤ ‖A‖/c for all n ∈ IN, and the statement follows from

Theorem 13.6. �

Theorems 13.29 and 13.30 can be extended to strictly coercive operators mappinga Hilbert space X into its dual space X∗ (see Problem 13.4).

We conclude our presentation of the Galerkin method by describing one morespecial case.

Theorem 13.31. Let X and Y be Hilbert spaces and let A : X → Y be an injectivebounded linear operator. Let Xn ⊂ X be a finite-dimensional subspace. Then foreach f ∈ Y there exists a unique element ϕn ∈ Xn such that

‖Aϕn − f ‖ = infψ∈Xn

‖Aψ − f ‖.

It is called the least squares solution of Aϕ = f with respect to Xn and it coincideswith the Petrov–Galerkin solution for the subspaces Xn and Yn := A(Xn).

Proof. Obviously, ϕn is a least squares solution of Aϕ = f with respect to Xn if andonly if Aϕn is a best approximation of f with respect to Yn. By Theorem 1.26, thebest approximation exists and is unique. The injectivity of A then implies unique-ness for the least squares solution. By Theorem 1.25, the best approximation Aϕn ischaracterized by the orthogonality condition (Aϕn − f , g) = 0 for all g ∈ Yn. But thisin turn is equivalent to the Galerkin equation (13.47) for the subspaces Xn and Yn. �

Note that the numerical implementation of the least squares solution in the caseof integral equations requires a triple integration for each matrix element and a dou-ble integration for each right-hand side of the linear system (13.51), since g j = Au j

requires an additional integration. Therefore, in general, it cannot compete in effi-ciency with other methods.

We now outline the use of the Galerkin method for integral equations. We startwith the Bubnov–Galerkin method for equations of the second kind

ϕ(x) −∫

GK(x, y)ϕ(y) dy = f (x), x ∈ G,

with continuous or weakly singular kernel K in the Hilbert space L2(G). LetXn = span{u1, . . . , un} be an n-dimensional subspace of L2(G). Then we writeϕ =

∑nk=1 γkuk and the Bubnov–Galerkin equations assume the form

n∑k=1

γk

{∫G

uk(x)u j(x) dx −∫

G

∫G

K(x, y)uk(y)u j(x) dxdy

}

=

∫G

f (x)u j(x) dx, j = 1, . . . , n,

(13.55)

for the coefficients γ1, . . . , γn. For this system, the first term in the expression for thematrix elements can usually be evaluated in closed form, whereas the second term


needs a double numerical integration. Before we proceed, we wish to point out thatthe matrix of the system (13.55) coincides with the system (11.36) obtained by thedegenerate kernel method via orthogonal expansion.

Without entering too deeply into the numerical implementation, we consider asan example the Galerkin method using linear splines. If we pursue the same ideaas in the collocation method and use interpolatory quadratures with respect to bothvariables, it turns out that we solve the same approximate finite-dimensional equa-tion on the space of spline functions as in the degenerate kernel and in the collo-cation method via spline interpolation. Therefore the numerical results will be thesame. Only the method of solution for this equation is different: In the collocationmethod we equate spline functions by requiring them to coincide at the interpolationgrid; in the Galerkin method we require equality of scalar products with respect toa basis. In general, we may say that most of the fully discrete implementations ofGalerkin methods may be interpreted as implementations of a related collocationmethod.

For integral equations for periodic functions using trigonometric polynomials wenote that instead of setting up the Bubnov–Galerkin method in L2[0, 2π] we may usethe Sobolev spaces Hp[0, 2π] with p ≥ 0. Since the trigonometric monomials areorthogonal in each of these Sobolev spaces, the orthogonal projection onto trigono-metric polynomials is always given through truncation of the Fourier series. There-fore, the Bubnov–Galerkin equations are the same for all p ≥ 0. In particular, thismeans that we automatically have convergence with respect to all Sobolev norms inthe case of infinitely differentiable kernels and right-hand sides.

In principle, for the Galerkin method for equations of the second kind the sameremarks as for the collocation method apply. As long as numerical quadratures areavailable, in general, the Galerkin method cannot compete in efficiency with theNystrom method. Compared with the collocation method, it is less efficient, sinceits matrix elements require double integrations. Therefore, in practical problems, thecollocation method is the most widely used projection method for solving integralequations of the second kind, despite the fact that its error analysis is often lesssatisfactory than the error analysis for the Galerkin method.

Finally, we give an indication of the Petrov–Galerkin method for equations ofthe form (13.25) and (13.30). From Section 13.4 we recall that for both equationsthe leading parts R : Hp[0, 2π] → Hp[0, 2π] and S0 : Hp[0, 2π] → Hp+1[0, 2π] areisomorphisms for each p ≥ 0.

For the first equation, as subspaces Xn and Yn of Hp[0, 2π], we choose trigono-metric polynomials of the form (13.26) and (13.27), respectively. Because R mapsXn bijectively onto Yn, the Petrov–Galerkin equation

PnRϕn = Pn f

has a unique solution ϕn ∈ Xn given by ϕn = R−1Pn f . Therefore, the Petrov–Galerkin method for R converges, and by the stability Theorem 13.7 it also con-verges for equation (13.25). The equation (13.30) is treated analogously, using as


subspaces Xn of Hp[0, 2π] and Yn of Hp+1[0, 2π] trigonometric polynomials of theform (13.26).

We summarize these observations in the following theorem.

Theorem 13.32. The Petrov–Galerkin method using trigonometric polynomials con-verges in Hp[0, 2π] for equations of the first kind with Hilbert or logarithmic kernels.

13.7 The Lippmann–Schwinger Equation

We will conclude this chapter by an example for an application of a projectionmethod to a volume integral equation and begin with presenting some basic analysison the scattering of time harmonic waves from an inhomogeneous medium. For amore comprehensive study of this field we refer the reader to Colton and Kress [32].

The propagation of time harmonic acoustic waves in three-dimensional space isgoverned by the Helmholtz equation for the inhomogeneous medium

Δu + κ2nu = 0 in IR3 (13.56)

where u describes the pressure p of the sound wave, depending on the space variablex and the time variable t, via p(x, t) = Re {u(x) e−iωt} with a positive frequencyω. Following standard notations from scattering theory, here by n we denote therefractive index of the inhomogeneous medium given by n(x) := c2

0/c2(x) in terms of

the constant speed of sound c0 in the background medium and the space dependentspeed of sound c in the inhomogeneity. The positive wave number κ is given by κ =ω/c0. We assume that the contrast a := 1−n has compact support and, for simplicity,is continuously differentiable in IR3. The function u is complex-valued and describesboth the amplitude and the phase of the time harmonic wave. It is decomposed intothe sum u = ui + us of the incident wave ui satisfying the free space Helmholtzequation Δui + κ2ui = 0 in IR3 and the scattered wave us. A typical example foran incident wave is a plane wave ui(x) = eiκ x·d with a unit vector d giving thepropagation direction. In order to ensure that the scattered wave is outgoing it hasto satisfy the Sommerfeld radiation condition

r

(∂us

∂r− iκus

)→ 0, r = |x| → ∞, (13.57)

uniformly for all directions.By

Φ(x, y) :=1

4πeiκ|x−y|

|x − y| , x � y, (13.58)

we now denote the fundamental solution to the Helmholtz equation Δu + κ2u = 0 inIR3. Using Green’s theorem and properties of volume potentials it can be shown thatthe scattering problem (13.56)–(13.57) is equivalent to solving the integral equation

13.7 The Lippmann–Schwinger Equation 273

of the second kind

u(x) = ui(x) − κ2∫

IR3Φ(x, y)a(y)u(y) dy, x ∈ IR3, (13.59)

for u which is known as the Lippmann–Schwinger equation. For a proof of the fol-lowing theorem we refer to [32].

Theorem 13.33. If u ∈ C2(IR3) is a solution of the scattering problem (13.56)–(13.57), then u solves the integral equation (13.59). Conversely, if u ∈ C(IR3) is asolution of (13.59) then u ∈ C2(IR3) and u solves (13.56)–(13.57).

We note that in (13.59) we can replace the region of integration by any domainD such that the support of a is contained in D and look for functions in C(D) thatsatisfy (13.59) for x ∈ D. Then for x ∈ IR3 \ D we define u(x) by the right-hand sideof (13.59) and obtain a continuous solution u to the Lippmann–Schwinger equationin all of IR3.

The integral operator in (13.59) has a weakly singular kernel and hence theRiesz theory can be applied to the integral equation in D, i.e., we need to showthat the homogeneous integral equation only allows the trivial solution. However,this is a nontrivial task since uniqueness of the solution to the scattering problem(13.56)–(13.57) is based on a unique continuation principle for solutions of (13.56).For a detailed analysis establishing existence and uniqueness of a solution to theLippmann–Schwinger equation we refer to [32]. Here, we confine ourselves to proveexistence of a unique solution for κ sufficiently small by the Neumann series.

Theorem 13.34. Suppose that the support of a is contained in a ball of radius Rcentered at the origin and that κ2R2‖a‖∞ < 2. Then there exists a unique solution tothe Lippmann–Schwinger equation (13.59).

Proof. It suffices to solve (13.59) for u ∈ C(B) with the ball B := {x ∈ IR3 : |x| ≤ R}.On the Banach space C(B), we define the operator V : C(B)→ C(B) by

(Vu)(x) := κ2∫

BΦ(x, y)a(y)u(y) dy, x ∈ B. (13.60)

From Theorem 2.13 and Problem 2.4 we conclude that

‖V‖∞ ≤ κ2‖a‖∞4π

maxx∈B

∫B

1|x − y| dy.

Geometric considerations show that the maximum of the integral on the right-handside is obtained for x = 0 leading to ‖V‖∞ ≤ κ2R2‖a‖∞/2. Now the proof is com-pleted by using the Neumann series Theorem 2.14. �

Following Hohage [98, 99] and Saranen and Vainikko [213], we now presentthe main ideas of a fast numerical solution method for the Lippmann–Schwingerequation that was proposed by Vainikko [238]. In view of the possibility to scale the


geometry, without loss of generality, we assume that the support of a is containedin a ball of radius R ≤ π/2 centered at the origin. The main idea of Vainikko’sapproach is to periodize the integral equation in order to make it accessible to fastFourier transform techniques. Before doing so it is convenient to transform (13.59)by multiplication with a equivalently into

ϕ + aVϕ = f (13.61)

for the unknown ϕ := au and the right-hand side f := aui. (Recall that it suffices tosolve (13.59) on the support of a.)

Obviously the solution of the Lippmann–Schwinger equation does not change ifwe modify its kernel outside a ball centered at the origin with radius larger than orequal to the diameter 2R of the support of m. We set B := B[0; π] and W := [−π, π]3

and define a new kernel function by

K(x) :=

⎧⎪⎪⎪⎨⎪⎪⎪⎩κ2Φ(x, 0), x ∈ B,

0, x ∈ W \ B.

Then we extend the functions K, f , a and ϕ from W to all of IR3 as 2π-periodicfunctions with respect to the three variables in x = (x1, x2, x3) and arrive at theperiodic integral equation

ϕ(x) + a(x)∫

WK(x − y)ϕ(y) dy = f (x), x ∈ W, (13.62)

that is equivalent to the Lippmann–Schwinger equation. We define an operator A :L2(W)→ L2(W) by

(Aϕ)(x) :=∫

WK(x − y)ϕ(y) dy, x ∈ W, (13.63)

and rewrite (13.62) in operator form as

ϕ + aAϕ = f (13.64)

with a interpreted as a multiplication operator.Analogous to Chapter 8, for 0 ≤ p < ∞ we introduce the Sobolev space Hp(W)

as the Hilbert space of all functions ϕ ∈ L2(W) with∑

m∈ZZ3

(1 + |m|2)p|ϕm|2 < ∞

for the Fourier coefficients

ϕm :=1

(2π)3

∫Wϕ(x)e−i m·xdx, m ∈ ZZ3,


of ϕ. The scalar product and the norm on Hp(W) are defined analogous to that onHp[0, 2π] in Theorem 8.2. We leave it as an exercise to formulate and prove theanalogues of Theorems 8.3, 8.4 (for p > 3/2) and 8.5 and their corollaries for thethree-dimensional case.

To establish the mapping properties of the above operator A in these Sobolevspaces we require the following lemma on the Fourier coefficients of the kernelfunction K.

Lemma 13.35. The Fourier coefficients Km of the kernel function K are given by

8π3

κ2(κ2 − |m|2)Km =

{cos(π|m|) − iκ

|m| sin(π|m|)}

eiκπ − 1

for m � 0 and8π3K0 = (1 − iκπ)eiκπ − 1.

Proof. With essentially the same proof as in Theorem 6.5, for a function u ∈ C2(B)the Green’s representation formula

u(x) =∫∂B

{∂u∂ν

(y)Φ(x, y) − u(y)∂Φ(x, y)∂ν(y)

}ds(y)

−∫

B

{Δu(y) + κ2u(y)

}Φ(x, y) dy, x ∈ B,

in terms of the fundamental solution (13.58) to the Helmholtz equation can be es-tablished (see also [32]). As usual ν denotes the unit normal to ∂B directed into theexterior of B. From this, inserting u = f−m with the trigonometric basis functions

fm(z) := ei m·z, z ∈ IR3, m ∈ ZZ3,

and setting x = 0 we obtain that

1κ2

(κ2 − |m|2)∫

BK f−m dy =

eiκπ

4π2

∫∂B

{∂ f−m

∂ν−

[iκ − 1

π

]f−m

}ds − 1.

Utilizing symmetry, elementary integrations yield

∫∂B

f−m ds =4π2

|m| sin(π|m|)

and ∫∂B

∂ f−m

∂νds = 4π2 cos(π|m|) − 4π

|m| sin(π|m|)

for m � 0 and ∫∂B

f0 ds = 4π3 and∫∂B

∂ f0∂ν

ds = 0.

Piecing the above together yields the assertion of the lemma. �


Theorem 13.36. For p ≥ 0, the operator A defined by (13.63) is a bounded linearoperator from Hp(W) into Hp+2(W).

Proof. Writing∫

WK(x − y)ei m·y dy = ei m·x

∫W

K(z)e−i m·z dz

we observe thatA fm = (2π)3Km fm, m ∈ ZZ3. (13.65)

Then from Lemma 13.35 we conclude that

|Km| ≤ c1 + |m|2

for all m ∈ ZZ and some positive constant c whence the statement of the theoremfollows. �

We now proceed to describe the approximate solution of (13.64) by a colloca-tion method using trigonometric polynomials. To this end we extend the trigono-metric interpolation as considered in Section 11.3 for 2π-periodic functions of onevariable to the three-dimensional case. Slightly deviating from the presentation inSection 11.3, where for the case of an even number 2n of interpolation points wewanted the interpolation of real-valued functions to remain real-valued, for n ∈ INwe define the n3-dimensional space of trigonometric polynomials

Tn := span{fm : m ∈ ZZ3

n

}

whereZZ3

n :={m = (m1,m2,m3) ∈ ZZ3 : −n

2≤ m1,m2,m3 <

n2

}.

The trigonometric interpolation operator Pn from the space of 2π-periodic continu-ous functions C2π into Tn is well defined by the interpolation property

(Png)(mh) = g(mh), m ∈ ZZ3n,

where h := 2π/n. Its Fourier coefficients (Png) j are given in terms of the nodalvalues of g by

(Png)m =1n3

∑k∈ZZ3

n

g(kh)e−ih k·m, (13.66)

i.e., by a discrete Fourier transform that can be implemented by fast Fouriertransform techniques using O(n3 log n) arithmetic operations (see [46, 186, 202]).Obviously, evaluating the nodal values (Png)(mh) of Png from the Fourier coeffi-cients (Png) j again amounts to performing a discrete Fourier transform.


Analogously to Theorem 11.8 the interpolation error can be estimated by

‖Png − g‖q ≤ Cnp−q

‖g‖p, 0 ≤ q ≤ p,32< p, (13.67)

for all g ∈ Hp(W) and some constant C depending on p and q (see [213]). As pointedout above, each function g ∈ Hp(W) is continuous if p > 3/2.

Now we are in the position to apply trigonometric collocation to the periodizedLippmann–Schwinger equation (13.64) and approximate its solution ϕ by solving

ϕn + PnaAϕn = Pn f (13.68)

for ϕn ∈ Tn. In view of Lemma 13.35 the approximating equation (13.68) is fullydiscrete since for ψ ∈ Tn from (13.65) we have that

(Aψ)( jh) = (2π)3∑

m∈ZZ3n

eih j·mψmKm, j ∈ ZZ3n, (13.69)

in terms of the Fourier coefficients ψm of ψ. Putting (13.66) and (13.69) together,we observe that solving (13.68) for the trigonometric polynomial ϕn is equivalent tosolving the linear system

ϕn( jh) + a( jh)(2π)3

n3

∑m∈ZZ3

n

∑k∈ZZ3

n

eih ( j−k)·mKmϕn(kh) = f ( jh), j ∈ ZZ3n, (13.70)

for the nodal values of ϕn. Because of the size of the linear system (13.70) that is re-quired for a reasonable accuracy of the approximation, iterative solution techniquesneed to be applied such as the two-grid iterations as described in the followingChapter 14. From the above it is obvious that in each iteration step the evaluationof PnaAϕn amounts to two consecutive discrete Fourier transforms that can be per-formed with O(n3 log n) arithmetic operations.

We conclude with the following convergence result.

Theorem 13.37. Assume that the contrast a is twice continuously differentiable.Then the collocation method (13.68) with trigonometric polynomials for the Lipp-mann–Schwinger equation (13.64) converges uniformly.

Proof. From the error estimate (13.67), the three-dimensional versions of Theorem8.4 and Corollary 8.8, and Theorem 13.36 it follows that

‖Pn(aAϕ) − aAϕ‖∞ ≤ c1‖Pn(aAϕ) − aAϕ‖Hq ≤ c2nq−2‖aAϕ‖H2

≤ c3nq−2‖Aϕ‖H2 ≤ c4nq−2‖ϕ‖H0 ≤ c5nq−2‖ϕ‖∞for all ϕ ∈ C2π, all 3/2 < q < 2, and some constants c1, c2, c3, c4, and c5 dependingon q. Hence ‖PnA − A‖∞ ≤ c5 nq−2 for all 3/2 < q < 2 and the proof is completedusing Theorem 13.10. �


Problems

13.1. Show that orthogonal projection operators are self-adjoint.

13.2. Let A : X → X be a bounded positive self-adjoint operator in a Hilbert space X. Chooseu0 ∈ X and define uj = Auj−1 for j = 1, . . . , n − 1. Show that the Bubnov–Galerkin equationsfor Aϕ = f with respect to the so-called Krylov subspaces Xn = span{u0, . . . , un−1} are uniquelysolvable for each n ∈ IN. Moreover, if f is in the closure of span{Aju0 : j = 0, 1, . . .}, then theBubnov–Galerkin approximation ϕn converges to the solution of Aϕ = f .

Show that in the special case u0 = f the approximations ϕn can be computed iteratively by theformulas ϕ0 = 0, p0 = f and

ϕn+1 = ϕn − αn pn,

pn = rn + βn−1 pn−1 ,

rn = rn−1 − αn−1Apn−1 ,

αn−1 = (rn−1 , pn−1)/(Apn−1 , pn−1),

βn−1 = −(rn , Apn−1)/(Apn−1 , pn−1).

Here rn is the residual rn = Aϕn − f . This is the conjugate gradient method of Hestenes andStiefel [92].

13.3. Under the assumptions of Theorem 13.27 let ϕn be the Bubnov–Galerkin solution. Considerthe iterated Bubnov–Galerkin solution ϕn, defined by

ϕn := Aϕn + f

and show that‖ϕn − ϕ‖ ≤ cn‖ϕn − ϕ‖

where cn → 0, n → ∞. This more rapid convergence of the iterated Bubnov–Galerkin solution iscalled superconvergence (see Sloan [221]).Hint: Show first that ϕn − APnϕn = f , then that ‖APn − A‖ → 0, n → ∞, and finally that ϕn − ϕ =(I − APn)−1(APn − A)(ϕ − Pnϕ).

13.4. A bounded linear operator A : X → X∗ mapping a Hilbert space X into its dual space X∗ iscalled strictly coercive if there exists a constant c > 0 such that

Re(Aϕ)(ϕ) ≥ c ‖ϕ‖2

for all ϕ ∈ X. Formulate and prove extensions of Theorems 13.29 and 13.30.

13.5. Apply the results of Problem 13.4 to the integral equation (13.30) in the Sobolev spaceH−1/2[0, 2π].

Chapter 14Iterative Solution and Stability

The approximation methods for integral equations described in Chapters 11–13 leadto full linear systems. Only if the number of unknowns is reasonably small may theseequations be solved by direct methods like Gaussian elimination. But, in general, asatisfying accuracy of the approximate solution to the integral equation will requirea comparatively large number of unknowns, in particular for integral equations inmore than one dimension. Therefore iterative methods for the resulting linear sys-tems will be preferable. For this, in principle, in the case of positive definite symmet-ric matrices the classical conjugate gradient method (see Problem 13.2) can be used.In the general case, when the matrix is not symmetric more general Krylov subspaceiterations may be used among which a method called generalized minimum residualmethod (GMRES) due to Saad and Schultz [211] is widely used. Since there is alarge literature on these and other general iteration methods for large linear systems(see Freud, Golub, and Nachtigal [56], Golub and van Loan [67], Greenbaum [69],Saad [210], and Trefethen and Bau [235], among others), we do not intend to presentthem in this book. At the end of this chapter we will only briefly describe the mainidea of the panel clustering methods and the fast multipole methods based on iter-ative methods and on a speed-up of matrix-vector multiplications for the matricesarising from the discretization of integral equations.

Instead of describing general iterative methods for linear systems, however, wewill discuss two-grid and multigrid iterations that are especially suitable for solvingthe linear systems arising from the numerical solution of integral equations of thesecond kind.

14.1 Stability of Linear Systems

We will start this chapter by briefly considering the question of stability of the linearsystems arising in the discretization of integral equations. Usually, in the course ofany numerical procedure for solving linear systems, errors will be introduced. Forexample, in the collocation and Galerkin method, errors occur through the numerical


279

280 14 Iterative Solution and Stability

approximations of the matrix elements and the right-hand side of the linear systems.A linear system is called stable if small changes in the data of the system cause onlysmall changes in the solution. We will make this notion more precise by introducingthe concept of a condition number.

Definition 14.1. Let X and Y be normed spaces and let A : X → Y be a boundedlinear operator with a bounded inverse A−1 : Y → X. Then

cond(A) := ‖A‖ ‖A−1‖is called the condition number of A.

Since 1 = ‖I‖ = ‖AA−1‖ ≤ ‖A‖ ‖A−1‖, we always have cond(A) ≥ 1. The follow-ing theorem shows that the condition number may serve as a measure for stability.

Theorem 14.2. Let X and Y be Banach spaces, let A : X → Y be a bijectivebounded linear operator, and let Aδ : X → Y be a bounded linear operator suchthat ‖A−1‖ ‖Aδ − A‖ < 1. Assume that ϕ and ϕδ are solutions of the equations

Aϕ = f (14.1)

andAδϕδ = f δ, (14.2)

respectively. Then

‖ϕδ − ϕ‖‖ϕ‖ ≤ cond(A)

1 − cond(A)‖Aδ − A‖‖A‖

{ ‖ f δ − f ‖‖ f ‖ +

‖Aδ − A‖‖A‖

}. (14.3)

Proof. Note that the inverse operator A−1 is bounded by the open mapping Theorem10.8. Writing Aδ = A[I+A−1(Aδ−A)], by Theorem 10.1 we observe that the inverseoperator (Aδ)−1 = [I + A−1(Aδ − A)]−1A−1 exists and is bounded by

‖(Aδ)−1‖ ≤ ‖A−1‖1 − ‖A−1‖ ‖Aδ − A‖ . (14.4)

From (14.1) and (14.2) we find that

Aδ(ϕδ − ϕ) = f δ − f − (Aδ − A)ϕ,

whenceϕδ − ϕ = (Aδ)−1{ f δ − f − (Aδ − A)ϕ}

follows. Now we can estimate

‖ϕδ − ϕ‖ ≤ ‖(Aδ)−1‖{‖ f δ − f ‖ + ‖Aδ − A‖ ‖ϕ‖

}

14.1 Stability of Linear Systems 281

and insert (14.4) to obtain

‖ϕδ − ϕ‖‖ϕ‖ ≤ cond(A)

1 − ‖A−1‖ ‖Aδ − A‖{ ‖ f δ − f ‖‖A‖ ‖ϕ‖ +

‖Aδ − A‖‖A‖

}.

From this the assertion follows with the aid of ‖A‖ ‖ϕ‖ ≥ ‖ f ‖. �

For finite-dimensional approximations of a given operator equation we have todistinguish three condition numbers: namely, the condition numbers of the originaloperator, of the approximating operator as mappings from the Banach space X intothe Banach space Y, and of the finite-dimensional system that has to be set up for theactual numerical solution. This latter system we can influence, for example, in thecollocation and Galerkin methods by the choice of the basis for the approximatingsubspaces.

Consider an equation of the second kind ϕ − Aϕ = f in a Banach space X andapproximating equations ϕn − Anϕn = fn under the assumptions of Theorem 10.1,i.e., norm convergence, or Theorem 10.12, i.e., collective compactness and point-wise convergence. Then from Theorems 10.1, 10.6, and 10.12 it follows that thecondition numbers cond(I −An) are uniformly bounded. Hence, for the condition ofthe approximating scheme, we mainly have to be concerned with the condition ofthe linear system for the actual computation of the solution of ϕn − Anϕn = fn.

For the discussion of the condition number for the Nystrom method we recall thenotations of Section 12.2. We have to relate the condition number for the numericalquadrature operator

(Anϕ)(x) :=n∑

k=1

αkK(x, xk)ϕ(xk), x ∈ G,

as operator An : C(G) → C(G) in the Banach space C(G) to the condition numberfor the matrix operator An : Cn → Cn given by

(AnΦ) j :=n∑

k=1

αkK(x j, xk)Φk, j = 1, . . . , n,

for Φ = (Φ1, . . . , Φn) ∈ Cn. We choose functions L j ∈ C(G) with the properties‖L j‖∞ = 1 and L j(x j) = 1 for j = 1, . . . , n, such that L j(xk) = 0 for j � k. Then weintroduce linear operators Rn : C(G)→ Cn and Mn : Cn → C(G) by

Rn f := ( f (x1), . . . , f (xn)), f ∈ C(G),

and

MnΦ :=n∑

j=1

Φ jL j, Φ = (Φ1, . . . , Φn) ∈ Cn,


and have ‖Rn‖∞ = ‖Mn‖∞ = 1 (see Problem 14.1). From Theorem 12.7 we concludethat

(I − An) = Rn(I − An)Mn

and(I − An)−1 = Rn(I − An)−1Mn.

From these relations we immediately obtain the following theorem.

Theorem 14.3. For the Nystrom method the condition numbers for the linear systemare uniformly bounded.

This theorem states that the Nystrom method essentially preserves the stabilityof the original integral equation.

For the collocation method we recall Section 13.3 and introduce operatorsEn, An : Cn → Cn by

(EnΦ) j :=n∑

k=1

uk(x j)Φk, j = 1, . . . , n,

and

(AnΦ) j :=n∑

k=1

(Auk)(x j)Φk, j = 1, . . . , n,

forΦ = (Φ1, . . . , Φn) ∈ Cn. Since Xn = span{u1, . . . , un} is assumed to be unisolvent,the operator En is invertible. In addition, let the operator Wn : Cn → C(G) bedefined by

Wnγ :=n∑

k=1

γkuk

for γ = (γ1, . . . , γn) and recall the operators Rn and Mn from above. Then we haveWn = PnMnEn. From (13.19) and (13.21) we can conclude that

(En − An) = RnPn(I − A)Wn

and(En − An)−1 = E−1

n Rn(I − PnA)−1PnMn.

From these three relations and the fact that by Theorems 10.6 and 13.10 the se-quence of operators (I − PnA)−1Pn is uniformly bounded, we obtain the followingtheorem.

Theorem 14.4. Under the assumptions of Theorem 13.10, for the collocationmethod the condition number of the linear system satisfies

cond(En − An) ≤ C‖Pn‖2∞ cond En


14.2 Two-Grid Methods 283

This theorem suggests that the basis functions must be chosen with caution. Fora poor choice, like monomials, the condition number of En can grow quite rapidly.However, for the Lagrange basis, En is the identity matrix with condition num-ber one. In addition, ‖Pn‖ enters in the estimate on the condition number of thelinear system, and, for example, for polynomial or trigonometric polynomial inter-polation we have ‖Pn‖ → ∞, n → ∞ (see the estimate (11.15)). A discussion ofthe condition number for the Galerkin method can be carried out quite similarly(see Problem 14.2).

Summarizing, we can state that for equations of the second kind the Nystrommethod is generically stable whereas the collocation and Galerkin methods maysuffer from instabilities due to a poor choice of the basis for the approximatingsubspaces.

14.2 Two-Grid Methods

Let X be a Banach space and let A : X → X be a bounded linear operator such thatI−A is bijective. As described in previous chapters we replace the operator equationof the second kind

ϕ − Aϕ = f (14.5)

by a sequence of approximating equations

ϕn − Anϕn = fn (14.6)

leading to finite-dimensional linear systems. We assume the sequence (An) ofbounded linear operators An : X → X to be either norm convergent or collectivelycompact and pointwise convergent such that either Theorem 10.1 or Theorem 10.12may be applied to yield existence and uniqueness of a solution to the approximatingequation (14.6) for all sufficiently large n. Our analysis includes projection methodsin the setting of Theorem 13.10.

We note that the index n indicates different levels of discretization, i.e., differentnumbers of quadrature points in the Nystrom method, or different dimensions of theapproximating subspaces in the degenerate kernel or projection methods. Referringto the quadrature and collocation methods, we will use the term grid instead of level.Deviating from the notation in preceding chapters, here the number of unknowns inthe linear system on the level n will be denoted by zn and will be different from n.Frequently, in particular for one-dimensional integral equations, the number of un-knowns zn on the level n will double the number of unknowns zn−1 on the precedinglevel n − 1 as in all our numerical tables in Chapters 11–13.

Assume that we already have an approximate solution ϕn,0 of (14.6) with aresidual

rn := fn − (I − An)ϕn,0.


Then we try to improve on the accuracy by writing

ϕn,1 = ϕn,0 + δn. (14.7)

For ϕn,1 to solve equation (14.6) the correction term δn has to satisfy the residualcorrection equation

δn − Anδn = rn.

We observe that the correction term δn, in general, will be small compared withϕn,0, and therefore it is unnecessary to solve the residual correction equation exactly.Hence we write

δn = Bnrn,

where the bounded linear operator Bn is some approximation for the inverse operator(I − An)−1 of I − An. Plugging this into (14.7) we obtain

ϕn,1 = [I − Bn(I − An)]ϕn,0 + Bn fn

as our new approximate solution to (14.6). Repeating this procedure yields the defectcorrection iteration defined by

ϕn,i+1 := [I − Bn(I − An)]ϕn,i + Bn fn, i = 0, 1, 2, . . . , (14.8)

for the solution of (14.6). By Theorem 2.15, the iteration (14.8) converges to theunique solution ψn of Bn(I − An)ψn = Bn fn provided

‖I − Bn(I − An)‖ < 1,

or by Theorem 10.16, if the spectral radius of I − Bn(I − An) is less than one. Sincethe unique solution ϕn of ϕn−Anϕn = fn trivially satisfies Bn(I−An)ϕn = Bn fn, if theiteration scheme (14.8) converges it converges to the unique solution of (14.6). Fora rapid convergence it is desirable that the norm or spectral radius be close to zero.For a more complete introduction into this residual correction principle, we refer toStetter [227].

Keeping in mind that the operators An approximate the operator A, an obviouschoice for an approximate inverse Bn is given by the correct inverse Bn = (I −Am)−1

of I − Am for some level m < n corresponding to a coarser discretization. For thisso-called two-grid method, we can simplify (14.8) using

I − (I − Am)−1(I − An) = (I − Am)−1(An − Am).

Following Brakhage [19] and Atkinson [10], we will consider the two special caseswhere we use either the preceding level m = n − 1 or the coarsest level m = 0.

Theorem 14.5. Assume that the sequence of operators An : X → X is either normconvergent ‖An−A‖ → 0, n→ ∞, or collectively compact and pointwise convergent

14.2 Two-Grid Methods 285

Anϕ→ Aϕ, n→ ∞, for all ϕ ∈ X. Then the two-grid iteration

ϕn,i+1 := (I − An−1)−1{(An − An−1)ϕn,i + fn}, i = 0, 1, 2, . . . , (14.9)

using two consecutive grids converges, provided n is sufficiently large. The two-griditeration

ϕn,i+1 := (I − A0)−1{(An − A0)ϕn,i + fn}, i = 0, 1, 2, . . . , (14.10)

using a fine and a coarse grid converges, provided the approximation A0 is alreadysufficiently close to A.

Proof. Let (An) be norm convergent. Then from the estimate (10.1) we observe that‖(I−An)−1‖ ≤ C for all sufficiently large n with some constant C. Then the statementon the scheme (14.9) follows from

∥∥∥(I − An−1)−1(An − An−1)∥∥∥ ≤ C‖An − An−1‖ → 0, n→ ∞. (14.11)

For the scheme (14.10) we assume that the coarsest grid is chosen such that

‖An − A0‖ < 1/2C

for all n ≥ 0. Then ‖(I − A0)−1(An − A0)‖ < 1 for all n ∈ IN. Note that∥∥∥(I − A0)−1(An − A0)

∥∥∥→ ∥∥∥(I − A0)−1(A − A0)∥∥∥ > 0, n→ ∞. (14.12)

Now assume that the sequence (An) is collectively compact and pointwise conver-gent. Then the limit operator A is also compact and, by Theorem 10.12, the inverseoperators (I − An)−1 exist and are uniformly bounded for sufficiently large n. Hencethe sequence (An) defined by

An := (I − An−1)−1(An − An−1)

is collectively compact. From the pointwise convergence Anϕ−An−1ϕ→ 0, n→ ∞,for all ϕ ∈ X, by Theorem 10.10, we conclude that

∥∥∥{(I − An−1)−1(An − An−1)}2∥∥∥→ 0, n→ ∞. (14.13)

Again by Theorem 10.10, we may choose the coarsest grid such that

∥∥∥(I − A0)−1(Am − A)(I − A0)−1(An − A0)∥∥∥ < 1

2

for all m, n ≥ 0. This implies∥∥∥∥{(I − A0)−1(An − A0)

}2∥∥∥∥ < 1 (14.14)


for all n ∈ IN. Now the statement follows from (14.13) and (14.14) and the variantof Theorem 2.15 discussed in Problem 2.2. �

Note that, in view of Theorem 12.8, for the Nystrom method we cannot expectthat ∥∥∥(I − An−1)−1(An − An−1)

∥∥∥→ 0, n→ ∞,instead of (14.13) or that

∥∥∥(I − A0)−1(An − A0)∥∥∥ < 1

instead of (14.14). Based on this observation, Brakhage [19] suggested a furthervariant of the defect correction iteration (see Problem 14.3).

Comparing the two variants described in Theorem 14.5, we observe from (14.11)to (14.14) that, in general, we have a more rapid convergence in the first case. Ac-tually the convergence rate for (14.9) tends to zero when n → ∞. But for a valuejudgment we also have to take into account the computational effort for each itera-tion step. For the actual computation we write the defect correction equation in theform

ϕn,i+1 − Amϕn,i+1 = (An − Am)ϕn,i + fn, (14.15)

indicating that, given ϕn,i, we have to solve a linear system for the unknown ϕn,i+1.From this we observe that for the evaluation of the right-hand side in (14.15) weneed to multiply a zn × zn matrix with a zn vector requiring, in general, O(z2

n) oper-ations. Then solving (14.15) for ϕn,i+1 means directly solving a linear system on themth level with zm unknowns and, in general, requires O(z3

m) operations. Therefore,for the second variant, working with the coarsest grid we may neglect the computa-tional effort for the solution of the linear system and remain with O(z2

n) operationsfor each iteration step. However, for the first variant, the effort for the solution of thelinear system dominates: Each iteration step needs O(z3

n−1) operations. In particular,when the discretization is set up such that the number of unknowns is doubled whenwe proceed from the level n − 1 to the next level n, then the computational effortfor each iteration step in (14.9) is roughly 1/8th of the effort for solving the linearsystem on the level n directly. Hence, we have rapid convergence but each iterationstep is still very costly.

We indicate the numerical implementation of the defect correction iteration forthe Nystrom method using the notations introduced in Chapter 12. Here the approx-imate operators are given by

(Anϕ)(x) =zn∑

k=1

α(n)k K(x, x(n)

k )ϕ(x(n)k ), x ∈ G.

Each iteration step first requires the evaluation of the right-hand side

gn,i := fn + (An − Am)ϕn,i

14.3 Multigrid Methods 287

of (14.15) at the zm quadrature points x(m)j , j = 1, . . . , zm, on the level m and at the zn

quadrature points x(n)j , j = 1, . . . , zn, on the level n. The corresponding computations

gn,i(x(m)j ) = fn(x(m)

j ) +zn∑

k=1

α(n)k K(x(m)

j , x(n)k )ϕn,i(x(n)

k )

−zm∑

k=1

α(m)k K(x(m)

j , x(m)k )ϕn,i(x(m)

k ), j = 1, . . . , zm,

and

gn,i(x(n)j ) = fn(x(n)

j ) +zn∑

k=1

α(n)k K(x(n)

j , x(n)k )ϕn,i(x(n)

k )

−zm∑

k=1

α(m)k K(x(n)

j , x(m)k )ϕn,i(x(m)

k ), j = 1, . . . , zn,

require O(z2n) operations. Then, according to Theorem 12.7, we have to solve the

linear system

ϕn,i+1(x(m)j ) −

zm∑k=1

α(m)k K(x(m)

j , x(m)k )ϕn,i+1(x(m)

k ) = gn,i(x(m)j )

for the values ϕn,i+1(x(m)j ) at the zm quadrature points x(m)

j , j = 1, . . . , zm. The direct

solution of this system needs O(z3m) operations. Finally, the values at the zn quadra-

ture points x(n)j , j = 1, . . . , zn, are obtained from the Nystrom interpolation

ϕn,i+1(x(n)j ) =

zm∑k=1

α(m)k K(x(n)

j , x(m)k )ϕn,i+1(x(m)

k ) + gn,i(x(n)j )

for j = 1, . . . , zn, requiring O(znzm) operations. All together, in agreement with ourprevious operation count, we need O(z2

n) +O(z3m) operations. It is left as an exercise

to set up the corresponding equations for the degenerate kernel and the collocationmethod (see Problem 14.4).

14.3 Multigrid Methods

Multigrid methods have been developed as a very efficient iteration scheme forsolving the sparse linear systems that arise from finite difference or finite ele-ment discretization of elliptic boundary value problems. Following Hackbusch [75]and Schippers [216] we now briefly sketch how multigrid methods can also be


applied for integral equations of the second kind. The two-grid methods described inTheorem 14.5 use only two levels; the multigrid methods use n + 1 levels.

Definition 14.6. The multigrid iteration is a defect correction iteration of the form(14.8) with the approximate inverses defined recursively by

B0 := (I − A0)−1,

Bn :=p−1∑m=0

[I − Bn−1(I − An−1)]mBn−1, n = 1, 2, . . . ,(14.16)

with some p ∈ IN.

Apparently, B0 stands for the exact solution on the coarsest grid. The approxi-mate inverse Bn for n+1 levels represents the application of p steps of the multigriditeration Bn−1 on n levels for approximately solving the residual correction equationstarting with initial element zero, since it is given by the pth partial sum of the Neu-mann series. This observation, simultaneously, is the motivation for the definitionof the multigrid method and the basis for its actual recursive numerical implemen-tation. In practical calculations p = 1, 2, or 3 are appropriate values for the iterationnumber.

It is our aim to illustrate that the multigrid iteration combines the advantagesof the two-grid methods of Theorem 14.5. It has the fast convergence rate of thescheme (14.9) and the computational effort for one step is essentially of the sameorder as for one step of the scheme (14.10). First we note that the approximateinverses satisfy

I − Bn(I − An−1) = [I − Bn−1(I − An−1)]p (14.17)

for all n ∈ IN. For the iteration operator Mn corresponding to one step of the multi-grid iteration using n + 1 levels, from Definition 14.6 and from (14.17), we deducethat

Mn = I − Bn(I − An) = I − Bn(I − An−1)(I − An−1)−1(I − An)

= I − (I−An−1)−1(I−An)+{I−Bn(I−An−1)}(I−An−1)−1(I−An)

= (I − An−1)−1(An − An−1) + Mpn−1(I − An−1)−1(I − An).

After introducing the two-grid iteration operator

Tn := (I − An−1)−1(An − An−1), (14.18)

we can write the recursion for the iteration operator Mn in the form

M1 = T1,

Mn+1 = Tn+1 + Mpn (I − Tn+1), n = 1, 2, . . . .

(14.19)


Theorem 14.7. Assume that‖Tn‖ ≤ qn−1C (14.20)

for all n ∈ IN and some constants q ∈ (0, 1] and C > 0 satisfying

C ≤ 12q

(√1 + q2 − 1

). (14.21)

Then, if p ≥ 2, we have

‖Mn‖ ≤ 2qn−1C < 1, n ∈ IN.

Proof. Note that√

1 + q2−1 < q for all q > 0. Therefore the statement is correct forn = 1. Assume that it is proven for some n ∈ IN. Then, using the recurrence relation(14.19), we can conclude that

‖Mn+1‖ ≤ ‖Tn+1‖ + ‖Mn‖p(1 + ‖Tn+1‖)

≤ qnC + (2qn−1C)2(1 + qnC)

≤ qnC

{1 +

4q

C(1 + qC)

}≤ 2qnC < 1,

since 4t(1 + qt) ≤ q for all 0 ≤ t ≤( √

1 + q2 − 1)/2q. �

In particular, setting q = 1, we obtain that

supn∈IN‖Tn‖ ≤ 1

2

(√2 − 1

)= 0.207 . . .

implies that ‖Mn‖ ≤√

2 − 1 < 1 for all n ∈ IN and all p ≥ 2. Roughly speaking,this ensures convergence of the multigrid method, provided the approximation onthe coarsest level is sufficiently accurate.

More generally, Theorem 14.7 implies that the convergence rate of the multigridmethod has the same behavior as the two-grid method based on the two finest grids.Consider as a typical case an approximation of order s in the number zn of gridpoints on the nth level

‖An − A‖ = O

(1zs

n

)

and assume a relationzn−1 = ζzn (14.22)

between the number of grid points on two consecutive levels with a constant ζ < 1.Then

‖An − An−1‖ = O(ζns).


Hence, in this case, (14.20) is satisfied with q = ζ s. In one dimension, typicallythe number of grid points on each level will be doubled, i.e., ζ = 1/2. In this casefor an approximation of second order s = 2 we have q = 1/4 and (14.21) requiresC ≤

(√17 − 4

)/2 = 0.061 . . . .

For a discussion of the computational complexity, let an denote the number ofoperations necessary for the matrix-vector multiplication involved in the applicationof An. In most cases we will have an = az2

n, where a is some constant. Denote bybn the number of operations for one step of the multigrid iteration with n + 1 levels.Then, from the recursive definition, we have

bn = an + pbn−1, (14.23)

since each step first requires the evaluation of the residual on the level n and thenperforms p steps of the multigrid iteration with n levels. Neglecting the effort forthe direct solutions on the coarsest grid, from (14.22) and (14.23) we obtain that

bn ≤ a1 − pζ2

z2n (14.24)

provided pζ2 < 1. In particular, for the canonical case where p = 2 and ζ = 1/2 wehave bn ≤ 2az2

n. In general, provided the number of grid points grows fast enoughin relation to the iteration number p, the computational complexity for the multigridmethod is of the same order as for the two-grid method using the finest and coarsestgrids.

Using iterative methods for the approximate solution of ϕn − Anϕn = fn, in gen-eral, it is useless to try and make the iteration error ‖ϕn,i − ϕn‖ smaller than thediscretization error ‖ϕn − ϕ‖ to the exact solution of ϕ − Aϕ = f . Therefore, thenumber of iterations should be chosen such that both errors are of the same magni-tude. Unfortunately, the quantitative size of the discretization error, in most practicalsituations, is not known beforehand, only its order of convergence is known. How-ever, the nested iteration, or full multigrid scheme, which we will describe briefly,constructs approximations with iteration errors roughly of the same size as the dis-cretization error. The basic idea is to provide a good initial element ϕn,0 for themultigrid iteration on n + 1 levels by multigrid iterations on coarser levels as de-scribed in the following definition.

Definition 14.8. Starting with ϕ0 := (I − A0)−1 f0, the full multigrid scheme con-structs a sequence (ϕn) of approximations by performing k steps of the multigriditeration on n + 1 levels using the preceding ϕn−1 as initial element.

First, we note that the computational complexity of the full multigrid method isstill of order O(z2

n). Using (14.23) and (14.24), the total number cn of operations upto n + 1 levels can be estimated by

cn = kn∑

m=1

bm ≤ ka

1 − pζ2

11 − ζ2

z2n.


Theorem 14.9. Assume that the discretization error satisfies

‖ϕn − ϕ‖ ≤ Cqn (14.25)

for some constants 0 < q < 1 and C > 0, and that t := supn∈IN ‖Mn‖ satisfies

tk < q. (14.26)

Then for the approximation ϕn obtained by the full multigrid method we have that

‖ϕn − ϕn‖ ≤ C(q + 1)tk

q − tkqn, n ∈ IN. (14.27)

Proof. We set

α :=(q + 1)tk

q − tk

and note that(α + 1 + q)tk = qα.

Trivially, (14.27) is true for n = 0. Assume that it has been proven for some n ≥ 0.Since ϕn+1 is obtained through k steps of the defect correction iteration (14.8) on thelevel n + 1 with ϕn as initial element, we can write

ϕn+1 = Mkn+1ϕn +

k−1∑m=0

Mmn+1Bn+1 fn+1.

From ϕn+1 − An+1ϕn+1 = fn+1 we deduce that

Bn+1 fn+1 = Bn+1(I − An+1)ϕn+1 = ϕn+1 − Mn+1ϕn+1

and can insert this into the previous equation to arrive at

ϕn+1 − ϕn+1 = Mkn+1(ϕn − ϕn+1).

Then we can estimate

‖ϕn+1 − ϕn+1‖ = ‖Mkn+1(ϕn − ϕn+1)‖ ≤ tk‖ϕn − ϕn+1‖

≤ tk (‖ϕn − ϕn‖ + ‖ϕn − ϕ‖ + ‖ϕ − ϕn+1‖)

≤ tkC(α + 1 + q)qn = αCqn+1,


Now, indeed, Theorem 14.9 implies that the iteration error, even for k = 1,is of the same size as the discretization error, provided the approximation on the


coarsest level is sufficiently accurate to ensure (14.26) by Theorem 14.7. In thetypical situation where s = 2 and ζ = 1/2, an estimate of the form (14.25) is satis-fied with q = 1/4.

In connection with the two-grid iterations of Theorem 14.5 we observed alreadythat for the Nystrom method we cannot expect ‖Tn‖ → 0, n → ∞. Therefore, inview of Theorem 14.7, modifications of the multigrid scheme are necessary for thequadrature method. The multigrid iteration proposed by Hemker and Schippers [90]replaces (14.16) by

B0 := (I − A0)−1

Bn :=p−1∑m=0

[I − Bn−1(I − An−1)]mBn−1(I − An−1 + An), n = 1, 2, . . . .

This means that at each level a smoothing operation I −An−1 +An is included beforethe application of p steps of the multigrid iteration at the preceding level. We write

Bn = Qn(I − An−1 + An)

and, as above, we have the property

I − Qn(I − An−1) = [I − Bn−1(I − An−1)]p.

Proceeding as in the derivation of (14.19), it can be shown that the iteration operatorssatisfy the recurrence relation M1 = T1 and

Mn+1 = Tn+1 + Mpn (I − Tn+1), n = 1, 2, . . . ,

where the two-grid iteration operator Tn is defined by

Tn := (I − An−1)−1(An − An−1)An, n = 1, 2, . . . .

Now Theorem 14.7 holds with Tn and Mn replaced by Tn and Mn. From Corollary10.11 we have that ‖Tn‖ → 0, n→ ∞.

For further variants and a more detailed study of multigrid methods for equa-tions of the second kind including numerical examples, we refer the reader to Hack-busch [75].

14.4 Fast Matrix-Vector Multiplication

If A is a compact operator then the eigenvalues of I − A are clustered at one (seeTheorem 3.9). Furthermore, from Theorems 14.3 and 14.4, we observe that the con-dition number of the linear systems arising from the Nystrom method (or the collo-cation method with piecewise linear interpolation) is independent of the size n of thematrix, i.e., of the degree of discretization. Therefore, using conjugate gradient-type

14.4 Fast Matrix-Vector Multiplication 293

methods like GMRES (or other Krylov subspace methods) for the iterative solutionof these systems, we can expect that the number of iterations required for a certainaccuracy is independent of n (see [70]). Hence, the computational complexity of theiterative solution of integral equations of the second kind by these iteration methodsis determined by the amount of work required for the computation of matrix-vectorproducts. Note that these and subsequent considerations also apply to the matrix-vector multiplications occurring in the two-grid and multigrid methods of the twoprevious sections.

Since the discretization of integral equations leads to dense matrices, eachmatrix-vector multiplication, if it is performed in the naive way, requires O(n2) op-erations. Therefore, even solving the linear systems by iteration methods such as theconjugate gradient method or GMRES might become too expensive for realistic ap-plied problems. A remedy is to try and perform the matrix-vector multiplication onlyapproximately within some tolerance. Indeed, in many cases this concept allows areduction of the computational complexity to almost linear growth of order O(n).Examples of this type are the panel clustering methods suggested by Hackbuschand Nowak [77] and the closely related fast multipole methods of Rokhlin [208] andGreengard and Rokhlin [71]. Another approach has been initiated through Beylkin,Coifman, and Rokhlin [18] by using a wavelet basis for spline spaces in the Galerkinmethod for integral equations of the second kind. In the latter method, the main ideais to replace the exact Galerkin matrix through a matrix that is nearly sparse by set-ting all entries below a certain threshold equal to zero. For a survey on these waveletmethods we refer to Dahmen [38]. Since developing the main ideas of wavelets isbeyond the aim of this introduction, we will confine ourselves to explaining the ba-sic principle of the panel clustering and fast multipole methods by considering asimple model problem of a one-dimensional integral equation.

For one-dimensional integral equations, in general, the discrete linear system canbe solved efficiently by either elimination methods or the two-grid and multigridmethods with traditional matrix-vector multiplication. Nevertheless, the followingexample is suitable for explaining the basic ideas of panel clustering and fast multi-pole methods. This is due to the fact that in higher dimensions there are more tech-nical details to consider, which distract from the basic principles. However, thesebasic principles do not depend on the dimension of the integral equation.

Denote by Γ the unit circle in IR2 and consider an integral equation of the secondkind

ϕ(x) −∫Γ

K(x, y)ϕ(y) ds(y) = f (x), x ∈ Γ, (14.28)

with sufficiently smooth kernel K and right-hand side f . Using the composite trape-zoidal rule in the Nystrom method the integral equation (14.28) is approximated bythe linear system

ϕ j − 2πn

n∑k=1

K(x j, xk)ϕk = f (x j), j = 1, . . . , n, (14.29)


with the equidistant discretization points x j = (cos(2 jπ/n), sin(2 jπ/n)), j = 1, . . . , n.Assume that for each z ∈ Γ we have degenerate kernels

KM(x, y; z) =M∑

m=1

am(x; z)bm(y; z) (14.30)

available such that|K(x, y) − KM(x, y; z)| ≤ c

aM(14.31)

for |y−z| ≥ 2|x−z| and some constants c > 0 and a > 1. Such degenerate kernels canbe obtained, for example, by Taylor series expansions. In the fast multipole methodfor the kernel of the logarithmic double-layer potential operator the approximatingkernels are obtained by taking the real part of Taylor expansions of the complexlogarithm in C. Now the main idea of both the panel clustering and the fast multipolemethods consists in approximating the exact kernel in (14.29) by the degeneratekernels (14.30) for xk away from x j. As a result, large submatrices of the n × nmatrix K(x j, xk) are well approximated by matrices of low rank. Multiplying an n×nmatrix of rank M with a vector requires only O(Mn) operations as opposed to O(n2)for a matrix of full rank. This observation leads to a scheme for the approximateevaluation of the matrix-vector multiplication in (14.29) with the computational costreduced to order O(Mn log n).

To explain this scheme we assume that n = 2p with an integer p ≥ 3. Forq = 1, . . . , p − 1 and � = 1, 2, . . . , 2q+1 we introduce subsets Aq

�, Bq

�⊂ Γ by set-

ting

Aq�

:=

{(cosϕ, sinϕ) :

∣∣∣∣∣ϕ − �π2q

∣∣∣∣∣ < π

2q

}and Bq

�:= Γ \ Aq

�.

Then we have that Aq2� ⊂ Aq−1

� and Aq2�−1 ⊂ Aq−1

� and also that Bq2� ⊃ Bq−1

� and

Bq2�−1 ⊃ Bq−1

�for q = 2, . . . , p − 1 and � = 1, 2, . . . , 2q. Each set Aq

�contains a total

of 2p−q − 1 of the discretization points centered around

zq� := x2p−q−1�.

Correspondingly, the complement set Bq�

contains 2p−2p−q+1 discretization points.For notational convenience we also define Ap

�:= {x�} for � = 1, . . . , n. Finally, we

set C1� := B1

� for � = 1, . . . , 4, and recalling that Bq2� ⊃ Bq−1

� and Bq2�−1 ⊃ Bq−1

� , wefurther define

Cq2� := Bq

2� \ Bq−1�, Cq

2�−1 := Bq2�−1 \ Bq−1

�

for q = 2, . . . , p − 1 and � = 1, 2, . . . , 2q. Then, for q ≥ 2, the set Cq� contains 2p−q

discretization points.Now, we set S0(x j) := 0, j = 1, . . . , n, and compute recursively

Sq(x j) := Sq−1(x j) +M∑

m=1

am(x j; zq�)∑

xk∈Cq�

bm(xk; zq�)ϕk, x j ∈ Aq

�, (14.32)

14.4 Fast Matrix-Vector Multiplication 295

for q = 1, . . . , p − 1 and � = 1, 2, . . . , 2q+1. Since each of the sets Cq�

contains 2p−q

discretization points and since we have 2q+1 sets Cq�, the computations of the Sq(x j)

from the preceding Sq−1(x j) requires a total of O(Mn) operations. Therefore, thetotal cost to compute Sp−1(x j) for j = 1, . . . , n is of order O(Mpn), i.e., of orderO(Mn log n).

Since 2|x j − zq�| ≤ |xk − zq

�| for x j ∈ Aq+1

2� and xk ∈ Cq�, from (14.31) we can

conclude that∣∣∣∣∣∣∣∣∣∑

xk∈Cq�

⎧⎪⎪⎨⎪⎪⎩M∑

m=1

am(x j; zq� )bm(xk; zq

� ) − K(x j, xk)

⎫⎪⎪⎬⎪⎪⎭ϕk

∣∣∣∣∣∣∣∣∣≤ cn

2qaM‖ϕ‖∞

for x j ∈ Aq+12� and � = 1, . . . , 2q+1. From this, by induction, we obtain that

∣∣∣∣∣∣∣∣∣Sq(x j) −

∑xk∈Bq

�

K(x j, xk)ϕk

∣∣∣∣∣∣∣∣∣≤ cn

aM‖ϕ‖∞

q∑r=1

12r

for x j ∈ Aq+12� and � = 1, . . . , 2q+1. In particular, for q = p − 1, we have that

∣∣∣∣∣∣∣∣∣∣∣1n

Sp−1(x j) − 1n

n∑k=1k� j

K(x j, xk)ϕk

∣∣∣∣∣∣∣∣∣∣∣≤ c

aM‖ϕ‖∞, j = 1, . . . , n, (14.33)

since Ap−1�= {x�} for � = 1, . . . , n. Therefore, the above scheme indeed computes

the required matrix-vector multiplication with a total computational cost of orderO(Mn log n).

From (14.33) we also observe that the computations correspond to replacing thematrix with the entries K(x j, xk)/n by an approximating matrix such that the differ-ence has maximum norm less than or equal to ca−M. This, by Theorem 14.2, impliesthat the resulting solution of the perturbed linear systems differs from the solutionof (14.29) by an error of order O(a−M). We want to ensure that the additional er-ror induced by the approximation of the matrix elements is of the same magnitudeas the error for the approximation of the integral equation (14.28) by the Nystrommethod (14.29). If for the latter we assume a convergence order O(n−m) for somem ∈ IN, then we need to make sure that a−M = O(n−m), i.e., we have to chooseM = O(m log n). Therefore, we have achieved a computational scheme preservingthe accuracy of the numerical solution of the integral equation with the computa-tional cost of order O(m n(log n)2).

For analytic kernels and right-hand sides, for the Nystrom method, we have ex-ponential convergence O(e−sn) for some s > 0. Therefore, in this case, the above ar-gument leads to the requirement that M = O(n), i.e., we do not obtain a reduction of


the computational cost compared to the naive matrix-vector multiplications. Hence,as a rule of thumb, panel clustering and fast multipole methods are only efficient forlow-order approximations.

For further studies of panel clustering and fast multipole methods, we refer toGreengard and Rokhlin [72], Hackbusch [76], Hackbusch and Sauter [78], andLiu [159], and the references therein.

Problems

14.1. For the operator Mn, from the proof of Theorem 14.3, show that

‖Mn‖∞ = sup‖Φ‖∞=1

‖MnΦ‖∞ = 1

and verify the existence of the functions Lj used in the definition of Mn.

14.2. For the condition number of the Bubnov–Galerkin method for an equation of the secondkind, define matrix operators En, An : Cn → Cn by

(Enγ) j :=n∑

k=1

(uj, uk)γk

and

(Anγ) j :=n∑

k=1

(uj, Auk)γk

for j = 1, . . . , n. Proceding as in Theorem 14.4, show that in the Euclidean norm the conditionnumber of the linear system satisfies

cond(En − An) ≤ C cond En


14.3. Brakhage [19] suggested using Bn = I + (I − Am)−1An as an approximate inverse for theNystrom method in the defect correction method (compare also the proof of Theorem 10.12).Prove a variant of Theorem 14.5 for this case.

14.4. Set up the equations for one step of the two-grid iteration for the collocation method.

14.5. Formulate and prove an analog of Theorem 14.7 for the case of exponential convergence‖Tn‖ = O(exp(−szn−1)) with some s > 0.

Chapter 15Equations of the First Kind

Compact operators cannot have a bounded inverse. Therefore, equations of the firstkind with a compact operator provide a typical example for so-called ill-posedproblems. This chapter is intended as an introduction into the basic ideas on ill-posed problems and regularization methods for their stable solution. We mainlyconfine ourselves to linear equations of the first kind with compact operators ina Hilbert space setting and will base our presentation on the singular value de-composition. For a more comprehensive study of ill-posed problems, we refer toBaumeister [15], Engl, Hanke, and Neubauer [49], Groetsch [74], Kabanikhin [113],Kaltenbacher, Neubauer and Scherzer [115], Kirsch [126], Louis [160], Morozov[176], Rieder [204], Tikhonov and Arsenin [234] and Wang, Yagola and Yang [243].

15.1 Ill-Posed Problems

For problems in mathematical physics, in particular for initial and boundary valueproblems for partial differential equations, Hadamard [79] postulated threeproperties:

(1) Existence of a solution.(2) Uniqueness of the solution.(3) Continuous dependence of the solution on the data.The third postulate is motivated by the fact that in all applications the data will

be measured quantities. Therefore, one wants to make sure that small errors in thedata will cause only small errors in the solution. A problem satisfying all threerequirements is called well-posed. The potential theoretic boundary value prob-lems of Chapter 6 and the initial boundary value problem for the heat equation ofChapter 9 are examples of well-posed problems. We will make Hadamard’s conceptof well-posedness more precise through the following definition.


297

298 15 Equations of the First Kind

Definition 15.1. Let A : U → V be an operator from a subset U of a normed spaceX into a subset V of a normed space Y. The equation

Aϕ = f (15.1)

is called well-posed or properly posed if A : U → V is bijective and the inverseoperator A−1 : V → U is continuous. Otherwise, the equation is called ill-posed orimproperly posed.

According to this definition we may distinguish three types of ill-posedness. IfA is not surjective, then equation (15.1) is not solvable for all f ∈ V (nonexis-tence). If A is not injective, then equation (15.1) may have more than one solution(nonuniqueness). Finally, if A−1 : V → U exists but is not continuous, then the solu-tion ϕ of (15.1) does not depend continuously on the data f (instability). The lattercase of instability is the one of primary interest in the study of ill-posed problems.We note that the three properties, in general, are not independent. For example, ifA : X → Y is a bounded linear operator mapping a Banach space X bijectivelyonto a Banach space Y, then by the Banach open mapping Theorem 10.8 the inverseoperator A−1 : Y → X is bounded and therefore continuous. For a long time the re-search on improperly posed problems was neglected, since they were not consideredrelevant to the proper treatment of applied problems.

Note that the well-posedness of a problem is a property of the operator A togetherwith the solution space X and the data space Y including the norms on X and Y.Therefore, if an equation is ill-posed one could try to restore stability by changingthe spaces X and Y and their norms. But, in general, this approach is inadequate,since the spaces X and Y, including their norms are determined by practical needs.In particular, the space Y and its norm must be suitable to describe the measureddata and, especially, the measurement errors.

Example 15.2. A classic example of an ill-posed problem, given by Hadamard, isthe following Cauchy problem (or initial value problem) for the Laplace equation:Find a harmonic function u in IR × [0,∞) satisfying the initial conditions

u(· , 0) = 0,∂

∂yu(· , 0) = f ,

where f is a given continuous function. By Holmgren’s Theorem 6.7 we haveuniqueness of the solution to this Cauchy problem. If we choose as data

fn(x) =1n

sin nx, x ∈ IR,

for n ∈ IN, then we obtain the solution

un(x, y) =1n2

sin nx sinh ny, (x, y) ∈ IR × [0,∞).

Obviously, the data sequence ( fn) converges uniformly to zero, whereas the solutionsequence (un) does not converge in any reasonable norm. Therefore, the solution

15.1 Ill-Posed Problems 299

to the Cauchy problem for harmonic functions is ill-posed. But a number of appli-cations lead to such a Cauchy problem, for example, the extrapolation of measureddata for the gravity potential from parts of the surface of the earth to the space aboveor below this surface. �

Example 15.3. A second typical example of an ill-posed problem is the backwardheat conduction. Consider the heat equation

∂u∂t=∂2u∂x2

in a rectangle [0, π] × [0, T ] subject to the boundary conditions

u(0, t) = u(π, t) = 0, 0 ≤ t ≤ T,

and the initial condition

u(x, 0) = ϕ(x), 0 ≤ x ≤ π,where ϕ is a given function. For existence and uniqueness, we refer back toChapter 9. Here, the solution can be obtained by separation of variables in the form

u(x, t) =∞∑

n=1

ϕne−n2t sin nx (15.2)

with the Fourier coefficients

ϕn =2π

∫ π

0ϕ(y) sin ny dy (15.3)

of the given initial values. This initial value problem is well-posed: The final tem-perature f := u(· , T ) depends continuously on the initial temperature. For example,in an L2 setting, using Parseval’s equality, we have

‖ f ‖22 =π

2

∞∑n=1

ϕ2ne−2n2T ≤ π

2e−2T

∞∑n=1

ϕ2n = e−2T ‖ϕ‖22.

However, the inverse problem, i.e., the determination of the initial temperature ϕfrom the knowledge of the final temperature f , is ill-posed. Here, we can write

u(x, t) =∞∑

n=1

fnen2(T−t) sin nx

with the Fourier coefficients

fn =2π

∫ π

0f (y) sin ny dy


of the given final temperature. Then we have

‖ϕ‖22 =π

2

∞∑n=1

f 2n e2n2T

and there exists no positive constant C such that

‖ϕ‖22 ≤ C‖ f ‖22 = Cπ

2

∞∑n=1

f 2n .

We may interpret this inverse problem as an initial value problem for the backwardheat equation, where t is replaced by −t. The ill-posedness reflects the fact that heatconduction is an irreversible physical process. �

Inserting (15.3) into (15.2), we see that the last example can be put into the formof an integral equation of the first kind

∫ π

0K(x, y)ϕ(y) dy = f (x), 0 ≤ x ≤ π,

where the kernel is given by

K(x, y) =2π

∞∑n=1

e−n2T sin nx sin ny, 0 ≤ x, y ≤ π.

In general, linear integral equations of the first kind with continuous or weakly sin-gular kernels provide typical examples for ill-posed problems with respect to boththe maximum and the mean square norm. They are special cases of the followingtheorem which is a reformulation of Theorem 2.26.

Theorem 15.4. Let X and Y be normed spaces and let A : X → Y be a compactlinear operator. Then the equation of the first kind Aϕ = f is improperly posed if Xis not of finite dimension.

The ill-posed nature of an equation, of course, has consequences for its numericaltreatment. We may view a numerical solution of a given equation as the solutionto perturbed data. Therefore, straightforward application of the methods developedin previous chapters to ill-posed equations of the first kind will usually generatenumerical nonsense. In terms of the concepts introduced in Section 14.1, the factthat an operator does not have a bounded inverse means that the condition numbersof its finite-dimensional approximations grow with the quality of the approximation.Hence, a careless discretization of ill-posed problems leads to a numerical behaviorthat only at first glance seems to be paradoxical. Namely, increasing the degree ofdiscretization, i.e., increasing the accuracy of the approximation for the operator A,will cause the approximate solution to the equation Aϕ = f to become less reliable.

15.2 Regularization of Ill-Posed Problems 301

15.2 Regularization of Ill-Posed Problems

Methods for constructing a stable approximate solution of ill-posed problems arecalled regularization methods. Note that in the context of improperly posed prob-lems the term regularization has a different meaning than in the theory of singularequations described in Chapter 5. It is our aim to introduce a few of the classicalregularization concepts for linear equations of the first kind.

In the sequel, we will mostly assume that the linear operator A is injective. This isnot a principal loss of generality, since uniqueness for a linear equation can alwaysbe achieved by a suitable modification of the solution space X. We wish to approx-imate the solution ϕ to the equation Aϕ = f from the knowledge of a perturbedright-hand side f δ with a known error level

‖ f δ − f ‖ ≤ δ. (15.4)

When f belongs to the range A(X), there exists a unique solution ϕ of Aϕ = f .For the perturbed right-hand side, in general, we cannot expect f δ ∈ A(X). Usingthe erroneous data f δ we want to construct a reasonable approximation ϕδ to theexact solution ϕ of the unperturbed equation Aϕ = f . Of course, we want this ap-proximation to be stable, i.e., we want ϕδ to depend continuously on the actual dataf δ. Therefore our task requires finding an approximation of the unbounded inverseoperator A−1 : A(X)→ X by a bounded linear operator R : Y → X.

Definition 15.5. Let X and Y be normed spaces and let A : X → Y be an injectivebounded linear operator. Then a family of bounded linear operators Rα : Y → X,α > 0, with the property of pointwise convergence

limα→0

RαAϕ = ϕ, ϕ ∈ X, (15.5)

is called a regularization scheme for the operator A. The parameter α is called theregularization parameter.

Of course, (15.5) is equivalent to Rα f → A−1 f , α → 0, for all f ∈ A(X).Occasionally, we will use regularization parameter sets other than the positive realnumbers.

The following theorem illustrates properties that cannot be fulfilled by regular-ization schemes for compact operators.

Theorem 15.6. Let X and Y be normed spaces, A : X → Y be a compact linearoperator, and dim X = ∞. Then for a regularization scheme the operators Rα cannotbe uniformly bounded with respect to α, and the operators RαA cannot be normconvergent as α→ 0.

Proof. For the first statement, assume that ‖Rα‖ ≤ C for all α > 0 with some constantC. Then from Rα f → A−1 f , α→ 0, for all f ∈ A(X) we deduce ‖A−1 f ‖ ≤ C‖ f ‖, i.e.,A−1 : A(X)→ X is bounded. By Theorem 15.4 this is a contradiction to dim X = ∞.


For the second statement, assume that we have norm convergence. Then thereexists α > 0 such that ‖RαA − I‖ < 1/2. Now for all f ∈ A(X) we can estimate

‖A−1 f ‖ ≤ ‖A−1 f − RαAA−1 f ‖ + ‖Rα f ‖ ≤ 12‖A−1 f ‖ + ‖Rα‖ ‖ f ‖,

whence ‖A−1 f ‖ ≤ 2‖Rα‖ ‖ f ‖ follows. Therefore A−1 : A(X) → X is bounded, andwe have the same contradiction as above. �

The regularization scheme approximates the solution ϕ of Aϕ = f by the regu-larized solution

ϕδα := Rα f δ. (15.6)

Then, for the total approximation error, writing

ϕδα − ϕ = Rα f δ − Rα f + RαAϕ − ϕ,by the triangle inequality we have the estimate

‖ϕδα − ϕ‖ ≤ δ‖Rα‖ + ‖RαAϕ − ϕ‖. (15.7)

This decomposition shows that the error consists of two parts: the first term reflectsthe influence of the incorrect data and the second term is due to the approxima-tion error between Rα and A−1. Under the assumptions of Theorem 15.6, the firstterm cannot be estimated uniformly with respect to α and the second term cannotbe estimated uniformly with respect to ϕ. Typically, the first term will be increas-ing as α → 0 due to the ill-posed nature of the problem whereas the second termwill be decreasing as α → 0 according to (15.5). Every regularization scheme re-quires a strategy for choosing the parameter α in dependence on the error level δand on the given data f δ in order to achieve an acceptable total error for the regu-larized solution. On one hand, the accuracy of the approximation asks for a smallerror ‖RαAϕ − ϕ‖, i.e., for a small parameter α. On the other hand, the stability re-quires a small ‖Rα‖, i.e., a large parameter α. An optimal choice would try and makethe right-hand side of (15.7) minimal. The corresponding parameter effects a com-promise between accuracy and stability. For a reasonable regularization strategy weexpect the regularized solution to converge to the exact solution when the error leveltends to zero. We express this requirement through the following definition.

Definition 15.7. A strategy for a regularization scheme Rα, α > 0, i.e., the choiceof the regularization parameter α = α(δ) depending on the error level δ and on f δ,is called regular if for all f ∈ A(X) and all f δ ∈ Y with ‖ f δ − f ‖ ≤ δ we have

Rα(δ) f δ → A−1 f , δ→ 0.

In the discussion of regularization schemes, one usually has to distinguish be-tween an a priori and an a posteriori choice of the regularization parameter α.An a priori choice would be based on some information on smoothness proper-ties of the exact solution that, in practical problems, will not generally be available.

15.3 Compact Self-Adjoint Operators 303

Therefore, a posteriori strategies based on some considerations of the data errorlevel δ are more practical.

A natural a posteriori strategy is given by the discrepancy or residue principleintroduced by Morozov [174, 175]. Its motivation is based on the consideration that,in general, for erroneous data the residual ‖Aϕδα − f δ‖ should not be smaller than theaccuracy of the measurements of f , i.e., the regularization parameter α should bechosen such that

‖ARα f δ − f δ‖ = γδwith some fixed parameter γ ≥ 1 multiplying the error level δ. In the case of aregularization scheme Rm with a regularization parameter m = 1, 2, 3, . . . takingonly discrete values, m should be chosen as the smallest integer satisfying

‖ARm f δ − f δ‖ ≤ γδ.Finally, we also need to note that quite often the only choice for selecting the

regularization parameter will be trial and error, i.e., one uses a few different param-eters α and then picks the most reasonable result based on appropriate informationon the expected solution.

In the sequel we will describe some regularization schemes including regularstrategies in a Hilbert space setting. Our approach will be based on the singularvalue decomposition.

15.3 Compact Self-Adjoint Operators

Throughout the remainder of this chapter, X and Y will always denote Hilbertspaces. From Theorem 4.11 recall that every bounded linear operator A : X → Ypossesses an adjoint operator A∗ : Y → X, i.e.,

(Aϕ, ψ) = (ϕ, A∗ψ)

for all ϕ ∈ X and ψ ∈ Y. For the norms we have ‖A‖ = ‖A∗‖.Theorem 15.8. For a bounded linear operator we have

A(X)⊥ = N(A∗) and N(A∗)⊥ = A(X).

Proof. g ∈ A(X)⊥ means (Aϕ, g) = 0 for all ϕ ∈ X. This is equivalent to (ϕ, A∗g) = 0for all ϕ ∈ X, which is equivalent to A∗g = 0, i.e., g ∈ N(A∗). Hence, A(X)⊥ = N(A∗).We abbreviate U = A(X) and, trivially, have U ⊂ (U⊥)⊥. Denote by P : Y → U theorthogonal projection operator. For arbitrary ϕ ∈ (U⊥)⊥, by Theorem 1.25, we haveorthogonality Pϕ − ϕ ⊥ U. But we also have Pϕ − ϕ ⊥ U⊥, because we alreadyknow that U ⊂ (U⊥)⊥. Therefore it follows that ϕ = Pϕ ∈ U, whence U = (U⊥)⊥,i.e., A(X) = N(A∗)⊥. �


An operator A : X → X mapping a Hilbert space X into itself is called self-adjointif A = A∗, i.e., if

(Aϕ, ψ) = (ϕ, Aψ)

for all ϕ, ψ ∈ X. Note that for a self-adjoint operator the scalar product (Aϕ, ϕ) isreal-valued for all ϕ ∈ X, since (Aϕ, ϕ) = (ϕ, Aϕ) = (Aϕ, ϕ).

Theorem 15.9. For a bounded self-adjoint operator we have

‖A‖ = sup‖ϕ‖=1|(Aϕ, ϕ)|. (15.8)

Proof. We abbreviate the right-hand side of (15.8) by q. By the Cauchy–Schwarzinequality we find

|(Aϕ, ϕ)| ≤ ‖Aϕ‖ ‖ϕ‖ ≤ ‖A‖for all ‖ϕ‖ = 1, whence q ≤ ‖A‖ follows. On the other hand, for all ϕ, ψ ∈ X we have

(A(ϕ + ψ), ϕ + ψ) − (A(ϕ − ψ), ϕ − ψ) = 2 {(Aϕ, ψ) + (Aψ, ϕ)} = 4 Re(Aϕ, ψ).

Therefore we can estimate

4 Re(Aϕ, ψ) ≤ q{‖ϕ + ψ‖2 + ‖ϕ − ψ‖2

}= 2q

{‖ϕ‖2 + ‖ψ‖2

}.

Now let ‖ϕ‖ = 1 and Aϕ � 0. Then choose ψ = ‖Aϕ‖−1Aϕ to obtain

‖Aϕ‖ = Re(Aϕ, ψ) ≤ q.

This implies ‖A‖ ≤ q and completes the proof. �

Recall the spectral theoretic notions of Definition 3.8.

Theorem 15.10. All eigenvalues of a self-adjoint operator are real and eigenele-ments to different eigenvalues are orthogonal.

Proof. Aϕ = λϕ and ϕ � 0 imply λ(ϕ, ϕ) = (Aϕ, ϕ) ∈ IR, whence λ ∈ IR. LetAϕ = λϕ and Aψ = μψ with λ � μ. Then, from

(λ − μ) (ϕ, ψ) = (Aϕ, ψ) − (ϕ, Aψ) = 0

it follows that (ϕ, ψ) = 0. �

Theorem 15.11. The spectral radius of a bounded self-adjoint operator A satisfies

r(A) = ‖A‖. (15.9)

If A is compact then there exists at least one eigenvalue with |λ| = ‖A‖.Proof. For each bounded linear operator we have r(A) ≤ ‖A‖, since all λ ∈ C with|λ| > ‖A‖ are regular by the Neumann series Theorem 2.14. By Theorem 15.9 there


exists a sequence (ϕn) in X with ‖ϕn‖ = 1 such that

|(Aϕn, ϕn)| → ‖A‖, n→ ∞.We may assume that (Aϕn, ϕn)→ λ, n→ ∞, where λ is real and |λ| = ‖A‖. Then

0 ≤ ‖Aϕn − λϕn‖2

= ‖Aϕn‖2 − 2λ(Aϕn, ϕn) + λ2‖ϕn‖2

≤ ‖A‖2 − 2λ(Aϕn, ϕn) + λ2

= 2λ{λ − (Aϕn, ϕn)} → 0, n→ ∞.Therefore

Aϕn − λϕn → 0, n→ ∞. (15.10)

This implies that λ is not a regular value because if it was we would have the con-tradiction

1 = ‖ϕn‖ = ‖(λI − A)−1(λϕn − Aϕn)‖ → 0, n→ ∞.Hence, r(A) ≥ |λ| = ‖A‖.

If A is compact, then the bounded sequence (ϕn) contains a subsequence (ϕn(k))such that Aϕn(k) → ψ, k → ∞, for some ψ ∈ X. We may assume that A � 0, sincefor A = 0 the statement of the theorem is trivial. Then, from (15.10) it follows thatϕn(k) → ϕ, k → ∞, for some ϕ ∈ X, and ‖ϕn(k)‖ = 1 for all k implies that ‖ϕ‖ = 1.Finally, again from (15.10), by the continuity of A, we obtain Aϕ = λϕ, and theproof is finished. �

Now we are ready to summarize our results into the following spectral theoremfor self-adjoint compact operators.

Theorem 15.12. Let X be a Hilbert space and let A : X → X be a self-adjointcompact operator (with A � 0). Then all eigenvalues of A are real. A has at leastone eigenvalue different from zero and at most a countable set of eigenvalues ac-cumulating only at zero. All eigenspaces N(λI − A) for nonzero eigenvalues λ havefinite dimension and eigenspaces to different eigenvalues are orthogonal. Assumethe sequence (λn) of the nonzero eigenvalues to be ordered such that

|λ1| ≥ |λ2| ≥ |λ3| ≥ · · ·and denote by Pn : X → N(λnI − A) the orthogonal projection operator onto theeigenspace for the eigenvalue λn. Then

A =∞∑

n=1

λnPn (15.11)


in the sense of norm convergence. Let Q : X → N(A) denote the orthogonal projec-tion operator onto the nullspace N(A). Then

ϕ =∞∑

n=1

Pnϕ + Qϕ (15.12)

for all ϕ ∈ X. (When there are only finitely many eigenvalues, the series (15.11) and(15.12) degenerate into finite sums.)

Proof. The first part of the statement is a consequence of Theorems 3.1, 3.9, 15.10,and 15.11. The orthogonal projection operators Pn are self-adjoint by Problem 13.1and bounded with ‖Pn‖ = 1 by Theorem 13.3. Therefore, by Theorem 2.23, theyare compact, since the eigenspaces N(λnI − A) have finite dimension. Hence theoperators Am := A −∑m

n=1 λnPn are self-adjoint and compact.Let λ � 0 be an eigenvalue of Am with eigenelement ϕ, i.e., Amϕ = λϕ. Then for

1 ≤ n ≤ m we haveλPnϕ = PnAmϕ = Pn(Aϕ − λnϕ),

since PnPk = 0 for n � k. From this it follows that

λ2‖Pnϕ‖2 = (Aϕ − λnϕ, Pn(Aϕ − λnϕ)) = (ϕ, (A − λnI)Pn(Aϕ − λnϕ)) = 0,

since Pn(Aϕ−λnϕ) ∈ N(λnI−A). Therefore Pnϕ = 0, whence Aϕ = λϕ. Conversely,let ϕ ∈ N(λnI − A). Then Amϕ = λnϕ if n > m and Amϕ = 0 if n ≤ m. Therefore, theeigenvalues of Am different from zero are given by λm+1, λm+2, . . . . Now Theorem15.11 yields ‖Am‖ = |λm+1|, whence (15.11) follows.

From ∥∥∥∥∥∥∥ϕ −m∑

n=1

Pnϕ

∥∥∥∥∥∥∥2

= ‖ϕ‖2 −m∑

n=1

‖Pnϕ‖2

we observe that∑∞

n=1 ‖Pnϕ‖2 converges in IR. As in the proof of Theorem 8.2 thisimplies convergence of the series

∑∞n=1 Pnϕ in the Hilbert space X. Then, by the

continuity of A and (15.11), we obtain

A

⎛⎜⎜⎜⎜⎜⎝ϕ −∞∑

n=1

Pnϕ

⎞⎟⎟⎟⎟⎟⎠ = Aϕ −∞∑

n=1

λnPnϕ = 0.

Observing that QPn = 0 for all n completes the proof of (15.12). �

Let U be a finite-dimensional subspace of X and let ϕ1, . . . , ϕm be an orthonormalbasis, i.e., (ϕn, ϕk) = 0, n, k = 1, . . . ,m, n � k and ‖ϕn‖ = 1, n = 1, . . . ,m. Then theorthogonal projection operator P : X → U has the representation

Pϕ =m∑

n=1

(ϕ, ϕn)ϕn.


Hence, we can rewrite the series (15.11) and (15.12) into a more explicit form. Forthis, deviating from the numbering in Theorem 15.12, we repeat each eigenvalue inthe sequence (λn) according to its multiplicity, i.e., according to the dimension of theeigenspace N(λnI−A). Assume (ϕn) to be a sequence of corresponding orthonormaleigenelements. Then for each ϕ ∈ X we can expand

Aϕ =∞∑

n=1

λn(ϕ, ϕn)ϕn (15.13)

and

ϕ =

∞∑n=1

(ϕ, ϕn)ϕn + Qϕ. (15.14)

By Theorem 15.12, the orthonormal eigenelements of a compact self-adjoint oper-ator, including those for the possible eigenvalue zero, are complete in the sense ofTheorem 1.28.

Example 15.13. Consider the integral operator A : L2[0, π]→ L2[0, π] with contin-uous kernel

K(x, y) :=

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

1π

(π − x) y, 0 ≤ y ≤ x ≤ π,

1π

(π − y) x, 0 ≤ x ≤ y ≤ π.This kernel is the so-called Green’s function to the boundary value problem for thesimple ordinary differential equation

ϕ′′(x) = − f (x), 0 ≤ x ≤ π,with homogeneous boundary condition ϕ(0) = ϕ(π) = 0. To each f ∈ C[0, π]there exists a unique solution ϕ ∈ C2[0, π] of the boundary value problem, which isgiven by

ϕ(x) =∫ π

0K(x, y) f (y) dy, 0 ≤ x ≤ π. (15.15)

Uniqueness of the solution is obvious, and straightforward differentiation shows that(15.15) indeed represents the solution. The compact integral operator A with thisso-called triangular kernel is self-adjoint, since its real-valued kernel is symmetricK(x, y) = K(y, x). The eigenvalue equation Aϕ = λϕ is equivalent to the differentialequation

λϕ′′(x) + ϕ(x) = 0, 0 ≤ x ≤ π,with homogeneous boundary condition ϕ(0) = ϕ(π) = 0. In particular, taking λ = 0yields a trivial nullspace N(A) = {0}. The only nontrivial solutions to the boundaryvalue problem are given through

λn =1n2

and ϕn(x) =

√2π

sin nx, 0 ≤ x ≤ π,


for n ∈ IN. Therefore, in this example, (15.13) becomes

∫ π

0K(x, y)ϕ(y) dy =

2π

∞∑n=1

1n2

sin nx∫ π

0sin ny ϕ(y) dy

corresponding to the Fourier expansion of the kernel K. �

We now wish to characterize the eigenvalues of compact self-adjoint operatorsby a minimum-maximum principle. We confine our presentation to the case of anonnegative operator. A self-adjoint operator A : X → X is called nonnegative if

(Aϕ, ϕ) ≥ 0, ϕ ∈ X.

Note that nonnegative operators only have nonnegative eigenvalues.

Theorem 15.14 (Courant). Let X be a Hilbert space, A : X → X be a nonnegativeself-adjoint compact operator, and (λn) denote the nonincreasing sequence of thenonzero eigenvalues repeated according to their multiplicity. Then

λ1 = ‖A‖ = sup‖ϕ‖=1

(Aϕ, ϕ) (15.16)

andλn+1 = inf

ψ1,...,ψn∈Xsup

ϕ⊥ψ1 ,...,ψn‖ϕ‖=1

(Aϕ, ϕ), n = 1, 2, . . . . (15.17)

Proof. We need only to prove (15.17). To this end, we abbreviate the right-hand sideof (15.17) by μn+1. If we choose ψ1 = ϕ1, . . . , ψn = ϕn, then for all ϕ ⊥ ψ1, . . . , ψn

from (15.13) and (15.14) we obtain that

(Aϕ, ϕ) =∞∑

k=n+1

λk |(ϕ, ϕk)|2 ≤ λn+1

∞∑k=n+1

|(ϕ, ϕk)|2 ≤ λn+1 ‖ϕ‖2,

with equality holding for ϕ = ϕn+1. Therefore μn+1 ≤ λn+1.On the other hand, to each choice ψ1, . . . , ψn ∈ X there exists an element

ϕ ∈ span{ϕ1, . . . , ϕn+1}with ‖ϕ‖ = 1 and ϕ ⊥ ψ1, . . . , ψn. To see this we write ϕ =

∑n+1k=1 γkϕk and have to

solve the linear system of n equations

n+1∑k=1

γk(ϕk, ψ j) = 0, j = 1, . . . , n,

for the n + 1 unknowns γ1, . . . , γn+1. This system always allows a solution that canbe normalized such that

‖ϕ‖2 =n+1∑k=1

|γk|2 = 1.

15.4 Singular Value Decomposition 309

Then, again from (15.13) and (15.14), it follows that

(Aϕ, ϕ) =n+1∑k=1

λk |(ϕ, ϕk)|2 ≥ λn+1

n+1∑k=1

|(ϕ, ϕk)|2 = λn+1‖ϕ‖2 = λn+1.

Hence μn+1 ≥ λn+1, and the proof is complete. �

15.4 Singular Value Decomposition

We now will describe modified forms of the expansions (15.11) and (15.12) forarbitrary compact operators in Hilbert spaces. From Theorem 4.12 we recall thatthe adjoint operator of a compact linear operator is compact.

Definition 15.15. Let X and Y be Hilbert spaces, A : X → Y be a compact linearoperator, and A∗ : Y → X be its adjoint. The nonnegative square roots of the eigen-values of the nonnegative self-adjoint compact operator A∗A : X → X are calledsingular values of A.

Theorem 15.16. Let (μn) denote the sequence of the nonzero singular values of thecompact linear operator A (with A � 0) repeated according to their multiplicity,i.e., according to the dimension of the nullspaces N(μ2

nI − A∗A). Then there existorthonormal sequences (ϕn) in X and (gn) in Y such that

Aϕn = μngn, A∗gn = μnϕn (15.18)

for all n ∈ IN. For each ϕ ∈ X we have the singular value decomposition

ϕ =

∞∑n=1

(ϕ, ϕn)ϕn + Qϕ (15.19)

with the orthogonal projection operator Q : X → N(A) and

Aϕ =∞∑

n=1

μn(ϕ, ϕn)gn. (15.20)

Each system (μn, ϕn, gn), n ∈ IN, with these properties is called a singular system ofA. When there are only finitely many singular values, the series (15.19) and (15.20)degenerate into finite sums. (Note that for an injective operator A the orthonormalsystem {ϕn : n ∈ IN} provided by the singular system is complete in X.)

Proof. Let (ϕn) denote an orthonormal sequence of the eigenelements of A∗A, i.e.,

A∗Aϕn = μ2nϕn


and define a second orthonormal sequence by

gn :=1μn

Aϕn.

Straightforward computations show that the system (μn, ϕn, gn), n ∈ IN, satisfies(15.18). Application of the expansion (15.14) to the self-adjoint compact operatorA∗A yields

ϕ =

∞∑n=1

(ϕ, ϕn)ϕn + Qϕ, ϕ ∈ X,

where Q denotes the orthogonal projection operator from X onto N(A∗A). Let ψ ∈N(A∗A). Then

(Aψ, Aψ) = (ψ, A∗Aψ) = 0,

and this implies that N(A∗A) = N(A). Therefore, (15.19) is proven and (15.20) fol-lows by applying A to (15.19). �

Note that the singular value decomposition implies that for all ϕ ∈ X we have

‖ϕ‖2 =∞∑

n=1

|(ϕ, ϕn)|2 + ‖Qϕ‖2 (15.21)

and

‖Aϕ‖2 =∞∑

n=1

μ2n|(ϕ, ϕn)|2. (15.22)

Theorem 15.17. Let A, B : X → Y be compact linear operators. Then for the non-increasing sequence of singular values we have

μ1(A) = ‖A‖ = sup‖ϕ‖=1‖Aϕ‖ (15.23)

andμn+1(A) = inf

ψ1,...,ψn∈Xsup

ϕ⊥ψ1,...,ψn‖ϕ‖=1

‖Aϕ‖, n = 1, 2, . . . . (15.24)

Furthermore

μn+m+1(A + B) ≤ μn+1(A) + μm+1(B), n,m = 0, 1, 2, . . . . (15.25)

Proof. (15.23) and (15.24) follow immediately from Theorem 15.14, since thesquares μ2

n of the singular values of A are given by the eigenvalues of the nonnegative

15.4 Singular Value Decomposition 311

self-adjoint operator A∗A, and since (A∗Aϕ, ϕ) = ‖Aϕ‖2 for all ϕ ∈ X. The inequality(15.25) is a consequence of

infψ1,...,ψn+m∈X

supϕ⊥ψ1,...,ψn+m‖ϕ‖=1

‖(A + B)ϕ‖

≤ infψ1,...,ψn+m∈X

supϕ⊥ψ1,...,ψn‖ϕ‖=1

‖Aϕ‖ + infψ1,...,ψn+m∈X

supϕ⊥ψn+1,...,ψn+m‖ϕ‖=1

‖Bϕ‖

≤ infψ1,...,ψn∈X

supϕ⊥ψ1,...,ψn‖ϕ‖=1

‖Aϕ‖ + infψn+1,...,ψn+m∈X

supϕ⊥ψn+1,...,ψn+m‖ϕ‖=1

‖Bϕ‖

and (15.24). �

In the following theorem, we express the solution to an equation of the first kindwith a compact operator in terms of a singular system.

Theorem 15.18 (Picard). Let A : X → Y be a compact linear operator with singu-lar system (μn, ϕn, gn). The equation of the first kind

Aϕ = f (15.26)

is solvable if and only if f belongs to the orthogonal complement N(A∗)⊥ andsatisfies

∞∑n=1

1μ2

n|( f , gn)|2 < ∞. (15.27)

In this case a solution is given by

ϕ =

∞∑n=1

1μn

( f , gn)ϕn. (15.28)

Proof. The necessity of f ∈ N(A∗)⊥ follows from Theorem 15.8. If ϕ is a solutionof (15.26), then

μn(ϕ, ϕn) = (ϕ, A∗gn) = (Aϕ, gn) = ( f , gn),

and (15.21) implies

∞∑n=1

1μ2

n|( f , gn)|2 =

∞∑n=1

|(ϕ, ϕn)|2 ≤ ‖ϕ‖2,

whence the necessity of (15.27) follows.Conversely, assume that f ⊥ N(A∗) and (15.27) is fulfilled. Then, as in the proof

of Theorems 8.2 and 15.12, the convergence of the series (15.27) in IR impliesconvergence of the series (15.28) in the Hilbert space X. We apply A to (15.28),


use (15.19) with the singular system (μn, gn, ϕn) of the operator A∗, and observef ∈ N(A∗)⊥ to obtain

Aϕ =∞∑

n=1

( f , gn)gn = f .

This ends the proof. �

Picard’s theorem demonstrates the ill-posed nature of the equation Aϕ = f . If weperturb the right-hand side f by f δ = f +δgn we obtain the solution ϕδ = ϕ+δμ−1

n ϕn.Hence, the ratio ‖ϕδ − ϕ‖/‖ f δ − f ‖ = 1/μn can be made arbitrarily large due to thefact that the singular values tend to zero. The influence of errors in the data f isobviously controlled by the rate of this convergence. In this sense we may say thatthe equation is mildly ill-posed if the singular values decay slowly to zero and thatit is severely ill-posed if they decay very rapidly.

Example 15.19. Consider the integral operator A : L2[0, 1]→ L2[0, 1] defined by

(Aϕ)(x) :=∫ x

0ϕ(y) dy, 0 ≤ x ≤ 1.

Then the inverse operator A−1 corresponds to differentiation. The adjoint operator isgiven by

(A∗ψ)(x) =∫ 1

xψ(y) dy, 0 ≤ x ≤ 1.

Hence

(A∗Aϕ)(x) =∫ 1

x

∫ y

0ϕ(z) dzdy, 0 ≤ x ≤ 1,

and the eigenvalue equation A∗Aϕ = μ2ϕ is equivalent to the boundary value prob-lem for the ordinary differential equation

μ2ϕ′′ + ϕ = 0

with homogeneous boundary conditions ϕ(1) = ϕ′(0) = 0. The nontrivial solutionsare given by

μn =2

(2n − 1)π, ϕn(x) =

√2 cos

(2n − 1)πx2

, n ∈ IN.

The singular system is completed by gn = μ−1n Aϕn through

gn(x) =√

2 sin(2n − 1)πx

2.

In this example we have a decay μn = O(1/n) for an integral operator with a discon-tinuity of the kernel along the diagonal x = y. The kernel of the integral operatorin Example 15.13 has a discontinuity in the first derivatives at the diagonal and thesingular values decay μn = O(1/n2). �

15.5 Regularization Schemes 313

In general, for compact integral operators the smoothness of the kernel controlsthe degree of ill-posedness. Roughly speaking, the smoothness of the kernel of theoperator A determines the smoothness of the range of A, and this effects the regular-ity condition on f , which is required for the solvability of Aϕ = f . We illustrate thisstatement with the following result due to Little and Reade [158].

Theorem 15.20. Let A : L2[−1, 1] → L2[−1, 1] be an integral operator with ana-lytic kernel on [−1, 1] × [−1, 1]. Then the singular values of A decay at least expo-nentially μn = O(R−n) for some constant R > 1.

Proof. Let K denote the kernel of A and

Kn(x, y) :=12

T0(x)a0(y) +n∑

m=1

Tm(x)am(y)

its approximation by the orthonormal Chebyshev expansion as considered in Section11.5. For the integral operator An with the degenerate kernel Kn, by (11.40) we have‖An − A‖∞ = O(R−n) with some R > 1. Since dim An(X) ≤ n + 1, from the singularvalue decomposition (15.20) we observe that An has at most n + 1 nonzero singularvalues. Therefore we can apply Theorem 15.17 to obtain

μn+2(A) ≤ μ1(A − An) + μn+2(An) = ‖A − An‖2 = O(R−n),


15.5 Regularization Schemes

As already pointed out, Picard’s Theorem 15.18 illustrates the fact that the ill-posedness of an equation of the first kind with a compact operator stems from thebehavior of the singular values μn → 0, n → ∞. This suggests trying to regular-ize the equation by damping or filtering out the influence of the factor 1/μn in thesolution formula (15.28).

Theorem 15.21. Let A : X → Y be an injective compact linear operator with sin-gular system (μn, ϕn, gn), n ∈ IN, and let q : (0,∞) × (0, ‖A‖] → IR be a boundedfunction such that for each α > 0 there exists a positive constant c(α) with

|q(α, μ)| ≤ c(α)μ, 0 < μ ≤ ‖A‖, (15.29)

andlimα→0

q(α, μ) = 1, 0 < μ ≤ ‖A‖. (15.30)

Then the bounded linear operators Rα : Y → X, α > 0, defined by

Rα f :=∞∑

n=1

1μn

q(α, μn) ( f , gn)ϕn, f ∈ Y, (15.31)


describe a regularization scheme with

‖Rα‖ ≤ c(α). (15.32)

Proof. From (15.21) and (15.29) we have

‖Rα f ‖2 =∞∑

n=1

1μ2

n[q(α, μn)]2 |( f , gn)|2 ≤ [c(α)]2

∞∑n=1

|( f , gn)|2 ≤ [c(α)]2‖ f ‖2

for all f ∈ Y. Therefore, the series (15.31) converges, i.e., the operator Rα is welldefined, and Rα is bounded by (15.32). With the aid of

(RαAϕ, ϕn) =1μn

q(α, μn) (Aϕ, gn) = q(α, μn) (ϕ, ϕn)

and the singular value decomposition for RαAϕ − ϕ we obtain

‖RαAϕ − ϕ‖2 =∞∑

n=1

|(RαAϕ − ϕ, ϕn)|2 =∞∑

n=1

[q(α, μn) − 1]2 |(ϕ, ϕn)|2.

Here we have used the fact that A is injective. Let ϕ ∈ X with ϕ � 0 and ε > 0 begiven and let M denote a bound for q. Then there exists N(ε) ∈ IN such that

∞∑n=N+1

|(ϕ, ϕn)|2 < ε

2(M + 1)2.

By the convergence condition (15.30), there exists α0(ε) > 0 such that

[q(α, μn) − 1]2 <ε

2‖ϕ‖2

for all n = 1, . . . ,N and all 0 < α ≤ α0. Splitting the series in two parts and using(15.21), it follows that

‖RαAϕ − ϕ‖2 < ε

2‖ϕ‖2N∑

n=1

|(ϕ, ϕn)|2 + ε2≤ ε

for all 0 < α ≤ α0. Thus we have established that RαAϕ→ ϕ, α → 0, for all ϕ ∈ X,and the proof is complete. �

Remark 15.22. If we replace condition (15.29) by the stronger condition

q(α, μ) ≤ c(α)μ2, 0 < μ ≤ ‖A‖, (15.33)

and denote by M a bound on q, then instead of (15.32) we have

‖Rα‖ ≤√

Mc(α). (15.34)


Proof. From

ARα f =∞∑

n=1

q(α, μn) ( f , gn)gn

we find

‖ARα f ‖2 =∞∑

n=1

[q(α, μn)]2 |( f , gn)|2 ≤ M2∞∑

n=1

|( f , gn)|2 ≤ M2‖ f ‖2,

whence ‖ARα‖ ≤ M follows. For the operators Rα : Y → Y, α > 0, defined by

Rα f :=∞∑

n=1

1μ2

nq(α, μn) ( f , gn)gn, f ∈ Y,

as in the proof of the previous Theorem 15.21, the condition (15.33) implies thebound ‖Rα‖ ≤ c(α). Now, using A∗Rα = Rα, by the Cauchy–Schwarz inequality weobtain

‖Rα f ‖2 = (Rα f , A∗Rα f ) = (ARα f , Rα f ) ≤ ‖ARα f ‖ ‖Rα f ‖ ≤ Mc(α)‖ f ‖2

for all f ∈ Y, whence (15.34) follows. �

We now describe some classical regularization schemes by choosing the dampingor filter function q appropriately.

Theorem 15.23. Let A : X → Y be a compact linear operator. Then for each α > 0the operator αI+A∗A : X → X has a bounded inverse. Furthermore, if A is injectivethen

Rα := (αI + A∗A)−1A∗

describes a regularization scheme with ‖Rα‖ ≤ 1/2√α.

Proof. Fromα‖ϕ‖2 ≤ (αϕ + A∗Aϕ, ϕ), ϕ ∈ X,

we conclude that for α > 0 the operator αI + A∗A is injective. Hence, by the RieszTheorem 3.4 we have a bounded inverse (αI + A∗A)−1, because A∗A is compact.

Now assume that A is injective and let (μn, ϕn, gn), n ∈ IN, be a singular systemfor A. Then the unique solution ϕα of

αϕα + A∗Aϕα = A∗ f

can be written in the form

ϕα =∞∑

n=1

μn

α + μ2n

( f , gn)ϕn.


Indeed, using A∗Aϕn = μ2nϕn and the singular value decomposition (15.20) applied

to A∗ f , we find

(αI + A∗A)ϕα =∞∑

n=1

μn( f , gn)ϕn = A∗ f .

Hence, Rα can be brought into the form (15.31) with

q(α, μ) =μ2

α + μ2.

This function q is bounded by 0 < q(α, μ) < 1 and satisfies the conditions (15.29)and (15.30) with

c(α) =1

2√α

because of the arithmetic-geometric mean inequality

√αμ ≤ α + μ2

2.

The statement of the theorem now follows from Theorem 15.21. �

The regularization described in Theorem 15.23 is called Tikhonov regularization,since it was introduced by Tikhonov [232]. We will analyze it in more detail inChapter 16.

Theorem 15.24. Let A : X → Y be an injective compact linear operator with sin-gular system (μn, ϕn, gn), n ∈ IN. Then the spectral cut-off

Rm f :=∑μn≥μm

1μn

( f , gn)ϕn (15.35)

describes a regularization scheme with regularization parameter m→ ∞ and

‖Rm‖ = 1μm.

Proof. The function q with q(m, μ) = 1 for μ ≥ μm and q(m, μ) = 0 otherwisesatisfies the conditions (15.29) and (15.30) with q(m, μ) ≤ μ/μm. For the norm wecan estimate

‖Rm f ‖2 =∑μn≥μm

1μ2

n|( f , gn)|2 ≤ 1

μ2m

∑μn≥μm

|( f , gn)|2 ≤ 1μ2

m‖ f ‖2,

whence ‖Rm‖ ≤ 1/μm. Equality follows from Rm(gm) = ϕm/μm. �

Here, the regularization parameter m determines the number of terms in the sum(15.35). Accuracy of the approximation requires this number to be large, and sta-bility requires it to be small. In particular, the following discrepancy principle turns


out to be a regular a posteriori strategy for determining the stopping point for thespectral cut-off.

Theorem 15.25. Let A : X → Y be an injective compact linear operator with denserange, f ∈ Y, and let δ > 0. Then, for the spectral cut-off, there exists a smallestinteger m, depending on f and δ, such that

‖ARm f − f ‖ ≤ δ.Proof. By Theorem 15.8, the dense range A(X) = Y implies that A∗ is injec-tive. Hence, the singular value decomposition (15.19) with the singular system(μn, gn, ϕn) for the adjoint operator A∗ yields

f =∞∑

n=1

( f , gn)gn, f ∈ Y, (15.36)

and consequently

‖(ARm − I) f ‖2 =∑μn<μm

|( f , gn)|2 → 0, m→ ∞. (15.37)

From this we now can conclude that there exists a smallest integer m such that‖ARm f − f ‖ ≤ δ. �

From (15.36) and (15.37), we see that

‖ARm f − f ‖2 = ‖ f ‖2 −∑μn≥μm

|( f , gn)|2.

This allows a stable determination of the stopping parameter m(δ) by terminatingthe sum when the right-hand side becomes smaller than or equal to δ2 for the firsttime.

The regularity of the discrepancy principle for the spectral cut-off describedthrough Theorem 15.25 is established in the following theorem.

Theorem 15.26. Let A : X → Y be an injective compact linear operator with denserange. Let f ∈ A(X) and f δ ∈ Y satisfy ‖ f δ − f ‖ ≤ δ with δ > 0 and let γ > 1. Then,for the spectral cut-off, there exists a smallest integer m = m(δ), depending on f δ

and δ, such that‖ARm(δ) f δ − f δ‖ ≤ γδ (15.38)

is satisfied andRm(δ) f δ → A−1 f , δ→ 0. (15.39)

Proof. In view of Theorem 15.25, we only need to establish the convergence (15.39).We first note that (15.37) implies ‖I − ARm‖ = 1 for all m ∈ IN. Therefore, writing

(ARm f δ − f δ) − (ARm f − f ) = (ARm − I)( f δ − f )


we have the triangle inequalities

‖ARm f − f ‖ ≤ δ + ‖ARm f δ − f δ‖ (15.40)

and‖ARm f δ − f δ‖ ≤ δ + ‖ARm f − f ‖. (15.41)

From (15.38) and (15.40) we obtain

‖ARm(δ) f − f ‖ ≤ δ + ‖ARm(δ) f δ − f δ‖ ≤ (1 + γ)δ→ 0, δ→ 0. (15.42)

Therefore, from the expansion (15.37), we conclude that either the cut-off numberm(δ)→ ∞, δ→ 0, or the expansion for f degenerates into a finite sum

f =∑

μn≥μm0

( f , gn)gn

and m(δ) ≥ m0. In the first case, from

‖ARm(δ)−1 f δ − f δ‖ > γδand (15.41), we conclude that

γδ < δ + ‖A(Rm(δ)−1 f − A−1 f )‖,whence

δ <1

γ − 1‖A(Rm(δ)−1 f − A−1 f )‖ (15.43)

follows. In order to establish the convergence (15.39), in this case, in view of (15.7)and (15.43), it suffices to show that

‖Rm‖ ‖A(Rm−1Aϕ − ϕ)‖ → 0, m→ ∞,for all ϕ ∈ X. But the latter property is obvious from

‖Rm‖2 ‖A(Rm−1Aϕ − ϕ)‖2 = 1μ2

m

∑μn<μm−1

μ2n|(ϕ, ϕn)|2 ≤

∑μn≤μm

|(ϕ, ϕn)|2.

In the case where f has a finite expansion then clearly

A−1 f =∑

μn≥μm0

1μn

( f , gn)ϕn = Rm f

for all m ≥ m0. Hence

‖ARm f δ − f δ‖ = ‖(ARm − I)( f δ − f )‖ ≤ ‖ f δ − f ‖ ≤ δ < γδ


for all m ≥ m0, and therefore m(δ) ≤ m0. This implies m(δ) = m0 since m(δ) ≥ m0

as noted above. Now observing

‖Rm(δ) f δ − A−1 f ‖ = ‖Rm0 ( f δ − f )‖ ≤ δ

μm0

→ 0, δ→ 0,

the proof is finished. �

Theorem 15.27. Let A : X → Y be an injective compact linear operator and let0 < a < 1/‖A‖2. Then the bounded linear operators

Rm := am∑

k=0

(I − aA∗A)kA∗

describe a regularization scheme with regularization parameter m→ ∞ and

‖Rm‖ ≤√

a(m + 1).

Proof. Using a singular system for A and the singular value decomposition (15.20)applied to A∗ f , with the aid of

aμ2m∑

k=0

(1 − aμ2)k = 1 − (1 − aμ2)m+1,

we can write

Rm f =∞∑

n=1

1μn

{1 − (1 − aμ2

n)m+1}

( f , gn)ϕn. (15.44)

The corresponding function q is given by

q(m, μ) = 1 − (1 − aμ2)m+1.

It is bounded by 0 ≤ q(m, μ) ≤ 1 for 0 < μ ≤ ‖A‖, and it satisfies the conditions(15.33) and (15.30) with q(m, μ) ≤ a(m + 1)μ2. Hence, the proof is completed byRemark 15.22. �

The evaluation of the approximation ψm = Rm f corresponds to m steps of theiteration scheme

ψk := (I − a A∗A)ψk−1 + a A∗ f , k = 1, 2, . . . ,m, (15.45)

starting with ψ0 := a A∗ f . This scheme goes back to Landweber [155] and Frid-man [57] and consequently is known as the Landweber–Fridman iteration. The reg-ularization parameter is given by the number m of iteration steps. Accuracy of theapproximation requires m to be large, and stability requires m to be small. Again,we consider the discrepancy principle as a strategy for terminating the iterations.


Theorem 15.28. Let A : X → Y be an injective compact linear operator with denserange, f ∈ Y, and δ > 0. Then, for the Landweber–Fridman iterations, there existsa smallest integer m, depending on f and δ, such that

‖ARm f − f ‖ ≤ δ.Proof. (Compare the proof of Theorem 15.25.) From (15.44) it follows that

‖ARm f − f ‖2 =∞∑

n=1

(1 − aμ2n)2m+2|( f , gn)|2. (15.46)

Now, because of aμ2n ≤ a‖A‖2 < 1 for n ∈ IN, we can choose N ∈ IN such that

∞∑n=N+1

(1 − aμ2n)2m+2|( f , gn)|2 ≤

∞∑n=N+1

|( f , gn)|2 ≤ δ2

2.

Since (1 − aμ2n)2m+2 → 0, m→ ∞, for n = 1, . . . ,N, there exists m0 ∈ IN such that

N∑n=1

(1 − aμ2n)2m+2|( f , gn)|2 ≤ δ2

2

for all m ≥ m0. In view of (15.46), the last two inequalities imply that ‖ARm f− f ‖ ≤ δfor all m ≥ m0, and the theorem is proven. �

In the next theorem we establish the regularity of this discrepancy principle forthe Landweber–Fridman iteration.

Theorem 15.29. Let A : X → Y be an injective compact linear operator with denserange. Let f ∈ A(X) and f δ ∈ Y satisfy ‖ f δ − f ‖ ≤ δ with δ > 0 and let γ > 1. Then,for the Landweber–Fridman iterations, there exists a smallest integer m = m(δ),depending on f δ and δ, such that

‖ARm(δ) f δ − f δ‖ ≤ γδ (15.47)

is satisfied andRm(δ) f δ → A−1 f , δ→ 0. (15.48)

Proof. We only need to establish the convergence (15.48). From (15.46) we concludethat ‖ARm − I‖ ≤ 1 for all m ∈ IN. Therefore, proceeding as in the proof of Theorem15.26 (see 15.42) we obtain that ‖ARm(δ) f − f ‖ → 0 as δ → 0. Hence, from (15.46)we conclude that either m(δ) → ∞, δ → 0, or f = 0. Since in the latter case thestatement of the theorem is obvious, we only need to be concerned with the casewhere m(δ)→ ∞, δ→ 0. Analogously to (15.43) we obtain

(γ − 1)δ < ‖ARm(δ)−1 f − f ‖,

Problems 321

and hence(γ − 1) ‖Rm(δ)‖δ ≤

√a(m(δ) + 1) ‖ARm(δ)−1 f − f ‖.

Therefore, in view of (15.7) the proof is completed by showing that

√m + 1 ‖ARm−1Aϕ − Aϕ‖ → 0, m→ ∞, (15.49)

for all ϕ ∈ X. From (15.46) we conclude that

(m + 1) ‖ARm−1Aϕ − Aϕ‖2 =∞∑

n=1

(m + 1)(1 − aμ2n)2mμ2

n|(ϕ, ϕn)|2.

By induction, it can be seen that

(m + 1)(1 − aμ2)2mμ2 ≤ 12a

, 0 < aμ2 ≤ 12,

for all m ∈ IN. Proceeding as in the proof of Theorem 15.21, with the help of

(m + 1)(1 − aμ2)2m → 0, m→ ∞,for all 0 < aμ2 < 1, we can establish (15.49). �

Problems

15.1. Prove the Riemann–Lebesgue lemma∫ π

0K(· , y) sin ny dy→ 0, n→∞,

in the mean square norm for a kernel K ∈ L2([0, π]×[0, π]). How can this result be used to illustratethe ill-posedness of integral equations of the first kind?

15.2. Let X be a Hilbert space and A : X → X be a compact self-adjoint operator. With the aid ofthe spectral Theorem 15.12, solve the equation of the second kind

λϕ − Aϕ = f , λ � 0.

15.3. The Poisson integral

u(ρ cos t, ρ sin t) =1

2π

∫ 2π

0

1 − ρ2

1 + ρ2 − 2ρ cos(t − τ)ϕ(τ) dτ, 0 ≤ ρ < 1, 0 ≤ t ≤ 2π,

gives the solution to the Dirichlet problem for the Laplace equation �u = 0 in the unit disk

D = {x ∈ IR2 : |x| < 1}with boundary values

u(cos t, sin t) = ϕ(t), 0 ≤ t ≤ 2π,


on ∂D in polar coordinates. Therefore, the continuation of a harmonic function given on a concen-tric disk of radius r < 1 to a harmonic function on the unit disk is equivalent to the solution of theintegral equation of the first kind

12π

∫ 2π

0

1 − r2

1 + r2 − 2r cos(t − τ)ϕ(τ) dτ = f (t), 0 ≤ t ≤ 2π,

with f (t) = u(r cos t, r sin t), 0 ≤ t ≤ 2π. Determine the singular values.

15.4. Determine the singular values of the operator A : L2[0, 1] → L2[0, 1] given by

(Aϕ)(x) :=∫ x

0(x − y)ϕ(y) dy, 0 ≤ x ≤ 1.

What is the inverse of A?

15.5. Consider the central difference quotient

(Rh f )(x) :=1h

{f

(x +

h2

)− f

(x − h

2

)}, 0 ≤ x ≤ 1,

for functions f that are odd with respect to x = 0 and even with respect to x = 1. Show that Rh,h > 0, can be interpreted as a regularization scheme in the sense of Theorem 15.21 for the integraloperator of Example 15.19.

Chapter 16Tikhonov Regularization

This chapter will continue the study of Tikhonov regularization and will be basedon its classical interpretation as a penalized residual minimization. For this we willconsider the more general case of merely bounded linear operators. In particular,we shall explain the concepts of quasi-solutions and minimum norm solutions asstrategies for the selection of the regularization parameter. We then proceed witha discussion of the classical regularization of integral equations of the first kind asintroduced by Tikhonov [232] and Phillips [189]. The final section of this chapteris devoted to an application of Tikhonov regularization to severely ill-posed integralequations arising in a solution method for boundary value problems for the Laplaceequation due to Kupradze [153].

16.1 Weak Convergence

As a prerequisite of the subsequent analysis we need to introduce the notion of weakconvergence.

Definition 16.1. A sequence (ϕn) of elements from a Hilbert space X is calledweakly convergent to an element ϕ ∈ X if

limn→∞(ψ, ϕn) = (ψ, ϕ)

for all ψ ∈ X.

For a weakly convergent sequence we will write ϕn ⇀ ϕ, n→ ∞. Note that normconvergence ϕn → ϕ, n → ∞, always implies weak convergence ϕn ⇀ ϕ, n→ ∞,whereas simple examples show that the converse of this statement is generally false.In particular, the elements un of a complete orthonormal system (see Theorem 1.28)converge weakly to the zero element as a consequence of Parseval’s equality. But


323

324 16 Tikhonov Regularization

they are not norm convergent to the zero element since they all have norm one.We leave it to the reader to verify that a sequence in a Hilbert space cannot weaklyconverge to two different elements.

Theorem 16.2. A weakly convergent sequence in a Hilbert space is bounded.

Proof. Let (ϕn) be a weakly convergent sequence. Then the sequence of boundedlinear functionals Fn : X → C, defined by Fn(ψ) := (ψ, ϕn) for ψ ∈ X, is pointwiseconvergent. Therefore, by the uniform boundedness principle Theorem 10.6 it isuniformly bounded, i.e., ‖Fn‖ ≤ C for all n ∈ IN and some constant C. Hence from‖ϕn‖2 = Fn(ϕn) ≤ ‖Fn‖‖ϕn‖ ≤ C‖ϕn‖ we have ‖ϕn‖ ≤ C for all n ∈ IN. �

Theorem 16.3. Every bounded sequence in a Hilbert space contains a weakly con-vergent subsequence.

Proof. Let (ϕn) be a bounded sequence, i.e., ‖ϕn‖ ≤ C for all n ∈ IN and someconstant C. Then, for each i ∈ IN the sequence (ϕi, ϕn) is bounded in C. Therefore,by the standard diagonalization procedure (see the proof of Theorem 1.18) we canselect a subsequence (ϕn(k)) such that (ϕi, ϕn(k)) converges in C as k → ∞ for eachi ∈ IN. Hence, the linear functional F given by

F(ψ) := limk→∞

(ψ, ϕn(k))

is well defined on U := span{ϕi : i ∈ IN} and by continuity it is also well defined onthe closure U. By decomposing an arbitrary ψ ∈ X into the form

ψ = Pψ + ψ − Pψ

where P : X → U denotes the orthogonal projection operator, we finally find thatF is well defined on all of X. Furthermore, F is bounded by ‖F‖ ≤ C. Therefore,by the Riesz representation Theorem 4.10, there exists a unique element ϕ ∈ X suchthat F(ψ) = (ψ, ϕ) for all ψ ∈ X. Hence,

limk→∞

(ψ, ϕn(k)) = (ψ, ϕ)

for all ψ ∈ X, that is, (ϕn(k)) converges weakly to ϕ as k → ∞. �

In an obvious meaning we may reformulate Theorem 16.3 by saying that in aHilbert space each bounded set is relatively weakly sequentially compact (see Sec-tion 1.4).

16.2 The Tikhonov Functional

The following theorem presents another aspect of the Tikhonov regularization com-plementing its introduction in Theorem 15.23. Throughout this chapter X and Y willalways denote Hilbert spaces.

16.2 The Tikhonov Functional 325

Theorem 16.4. Let A : X → Y be a bounded linear operator and let α > 0. Thenfor each f ∈ Y there exists a unique ϕα ∈ X such that

‖Aϕα − f ‖2 + α‖ϕα‖2 = infϕ∈X

{‖Aϕ − f ‖2 + α‖ϕ‖2

}. (16.1)

The minimizer ϕα is given by the unique solution of the equation

αϕα + A∗Aϕα = A∗ f (16.2)

and depends continuously on f .

Proof. From the equation

‖Aϕ − f ‖2 + α‖ϕ‖2 = ‖Aϕα − f ‖2 + α‖ϕα‖2

+2 Re(ϕ − ϕα, αϕα + A∗(Aϕα − f ))

+‖A(ϕ − ϕα)‖2 + α‖ϕ − ϕα‖2,which is valid for all ϕ ∈ X, we observe that the condition (16.2) is necessary andsufficient for ϕα to minimize the Tikhonov functional defined by (16.1).

Consider the operator Tα : X → X, given by Tα := αI + A∗A. Since

α‖ϕ‖2 ≤ α‖ϕ‖2 + ‖Aϕ‖2 = Re(Tαϕ, ϕ), ϕ ∈ X, (16.3)

the operator Tα is strictly coercive and therefore, by the Lax–Milgram Theorem13.29, has a bounded inverse T−1

α : X → X. �

The equation (16.2), of course, coincides with the Tikhonov regularization in-troduced in Theorem 15.23. The interpretation of the Tikhonov regularization as aminimizer of the Tikhonov functional allows us to extend the regularization propertyfrom Theorem 15.23 to the case of merely bounded operators.

Theorem 16.5. Let A : X → Y be an injective bounded linear operator. Then

Rα := (αI + A∗A)−1A∗

describes a regularization scheme with

‖Rα‖ ≤ ‖A‖α

.

Proof. The bound on Rα follows from (16.3) (see the proof of the Lax–MilgramTheorem 13.29) and the fact that ‖A∗‖ = ‖A‖. Considering the case A = I it can beseen that the order of this estimate is optimal.

Let f ∈ A(X) and setϕα := Rα f .


Since ϕα minimizes the Tikhonov functional we can estimate

α‖ϕα‖2 ≤ α‖ϕα‖2 + ‖Aϕα − f ‖2 ≤ α‖A−1 f ‖2.From this we conclude that

‖ϕα‖ ≤ ‖A−1 f ‖ (16.4)

for all α > 0 and‖Aϕα − f ‖ → 0, α→ 0.

Now let g ∈ Y be arbitrary. Then we have

|(ϕα − A−1 f , A∗g)| = |(Aϕα − f , g)| ≤ ‖Aϕα − f ‖ ‖g‖ → 0, α→ 0.

This implies weak convergence ϕα ⇀ A−1 f , α→ 0, since for the injective operatorA the range A∗(Y) is dense in X by Theorem 15.8 and since ϕα is bounded accordingto (16.4). Finally we can estimate

‖ϕα − A−1 f ‖2 = ‖ϕα‖2 − 2 Re(ϕα, A−1 f ) + ‖A−1 f ‖2

≤ 2 Re(A−1 f − ϕα, A−1 f ) → 0, α→ 0,


In general, convergence ϕα → A−1 f for α→ 0 will be slow. The optimal conver-gence rate is described in the following theorem.

Theorem 16.6. Let A : X → Y be an injective compact linear operator. Then forf ∈ A(X) the condition f ∈ AA∗A(X) is necessary and sufficient for

‖ϕα − A−1 f ‖ = O(α), α→ 0.

Proof. Let (μn, ϕn, gn), n = 1, 2, . . . , be a singular system for the operator A. Thenby Picard’s Theorem 15.18 we have

A−1 f =∞∑

n=1

1μn

( f , gn)ϕn.

From the proof of Theorem 15.23 we recall that

ϕα =

∞∑n=1

μn

α + μ2n

( f , gn)ϕn. (16.5)

The expansions for A−1 f and ϕα together imply that

‖ϕα − A−1 f ‖2 =∞∑

n=1

α2

μ2n(α + μ2

n)2|( f , gn)|2. (16.6)

16.2 The Tikhonov Functional 327

Now let f = AA∗Ag for some g ∈ X. Then

( f , gn) = (AA∗Ag, gn) = (g, A∗AA∗gn) = μ3n(g, ϕn),

and from (16.6) it follows that

‖ϕα − A−1 f ‖2 =∞∑

n=1

α2μ4n

(α + μ2n)2|(g, ϕn)|2 ≤ α2

∞∑n=1

|(g, ϕn)|2 = α2‖g‖2,

that is, ‖ϕα − A−1 f ‖ = O(α), α→ 0.Conversely, assume that ‖ϕα − A−1 f ‖ = O(α), α → 0. Then from (16.6) we

observe that there exists a constant M > 0 such that

∞∑n=1

|( f , gn)|2μ2

n(α + μ2n)2≤ M

for all α > 0. Passing to the limit α→ 0, this implies that

∞∑n=1

1

μ6n

|( f , gn)|2 ≤ M.

Hence,

g :=∞∑

n=1

1

μ3n

( f , gn)ϕn ∈ X

is well defined and AA∗Ag = f . �

The condition f ∈ AA∗A(X) of the preceding theorem can be interpreted as a reg-ularity assumption on f because it controls how fast the Fourier coefficients ( f , gn)tend to zero as n→ ∞.

Remark 16.7. For an injective bounded linear operator A : X → Y with denserange we have convergence Rα f → ϕ ∈ X as α→ 0 if and only if f ∈ A(X).

Proof. On one hand, f ∈ A(X) implies convergence since Rα is a regularizationscheme by Theorem 16.5. On the other hand, convergence implies that the limit ϕsatisfies A∗Aϕ = A∗ f . From this if follows that Aϕ = f since A∗ is injective as aconsequence of the dense range of A. �

However, in the image space we have the following convergence results.

Theorem 16.8. Let A : X → Y be an injective bounded linear operator with denserange. Then

‖Aϕα − f ‖ → 0, α→ 0,

for all f ∈ Y.


Proof. Since A(X) is dense in Y, for every ε > 0 there exists an element ϕε ∈ X suchthat ‖Aϕε − f ‖2 < ε/2. Choose δ such that δ‖ϕε‖2 ≤ ε/2. Then, using Theorem 16.4,for all α < δ we have

‖Aϕα − f ‖2 ≤ ‖Aϕα − f ‖2 + α‖ϕα‖2 ≤ ‖Aϕε − f ‖2 + α‖ϕε‖2 < ε.This implies convergence Aϕα → f , α→ 0. �

Theorem 16.9. Under the assumptions of Theorem 16.8 assume that f ∈ A(X).Then

‖Aϕα − f ‖ = o (√α ), α→ 0.

Proof. We write f = Aϕ and, by Theorem 16.5, we have ϕα → ϕ, α→ 0. Then

‖Aϕα − f ‖2 = (ϕα − ϕ, A∗[Aϕα − f ]) = α(ϕ − ϕα, ϕα), (16.7)

which implies ‖Aϕα − f ‖2 = o (α), and the proof is complete. �

The optimal rate of convergence is described in the following theorem.

Theorem 16.10. Let A : X → Y be an injective compact linear operator with denserange. Then the condition f ∈ AA∗(Y) is necessary and sufficient for

‖Aϕα − f ‖ = O(α), α→ 0.

Proof. Let (μn, ϕn, gn), n = 1, 2, . . . , be a singular system for the operator A. SinceA∗ is injective, by the singular value decomposition Theorem 15.16, the {gn : n ∈ IN}form a complete orthonormal system in Y, i.e., we can expand

f =∞∑

n=1

( f , gn)gn.

This expansion of f and the expansion (16.5) for ϕα together imply that

‖Aϕα − f ‖2 =∞∑

n=1

α2

(α + μ2n)2|( f , gn)|2. (16.8)

Now let f = AA∗g for some g ∈ X. Then

( f , gn) = (AA∗g, gn) = (g, AA∗gn) = μ2n(g, gn), (16.9)

and from (16.8), as in the proof of Theorem 16.6, it follows that ‖Aϕα− f ‖2 ≤ α2‖g‖2.Conversely, assume that ‖Aϕα − f ‖ = O(α), α → 0. Then from (16.8), again as

in the proof of Theorem 16.6, we obtain that there exists a constant M > 0 such that

∞∑n=1

1μ4

n|( f , gn)|2 ≤ M.

16.3 Quasi-Solutions 329

Hence,

g :=∞∑

n=1

1μ2

n( f , gn)gn ∈ Y

is well defined and AA∗g = f . �

We note that under the assumptions of Theorem 16.10 as a consequence of (16.6),(16.9) and the arithmetic-geometric mean inequality we have that

‖ϕα − A−1 f ‖ = O(√α), α→ 0,

for all f ∈ AA∗(Y) as an intermediate convergence order between Theorems 16.5and 16.6.

By the interpretation of the Tikhonov regularization as minimizer of the Tikhonovfunctional, its solution keeps the residual ‖Aϕα − f ‖2 small and is stabilized throughthe penalty term α‖ϕα‖2. Although Tikhonov regularization itself is not a penaltymethod, such a view nevertheless suggests the following constraint optimizationproblems:(a) For given ρ > 0, minimize the defect ‖Aϕ − f ‖ subject to the constraint that the

norm is bounded by ‖ϕ‖ ≤ ρ.(b) For given δ > 0, minimize the norm ‖ϕ‖ subject to the constraint that the defect

is bounded by ‖Aϕ − f ‖ ≤ δ.The first interpretation leads to the concept of quasi-solutions and the second to

the concept of minimum norm solutions and the discrepancy principle.

16.3 Quasi-Solutions

The principal idea underlying the concept of quasi-solutions as introduced byIvanov [106] is to stabilize an ill-posed problem by restricting the solution set tosome subset U ⊂ X exploiting suitable a priori informations on the solution ofAϕ = f . For perturbed right-hand sides, in general, we cannot expect a solutionin U. Therefore, instead of trying to solve the equation exactly, we minimize theresidual. For simplicity we restrict our presentation to the case where U = B[0; ρ]is a closed ball of radius ρ with some ρ > 0. This choice requires some a prioriknowledge on the norm of the solution.

Definition 16.11. Let A : X → Y be a bounded injective linear operator and letρ > 0. For a given f ∈ Y an element ϕ0 ∈ X is called a quasi-solution of Aϕ = fwith constraint ρ if ‖ϕ0‖ ≤ ρ and

‖Aϕ0 − f ‖ = inf‖ϕ‖≤ρ‖Aϕ − f ‖.

Note that ϕ0 is a quasi-solution to Aϕ = f with constraint ρ if and only if Aϕ0

is a best approximation to f with respect to the set V := A(B[0; ρ]). It is obvious


how the definition of a quasi-solution can be extended to more general constraintsets. The injectivity of the operator A is essential for uniqueness properties of thequasi-solution. For the sake of clarity, in the following analysis on quasi-solutions,we confine ourselves to the case where the operator A has dense range in Y. ByTheorem 15.8 this is equivalent to assuming that the adjoint operator A∗ : Y → X isinjective.

Theorem 16.12. Let A : X → Y be a bounded injective linear operator with denserange and let ρ > 0. Then for each f ∈ Y there exists a unique quasi-solution ofAϕ = f with constraint ρ.

Proof. Since A is linear, the set V = A(B[0; ρ]) clearly is convex. By Theorem1.27 there exists at most one best approximation to f with respect to V . Since A isinjective, this implies uniqueness of the quasi-solution.

If f ∈ A(B[0; ρ]), then there exists ϕ0 with ‖ϕ0‖ ≤ ρ and Aϕ0 = f . Clearly, ϕ0

is a quasi-solution. Therefore, we only need to be concerned with the case wheref � A(B[0; ρ]). We will establish existence of the quasi-solution by constructing anelement ϕ0 that satisfies the sufficient condition for the best approximation given inTheorem 1.27. In the case of approximation with respect to V , this condition readsRe( f − Aϕ0, Aϕ − Aϕ0) ≤ 0, that is,

Re(A∗( f − Aϕ0), ϕ − ϕ0) ≤ 0 (16.10)

for all ϕ with ‖ϕ‖ ≤ ρ. Obviously, any element ϕ0 with ‖ϕ0‖ = ρ satisfying

αϕ0 + A∗Aϕ0 = A∗ f , (16.11)

for some α > 0, fulfills the condition (16.10) and therefore provides a quasi-solution.For f � A(B[0; ρ]) we will show that α can be chosen such that the unique solutionϕ0 of (16.11) (see Theorem 16.4) satisfies ‖ϕ0‖ = ρ.

Define a function F : (0,∞)→ IR by

F(α) := ‖ϕα‖2 − ρ2,

where ϕα denotes the unique solution of (16.2). We have to show that F has a zero.By a Neumann series argument, using Theorem 10.1, the function F can be seen tobe continuous. From Theorem 16.5 we have that ‖Rα‖ ≤ ‖A‖/α whence

ϕα = Rα f → 0, α→ ∞, (16.12)

follows. This implies F(α)→ −ρ2 < 0, α→ ∞.Now assume that F(α) ≤ 0, i.e., ‖ϕα‖ ≤ ρ for all α > 0. Then, by Theorem

16.3, we can choose a sequence (αn) with αn → 0, n→ ∞, such that we have weakconvergence ϕn := ϕαn ⇀ ϕ, n→ ∞, with some ϕ ∈ X. From

‖ϕ‖2 = limn→∞(ϕn, ϕ) ≤ ρ‖ϕ‖


we obtain ‖ϕ‖ ≤ ρ. Writing (Aϕn, ψ) = (ϕn, A∗ψ) we conclude weak convergenceAϕn ⇀ Aϕ, n→ ∞. Finally, we can use Theorem 16.8 to find

‖Aϕ − f ‖2 = limn→∞(Aϕn − f , Aϕ − f ) ≤ lim

n→∞ ‖Aϕn − f ‖ ‖Aϕ − f ‖ = 0,

which is a contradiction to f � A(B[0; ρ]). Therefore there exists α such thatF(α) = ‖ϕα‖2 − ρ2 > 0. Now the continuity of F implies the existence of a zeroof F, and the proof is completed. �

We note that in the case when A is compact a simpler proof of Theorem 16.12based on the singular value decomposition can be given (see [32, Theorem 4.18]).

The quasi-solution can be shown to restore stability in the sense that it dependsweakly continuously on the right-hand side f (see Problem 16.1).

In applications, errors in the data f generally ensure that f � A(B[0; ρ]). Then thequasi-solution with constraint ρ can be obtained numerically by Newton’s methodfor solving F(α) = 0. By writing

1h

(ϕα+h − ϕα) = −(αI + A∗A)−1ϕα+h

it can be seen that the solution ϕα to (16.2) is differentiable with respect to α and thederivative dϕα/dα satisfies the equation

αdϕαdα+ A∗A

dϕαdα= −ϕα. (16.13)

Then the derivative of F is given by

F′(α) = 2 Re

(dϕαdα

, ϕα

).

We may view the quasi-solution as described in Theorem 16.12 as an a posterioristrategy for the choice of the parameter α in the Tikhonov regularization: Given aperturbed right-hand side f δ of f ∈ A(X) with ‖ f δ − f ‖ ≤ δ, we choose α such that

ϕδα = (αI + A∗A)−1A∗ f δ

satisfies ‖ϕδα‖ = ρ with some a priori known bound ρ on the norm of the exactsolution. Then we can deduce that

αρ2 = α(ϕδα, ϕδα) = (Aϕδα, f δ − Aϕδα)

≤ ρ ‖A‖ ‖ f δ − Aϕδα‖

≤ ρ‖A‖ ‖ f δ − AA−1 f ‖

≤ ρ ‖A‖ δ


provided ‖A−1 f ‖ ≤ ρ. Therefore, in this case, we have the estimate

αρ ≤ ‖A‖ δ, (16.14)

which may serve as a starting value for the Newton iteration to find the zero of F.The following theorem answers the question for regularity in the sense of Definition15.7 for this strategy.

Theorem 16.13. Let A : X → Y be a bounded injective linear operator with denserange and let f ∈ A(X) and ρ ≥ ‖A−1 f ‖. For f δ ∈ Y with ‖ f δ − f ‖ ≤ δ, let ϕδ denotethe quasi-solution to Aϕ = f δ with constraint ρ. Then we have weak convergence

ϕδ ⇀ A−1 f , δ→ 0.

If ρ = ‖A−1 f ‖, then we have norm convergence

ϕδ → A−1 f , δ→ 0.

Proof. Let g ∈ Y be arbitrary. Then, since ‖A−1 f ‖ ≤ ρ, we can estimate

|(Aϕδ − f , g)| ≤{‖Aϕδ − f δ‖ + ‖ f δ − f ‖

}‖g‖

≤{‖AA−1 f − f δ‖ + ‖ f δ − f ‖

}‖g‖

≤ 2δ ‖g‖.

(16.15)

Hence, (ϕδ − A−1 f , A∗g)→ 0, δ→ 0, for all g ∈ Y. This implies weak convergenceϕδ ⇀ A−1 f , δ→ 0, because for the injective operator A the range A∗(Y) is dense inX by Theorem 15.8 and because ϕδ is bounded by ‖ϕδ‖ ≤ ρ.

When ρ = ‖A−1 f ‖, we have

‖ϕδ − A−1 f ‖2 = ‖ϕδ‖2 − 2 Re(ϕδ, A−1 f ) + ‖A−1 f ‖2

≤ 2 Re(A−1 f − ϕδ, A−1 f ) → 0, δ→ 0,


Note that we cannot expect weak convergence if ρ < ‖A−1 f ‖, since then wewould have the contradiction

‖A−1 f ‖2 = limδ→0

(ϕδ, A−1 f ) ≤ ρ‖A−1 f ‖ < ‖A−1 f ‖2.

In general, we also cannot expect norm convergence if ρ > ‖A−1 f ‖ because generi-cally we will have ‖ϕδ‖ = ρ for all δ. Thus, for regularity we need an exact a prioriinformation on the norm of the exact solution.

Under additional conditions on f , which may be interpreted as regularity condi-tions, we can obtain results on the order of convergence.


Theorem 16.14. Under the assumptions of Theorem 16.13, let f ∈ AA∗(Y) andρ = ‖A−1 f ‖. Then

‖ϕδ − A−1 f ‖ = O(δ1/2), δ→ 0.

Proof. We can write A−1 f = A∗g with some g ∈ Y. Therefore, the last inequality inthe proof of Theorem 16.13, together with (16.15), yields

‖ϕδ − A−1 f ‖2 ≤ 2 Re( f − Aϕδ, g) ≤ 4δ‖g‖,and this is the desired result. �

Example 16.15. The following counterexample shows that the result of Theorem16.14 is optimal. Let A be a compact injective operator with dim A(X) = ∞ andsingular system (μn, ϕn, gn). Consider f = μ1g1 and f δn = μ1g1 + δngn with δn = μ

2n.

Then A−1 f = ϕ1 and

ϕδn = (αnI + A∗A)−1A∗(μ1g1 + δngn)

=μ2

1

αn + μ21

ϕ1 +δnμn

αn + μ2nϕn,

(16.16)

where αn must satisfyμ4

1

(αn + μ21)2+

δ2nμ

2n

(αn + μ2n)2= 1

so that ϕδn is the quasi-solution with constraint ρ = 1. Assume now that we haveconvergence order

‖ϕδn − A−1 f ‖ = o (δ1/2n ), n→ ∞.

Then, using δn = μ2n, from (16.16) we find

δn

αn + δn=

δ1/2n μn

αn + μ2n→ 0, n→ ∞,

whence αn/δn → ∞, n → ∞ follows. But this is a contradiction to the inequality(16.14). �

For a compact integral operator A : L2[a, b] → L2[a, b], the Tikhonov residualfunctional corresponds to

∫ b

a|(Aϕ)(x) − f (x)|2dx + α

∫ b

a|ϕ(x)|2dx. (16.17)

The regularized equation (16.2) is an integral equation of the second kind. For itsnumerical solution the methods of Chapters 12 and 13 are available.


∫ 1

0exyϕ(y) dy =

1x

(ex − 1) , 0 < x ≤ 1.


Its unique solution (see Problem 16.4) is given by ϕ(x) = 1. For the correspondingintegral operator A : L2[0, 1]→ L2[0, 1] elementary calculations yield

(A∗Aϕ)(x) =∫ 1

0H(x, y)ϕ(y) dy, 0 ≤ x ≤ 1,

where

H(x, y) =∫ 1

0e(x+y)z dz =

1x + y

(ex+y − 1

)

for x + y > 0 and H(0, 0) = 1. Our numerical results are obtained by discretizingthe integral equation of the second kind with α > 0 by Nystrom’s method usingSimpson’s rule with eight equidistant intervals. The integral for the right-hand sideis also evaluated numerically by Simpson’s rule. We have assumed a regular errordistribution f δi = fi + (−1)iδ at the grid points and used the norm ρ = 1 of the exactsolution. Table 16.1 gives the values of the regularization parameter α (obtained byNewton’s method) and the mean square error E := ‖ϕδα − ϕ‖L2 between the regular-ized solution ϕδα and the exact solution ϕ depending on the error level δ. In addition,the quotient q := E/δ1/2 is listed. �


δ α E q

0.02 0.000055 0.112 0.7950.04 0.000076 0.161 0.8050.06 0.000093 0.198 0.8090.08 0.000109 0.229 0.8120.10 0.000123 0.257 0.814

16.4 Minimum Norm Solutions

As already observed at the end of Section 15.2, the principal motivation for the dis-crepancy principle as introduced by Morozov [174, 175] is based on the observationthat, in general, for erroneous data it does not make too much sense to try and makethe residual ‖Aϕ− f ‖ smaller than the error in f . Assume that we have some a prioribound δ on the error in f . Then we look for elements ϕ satisfying ‖Aϕ − f ‖ ≤ δ andstabilize by making the norm ‖ϕ‖ small.

Definition 16.17. Let A : X → Y be a bounded linear operator and let δ > 0. For agiven f ∈ Y an element ϕ0 ∈ X is called a minimum norm solution of Aϕ = f withdiscrepancy δ if ‖Aϕ0 − f ‖ ≤ δ and

‖ϕ0‖ = inf‖Aϕ− f ‖≤δ

‖ϕ‖.

16.4 Minimum Norm Solutions 335

Note that ϕ0 is a minimum norm solution to Aϕ = f with discrepancy δ ifand only if ϕ0 is a best approximation to the zero element of X with respect toU f := {ϕ ∈ X : ‖Aϕ − f ‖ ≤ δ}. As in the discussion of quasi-solutions, for thesake of simplicity, in dealing with existence of minimum norm solutions we confineourselves to operators with dense range and note again that by Theorem 15.8 this isequivalent to assuming that the adjoint operator A∗ : Y → X is injective.

Theorem 16.18. Let A : X → Y be a bounded linear operator with dense rangeand let δ > 0. Then for each f ∈ Y there exists a unique minimum norm solution ofAϕ = f with discrepancy δ.

Proof. From

‖A(λϕ1 + (1 − λ)ϕ2) − f ‖ ≤ λ‖Aϕ1 − f ‖ + (1 − λ)‖Aϕ2 − f ‖for all ϕ1, ϕ2 ∈ X and all λ ∈ (0, 1), we observe that U f is convex. Then, by Theorem1.27, there exists at most one best approximation of the zero element with respectto U f .

If ‖ f ‖ ≤ δ, then clearly ϕ0 = 0 is the minimum norm solution with discrepancy δ.Therefore, we only need to consider the case where ‖ f ‖ > δ. Since A has dense rangein Y, the set U f is not empty and we will again establish existence of the minimumnorm solution by constructing an element ϕ0 that satisfies the sufficient conditionfor the best approximation given in Theorem 1.27. For approximation with respectto U f , this condition reads

Re(ϕ0, ϕ0 − ϕ) ≤ 0 (16.18)

for all ϕ ∈ X with ‖Aϕ − f ‖ ≤ δ. Assume that ϕ0 satisfies

αϕ0 + A∗Aϕ0 = A∗ f (16.19)

for some α > 0 and ‖Aϕ0 − f ‖ = δ. Then

αRe(ϕ0, ϕ0 − ϕ) = Re(A∗( f − Aϕ0), ϕ0 − ϕ)

= Re(Aϕ0 − f , Aϕ − f ) − ‖Aϕ0 − f ‖2

≤ δ (‖Aϕ − f ‖ − δ) ≤ 0.

Hence ϕ0 satisfies the condition (16.18) and therefore is a minimum norm solution.We will show that α can be chosen such that the unique solution ϕ0 of (16.19) (seeTheorem 16.4) satisfies ‖Aϕ0 − f ‖ = δ.

Define a function G : (0,∞)→ IR by

G(α) := ‖Aϕα − f ‖2 − δ2,

where ϕα denotes the unique solution of equation (16.2). We have to show that thecontinuous function G has a zero. From the convergence (16.12), we observe that


G(α) → ‖ f ‖2 − δ2 > 0, α → ∞. On the other hand, from Theorem 16.8 we obtainthat G(α)→ −δ2 < 0, α→ 0. This completes the proof. �

We note that in the case when A is compact a simpler proof of Theorem 16.18based on the singular value decomposition can be given (see [32, Theorem 4.15]).

The minimum norm solution can be proven to depend weakly continuously onthe right-hand side f (see Problem 16.2).

In general, we will have data satisfying ‖ f ‖ > δ, i.e., data exceeding the errorlevel. Then the minimum norm solution with discrepancy δ can be obtained numer-ically by Newton’s method for solving G(α) = 0. After rewriting

‖ f − Aϕα‖2 = ( f − Aϕα, f ) − (A∗( f − Aϕα), ϕα) = ‖ f ‖2 − (ϕα, A∗ f ) − α‖ϕα‖2,

we getG(α) = ‖ f ‖2 − (ϕα, A∗ f ) − α‖ϕα‖2 − δ2

and

G′(α) = −(dϕαdα

, A∗ f

)− ‖ϕα‖2 − 2αRe

(dϕαdα

, ϕα

),

where the derivative dϕα/dα is given by (16.13).We may look at the minimum norm solution as described in Theorem 16.18 as an

a posteriori strategy for the choice of the parameterα in the Tikhonov regularization:Given a perturbed right-hand side f δ of an element f ∈ A(X) with a known errorlevel ‖ f δ − f ‖ ≤ δ < ‖ f δ‖, we choose α such that

ϕδα = (αI + A∗A)−1A∗ f δ

satisfies ‖Aϕδα − f δ‖ = δ. Then, using αϕδα + A∗Aϕδα = A∗ f δ, we find

‖ f δ‖ − δ = ‖ f δ‖ − ‖Aϕδα − f δ‖ ≤ ‖Aϕδα‖ =1α‖AA∗( f δ − Aϕδα)‖ ≤ ‖A‖

2δ

α

provided ‖ f δ‖ > δ. Hence, we have established the estimate

α (‖ f δ‖ − δ) ≤ ‖A‖2δ, (16.20)

which we may use as a starting value for the Newton iteration to find the zero of G.The following theorem answers the question of regularity for this discrepancy

principle for the Tikhonov regularization.

Theorem 16.19. Let A : X → Y be a bounded injective linear operator with denserange and let δ > 0 and f ∈ A(X). For f δ ∈ Y with ‖ f δ − f ‖ ≤ δ and δ < ‖ f δ‖, let ϕδ

denote the minimum norm solution with discrepancy δ. Then

ϕδ → A−1 f , δ→ 0.

16.4 Minimum Norm Solutions 337

Proof. Since ‖ f δ‖ > δ, from the proof of Theorem 16.18 we know that ϕδ minimizesthe Tikhonov functional. Therefore,

δ2 + α‖ϕδ‖2 = ‖Aϕδ − f δ‖2 + α‖ϕδ‖2

≤ ‖AA−1 f − f δ‖2 + α‖A−1 f ‖2

≤ δ2 + α‖A−1 f ‖2,whence

‖ϕδ‖ ≤ ‖A−1 f ‖ (16.21)

follows. This inequality is also trivially satisfied when ‖ f δ‖ ≤ δ, since in this caseϕδ = 0.

Now let g ∈ Y be arbitrary. Then we can estimate

|(Aϕδ − f , g)| ≤{‖Aϕδ − f δ‖ + ‖ f δ − f ‖

}‖g‖ ≤ 2δ ‖g‖.

As in the proof of Theorem 16.13, this implies ϕδ ⇀ A−1 f , δ → 0. Then using(16.21), we obtain

‖ϕδ − A−1 f ‖2 = ‖ϕδ‖2 − 2 Re(ϕδ, A−1 f ) + ‖A−1 f ‖2

≤ 2{‖A−1 f ‖2 − Re(ϕδ, A−1 f )} → 0, δ→ 0,(16.22)

which finishes the proof. �

Theorem 16.20. Under the assumptions of Theorem 16.19, let f ∈ AA∗(Y). Then

‖ϕδ − A−1 f ‖ = O(δ1/2), δ→ 0.

Proof. Writing A−1 f = A∗g with some g ∈ Y from (16.22) we deduce

‖ϕδ − A−1 f ‖2 ≤ 2 Re(A−1 f − ϕδ, A−1 f )

= 2 Re ( f − Aϕδ, g)

≤ 2{‖ f − f δ‖ + ‖ f δ − Aϕδ‖}‖g‖ ≤ 4δ‖g‖,and the proof is finished. �

Using the same example as in connection with Theorem 16.14, it can be shownthat the result of Theorem 16.20 is optimal (see Problem 16.3).

Example 16.21. We apply the discrepancy principle to the integral equation ofExample 16.16. Table 16.2 gives the values of the regularization parameter α, themean square error E := ‖ϕδα − ϕ‖L2 , and the quotient q := E/δ1/2 depending on theerror level δ. �



δ α E q

0.02 0.0059 0.067 0.4790.04 0.0148 0.108 0.5420.06 0.0252 0.131 0.5360.08 0.0359 0.145 0.5140.10 0.0466 0.155 0.492

16.5 Classical Tikhonov Regularization

In our examples of integral equations of the first kind, so far, we have interpretedthe integral operator as a mapping A : L2[a, b] → L2[a, b] corresponding to theTikhonov residual functional in the form (16.17). Here, the choice of the data spaceY = L2[a, b], in general, is determined by the need to adequately measure the error inthe data. But we have more flexibility concerning the solution space X, in particular,when additional regularity properties of the exact solution are a priori known.

In his pioneering papers on integral equations of the first kind in 1963, Tikho-nov [232, 233] suggested damping out highly oscillating parts in the approxi-mate solution by incorporating the derivative into the penalty term, i.e., to replace(16.17) by

∫ b

a|(Aϕ)(x) − f (x)|2dx + α

∫ b

a

{|ϕ(x)|2 + |ϕ′(x)|2

}dx. (16.23)

To include this approach into our general theory we need a Hilbert space with normcorresponding to the penalty term in (16.23). Since in Chapter 8 we introduced theSobolev spaces only via Fourier expansion, we briefly discuss the definition for thespace H1[a, b] based on the concept of weak derivatives. Compare also to the notionof weak solutions to the Laplace equation from Section 8.3.

Definition 16.22. A function ϕ ∈ L2[a, b] is said to possess a weak derivativeϕ′ ∈ L2[a, b] if ∫ b

aϕψ′ dx = −

∫ b

aϕ′ψ dx (16.24)

for all ψ ∈ C1[a, b] with ψ(a) = ψ(b) = 0.

By partial integration, it follows that (16.24) is satisfied for ϕ ∈ C1[a, b]. Hence,weak differentiability generalizes classical differentiability.

From the denseness of{ψ ∈ C1[a, b] : ψ(a) = ψ(b) = 0

}in L2[a, b], or from the

Fourier series for the odd extension for ϕ, it can be seen that the weak derivative,if it exists, is unique. From the denseness of C[a, b] in L2[a, b], or from the Fourier

16.5 Classical Tikhonov Regularization 339

series for the even extension of ϕ, it follows that each function with vanishing weakderivative must be constant (almost everywhere). The latter, in particular, implies

ϕ(x) =∫ x

aϕ′(ξ)dξ + c (16.25)

for almost all x ∈ [a, b] and some constant c, since by Fubini’s theorem

∫ b

a

(∫ x

aϕ′(ξ) dξ

)ψ′(x) dx =

∫ b

aϕ′(ξ)

(∫ b

ξ

ψ′(x)dx

)dξ = −

∫ b

aϕ′(ξ)ψ(ξ) dξ

for all ψ ∈ C1[a, b] with ψ(a) = ψ(b). Hence both sides of (16.25) have the sameweak derivative.

Theorem 16.23. The linear space

H1[a, b] :={ϕ ∈ L2[a, b] : ϕ′ ∈ L2[a, b]

}

endowed with the scalar product

(ϕ, ψ)H1 :=∫ b

a

(ϕψ + ϕ′ψ′

)dx (16.26)

is a Hilbert space.

Proof. It is readily checked that H1[a, b] is a linear space and that (16.26) defines ascalar product. Let (ϕn) denote a H1 Cauchy sequence. Then (ϕn) and (ϕ′n) are bothL2 Cauchy sequences. From the completeness of L2[a, b] we obtain the existenceof ϕ ∈ L2[a, b] and χ ∈ L2[a, b] such that ‖ϕn − ϕ‖L2 → 0 and ‖ϕ′n − χ‖L2 → 0 asn→ ∞. Then for all ψ ∈ C1[a, b] with ψ(a) = ψ(b) = 0, we can estimate

∫ b

a

(ϕψ′ + χψ

)dx =

∫ b

a

{(ϕ − ϕn)ψ′ + (χ − ϕ′n)ψ

}dx

≤ ‖ϕ − ϕn‖L2‖ψ′‖L2+‖χ − ϕ′n‖L2 ‖ψ‖L2 → 0, n→ ∞.

Therefore, ϕ ∈ H1[a, b] with ϕ′ = χ and ‖ϕ − ϕn‖H1 → 0, n→ ∞, which completesthe proof. �

Theorem 16.24. C1[a, b] is dense in H1[a, b].

Proof. Since C[a, b] is dense in L2[a, b], for each ϕ ∈ H1[a, b] and ε > 0 there existsχ ∈ C[a, b] such that ‖ϕ′ − χ‖L2 < ε. Then we define ψ ∈ C1[a, b] by

ψ(x) := ϕ(a) +∫ x

aχ(ξ) dξ, x ∈ [a, b],


and using (16.25), we have

ϕ(x) − ψ(x) =∫ x

a

{ϕ′(ξ) − χ(ξ)

}dξ, x ∈ [a, b].

By the Cauchy–Schwarz inequality this implies ‖ϕ − ψ‖L2 < (b − a)ε, and the proofis complete. �

Theorem 16.25. H1[a, b] is contained in C[a, b] with compact imbedding.

Proof. From (16.25) we have

ϕ(x) − ϕ(y) =∫ x

y

ϕ′(ξ) dξ, (16.27)

whence by the Cauchy–Schwarz inequality,

|ϕ(x) − ϕ(y)| ≤ |x − y|1/2‖ϕ′‖L2 (16.28)

follows for all x, y ∈ [a, b]. Therefore, every function ϕ ∈ H1[a, b] belongs toC[a, b], or more precisely, it coincides almost everywhere with a continuous func-tion.

Choose y ∈ [a, b] such that |ϕ(y)| = mina≤x≤b |ϕ(x)|. Then from

(b − a) mina≤x≤b

|ϕ(x)|2 ≤∫ b

a|ϕ(x)|2dx

and (16.27), again by the Cauchy–Schwarz inequality, we find that

‖ϕ‖∞ ≤ C‖ϕ‖H1 (16.29)

for all ϕ ∈ H1[a, b] and some constant C. The latter inequality means that the H1

norm is stronger than the maximum norm (in one space dimension!). The inequal-ities (16.28) and (16.29), in particular, imply that each bounded set in H1[a, b] is abounded and equicontinuous subset of C[a, b]. Hence, by the Arzela–Ascoli Theo-rem 1.18, the imbedding operator from H1[a, b] into C[a, b] is compact. (See alsoTheorem 8.4.) �

Now we return to integral equations of the first kind and interpret the inte-gral operator with smooth kernel as a mapping from H1[a, b] into L2[a, b]. Thenour complete theory on regularization including convergence and regularity re-mains applicable in this setting. Since the imbedding from H1[a, b] into L2[a, b]clearly is bounded (by Theorem 16.25 it is even compact) the integral operatorA : H1[a, b] → L2[a, b] is compact by Theorem 2.21. Only the adjoint operatorA∗ : L2[a, b] → H1[a, b] looks different. We denote it by A∗ to distinguish it fromthe adjoint A∗ : L2[a, b]→ L2[a, b] of A : L2[a, b]→ L2[a, b]. We avoid its explicitcalculation through the observation that the regularized equation

αϕα + A∗Aϕα = A∗ f (16.30)

16.5 Classical Tikhonov Regularization 341

is equivalent toα(ϕα, ψ)H1 + (A∗Aϕα, ψ)L2 = (A∗ f , ψ)L2 (16.31)

for all ψ ∈ H1[a, b]. This follows from the fact that

(A∗χ, ψ)H1 = (χ, Aψ)L2 = (A∗χ, ψ)L2

for all χ ∈ L2[a, b] and ψ ∈ H1[a, b]. By Theorem 16.4, there exists a unique solutionϕα to (16.30). In the following theorem we will show that ϕα is the solution to aboundary value problem for an integro-differential equation.

Theorem 16.26. Assume that the integral operator A has continuous kernel. Thenthe unique solution ϕα of the regularized equation (16.30) belongs to C2[a, b] andsatisfies the integro-differential equation

α(ϕα − ϕ′′α

)+ A∗Aϕα = A∗ f (16.32)

and the boundary condition

ϕ′α(a) = ϕ′α(b) = 0. (16.33)

Proof. First we define g ∈ C1[a, b] by

g(x) := −∫ b

x{(A∗Aϕα)(ξ) + αϕα(ξ) − (A∗ f )(ξ)} dξ, x ∈ [a, b],

and then ψ ∈ H1[a, b] by

ψ(x) :=∫ x

a

{αϕ′α(ξ) − g(ξ)

}dξ, x ∈ [a, b].

Then, by partial integration, since g(b) = ψ(a) = 0, we get

‖αϕ′α − g‖2L2 =

∫ b

a

(αϕ′α − g

)ψ′ dx =

∫ b

a

(αϕ′αψ

′ + g′ψ)

dx

= α(ϕα, ψ)H1 + (A∗Aϕα, ψ)L2 − (A∗ f , ψ)L2 = 0.

Note that by the denseness result of Theorem 16.24 partial integration can be carriedover from C1[a, b] into H1[a, b]. Our last equation implies αϕ′α = g. From this,it follows that ϕα ∈ C2[a, b] and that the integro-differential equation (16.32) issatisfied. Inserting (16.32) into (16.31) and performing a partial integration yields

ϕ′α(b)ψ(b) − ϕ′α(a)ψ(a) = 0

for all ψ ∈ H1[a, b]. This implies that the boundary conditions (16.33) arefulfilled. �


Note that by partial integration a solution of (16.32) and (16.33) also solves(16.31). Hence, the boundary value problem for the integro-differential equationand the regularized equation (16.30) are equivalent.

Example 16.27. We compare the Tikhonov regularization based on the L2 normand the H1 norm penalty term for the integral equation of Example 16.16. For thediscretization of the regularized equation we use the Petrov–Galerkin method withlinear splines. Let x j = jh, j = 0, . . . , n, be an equidistant grid with step size h = 1/nand let L j, j = 0, . . . , n, denote the corresponding Lagrange basis for linear splineinterpolation as introduced by (11.8). Then straightforward calculations yield thetridiagonal matrices

V =1h

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 −1−1 2 −1−1 2 −1· · · · · ·

−1 2 −1−1 1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

for the weights vmk =∫ 1

0L′m(y)L′k(y) dy and

W =h6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

2 11 4 1

1 4 1· · · · · ·

1 4 11 2

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

for the weights wmk =∫ 1

0Lm(y)Lk(y) dy. Writing the approximate solution in the

form

ϕn =

n∑k=0

γkLk,

we have to solve the linear system

n∑k=0

γk

{α (w jk + v jk) +

∫ 1

0

∫ 1

0H(x, y)L j(x)Lk(y) dxdy

}

=

∫ 1

0

∫ 1

0K(y, x)L j(x) f (y) dxdy, j = 0, . . . , n,

in the case of the H1 norm penalty term. Here, K and H denote the kernels of theintegral operators A and A∗A, respectively (see Example 16.16). In the case of theL2 norm the matrix V has to be omitted. Note that the weight matrix V indicates howoscillations in the approximate solution are penalized by the regularization using thederivative. For numerical evaluation of the matrix elements we apply interpolatoryquadratures as described in detail in Chapters 11 and 13.

16.6 Ill-Posed Integral Equations in Potential Theory 343

Table 16.3 compares the mean square error between the exact and approximatesolutions for both regularizations for an error distribution as in Example 16.16 withδ = 0.01.


logα −7 −6 −5 −4 −3 −2 −1L2 penalty 41.83 6.541 3.268 1.567 0.250 0.040 0.107H1 penalty 3.503 2.353 0.550 0.063 0.015 0.015 0.019

In closing this section, we wish to point out that in a paper predating that ofTikhonov, Phillips [189] proposed penalizing only by the L2 norm of the derivative,i.e., to replace (16.23) by

∫ b

a|(Aϕ)(x) − f (x)|2dx + α

∫ b

a|ϕ′(x)|2dx. (16.34)

Note that this differs from the Tikhonov regularization because the penalty term isonly a semi-norm rather than a norm. For an analysis of this approach we refer toGroetsch [74]. In view of Theorem 16.26 it is no surprise that minimizing (16.34) isequivalent to the boundary value problem for the integro-differential equation

−αϕ′′α + A∗Aϕα = A∗ f

with boundary condition ϕ′α(a) = ϕ′α(b) = 0.

16.6 Ill-Posed Integral Equations in Potential Theory

In our treatment of the potential theoretic boundary value problems in Chapters 6–8we have derived integral equations by seeking the solutions in the form of single-or double-layer potentials over the boundary of the domain in which the boundaryvalue problem is posed. In the sequel we shall refer to the integral equations of thefirst and second kind obtained by this approach as boundary integral equations. Dueto the singularity of the fundamental solution, these boundary integral equations, ingeneral, are weakly singular. Since this causes some difficulties with their numeri-cal solution, it is certainly tempting to try and solve the boundary value problemsby using potentials with densities on curves or surfaces different from the actualboundary of the underlying domain. We shall illustrate this idea by looking at theinterior two-dimensional Dirichlet problem.


Let D ⊂ IR2 be a bounded domain with a connected boundary ∂D of class C2.Consider the Dirichlet problem for a solution u ∈ C2(D) ∩ C(D) of Laplace’s equa-tion

� u = 0 in D (16.35)

with boundary conditionu = f on ∂D, (16.36)

where f is a given continuous function. We choose a closed curve Γ of class C2

such that the closure D is contained in the interior of Γ and denote its outward unitnormal by ν. Then we seek the solution of the Dirichlet problem (16.35), (16.36) inthe form of a double-layer potential

u(x) =∫Γ

ϕ(y)∂Φ(x, y)∂ν(y)

ds(y), x ∈ D, (16.37)

where we recall the fundamental solution to Laplace’s equation

Φ(x, y) :=1

2πln

1|x − y| , x � y,

in IR2 and ϕ ∈ L2(Γ) is an unknown density. The harmonic function u solves theDirichlet problem provided the density ϕ is a solution to the integral equation of thefirst kind ∫

Γ


ds(y) = f (x), x ∈ ∂D. (16.38)

We introduce the double-layer operator A : L2(Γ)→ L2(∂D) by

(Aϕ)(x) :=∫Γ


ds(y), x ∈ ∂D, (16.39)

and rewrite the integral equation (16.38) in the short form Aϕ = f . Since the integraloperator A has a continuous kernel, it is compact, and therefore the integral equa-tion (16.38) is ill-posed. Furthermore, the kernel is analytic with respect to x, andtherefore (16.38) is severely ill-posed (compare Theorem 15.20). This ill-posednesscan also be seen from the observation that the double-layer potential (16.37) definesa harmonic function in all of the interior of Γ, and not only in D. Hence, for a givenfunction f , the integral equation (16.38) can have a solution only if the solutionto the Dirichlet problem in D with boundary values f can be continued across theboundary ∂D as a harmonic function into the interior of Γ. A closer examinationshows that this condition, together with the property that the boundary data of thecontinued function belong to L2(Γ), is also sufficient for the solvability. In general,however, there are no means available to decide for given boundary values whethersuch a continuation is possible.

In order to apply the regularization results of this chapter we establish the fol-lowing theorem.


Theorem 16.28. The compact integral operator A : L2(Γ) → L2(∂D), defined by(16.39), is injective and has dense range.

Proof. Let ϕ satisfy Aϕ = 0 and define u in IR2 \ Γ by (16.37). Then u = 0 on ∂Dand, by the uniqueness Theorem 6.12, it follows that u = 0 in D. The analyticity ofharmonic functions from Theorem 6.6 then yields u = 0 in the interior of Γ. Fromthe jump relation (6.52) for the double-layer potential with L2 densities we derivethat ϕ − Kϕ = 0, where K denotes the double-layer boundary operator on the curveΓ. Since the kernel of K is continuous (see Problem 6.1), from ϕ − Kϕ = 0 weconclude that ϕ ∈ C(Γ). Now Theorem 6.21 implies that ϕ = 0, hence A is injective.

The adjoint operator A∗ : L2(∂D)→ L2(Γ) of A is given by

(A∗ψ)(x) =∫∂Dψ(y)

∂Φ(x, y)∂ν(x)

ds(y), x ∈ Γ.

When A∗ψ = 0, then, with the aid of Example 6.17, we have

(1, ψ)L2(∂D) = −(A1, ψ)L2(∂D) = −(1, A∗ψ)L2(Γ) = 0,

i.e.,∫∂Dψ ds = 0. Hence, the single-layer potential

v(x) =∫∂Dψ(y)Φ(x, y) ds(y), x ∈ IR2,

vanishes at infinity. Because A∗ψ = 0 implies ∂v/∂ν = 0 on Γ, by the uniquenessTheorem 6.13, we have v = 0 in the exterior of Γ, and by analyticity it follows thatv = 0 in the exterior of ∂D. Since by Theorem 7.34 logarithmic single-layer poten-tials with L2 densities are continuous in all of IR2, the uniqueness for the interiorDirichlet problem tells us that v = 0 everywhere. Then the L2 jump relation (6.53)for the normal derivative of single-layer potentials implies ψ = 0 . Therefore A∗ isinjective, whence, by Theorem 15.8, the range of A is dense in L2(∂D). �

We now apply Tikhonov regularization, i.e., we approximate (16.38) by the reg-ularized equation

αϕα + A∗Aϕα = A∗ f (16.40)

with some regularization parameter α > 0. Note that for the solution of (16.40)we have ϕα ∈ C(Γ), since A∗ maps L2(∂D) into C(Γ). As is always the case withTikhonov regularization, we can expect convergence of the solution ϕα to (16.40)as α→ 0 only when the original equation (16.38) has a solution (see Remark 16.7).Fortunately, in our current situation, in order to obtain approximate solutions to theboundary value problem by

uα(x) :=∫Γ

ϕα(y)∂Φ(x, y)∂ν(y)

ds(y), x ∈ D,

we do not need convergence of the densities ϕα. In order to satisfy the boundarycondition approximately, what we actually want is convergence Aϕα → f as α→ 0


on ∂D which is ensured by Theorem 16.8. Then, since the Dirichlet problem itself iswell-posed, small deviations uα − f in the boundary values ensure small deviationsuα − u in the solution in all of D. The following remark makes this comment moreprecise.

Remark 16.29. For each compact set W ⊂ D there exists a constant C (dependingon D and W) such that

‖u‖C(W) ≤ C‖u‖L2(∂D)

for all harmonic functions u ∈ C(D).

Proof. We define the double-layer potential operator V : C(∂D)→ C(D) by

(Vψ)(x) :=∫∂Dψ(y)

∂Φ(x, y)∂ν(y)

ds(y), x ∈ D.

Then, by Theorems 6.22 and 6.23, we can write

u = −2V(I − K)−1u|∂D

with the double-layer integral operator K for the boundary ∂D. By the Cauchy–Schwarz inequality, there exists a constant c depending on D and W such that

‖Vψ‖C(W) ≤ c‖ψ‖L2(∂D)

for all ψ ∈ C(∂D). Since K has a continuous kernel, from Theorems 2.28 and 6.21,and the Riesz Theorem 3.4 it follows that (I − K)−1 is bounded with respect to theL2 norm. �

For the convergence on the boundary we can make use of Theorems 16.9 and16.10. As already mentioned, the conditions f ∈ A(L2(Γ)) and f ∈ AA∗(L2(∂D)) ofthese two theorems can be interpreted as regularity assumptions on f because theycontrol how fast the Fourier coefficients ( f , gn) tend to zero as n → ∞. They mayalso be considered as conditions on how far the solution to the Dirichlet problemcan be continued across the boundary ∂D. The following example will shed somelight on these remarks.

Example 16.30. Consider the simple situation where D is the unit disk and Γ aconcentric circle with radius R > 1. We parameterize x ∈ ∂D by

x(t) = (cos t, sin t), 0 ≤ t ≤ 2π,

and x ∈ Γ byx(t) = (R cos t,R sin t), 0 ≤ t ≤ 2π,

and writeψ(t) := ϕ(x(t)) and g(t) := f (x(t)). Then we transform the integral equation(16.38) into

R2π

∫ 2π

0

cos(t − τ) − R

1 − 2R cos(t − τ) + R2ψ(τ) dτ = g(t), 0 ≤ t ≤ 2π, (16.41)


which we rewrite in operator notation Aψ = g with an obvious meaning for theself-adjoint operator A : L2[0, 2π]→ L2[0, 2π]. By decomposing

2(cos t − R)eint

1 − 2R cos t + R2=

eint

eit − R+

eint

e−it − R,

we derive the integrals

R2π

∫ 2π

0

(cos t − R)eint

1 − 2R cos t + R2dt =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

−12R|n|

, n = ±1,±2, . . . ,

−1, n = 0.

Hence, for ψn(t) = eint we have

Aψn =−1

2R|n|(1 + δn0)ψn, n ∈ ZZ.

Therefore, in polar coordinates x = (r cos θ, r sin θ), a singular system (μn, ϕn, gn) ofA is given by

μn =1

2R|n|+1/2(1 + δn0), ϕn(x) =

−1√2πR

einθ, gn(x) =1√2π

einθ

for n = 0,±1,±2, . . . . The approximate solution to the Dirichlet problem obtainedby (16.40) can now be written in the form

uα(r, θ) =1√2π

∞∑n=−∞

μ2n

α + μ2n

( f , gn) r|n|einθ

compared with the exact separation of variables solution

u(r, θ) =1√2π

∞∑n=−∞

( f , gn) r|n|einθ. (16.42)

The solvability condition f ∈ A(L2(Γ)) now corresponds to

∞∑n=−∞

R2|n| |( f , gn)|2 < ∞,

and therefore, by the Cauchy–Schwarz inequality, the series (16.42) can be seen toconverge uniformly in each disk with radius ρ < R centered at the origin. Hence, inthis case the solution to the Dirichlet problem in the unit disk with boundary valuesf can be harmonically extended into the interior of the concentric circle of radius R.The stronger condition f ∈ AA∗(L2(∂D)) corresponds to

∞∑n=−∞

R4|n||( f , gn)|2 < ∞


and, similarly, implies a harmonic continuation into the interior of the concentriccircle of radius R2. �

To illustrate our results numerically we consider the boundary condition given byf (x) = ln |x − x0|, where x0 = (q, 0) with q > 1. Here, the solution to the boundaryvalue problem clearly allows a harmonic continuation into the open disk of radius qcentered at the origin. Table 16.4 gives the error ‖Aϕα− f ‖ between the exact solutionand the Tikhonov regularization for (16.41), approximated by Nystrom’s methodusing the trapezoidal rule with 32 knots. The radius of Γ is chosen to be R = 2.The dependence of the error on the parameters α and q confirms the convergencebehavior predicted by our theoretical results.


logα q = 10 q = 4 q = 3 q = 2 q = 1.5

−5 0.00006 0.00009 0.0001 0.001 0.01−4 0.00060 0.00091 0.0014 0.006 0.02−3 0.00642 0.00873 0.0125 0.030 0.08−2 0.06255 0.07370 0.0955 0.162 0.27

This approach, of course, can be extended to cover the interior and exteriorDirichlet and Neumann problem in two and three space dimensions. Also, insteadof the double-layer potential, a single-layer potential can be used. For an applicationof these ideas to the Helmholtz equation, see Kress and Mohsen [148].

In a discretized version of the approach for the solution of the interior Dirichletproblem in D with boundary values u = f via a single-layer potential on a curve Γlocated in IR2 \ D, in principle, one tries to approximate the solution u by a finitesum of fundamental solutions

un(x) =n∑

k=1

ϕkΦ(x, yk)

with source points yk ∈ IR2 \ D and real coefficients ϕk, k = 1, . . . , n. For a givenchoice of the source points, the collocation un(x j) = f (x j) of the boundary con-dition at collocation points x j ∈ ∂D for j = 1, . . . , n provides a linear system fordetermining the coefficients ϕk, k = 1, . . . , n. For obvious reasons this approach tothe approximate solution of boundary value problems is known as the method of fun-damental solutions. Because of its close relation to the integral equations discussedabove it suffers from severe ill-posedness which compensates for the advantage ofits computational simplicity (see [50, 66]).

By Green’s Theorem 6.3, for each harmonic function u in C2(D)⋂

C1(D) wehave that ∫

∂D

{∂u∂ν

(y)Φ(x, y) − u(y)∂Φ(x, y)∂ν(y)

}ds(y) = 0

Problems 349

for all x ∈ IR2 \ D. From this, we derive the ill-posed integral equation of the firstkind ∫

∂Dψ(y)Φ(x, y) ds(y) =

∫∂D

f (y)∂Φ(x, y)∂ν(y)

ds(y), x ∈ Γ,

for the unknown normal derivative ψ = ∂u/∂ν on ∂D. This method to solve the po-tential theoretic boundary value problems is due to Kupradze [153]. A fairly com-plete description and extensive references are given by Christiansen [29].

The integral equations described in this section are simple, since as opposed tothe boundary integral equations they possess smooth kernels. Hence, they allow theapplication of simple quadrature rules without the need to treat singular integrals. Onthe other hand, in general, we can expect only poor convergence as the regularizationparameter tends to zero. Therefore, this approach may serve as a very instructiveexample for the use of Tikhonov regularization but, though it has its supporters in theliterature, it cannot be considered as a serious competitor for the numerically well-established boundary integral equations. However, in Chapter 18 we will illustratethat integral equations of this nature have some merits in solving inverse problemswhere the boundary is not known.

Problems

16.1. Show that the quasi-solution given by Theorem 16.12 depends weakly continuously on theright-hand side, i.e., norm convergence fn → f , n → ∞, implies weak convergence for the cor-responding quasi-solutions ϕn ⇀ ϕ, n → ∞. Show that in the complement of A(B[0; ρ]) thequasi-solution depends continuously on the right-hand side. How are these results related to theregularity Theorem 16.13?Hint: By Theorem 16.3 the sequence (ϕn) contains a subsequence that converges weakly to someelement ψ in X. Establish that ‖ψ‖ ≤ ρ and ‖Aψ− f ‖ ≤ ‖Aϕ − f ‖, i.e., ψ is a quasi-solution, and byuniqueness it must coincide with ϕ. Use this result to show that the sequence (ϕn) itself convergesweakly to ϕ. Finally, conclude norm convergence from ‖ϕn‖ = ρ.

16.2. Show that the minimum norm solution given by Theorem 16.18 depends weakly continu-ously on the right-hand side. Show that in the complement of B[0; δ] ⊂ Y the minimum normsolution depends continuously on the right-hand side. How are these results related to the regular-ity Theorem 16.19?Hint: Show that there exists an element χ ∈ X with χ ∈ U fn for all sufficiently large n. Thereforethe sequence (ϕn) is bounded and contains a weakly convergent subsequence with limit ψ ∈ X.Show that ‖Aψ − f ‖ ≤ δ and ‖ψ‖ ≤ ‖ϕ‖, i.e., ψ is a minimum norm solution, and by uniqueness itmust coincide with ϕ. Use this result to show that the sequence (ϕn) itself converges weakly to ϕ.Finally, conclude norm convergence from ‖Aϕn − fn‖ = δ.

16.3. Show that the convergence order given in Theorem 16.20 is optimal.

16.4. Show that the solution of the integral equation in Example 16.16 is unique.

Hint: Show that∫ 1

0ynϕ(y) dy = 0, n = 0, 1, 2, . . . , for each solution ϕ of the homogeneous equation.

16.5. Show that a linear operator A : X → Y from a Hilbert space X into a Hilbert space Y iscompact if and only if weak convergence ϕn ⇀ ϕ for n→ ∞ implies norm convergence Aϕn → Aϕfor n→∞.

Chapter 17Regularization by Discretization

We briefly return to the study of projection methods and will consider theirapplication to ill-posed equations of the first kind. In particular we will present anexposition of the moment discretization method. For further studies of regulariza-tion through discretization, we refer to Baumeister [15], Kirsch [126], Louis [160],Natterer [183] and Rieder [204].

17.1 Projection Methods for Ill-Posed Equations

Solving an equation of the first kind

Aϕ = f (17.1)

with an injective compact operator A : X → Y from a Banach space X into a Banachspace Y numerically without regularization usually means approximately solving itby a projection method. The ill-posedness of the equation (17.1) then will cause thecondition number of the discrete linear system to grow with the dimension n of thesubspace used for the projection method. Increasing n will make the error due tothe discretization smaller, but due to ill-conditioning the computation will be con-taminated by errors in the given data f . Note that in actual numerical computationssuch errors will automatically occur because of round-off effects for the data. Onthe other hand, if n is small, then the approximation is robust against errors in fbut will be inaccurate due to a large discretization error. This dilemma calls for acompromise in the choice of the discretization parameter n.

Recalling the projection methods from Chapter 13, let Xn ⊂ X and Yn ⊂ Y betwo sequences of subspaces with dim Xn = dim Yn = n and let Pn : Y → Yn beprojection operators. For given f ∈ A(X) the projection method approximates thesolution ϕ ∈ X of Aϕ = f by the solution ϕn ∈ Xn of the projected equation

PnAϕn = Pn f . (17.2)


351

352 17 Regularization by Discretization

For the remainder of this chapter we assume that the projection method is conver-gent for exact right-hand sides f ∈ A(X). This, by Theorem 13.6, implies that thefinite-dimensional operators PnA : Xn → Yn are invertible and that the sequenceof operators (PnA)−1PnA : X → Xn is uniformly bounded. Then the operatorsRn := (PnA)−1Pn : Y → Xn provide a regularization scheme in the sense of Defini-tion 15.5 with regularization parameter n. The approximation

ϕδn := Rn f δ (17.3)

corresponding to (15.6) is obtained by numerically performing the projection methodfor the inexact right-hand side f δ.

Theorem 17.1. Let A : X → Y be an injective compact linear operator, f ∈ A(X),and f δ ∈ Y satisfy ‖ f δ− f ‖ ≤ δ. Then the approximationϕδn := Rn f δ by the projectionmethod satisfies

‖ϕδn − A−1 f ‖ ≤ ‖Rn‖ δ +C infψ∈Xn

‖ψ − A−1 f ‖ (17.4)

for some constant C.

Proof. This follows from Theorem 13.6 by decomposing

‖ϕδn − A−1 f ‖ ≤ ‖Rn‖ ‖ f δ − f ‖ + ‖Rn f − A−1 f ‖

≤ ‖Rn‖ δ + (1 + M) infψ∈Xn

‖ψ − A−1 f ‖,

where M is a bound on ‖RnA‖. �

The error estimate (17.4) corresponds to (15.7) and illustrates that the total errorconsists of two parts: the influence of the incorrect data and the discretization errorof the projection method. Recall that for the compact operator A, by Theorem 15.6,the operators Rn cannot be uniformly bounded.

At this point we wish to emphasize the fact that for the analysis of the discretiza-tion error we may use any Banach space structure for the range space. In particular,we may choose an image space such that the equation becomes well-posed. Hence,we can apply the convergence results on projection methods given in Chapter 13.But for the influence of the error in the right-hand side we have to stick with animage space as dictated by the needs of the particular problem. We want to makethese remarks more precise through the following theorem.

Theorem 17.2. Let A : X → Y be an injective compact operator from a Banachspace X into a Banach space Y and assume an additional Banach space Z ⊂ Ywith continuous imbedding such that A : X → Z is an isomorphism. Let Xn ⊂ Xand Yn ⊂ Z ⊂ Y be two sequences of subspaces with dim Xn = dim Yn = n and letPn : Y → Yn be projection operators (bounded with respect to the norm on Y) andassume that the projection method is convergent for all f ∈ A(X) = Z. Then the

17.1 Projection Methods for Ill-Posed Equations 353

operators Rn : Y → X satisfy

‖Rn‖ ≤ C ‖Pn‖ supf∈Yn , ‖ f ‖Y=1

‖ f ‖Z

for some constant C.

Proof. For abbreviation we set

γn := supf∈Yn , ‖ f ‖Y=1

‖ f ‖Z .

Since by Theorem 1.6 all norms on the finite-dimensional subspace Yn are equiva-lent, we have that γn is finite. Then, from ‖ f ‖Z ≤ γn‖ f ‖Y for all f ∈ Yn, it followsthat

‖Pn f ‖Z ≤ γn‖Pn f ‖Y ≤ γn‖Pn‖ ‖ f ‖Y ≤ Mγn‖Pn‖ ‖ f ‖Z (17.5)

for all f ∈ Z and some constant M, since the imbedding from Z into Y is assumedto be bounded. Hence, the operators Pn are also bounded with respect to the normon Z and we can interpret the projection scheme for A : X → Y also as a projectionscheme in the structure A : X → Z. Since we assume convergence for all f ∈ A(X),by Theorem 13.6, we know that the operators (PnA)−1PnA : X → X are uniformlybounded. Therefore, for all f ∈ Y with ‖ f ‖Y = 1 we can estimate

‖Rn f ‖ = ‖(PnA)−1Pn f ‖ = ‖(PnA)−1PnAA−1Pn f ‖

≤ ‖(PnA)−1PnA‖ ‖A−1‖Z→X ‖Pn f ‖Z ≤ Cγn‖Pn‖for some constant C. �

We illustrate the use of Theorem 17.2 by considering the following example.

Example 17.3. Let A : L2[0, 2π]→ L2[0, 2π] be given by

(Aϕ)(t) =1

2π

∫ 2π

0

{ln 4 sin2 t − τ

2+ K(t, τ)

}ϕ(τ)dτ, 0 ≤ t ≤ 2π, (17.6)

where we assume that K is infinitely differentiable and 2π-periodic with respectto both variables. As a consequence of Theorem 8.24, it can be seen that A :L2[0, 2π]→ H1[0, 2π] is an isomorphism, provided A is injective. In this case, fromTheorem 13.32 we can conclude that the Petrov–Galerkin method with the orthog-onal projection operator Pn from L2[0, 2π] onto the space Yn of trigonometric poly-nomials of degree less than or equal to n converges for A : L2[0, 2π] → H1[0, 2π].(Here we need to use the fact that by Theorem 1.25 it can be observed that Pn coin-cides with the orthogonal projection operator from H1[0, 2π] onto Yn.) In particular,if the right-hand side also is infinitely differentiable, we have a convergence orderO(n−m) for all m ∈ IN. On the other hand, the orthogonal projection operator hasnorm ‖Pn‖ = 1 and for the subspace Yn we readily see that γn =

√1 + n2. Hence, in

this case we have ‖Rn‖ = O(n) and the equation of the first kind with the operator


(17.6) is only mildly ill-posed. In particular, the decay of the approximation errorO(n−m) dominates the increase O(n) of the data error. Therefore, provided the errorof the right-hand side is reasonably small, we conclude that it is numerically safeto apply the Petrov–Galerkin method in this case because we need only a compar-atively small n to achieve an acceptable approximation error (see Example 13.23and Problem 17.1). For a more detailed discussion of the regularization of integralequations of the first kind with a logarithmic singularity and in particular Symm’sintegral equation we refer to [126]. �

To make the remarks in the preceding Example 17.3 more precise, followingideas developed by Kaltenbacher [114], we briefly consider the selection of the dis-cretization level via the discrepancy principle for a special type of mildly ill-posedequations. We note that in view of Theorem 11.8 the assumptions of the followingtheorem are satisfied for the integral equation (17.6) for a projection method with Xn

the space of trigonometric polynomials of the form (13.26) both for the collocationand the Petrov–Galerkin method.

Theorem 17.4. As in Theorem 17.2 let X, Y and Z be Banach spaces such that Z ⊂ Ywith continuous imbedding and let A : X → Y be an injective compact linearoperator of the form A = A0 + B such that A0 : X → Z is an isomorphism andB : X → Z is compact. Further let Xn ⊂ X be a sequence of finite-dimensionalsubspaces, define subspaces Yn ⊂ Y by Yn := A0(Xn) and let Pn : Y → Yn beprojection operators that are uniformly bounded. Assume that the correspondingprojection method converges for all f ∈ A(X) = Z and that

βn := supg∈Z, ‖g‖Z=1

‖Png − g‖Y and γn := supf∈Yn, ‖ f ‖Y=1

‖ f ‖Z

satisfy 1/M ≤ βnγn ≤ M for all n ∈ IN and some constant M > 1. Then wecan choose a positive constant τ such that for each f ∈ A(X) and f δ ∈ Y with‖ f δ − f ‖ ≤ δ and δ > 0 there exists a smallest integer n = n(δ), depending on f δ andδ, such that

‖ARn(δ) f δ − f δ‖ ≤ τδ. (17.7)

Proof. From the assumptions on A, A0 and B, by the Riesz theory we have thatA : X → Z is an isomorphism. By the definition of Yn, clearly, PnA0 : Xn → Yn

coincides with A0 : Xn → Yn and therefore (PnA0)−1 = A−10 . From the proof of

Theorem 13.7 we know that the inverse operators [I + (PnA0)−1PnB]−1 : Xn → Xn

exist and are uniformly bounded for all sufficiently large n and therefore (PnA)−1 :Yn → Xn is given by

(PnA)−1 = [I + (PnA0)−1PnB]−1A−10 .

From this, recalling that Rn = (PnA)−1Pn, as in the proof of Theorem 17.2 we inferthat

‖Rng‖ ≤ c1γn‖g‖ (17.8)

for all g ∈ Y, all sufficiently large n ∈ IN and some positive constant c1.


Now, in view of PnARn f δ = Pn f δ, we compute

ARn f δ − f δ = (I − Pn)(ARn f δ − f δ) = (I − Pn)[(ARn − I)( f δ − f ) + (ARn − I) f ].

From this, using the estimates

‖(I − Pn)ARn( f δ − f )‖ ≤ βn‖A‖X→Z‖Rn( f δ − f )‖ ≤ c1βnγn‖A‖X→Zδ

and‖(I − Pn)( f δ − f )‖ ≤ ‖I − Pn‖ δ

and the assumptions on the uniform boundedness of the Pn and the products βnγn,we obtain that

‖ARn f δ − f δ‖ ≤ Cδ + ‖(I − Pn)A(Rn f − A−1 f )‖ (17.9)

for all sufficiently large n and some positive constant C. Choosing τ > C, now theassertion of the theorem follows from the convergence Rn f → A−1 f as n → ∞ forf ∈ A(X). �

Theorem 17.5. Under the assumptions of Theorem 17.4 the discrepancy principleis a regular regularization strategy, i.e, we have that

Rn(δ) f δ → A−1 f , δ→ 0. (17.10)

Proof. We choose m(δ) as the smallest m ∈ IN with the property

βm‖A‖X→Z‖Rm f − A−1 f ‖ ≤ (τ − C)δ.

From (17.9) we observe that

(τ −C)δ < ‖(I − Pn)A(Rn f − A−1 f )‖ ≤ βn‖A‖X→Z‖Rn f − A−1 f ‖for all n < n(δ) and consequently n(δ) ≤ m(δ). After transforming

Rn(δ) f δ − A−1 f = Rm(δ)Pm(δ)(ARn(δ) f δ − f δ) + Rm(δ)( f δ − f ) + Rm(δ) f − A−1 f

and using (17.7) and (17.8), we can estimate

‖Rn(δ) f δ − A−1 f ‖ ≤ c1(τ + 1)γm(δ)δ + ‖Rm(δ) f − A−1 f ‖.From this, in view of the definition of m(δ) and the boundedness of the productsβnγn from below, we obtain that

‖Rn(δ) f δ − A−1 f ‖ ≤ c2γm(δ)δ (17.11)


with a positive constant c2. Using again the definition of m(δ) and the boundednessof the products βnγn, from (17.11) we further find that

‖Rn(δ) f δ − A−1 f ‖ ≤ c3‖Rm(δ)−1 f − A−1 f )‖ (17.12)

with a positive constants c3. Now have to distinguish two cases. When m(δ) → ∞for δ → 0 then from (17.12) it immediately follows that Rn(δ) f δ → A−1 f as δ → 0.If m(δ) does not tend to infinity as δ → 0 assume that there exists a null sequenceδk such that Rn(δk) f δk does not converge to A−1 f as k → ∞. Then by selecting asubsequence we may assume that there exists N ∈ IN such that m(δk) ≤ N for allk ∈ IN and from (17.11) we obtain the contradiction

‖Rn(δk) f δk − A−1 f ‖ ≤ c2γNδk → 0, k → ∞,and this ends the proof. �

For further analysis on the discrepancy principle for regularization via discretiza-tion we refer to [15, 85, 190, 204, 239].

For projection methods the norm on the image space Y might not always bethe most appropriate measure to describe the inaccuracy of the data. Consider thePetrov–Galerkin method for an operator A : X → Y between two Hilbert spacesX and Y. With subspaces Xn = span{u1, . . . , un} and Yn = span{v1, . . . , vn} we writeϕn =

∑nk=1 γkuk and then have to solve the Petrov–Galerkin equations

n∑k=1

γk(Auk, v j) = ( f , v j), j = 1, . . . , n, (17.13)

for the coefficients γ1, . . . , γn. Here, it makes more sense to measure the error of thediscrete data vector F = (( f , v1) , . . . , ( f , vn)) in the Euclidean norm on Cn (comparethe stability analysis of Section 14.1 for equations of the second kind). Let Fd ∈ Cn

be a perturbed right-hand side for (17.13) with

‖Fd − F‖2 ≤ d

and denote the Petrov–Galerkin solution to these inexact data by ϕdn. After introduc-

ing the matrix operators En, An : Cn → Cn by

(Enγ) j :=n∑

k=1

(uk, u j)γk, j = 1, . . . , n,

and

(Anγ) j :=n∑

k=1

(Auk, v j)γk, j = 1, . . . , n,

for γ = (γ1, . . . , γn) ∈ Cn we can formulate the following error estimate whichreplaces Theorem 17.1.


Theorem 17.6. The error in the Petrov–Galerkin method can be estimated by

‖ϕdn − A−1 f ‖ ≤ d√

λnνn+ C inf

ψ∈Xn

‖ψ − A−1 f ‖, (17.14)

where λn and νn denote the smallest singular values of An and AnE−1n , respectively,

and C is some positive constant.

Proof. Using the singular value decomposition Theorem 15.16, we can estimate

‖ϕn‖2 =n∑

j, k=1

γ jγk (u j, uk) = (Enγ, γ) = (EnA−1n F, A−1

n F)

≤ ‖EnA−1n ‖2 ‖A−1

n ‖2 ‖F‖22 ≤1

λnνn‖F‖22.

Now we apply this inequality to the difference ϕdn −ϕn and obtain (17.14) by decom-

posing the error as in the proof of Theorem 17.1. �

For the least squares method as described in Theorem 13.31, i.e., for Yn = A(Xn),the matrix operator An assumes the form

(Anγ) j :=n∑

k=1

(Auk, Au j)γk, j = 1, . . . , n,

and is self-adjoint and positive definite. In particular, for an orthonormal basisu1, . . . , un of Xn, we have En = I. Hence, in this case we only need to be con-cerned with the eigenvalues of An. Let (μm, ϕm, gm), m = 1, 2, . . . , be a singularsystem of the compact operator A. Then we can choose an element ϕ =

∑nj=1 γ ju j in

Xn such that (ϕ, ϕm) = 0, m = 1, . . . , n − 1, and ‖ϕ‖ = 1. Now, by the singular valuedecomposition Theorem 15.16, we obtain

μ2n ≥ (A∗Aϕ, ϕ) = (Anγ, γ) ≥ λn,

since ‖γ‖2 = ‖ϕ‖ = 1. Therefore,

1λn≥ 1μ2

n.

This again exhibits the influence of the singular values on the degree of ill-posedness.For the integral equation of Example 16.27, by Theorem 15.20, the singular val-

ues decay at least exponentially. Hence, projection methods cannot be recommendedfor this equation. More generally, for severely ill-posed equations projection meth-ods should only be applied in connection with further regularization, for example,by Tikhonov regularization.


17.2 The Moment Method

We proceed by describing the moment method to approximately solve an equationof the first kind. Let A : X → Y be an injective bounded linear operator from aHilbert space X into another Hilbert space Y. Choose a finite-dimensional subspaceYn ⊂ Y with dim Yn = n and, given an element f ∈ Y, define the affine linearsubspace Un ⊂ X by

Un := {ϕ ∈ X : (Aϕ, g) = ( f , g), g ∈ Yn}.We assume Un is nonempty. By the Riesz representation Theorem 4.10, boundedlinear functionals in a Hilbert space can be equivalently expressed by scalar prod-ucts. Therefore, we may interpret Un as the set of all elements ϕ ∈ X for which nlinearly independent functionals vanish when applied to the residual Aϕ − f .

Definition 17.7. An element ϕn ∈ Un is called a moment solution to Aϕ = f withrespect to Yn if

‖ϕn‖ = infϕ∈Un

‖ϕ‖.Note that ϕn is a moment solution with respect to Yn if and only if it is a best

approximation to the zero element in X with respect to the closed affine linear sub-space Un. Finding a best approximation with respect to an affine linear subspacecan be equivalently reduced to finding a best approximation with respect to a linearsubspace. Therefore, by Theorem 1.26 there exists a unique moment solution, andby Theorem 1.25 it can be characterized as the unique element ϕn ∈ Un satisfying

(ϕn − ϕ, ϕn) = 0 (17.15)

for all ϕ ∈ Un. The following theorem gives an interpretation of the moment methodas a projection method.

Theorem 17.8. Let A : X → Y be an injective bounded linear operator with denserange. Then the moment approximate solution of Aϕ = f with respect to a subspaceYn ⊂ Y coincides with the Petrov–Galerkin approximation corresponding to thesubspaces Xn := A∗(Yn) ⊂ X and Yn ⊂ Y.

Proof. By assumption we have A(X) = Y and therefore, by Theorem 15.8, the adjointoperator A∗ : Y → X is injective. Hence, dim Xn = dim Yn = n. Let Pn : Y → Yn bethe orthogonal projection operator. We will show that the Petrov–Galerkin operatorPnA : Xn → Yn is injective. Indeed, assume that ϕ ∈ Xn satisfies PnAϕ = 0. Thenwe can write ϕ = A∗g for some g ∈ Yn and have

‖ϕ‖2 = (A∗g, A∗g) = (AA∗g, g) = (PnAϕ, g) = 0.

The injectivity of PnA ensures that the Petrov–Galerkin approximation ϕn exists andis unique. It can be written in the form ϕn = A∗gn with gn ∈ Yn, and it satisfies

(AA∗gn, g) = ( f , g) (17.16)

17.2 The Moment Method 359

for all g ∈ Yn. This implies ϕn ∈ Un and

(ϕn − ϕ, ϕn) = (ϕn − ϕ, A∗gn) = (Aϕn − Aϕ, gn) = 0

for all ϕ ∈ Un. Therefore, by (17.15), the Petrov–Galerkin approximation ϕn is themoment solution with respect to Un. �

When f ∈ A(X) we write Aϕ = f and have (17.16) to be equivalent to

(A∗gn, A∗g) = (ϕ, A∗g)

for all g ∈ Yn. This implies that the moment method corresponds to the least squaresmethod for the adjoint (or dual) equation A∗h = ϕ with respect to Yn. Therefore itis also known as the dual least squares method (for the least squares method, recallTheorem 13.31).

We continue our analysis with a convergence result on the moment method. Notethat the least squares method, in general, will not converge (see the counterexamplein Problem 17.2).

Theorem 17.9. Let A be as in Theorem 17.8 and let f ∈ A(X). Assume that thesequence of subspaces Yn ⊂ Y has the denseness property

infh∈Yn

‖h − g‖ → 0, n→ ∞, (17.17)

for all g ∈ Y. Then the moment method converges, i.e., ϕn → A−1 f , n→ ∞.Proof. Let Pn : Y → Yn and Qn : X → Xn = A∗(Yn) be the orthogonal projectionoperators. We will show that (17.17) implies convergence

Qnϕ→ ϕ, n→ ∞,for all ϕ ∈ X. Since A is injective, by Theorem 15.8, the range A∗(Y) is dense in X.Therefore, given ϕ ∈ X and ε > 0, there exists g ∈ Y such that ‖A∗g − ϕ‖ < ε/2. Asa consequence of (17.17) we can choose N ∈ IN such that ‖Png − g‖ < ε‖A‖/2 forall n ≥ N. Since A∗Png ∈ Xn, for all n ≥ N we can estimate

‖Qnϕ − ϕ‖ ≤ ‖A∗Png − ϕ‖ ≤ ‖A∗(Png − g)‖ + ‖A∗g − ϕ‖ < ε.Now the assertion of the theorem follows from the fact that for f ∈ A(X) the dualleast squares solution ϕn is given by ϕn = QnA−1 f . Indeed, the orthogonal projectionsatisfies

(QnA−1 f , A∗g) = (A−1 f , A∗g)

for all g ∈ Yn. This is equivalent to (AQnA−1 f , g) = ( f , g) for all g ∈ Yn. HenceQnA−1 f solves the Petrov–Galerkin equation. �

For the dual least squares method, from Xn = A∗(Yn) we observe that the matrixoperators occurring in Theorem 17.6 are related by En = An. Therefore, in this case


the error estimate (17.14) can be simplified to

‖ϕdn − A−1 f ‖ ≤ d√

λn+C inf

ψ∈A∗(Yn)‖ψ − A−1 f ‖. (17.18)

17.3 Hilbert Spaces with Reproducing Kernel

For an application of the above results to collocation methods for equations of thefirst kind, we obviously need a Hilbert space in which the point evaluation function-als are bounded. Therefore, we briefly introduce the concept of Hilbert spaces withreproducing kernel.

Definition 17.10. Let H be a Hilbert space of real- or complex-valued functions fdefined on an interval [a, b]. A function M on [a, b] × [a, b] is called a reproducingkernel if hx := M(x, ·) belongs to H for all x ∈ [a, b] and

( f , hx) = f (x) (17.19)

for all x ∈ [a, b] and all f ∈ H.

Theorem 17.11. A Hilbert space H has a reproducing kernel if and only if the eval-uation functionals

f → f (x)

are bounded for all x ∈ [a, b].

Proof. If H has a reproducing kernel, then (17.19) implies continuity for the evalu-ation functionals. Conversely, if the evaluation functionals are bounded, then by theRiesz Theorem 4.10 for each x ∈ [a, b] there exists an element hx ∈ H such that(17.19) holds for all f ∈ H. Hence M(x, ·) := hx is a reproducing kernel. �

Obviously, L2[a, b] is not a Hilbert space with reproducing kernel, whereasH1[a, b] is, since by (16.29) the maximum norm is weaker than the H1 norm.

For each injective linear operator A : X → Y acting between two Hilbert spacesX and Y we can make the range H := A(X) a Hilbert space with the scalar productgiven by

( f , g)H = (A−1 f , A−1g)X , f , g ∈ A(X). (17.20)

Indeed, since A is injective, (17.20) defines a scalar product on A(X). Assume ( fn)is a Cauchy sequence in H. Then we can write fn = Aϕn with ϕn ∈ X and find that(ϕn) is a Cauchy sequence in X because

‖ϕn − ϕm‖X = ‖ fn − fm‖H .Since X is a Hilbert space, there exists ϕ ∈ X with ϕn → ϕ, n → ∞. Then we havef = Aϕ ∈ H and

‖ fn − f ‖H = ‖ϕn − ϕ‖X → 0, n→ ∞.

17.4 Moment Collocation 361

Hence, with the scalar product given by (17.20) the range A(X) is a Hilbert space.By (17.20), the linear operator A : X → H can be seen to be bounded with theadjoint operator A∗ : H → X given by A∗ = A−1. In the case of an integral operatorthis Hilbert space turns out to have a reproducing kernel.

Theorem 17.12. Let A : L2[a, b]→ L2[a, b] denote an injective integral operator

(Aϕ)(x) =∫ b

aK(x, y)ϕ(y) dy, a ≤ x ≤ b,

with continuous kernel K. Then the range H = A(L2[a, b]) furnished with the scalarproduct (17.20) is a Hilbert space with reproducing kernel

M(x, y) =∫ b

aK(x, z)K(y, z) dz, x, y ∈ [a, b]. (17.21)

Proof. Define M by (17.21) and set hx := M(x, ·) and kx := K(x, ·). Then

(Akx)(y) =∫ b

aK(y, z)K(x, z) dz = M(x, y),

i.e., Akx = hx for all x ∈ [a, b]. Therefore, writing f = Aϕ, we find

( f , hx)H = (A−1 f , A−1hx)L2 = (ϕ, kx)L2

=

∫ b

aK(x, y)ϕ(y) dy = (Aϕ)(x) = f (x).

Hence H is a Hilbert space with reproducing kernel M. �

The statement of Theorem 17.12 remains valid for a weakly singular kernel Kwith the property that M, defined by (17.21), is continuous.

For a detailed study of Hilbert spaces with reproducing kernels, we refer to Aron-szajn [8] and Meschkowski [168].

17.4 Moment Collocation

For the numerical solution of the integral equation of the first kind

∫ b

aK(x, y)ϕ(y) dy = f (x), a ≤ x ≤ b, (17.22)

with continuous kernel K by moment collocation we choose n collocation pointsx j ∈ [a, b], j = 1, . . . , n. We approximate the solution ϕ of (17.22) by a function ϕn


with minimal L2 norm satisfying the integral equation at the collocation points

∫ b

aK(x j, y)ϕn(y) dy = f (x j), j = 1, . . . , n. (17.23)

With the kernel function given in Theorem 17.12 we can rewrite (17.23) in the form

(Aϕn, hxj)H = ( f , hxj)H , j = 1, . . . , n. (17.24)

Hence, the moment collocation coincides with the moment method applied to theintegral operator A : L2[a, b] → H with the subspaces Hn ⊂ H given by Hn :=span{hx1 , . . . , hxn}. In particular, from the analysis in Sections 17.2 and 17.3, weconclude that a unique solution to (17.23) with minimal norm exists and is given bythe orthogonal projection of the exact solution ϕ onto the subspace

Xn = A∗(Hn) = A−1(Hn) = span{K(x1, ·), . . . ,K(xn, ·)}.We may assume that the kernel functions K(x1, ·), . . . ,K(xn, ·) are linearly inde-pendent, since otherwise the equations (17.23) would be linearly dependent. Then,writing

ϕn =

n∑k=1

γkK(xk, ·) (17.25)

we have to solve the linear system

n∑k=1

γk

∫ b

aK(xk, y)K(x j, y) dy = f (x j), j = 1, . . . , n, (17.26)

with a positive definite matrix for the coefficients γ1, . . . , γn.

Theorem 17.13. Assume that the sequence (xn) of collocation points is dense in theinterval [a, b]. Then the corresponding moment collocation solutions converge, i.e.,‖ϕn − A−1 f ‖L2 → 0, n→ ∞.Proof. By Theorem 17.9, applied to A : L2[a, b] → H, it suffices to show thatU := span{hxn : n ∈ IN} is dense in H. Note that each g ∈ H is continuous, since Ahas continuous kernel K. Denote by P : H → U the orthogonal projection operator.For g ∈ H, by Theorem 1.25, we have Pg − g ⊥ U. This implies (Pg − g, hxn)H = 0for all n ∈ IN, i.e., (Pg)(xn) = g(xn) for all n ∈ IN. From this, by the denseness of thecollocation points and the continuity of Pg and g, it follows that g = Pg ∈ U. �

For the error we can apply the special case (17.18) of Theorem 17.6. Let Fd ∈ Cn

be a perturbation of the data vector F = ( f (x1) , . . . , f (xn)) with ‖Fd − F‖2 ≤ d anddenote the moment collocation solution to these inexact data via (17.25) and (17.26)by ϕd

n. Then we have the following error estimate.

Problems 363

Theorem 17.14. The error in the moment collocation can be estimated by

‖ϕdn − A−1 f ‖L2 ≤ d√

λn+C inf

ψ∈Xn

‖ψ − A−1 f ‖L2 ,

where λn denotes the smallest eigenvalue of the positive definite matrix of the linearsystem (17.26) and C is some positive constant.

The quality of the discretization depends on how well the exact solution can beapproximated by the subspace Xn, i.e., (besides the smoothness of the exact solu-tion) it depends on the smoothness of the kernel: the smoother the kernel, the betterthe approximation. On the other hand, the smoothness of the kernel also controlsthe asymptotics of the singular values of the integral operator and, consequently, thebehavior of the eigenvalues λn (see Theorem 15.20). In general, the decay of thesingular values increases with the smoothness of the kernel. Therefore, we can ex-pect the moment collocation to deliver reliable approximations only for kernels withpoor smoothness. For an example see Problem 17.5.

For more details on the moment collocation, we refer to Engl [48], Groetsch [74],and Nashed and Wahba [181], and the literature therein.

Problems

17.1. Formulate and prove a variant of Theorem 17.6 for the collocation method of Section 13.4.Apply the result to equation (13.30).

17.2. Let A : X → X be a nonnegative self-adjoint compact operator with one-dimensionalnullspace N(A) = span{u0} and spectral decomposition

Aϕ =∞∑j=1

λ j(ϕ, uj)uj,

where λ j � 0 for all j ∈ IN. Consider the compact operator S : X → X with

Sϕ := Aϕ + (ϕ, u0)g,

where g ∈ X with (g, u0) = 1 and (g, uj) � 0 for all j ∈ IN. For the equation Sϕ = f with right-handside f = Ag + g and exact solution ϕ = g show that the coefficients γ j, j = 0, . . . , n, for the leastsquares approximation (see Theorem 13.31)

ϕn =

n∑j=0

γ ju j

with respect to the subspace Xn = span{u0 , . . . , un} satisfy

(γ0 − 1){1 + ‖g − Png‖2} = (Ag, g − Png)

andλ j{γ j − (g, uj)} = (1 − γ0)(g, uj), j = 1, . . . , n.


Here Pn : X → Xn denotes the orthogonal projection. Use these results to establish the inequalities

‖ϕn − g‖ ≥ |γn − (g, un)| ≥ |(Ag, g − Png)| |(g, un)|{1 + ‖g‖2}|λn | ≥ |λn+1 |

|λn | |(g, un+1)|2 |(g, un)|.

Use this to show that A and g can be chosen such that the least squares solutions ϕn do not convergeto the exact solution (see Seidman [218]).

17.3. Show that in a Hilbert space with a reproducing kernel a weakly convergent sequence ispointwise convergent. Show that a norm convergent sequence converges uniformly provided thereproducing kernel is continuous.

17.4. Determine the reproducing kernel of the Sobolev space H1[a, b].

17.5. Apply the moment collocation to the Volterra integral operator of Example 15.19 with anequidistant mesh xj = jh, j = 1, . . . , n, where h = 1/n. Show that the inverse matrix for the linearsystem (17.26) in this case is given by the tridiagonal matrix

n

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

2 −1−1 2 −1−1 2 −1· · · · · ·

−1 2 −1−1 1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

and that the total error in the spirit of Theorem 17.14 can be estimated by

‖ϕdn − A−1 f ‖L2 ≤ c1n1/2d +

c2

n

for some constants c1 and c2 provided the exact solution is twice continuously differentiable.

Chapter 18Inverse Boundary Value Problems

In this book, so far, we have considered only so-called direct boundary valueproblems where, given a differential equation, its domain, and a boundary condi-tion, we want to determine its solution. From Chapters 6–9 we know that the clas-sical boundary value problems for the Laplace equation and the heat equation arewell-posed. However, if for a given differential equation the problem consists of de-termining its domain, i.e., the boundary of the domain from a knowledge of the typeof boundary condition and information on one (or several) of its solutions, then thisinverse boundary value problem will often be ill-posed in any reasonable setting.Hence, the study of inverse boundary value problems is closely interlinked with theinvestigation of ill-posed problems.

Since integral equation methods are of fundamental importance in the solution ofdirect boundary value problems it is no surprise that they also play a prominent rolein the study of inverse problems. Thus an inclusion of a chapter on inverse boundaryvalue problems is imperative for a book on integral equations.

In the fifteen years since the second edition of this book was written the field ofinverse boundary value problems has experienced a number of new developments.To end this book, in order to acquaint the reader with current research directions ininverse boundary value problems, we shall study an inverse Dirichlet problem forthe Laplace equation as a model problem. To create more variety in the presentation,occasionally we also will consider the corresponding Neumann problem.

Of course, in a single chapter it is impossible to give a complete account of in-verse boundary value problems. Hence we shall content ourselves with developingsome of the main principles. For a detailed study of inverse boundary value prob-lems, we refer to Colton and Kress [32], Isakov [105] and Potthast [196].

18.1 An Inverse Problem for the Laplace Equation

To introduce the reader to inverse boundary value problems, for the sake of sim-plicity, we will confine ourselves to the presentation of an inverse boundary valueproblem for the Laplace equation. We recall the fundamental solution


365

366 18 Inverse Boundary Value Problems

Φ(x, y) :=1

2πln

1|x − y| , x � y,

to Laplace’s equation in IR2. Let D ⊂ IR2 be a bounded domain with a connectedboundary ∂D of class C2. Then we consider the two-dimensional exterior Dirichletproblem for a bounded function u ∈ C2(IR2 \ D) ∩ C(IR2 \ D) satisfying Laplace’sequation

Δu = 0 in IR2 \ D (18.1)

and the Dirichlet boundary condition

u = −Φ(· , ζ) on ∂D (18.2)

for some ζ ∈ IR2 \ D. From Chapter 6 we know that this exterior Dirichlet problemhas a unique solution and that this solution depends continuously on the boundarydata, i.e., on the location of the source point ζ. We note that analogous to Sec-tion 8.3, this boundary value problem can also be treated in a Sobolev space setting.On occasion we also will replace the Dirichlet condition (18.2) by the Neumanncondition

∂u∂ν= −∂Φ(· , ζ)

∂νon ∂D (18.3)

with the normal vector ν directed into the exterior of D together with the conditionu(x)→ 0 as |x| → ∞. Again from Chapter 6 we recall existence and uniqueness forthis exterior Neumann problem. Note that the sovability condition (6.49) is satisfied.

Now, let B ⊂ IR2 be an additional bounded domain with connected boundary ∂Bof class C2 containing the closure D. Then the inverse problem we want to consideris, given u|∂B on ∂B for a solution u of (18.1)–(18.2), determine the shape of D,i.e., we want to construct the shape of a perfectly conducting infinite cylinder withcross section D from the response to an incident static electric field of a line sourceparallel to the cylinder axis passing through a point ζ in the exterior of D. We mayview this inverse problem as a model problem for inverse boundary value problemsof similar structure occurring in various applications such as remote sensing, non-destructive testing, ultrasound medicine and seismic imaging. As we shall illustratein the sequel, this inverse boundary value problem associated with (18.1)–(18.2) isnonlinear and improperly posed. It is nonlinear, since the solution of (18.1)–(18.2)depends nonlinearly on the boundary ∂D. Roughly speaking, the inverse problem isill-posed, since perturbations of u|∂B in the maximum norm (or L2 norm) may lead tofunctions that are no longer analytic and therefore fail to be the restriction of a har-monic function on ∂B (in case of an analytic ∂B). Hence, for small perturbations ofu|∂B, in general, the inverse problem is not solvable, and if it is solvable the solutioncannot depend continuously on the data.

An important question to ask about the inverse problem is uniqueness, i.e., is theboundary curve ∂D uniquely determined by the values of the harmonic function u,solving (18.1)–(18.2), on the curve ∂B.

18.1 An Inverse Problem for the Laplace Equation 367

Theorem 18.1. Let D1 and D2 be two bounded domains with connected boundaries∂D1 and ∂D2 of class C2, respectively, such that D1 ⊂ B and D2 ⊂ B. Denote by u1

and u2 the solutions to the exterior Dirichlet problem (18.1)–(18.2) for the domainsD1 and D2, respectively, and assume that u1 = u2 on ∂B. Then D1 = D2.

Proof. Assume that D1 � D2. From u1 = u2 on ∂B, by the uniqueness of the solutionto the two-dimensional exterior Dirichlet problem (Theorem 6.12), it follows thatu1 = u2 in IR2 \ B. From this, since harmonic functions are analytic (Theorem 6.6),we can conclude that u1 = u2 in the unbounded component W of the complementof D1 ∪ D2. Without loss of generality, we can assume that W∗ := (IR2 \ W) \ D2

is nonempty. Then u2 is defined in W∗, since it describes the solution of (18.1)–(18.2) for D2. The function v := Φ(· , ζ) + u2 solves the Laplace equation in W∗, iscontinuous in W∗, and satisfies the homogeneous boundary condition v = 0 on ∂W∗.This boundary condition follows from the observation that each boundary point ofW∗ either belongs to ∂D2 or to ∂W ∩ ∂D1. For x ∈ ∂D2 we have Φ(x, ζ) + u2(x) = 0as a consequence of the boundary condition for u2, and for x ∈ ∂W ∩ ∂D1 we haveu2(x) = u1(x) and therefore

Φ(x, ζ) + u2(x) = Φ(x, ζ) + u1(x) = 0

as a consequence of the boundary condition for u1. Now, by the maximum-minimumprinciple for harmonic functions we can conclude that Φ(· , ζ) + u2 = 0 in W∗ andconsequently, by analyticity,Φ(· , ζ)+u2 = 0 in IR2 \ (D2∪{ζ}). Since u2 is harmonicin all of IR2 \ D2, this contradicts the singularity of the fundamental solution at thepoint ζ, and the proof is complete. �

The idea of the proof of Theorem 18.1 actually goes back to a private communi-cation of Schiffer referenced in [157]. Due to regularity issues, the proof does notcarry over to the Neumann boundary condition (18.3). However, in two dimensionsuniqueness can be settled by reducing the Neumann boundary condition for u to aDirichlet boundary condition for the conjugate harmonic v defined via the Cauchy–Riemann equations for the real and imaginary part of holomorphic functions by∂v/∂x1 := −∂u/∂x2 and ∂v/∂x2 := ∂u/∂x1 (see [80]).

Roughly speaking, most approaches for approximately solving an inverse bound-ary value problem like our model problem (18.1)–(18.2) belong to one group of thefollowing trilogy. In a first group of methods, denoted as direct methods or iterativemethods, the inverse boundary value problem is considered as an ill-posed nonlinearoperator equation and iterative techniques are employed for its solution. For fixedlocation ζ of the source point, the solution to the Dirichlet problem (18.1)–(18.2)defines an operator

F : ∂D → u|∂B (18.4)

that maps the boundary curve ∂D onto the trace of the solution u on ∂B. In terms ofthis operator F, which we also will denote as the boundary to data operator, givenu|∂B, the inverse boundary value problem consists in solving the operator equation

F(∂D) = u|∂B (18.5)


for the unknown boundary ∂D. For the approximate solution of (18.5), it seems nat-ural to apply iterative methods based on linearization, such as Newton’s method.However, since the linearized version of (18.5) inherits the ill-posedness from theinverse problem, the Newton iterations need to be regularized. This approach hasthe advantage that, in principle, it is conceptually simple and that it leads to highlyaccurate reconstructions. However, as disadvantage we note that the numerical im-plementation requires the forward solution of the Dirichlet problem (18.1)–(18.2)in each step of the Newton iteration and good a priori information is needed for theinitial approximation.

In the second group of methods, known as decomposition methods, the inverseproblem is decomposed or separated into a linear ill-posed part for the constructionof the harmonic function u from the given data u|∂B and a nonlinear and at mostmildly ill-posed part for finding the boundary as the location where the boundarycondition

u +Φ(· , ζ) = 0

is satisfied. The decomposition methods have the advantage that their numerical im-plementation does not require the solution for the boundary value problem (18.1)–(18.2) in each iteration step.

A third group of methods, denoted as sampling methods, is based on designing anappropriate indicator function f on B such that its value f (z) decides whether z ∈ Blies inside or outside the unknown domain D. Usually such an indicator function isdefined in terms of the solvability of a certain ill-posed linear integral equation ofthe first kind depending on the location parameter z. As a main advantage, samplingmethods do not need any a priori informations on the geometry of the unknown do-main D and on the boundary condition. In particular, sampling methods also can beapplied to the case where D has a finite number of connected components. How-ever, in general, they need more data such as the trace u|∂B for a large number ofsource points ζ whereas the iterative and decomposition methods, in general, workwith one source point only. In the remainder of this chapter we will consider typicalexamples for each of the three groups as applied to our model problem.

18.2 Decomposition Methods

We begin with a method belonging to the second group. In its first part, as in The-orem 6.24 we try to represent the solution u of (18.1)–(18.2) by a modified double-layer potential with a density ϕ ∈ L2(Γ) on some auxiliary closed curve Γ containedin the unknown domain D. The knowledge of such an internal curve Γ requires weaka priori information about D. Slightly deviating from the analysis of Theorem 6.24,we introduce an additional positive weight function ω ∈ C(Γ). In terms of a regular2π-periodic parameterization Γ = {z(t) : t ∈ [0, 2π]} we choose ω via

ω ◦ z =1|z′| , (18.6)

18.2 Decomposition Methods 369

i.e., ω(z(t)) = 1/|z′(t) for t ∈ [0, 2π], and seek the solution of exterior Dirichletproblems in the form

u(x) =∫Γ

ϕ(y)

{∂Φ(x, y)∂ν(y)

+ ω(y)

}ds(y), x ∈ IR2 \ Γ, (18.7)

where ν denotes the outward unit normal to Γ. The existence proof of Theorem6.25 carries over to this variant with the weight function which will lead to slightlysimpler parametrized versions of the potential and the integral equation. If we definean integral operator A : L2(Γ) → L2(∂B) by the restriction of the double-layerpotential (18.7) on ∂B, i.e., by

(Aϕ)(x) :=∫Γ

ϕ(y)

{∂Φ(x, y)∂ν(y)

+ ω(y)

}ds(y), x ∈ ∂B, (18.8)

then, given the values u|∂B on ∂B, we are looking for a density ϕ satisfying theequation

Aϕ = u|∂B. (18.9)

The integral operator A has a smooth kernel, and therefore equation (18.9) is aseverely ill-posed equation of the first kind. As for the integral equation (16.38),its solvability can again be related to the analytic continuation of the potential uacross the boundary ∂D. The integral equation (18.9) is solvable if and only if ucan be extended as a harmonic function into the exterior of Γ with boundary datain L2(Γ). As in Example 16.30 it can be shown that the singular values of A for thespecial case where Γ is the unit circle and B a concentric disk of radius R decayexponentially (see Problem 18.3).

Theorem 18.2. The integral operator A, defined by (18.8), is injective and has denserange.

Proof. The proof is analogous to that of Theorem 16.28. For the injectivity of A, theuniqueness part of the proof of Theorem 6.24 has to be used. For the dense range ofA we need to show injectivity of the adjoint operator A∗given by

(A∗ψ)(x) =∫∂Bψ(y)

{∂Φ(x, y)∂ν(x)

+ ω(x)

}ds(y), x ∈ Γ.

For this we observe that from A∗ψ = 0, with the aid of Example 6.17, it follows that

(1, ψ)L2(∂B)

∫Γ

ω ds = (A1, ψ)L2(∂B) = (1, A∗ψ)L2(Γ) = 0

and consequently∫∂Bψ ds = 0. Hence, the single-layer potential v with density ψ on

∂B satisfies ∂v/∂ν = 0 on Γ. Now the interior uniqueness Theorem 6.13 and ana-lyticity imply that v is constant in B. From the jump relation (6.53) for the normalderivative of single-layer potentials with L2 densities we derive ψ + K′ψ = 0, whereK′ denotes the normal derivative boundary operator on the curve ∂B. Now the con-


tinuity of the kernel of K′ (see Problem 6.1) and Theorem 6.21 imply that ψ = 0,since we already know that

∫∂Bψ ds = 0. Hence A∗ is injective. �

The Tikhonov regularization technique may be applied for the numerical solutionof the ill-posed equation (18.9). Recall that solving the regularized equation

αϕα + A∗Aϕα = A∗u|∂B (18.10)

with regularization parameter α > 0 is equivalent to minimizing the penalizedresidual

μ1(ϕ;α) := ‖Aϕ − u|∂B‖2L2(∂B) + α‖ϕ‖2L2 (Γ)

over all ϕ ∈ L2(Γ).After we have determined ϕα as the solution of (18.10), we use the corresponding

double-layer potential uα = Vϕα as an approximation for the solution u of (18.1)–(18.2). Here, for convenience, we use the abbreviation

(Vϕ)(x) :=∫Γ

ϕ(y)

{∂Φ(x, y)∂ν(y)

+ ω(y)

}ds(y), x ∈ IR2 \ Γ,

for the double-layer potential with density ϕ ∈ L2(Γ) as defined through (18.7).Then, in the second part of our method, we seek the boundary of D as the locationof the zeros of Φ(· , ζ) + uα in a minimum norm sense, i.e., we approximate ∂D byminimizing the defect

‖Φ(· , ζ) + uα‖L2(Λ)

over some suitable class U of admissible curves Λ. In the following, we will chooseU to be a compact subset (with respect to the C2

2π norm) of the set of all starlikeclosed C2 curves, described by

z(t) = r(t) (cos t, sin t), t ∈ [0, 2π], (18.11)

where r ∈ C22π satisfies the a priori assumption

0 < ri(t) ≤ r(t) ≤ re(t), t ∈ [0, 2π].

Here, ri and re are given functions representing curves Λi and Λe such that the in-ternal auxiliary curve Γ is contained in the interior of Λi, the boundary ∂D of theunknown scatterer D is contained in the annulus between Λi and Λe, and Λe is con-tained in B. For a sequence of curves we understand convergence Λn → Λ, n→ ∞,in the sense that ‖rn − r‖C2

2π→ 0, n → ∞, for the functions rn and r represent-

ing Λn and Λ via (18.11). We say that a sequence of functions fn from C(Λn) is L2

convergent to a function f in C(Λ) if

∫ 2π

0| fn(rn(t) (cos t, sin t)) − f (r(t) (cos t, sin t))|2 dt → 0, n→ ∞.


Now we introduce the functional

μ2(Λ, ϕ) := ‖Φ(· , ζ) + Vϕ‖2L2 (Λ).

Then, given ϕα, we seek an approximation to ∂D by minimizing the defect μ2(Λ, ϕα)over all Λ ∈ U. Due to the continuity of μ2(· , ϕ) and the compactness of U, thisminimization problem has a solution (see Problem 1.3).

Since, in general, we do not have convergence of the densities ϕα as α→ 0, fora theoretically satisfactory reformulation of the inverse boundary value problem asan optimization problem we combine the two steps and minimize the sum

μ(ϕ, Λ;α) := μ1(ϕ;α) + γμ2(Λ, ϕ) (18.12)

simultaneously over all ϕ ∈ L2(Γ) and Λ ∈ U, i.e., we call Λ0 ∈ U optimal if thereexists ϕ0 ∈ L2(Γ) such that μ(ϕ0, Λ0;α) = M(α), where

M(α) := infϕ∈L2(Γ), Λ∈U

μ(ϕ, Λ;α).

Here, γ > 0 denotes a coupling parameter, which has to be chosen appropriatelyfor the numerical implementation to make the two terms in (18.12) of the samemagnitude. In the sequel, for theoretical purposes we may assume that γ = 1. For ourreformulation of the inverse boundary value problem into a nonlinear optimizationproblem we can state the following results. Note that in the existence Theorem 18.3we need not assume that u|∂B is the exact restriction of a harmonic function.

Theorem 18.3. For each α > 0, the optimization formulation of the inverse bound-ary value problem given through (18.12) has a solution.

Proof. Let (ϕn, Λn) be a minimizing sequence in L2(Γ) × U, i.e.,

limn→∞ μ(ϕn, Λn;α) = M(α).

Since U is compact, we can assume that Λn → Λ ∈ U, n→ ∞. From α > 0 and

α‖ϕn‖2L2(Γ) ≤ μ(ϕn, Λn;α)→ M(α), n→ ∞,we conclude that the sequence (ϕn) is bounded, i.e., ‖ϕn‖L2 (Γ) ≤ c for all n and someconstant c. Hence, by Theorem 16.3, we can assume weak convergence ϕn ⇀ ϕwith some ϕ ∈ L2(Γ) as n→ ∞. Since A : L2(Γ)→ L2(∂B) and V : L2(Γ)→ L2(Λ)represent compact operators, by Problem 16.5 it follows that both Aϕn → Aϕ andVϕn → Vϕ as n → ∞. With functions rn and r representing Λn and Λ via (18.11),by Taylor’s formula we can estimate

∣∣∣∣∣∂Φ(rn(t) (cos t, sin t), y)∂ν(y)

− ∂Φ(r(t) (cos t, sin t), y)∂ν(y)

∣∣∣∣∣ ≤ L |rn(t) − r(t)|

for t ∈ [0, 2π] and y ∈ Γ. Here, L denotes a bound on gradx{∂Φ(x, y)/∂ν(y)} onW × Γ, where W is the closed annular domain between the two curves Λi and Λe.


Then, using the Cauchy–Schwarz inequality, we find that

|(Vϕn)(rn(t) (cos t, sin t)) − (Vϕn)(r(t) (cos t, sin t))| ≤ cL√|Γ| |rn(t) − r(t)|

for t ∈ [0, 2π]. Therefore, from ‖Vϕn − Vϕ‖2L2(Λ)→ 0, n→ ∞, we can deduce that

‖Φ(· , ζ) + Vϕn‖2L2 (Λn) → ‖Φ(· , ζ) + Vϕ‖2L2(Λ), n→ ∞.This now implies

α‖ϕn‖2L2(Γ) → M(α) − ‖Aϕ − u|∂B‖2L2(∂B) − ‖Φ(· , ζ) + Vϕ‖2L2 (Λ) ≤ α‖ϕ‖2L2 (Γ)

for n → ∞. Since we already know weak convergence ϕn ⇀ ϕ, n → ∞, it followsthat

limn→∞ ‖ϕn − ϕ‖2L2 (Γ) = lim

n→∞ ‖ϕn‖2L2(Γ) − ‖ϕ‖2L2(Γ) ≤ 0,

i.e., we also have norm convergence ϕn → ϕ, n → ∞. Finally, by continuity weobtain that

μ(ϕ, Λ;α) = limn→∞ μ(ϕn, Λn;α) = M(α),

and this completes the proof. �

In order to prove regularity (in the sense of Definition 15.7) of the regulariza-tion included in the above optimization reformulation of the inverse boundary valueproblem, we need the following two lemmas.

Lemma 18.4. Let u|∂B be the restriction to ∂B of the solution u to (18.1)–(18.2) fora domain D such that ∂D is contained in U. Then

limα→0

M(α) = 0. (18.13)

Proof. By Theorem 18.2, applied to the double-layer operator from L2(Γ) intoL2(∂D), given ε > 0 there exists ϕ ∈ L2(Γ) such that

‖Φ(· , ζ) + Vϕ‖L2(∂D) < ε.

Analogous to Remark 16.29 we can estimate

‖Aϕ − u‖L2(∂B) ≤ c‖Vϕ − u‖L2(∂D)

with some constant c. From Φ(· , ζ) + u = 0 on ∂D we then deduce that

μ(ϕ, ∂D;α) ≤ (1 + c2)ε2 + α‖ϕ‖2L2 (Γ) → (1 + c2)ε2, α→ 0.

Since ε is arbitrary, (18.13) follows. �

Lemma 18.5. Let (Λn) be a sequence of starlike curves of class C2 that convergesto a starlike curve Λ of class C2 and let un and u be bounded harmonic functions in


the exterior of Λn and Λ, respectively. Assume that un and u are continuous in theclosure of the exterior of Λn and Λ, respectively, and that the boundary values of un

on Λn are L2 convergent to the boundary values of u on Λ. Then ‖un − u‖C(∂B) → 0,n→ ∞.

Proof. We use the modified double-layer potential (18.7), i.e., we represent u in theform

u(x) =∫Λ

ϕ(y)

{∂Φ(x, y)∂ν(y)

+ ω(y)

}ds(y), x ∈ ∂B,

with a density ϕ ∈ C(Λ) and, analogously, we write un as a double-layer potentialwith density ϕn ∈ C(Λn). With the aid of the parametric representation (18.11) of Λwe transform the integral equation corresponding to (6.42) into the parameterizedform ψ+ Lψ = f , where ψ = ϕ ◦ z and f = 2u ◦ z. Analogously, we have an integralequation ψn + Lnψn = fn corresponding to un. By assumption, ‖ fn − f ‖L2 [0,2π] → 0as n→ ∞.

We denote the kernel functions of L and Ln by k and kn, respectively. By Problem6.1, the kernel k is given by

k(t, τ) =1π

[z′(τ)]⊥ · [z(t) − z(τ)]|z(t) − z(τ)|2 + 2, t � τ,

and kn is given by the analogous expression in terms of the parametric representa-tion zn(t) = rn(t) (cos t, sin t). As before, for a vector a = (a1, a2) in IR2 we denotea⊥ = (a2,−a1). From Taylor’s formula

z(t) − z(τ) = (t − τ)∫ 1

0z′(τ + λ(t − τ)) dλ (18.14)

we observe that the function

g(t, τ) :=

⎧⎪⎪⎪⎨⎪⎪⎪⎩|z(t) − z(τ)||t − τ| , t � τ,

|z′(t)|, t = τ,

is continuous in IR2. Therefore, in view of z(t) � z(τ) for 0 < |t − τ| ≤ π and|z′(t)|2 = [r(t)]2 + [r′(t)]2 > 0 for all t, there exists a constant c1, depending on r,such that

|z(t) − z(τ)| ≥ c1|t − τ|, |t − τ| ≤ π. (18.15)

From this, using (18.14) applied to zn − z, we deduce that

|[zn(t) − zn(τ)] − [z(t) − z(τ)]| ≤ c2‖rn − r‖C12π|z(t) − z(τ)| (18.16)

for |t − τ| ≤ π and some constant c2 depending on r. By Taylor’s formula we have∣∣∣∣∣ 1|xn|2 −

1|x|2

∣∣∣∣∣ ≤ 16|x|3 |xn − x|, x � 0, 2|xn − x| ≤ |x|.


From this, setting x = z(t) − z(τ), xn = zn(t) − zn(τ), and using (18.15) and (18.16)we find the estimate

∣∣∣∣∣ 1|zn(t) − zn(τ)|2 −

1|z(t) − z(τ)|2

∣∣∣∣∣ ≤ c3

‖rn − r‖C12π

(t − τ)2, (18.17)

which is valid for sufficiently small ‖rn − r‖C12π

, for |t− τ| ≤ π, and for some constantc3 depending on r. Using Taylor’s formula

z(t) − z(τ) = (t − τ)z′(τ) + (t − τ)2∫ 1

0(1 − λ)z′′(τ + λ(t − τ)) dλ,

we can write

[z′(τ)]⊥ · [z(t) − z(τ)] = (t − τ)2 [z′(τ)]⊥ ·∫ 1

0(1 − λ)z′′(τ + λ(t − τ)) dλ.

Hence, we can estimate

|[z′(τ)]⊥ · [z(t) − z(τ)]| ≤ c4(t − τ)2

and

|[z′n(τ)]⊥ · [zn(t) − zn(τ)] − [z′(τ)]⊥ · [z(t) − z(τ)]| ≤ c5(t − τ)2‖rn − r‖C22π

for all |t − τ| ≤ π and some constants c4 and c5 depending on r. Combining thesetwo estimates with (18.15)–(18.17), by decomposing

kn(t, τ) − k(t, τ) =1π

[z′n(τ)]⊥ · [zn(t) − zn(τ)]

{1

|zn(t) − zn(τ)|2 −1

|z(t) − z(τ)|2}

+1π

[z′n(τ)]⊥ · [zn(t) − zn(τ)] − [z′(τ)]⊥ · [z(t) − z(τ)]

|z(t) − z(τ)|2 ,

we finally obtain that

|kn(t, τ) − k(t, τ)| ≤ C‖rn − r‖C22π, |t − τ| ≤ π, (18.18)

for sufficiently small ‖rn − r‖C22π

and some constant C depending on r.The estimate (18.18) now implies norm convergence

‖Ln − L‖L2 [0,2π] ≤ 2πC‖rn − r‖C22π→ 0, n→ ∞.

Therefore, using Theorem 10.1 we can conclude that ‖ψn − ψ‖L2[0,2π] → 0, n → ∞.From this, the statement of the lemma follows by parameterizing the double-layerpotential representation for un−u and then using the Cauchy–Schwarz inequality. �


Now we are ready to establish regularity of the combined optimization schemethrough the following convergence result.

Theorem 18.6. Let (αn) be a null sequence and let (Λn) be a corresponding se-quence of solutions to the optimization problem with regularization parameter αn.Assume that u|∂B is the restriction to ∂B of the solution u to (18.1)–(18.2) for adomain D such that ∂D belongs to U. Then Λn → ∂D, n→ ∞.

Proof. Since by assumption U is compact, there exists a convergent subsequenceΛn(k) → Λ∗ ∈ U, k → ∞. Let u∗ denote the unique solution to the boundary valueproblem (18.1)–(18.2) for the exterior domain with boundary Λ∗, i.e., the boundarycondition reads

Φ(· , ζ) + u∗ = 0 on Λ∗. (18.19)

Since Λn(k) is optimal for the parameter αn(k), there exists ϕn(k) ∈ L2(Γ) such thatμ(ϕn(k), Λn(k);αn(k)) = M(αn(k)) for k = 1, 2, . . . . By Lemma 18.4, we have

‖Φ(· , ζ) + Vϕn(k)‖2L2(Λn(k))≤ M(αn(k))→ 0, k → ∞. (18.20)

Now, by Lemma 18.5, from (18.19) and (18.20), we deduce that

‖Aϕn(k) − u∗‖L2(∂B) → 0, k → ∞.By Lemma 18.4 we also have

‖Aϕn(k) − u‖2L2(∂B) ≤ M(αn(k))→ 0, k → ∞.Therefore, we conclude u = u∗ on ∂B, and by Theorem 18.1 we have ∂D = Λ∗.

Now, by a standard argument, assuming that the sequence (Λn) does not con-verge to ∂D leads to a contradiction, since by the above argument any subsequenceΛn contains a subsequence that converges to ∂D. �

The extension of the above method to inverse obstacle scattering problemsfor time-harmonic waves, i.e., for an inverse boundary value problems for theHelmholtz equation is due to Kirsch and Kress [128] and actually predated its ap-plication in potential theory. For a detailed discussion of the method as applied toinverse scattering theory and for numerical examples, we refer to [32, 129].

In a hybrid method combining ideas of the above decomposition method andNewton iterations as discussed in the following two sections the auxiliary internalcurve Γ is viewed as an approximation for the unknown boundary ∂D. Then, keep-ing the double-layer potential uα resulting via (18.7) from a regularized solution of(18.9) fixed, Γ is updated via linearization of the boundary conditionΦ(· , ζ)+uα = 0around Γ. If we assume again that Γ is starlike with radial function r and look foran update Γ = {z(t) : t ∈ [2π]} that is starlike with radial function r + q, the updateis found by solving the linear equation

{Φ(· , ζ) + uα}∣∣∣Γ+ grad {Φ(· , ζ) + uα}

∣∣∣Γ·((z − z) ◦ z−1

)= 0 (18.21)


for the radial function q determining the update z. In an obvious way, the two stepsof alternatingly solving (18.9) by Tikhonov regularization and solving (18.21) inthe least squares sense are iterated. For the numerical implementation, the termsuα|Γ and grad uα

∣∣∣Γ

in (18.21) are evaluated with the aid of the jump relations. Fromnumerical examples (see [27]) it can be concluded that the quality of the recon-structions is similar to that of Newton iterations as discussed in the next sections.Again the application of this approach in potential theory was predated by that ininverse obstacle scattering (see Kress [144]). For numerical examples in inverse ob-stacle scattering we refer to [144, 150, 219] in two dimensions and to [220] in threedimensions.

We conclude this section on decomposition methods by describing a complexanalysis approach that was developed in a series of papers by Akduman, Haddarand Kress [3, 80, 81, 145] for a slight modification of the inverse problem relatedto the Dirichlet problem (18.1)–(18.2). As we will see this approach can be viewedas a decomposition method with the sequential order of the nonlinear part and theill-posed linear part reversed.

For this we consider the inverse problem to determine the boundary curve ∂Dfrom the Cauchy data

f = u |∂B and g =∂u∂ν

∣∣∣∣∣∂B

of a harmonic function u in B \ D satisfying the homogeneous Dirichlet boundarycondition u = 0 on ∂D. Here, as usual, ν denotes the outward unit normal to ∂B.Clearly, after renaming the unknowns, the inverse problem related to (18.1)–(18.2)can be considered as a special case of this modification after determining the normalderivative on ∂B by additional measurements or by solving a Dirichlet problem inthe exterior of B. We note that the corresponding direct Dirichlet problem in B \ Dwith Dirichlet condition on both ∂D and ∂B is uniquely solvable (see Problem 18.1).Also the uniqueness Theorem 18.1 carries over to this case (see Problem 18.2).

We will identify R2 and C and introduce the annulus Ω bounded by two con-centric circles Γ0 with radius 0 < ρ < 1 and Γ1 with radius one centered at theorigin. By the Riemann conformal mapping theorem for doubly connected domains(see [244]) there exists a uniquely determined radius ρ and a holomorphic functionΨ that mapsΩ bijectively onto B\ D such that the boundaries Γ0 and Γ1 are mappedonto ∂D and ∂B, respectively, with all boundary curves in counterclockwise orien-tation. The function Ψ is unique up to a rotation of the annulus Ω. We represent ∂Bby a regular parameterization

∂B = {η(t) : t ∈ [0, 2π]}and fix the freedom in rotating Ω by prescribing Ψ (1) = η(0). Then we define aboundary correspondence function ϕ : [0, 2π]→ [0, 2π] by setting

ϕ(t) := η−1(Ψ (eit)), t ∈ [0, 2π]. (18.22)


Clearly, the boundary values ϕ uniquely determine Ψ as the solution to the Cauchyproblem for a holomorphic function with Ψ |Γ1 given by Ψ (eit) = η(ϕ(t)). The func-tion χ : [0, 2π]→ C defined by

χ(t) := Ψ (ρeit), t ∈ [0, 2π], (18.23)

describes ∂D and therefore determining χ solves the inverse problem.We now derive a non-local ordinary differential equation for the boundary corre-

spondence function ϕ. To this end, we denote by Aρ : H1/2[0, 2π] → H−1/2[0, 2π]the Dirichlet-to-Neumann operator for the annulus Ω given by

Aρ : F → ∂w

∂ν◦ η,

where w ∈ H1(Ω) is the unique harmonic function with boundary values in the tracesense given by

w ◦ η = F on Γ1

and w = 0 on Γ0 (see Section 8.3). Via v := u◦Ψ we associate the harmonic functionu in D with a harmonic function v in B. Locally, the conjugate harmonics u and v arewell defined and also related via v := u ◦ Ψ . Then

ddtv(eit) = ϕ′(t) |η′(ϕ(t))| ∂u

∂s(η(ϕ(t))

with the tangential derivative ∂u/∂s and from the Cauchy–Riemann equations for uand v and their harmonic conjugates u and v, respectively, we conclude that

∂v

∂ν(eit) = ϕ′(t) |η′(ϕ(t))| ∂u

∂ν(η(ϕ(t)), t ∈ [0, 2π].

Therefore we have the non-local differential equation

ϕ′ =Aρ( f ◦ η ◦ ϕ)

|η′ ◦ ϕ| g ◦ η ◦ ϕ (18.24)

for the boundary correspondence function ϕ in terms of the given Cauchy data fand g. The differential equation has to be complemented by the boundary conditionsϕ(0) = 0 and ϕ(2π) = 2π. It is non-local since the evaluation of its right-hand sideat a point t requires the knowledge of ϕ everywhere in [0, 2π].

Applying Green’s second integral theorem in Ω to v and x → ln |x| we obtain theequation

ρ = exp

⎛⎜⎜⎜⎜⎜⎜⎜⎝−∫ 2π

0f ◦ η ◦ ϕ dt∫∂Dg ds

⎞⎟⎟⎟⎟⎟⎟⎟⎠ (18.25)

for the radius ρ of the annulusΩ. In [3] it is shown that under appropriate conditionsthe two equations (18.24) and (18.25) can be solved simultaneously for the boundarycorrespondence function ϕ and the radius ρ by successive approximations.


The equations (18.24) and (18.25) suffer from two drawbacks. Zeros in theNeumann data g will cause difficulties through the appearance of g in the denomi-nator of (18.24), and (18.25) fails for g with mean value zero. Remedies for thesedeficiencies have been proposed in [80] by using a pair of Cauchy data and addi-tional formulas for the radius. Here we refrain from describing the details of thismodification.

Once the radius ρ and the boundary correspondence function ϕ are known weexpand η ◦ ϕ in a Fourier series to obtain

Ψ (eit) =∞∑

k=−∞ϕkeikt, t ∈ [0, 2π],

whence the Laurent series

Ψ (z) =∞∑

k=−∞ϕkzk, z ∈ Ω, (18.26)

follows. From this we find

χ(t) =∞∑

k=−∞ϕkρ

keikt, t ∈ [0, 2π], (18.27)

as a parameterization of the unknown boundary ∂D. The series (18.27) exhibits theill-posedness of the inverse problem, since small errors in the Fourier coefficientsϕk for k < 0 will be amplified by the exponentially increasing factors ρk. Thereforewe incorporate a Tikhonov type regularization replacing χ by

χα(t) :=∞∑

k=0

ϕkρkeikt +

∞∑k=1

ϕ−kρk

α + ρ2ke−ikt (18.28)

where α > 0 serves as a regularization parameter.We conclude this short outline on the conformal mapping method by noting that

it also has been used for the solution of an inverse scattering problem [82]. It alsocan be applied for the solution of the direct conformal mapping problem, namely theconstruction of a conformal mappingΨ : Ω→ B \ D for given B and D (see [145]).

18.3 Differentiability with Respect to the Boundary

We now turn to approximation methods of the first group mentioned at the end ofSection 18.1. The foundation of iterative methods for the solution of the nonlinearoperator equation (18.5) such as Newton methods requires the investigation of thedifferentiability of the boundary to data operator F with respect to the boundary andto characterize its derivative.

18.3 Differentiability with Respect to the Boundary 379

A mapping M : U ⊂ X → Y from an open subset of a normed space X into anormed space Y is said to be Frechet differentiable at the point ϕ ∈ U if there existsa bounded linear operator M′(ϕ; ·) : X → Y such that

limh→0

1‖h‖ ‖M(ϕ + h) − M(ϕ) − M′(ϕ; h)‖ = 0.

The bounded linear operator M′(ϕ; ·) is called the Frechet derivative of M at thepoint ϕ. The mapping M is called Frechet differentiable on U if it is Frechet differ-entiable for all ϕ ∈ U. Instead of M′(ϕ; ·) we also will write M′ϕ for the derivativeat ϕ ∈ U. For an introduction to the concept of the Frechet derivative and its basicproperties we refer to Berger [17].

So far we have not specified a domain of definition for the operator F. For this,a parameterization of the boundary curve is required. For the sake of simplicity, asin Section 18.2, we will restrict our analysis to the case of starlike domains of theform (18.11) and consider F as a mapping from the open set U :=

{r ∈ C2

2π : r > 0}

of positive functions in C22π into C(∂B). We will also write F(r) synonymously for

F(∂D). For notational convenience we associate with each 2π-periodic scalar func-tion q a vector function hq by setting

hq(t) := q(t) (cos t, sin t), t ∈ [0, 2π]. (18.29)

We note that the function hr corresponds to the parameterization (18.11) and maps[0, 2π) bijectively onto ∂D, that is, we can use its inverse h−1

r .We now establish differentiability of F with respect to the boundary. For this we

first need to examine the corresponding differentiability of the boundary integraloperators used in the existence proof for the solution of the Dirichlet problem. Werecall the double-layer approach for the exterior Dirichlet problem from Theorem6.25 in the modified form (18.7) with the weight function ω and, as in the proof ofTheorem 6.25, denote the corresponding integral operator by K : C(∂D)→ C(∂D).As in the proof of Lemma 18.5, via (18.11) and (18.29) we introduce its parameter-ized form

(Lrψ)(t) =∫ 2π

0k(t, τ; r)ψ(τ) dτ, t ∈ [0, 2π],

with kernel

k(t, τ; r) = 2 [h′r(τ)]⊥ · grady Φ(hr(t), hr(τ)) + 2, t � τ,

(see Problem 6.1). The subscript indicates the dependence of Lr on the boundarycurve.

Theorem 18.7. The mapping r → Lr is Frechet differentiable from U ⊂ C22π into

the space L(C2π,C2π) of bounded linear operators from C2π into C2π.

Proof. For t � τ, the Frechet derivative of r → k(· , · ; r) is given by

k′(t, τ; r, q)=2[h′r(τ)]⊥·Φ′′y (hr(t), hr(τ))[hq(τ)−hq(t)]+2[h′q(τ)]⊥·grady Φ(hr(t), hr(τ))


where Φ′′y (x, y) denotes the Hessian of Φ(x, y) with respect to y and the functionq ∈ C2

2π need to be such that r + q > 0. To establish this, proceeding as in the proofof the estimate (18.18) in Lemma 18.5, straightforward calculations using Taylor’sformula show that k′(· , · ; r, q) is continuous on IR2 and 2π-periodic with respect toboth variables and satisfies

‖k(· , · ; r + q) − k(· , · ; r) − k′(· , · ; r, q)‖∞ ≤ c ‖q‖2C2

2π(18.30)

for all sufficiently small ‖q‖2C2

2πand some constant c depending on r (see Problem

18.4). From this, it follows that the integral operator L′r,q with kernel k′(· , · ; r, q)satisfies

‖Lr+q − Lr − L′r,q‖∞ ≤ 2πc ‖q‖2C2

2π,

and therefore the linear mapping q → L′r,q is the Frechet derivative of r → Lr. �

The parameterized form of the double-layer operator is given by the integraloperator

(Vrψ)(x) =∫ 2π

0v(x, τ; r)ψ(τ) dτ, x ∈ IR2 \ D,

with kernel

v(x, τ; r) := [h′r(τ)]⊥ · grady Φ(x, hr(τ)) + 1, x ∈ IR2 \ D, τ ∈ [0, 2π].

We introduce the integral operator V ′r,q with kernel

v′(x, τ; r, q) = [h′r(τ)]⊥ ·Φ′′y (x, hr(τ)) hq(τ) + [h′q(τ)]⊥ · grady Φ(x, hr(τ)).

Then, since the kernel of Vr has no singularities, straightforward differentiationshows that for closed subsets W of IR2 \ D the Frechet derivative of the mappingr → Vr from U ⊂ C2

2π into the space L(C2π,C(W)) of bounded linear operators fromC2π into C(W) is given by q → V ′r,q.

Theorem 18.8. Assume that ψ ∈ C1,α2π . Then V ′r,qψ is a bounded harmonic function

in IR2 \ D that is continuous in IR2 \ D with boundary values

V ′r,qψ +(hq ◦ h−1

r

)· grad(Vrψ) =

12

(L′r,qψ) ◦ h−1r on ∂D. (18.31)

Proof. From the form of the kernel v′(· , · ; r, q), it is obvious that V ′r,qψ is bounded atinfinity and harmonic. To show that it is continuous up to the boundary ∂D and tocompute its boundary values we use the vector identity

grad divw = Δw + [grad divw⊥]⊥,

18.3 Differentiability with Respect to the Boundary 381

Laplace’s equation for Φ, and grady Φ = − gradx Φ to transform

[h′r(τ)]⊥ ·Φ′′y (x, hr(τ)) hq(τ) = [h′r(τ)]⊥ · gradx divx{Φ(x, hr(τ)) hq(τ)}

= − divx

{[hq(τ)]⊥

ddτ

Φ(x, hr(τ))

}

(compare the proof of Theorem 7.32). From this, by partial integration, we find that

(V ′r,qψ)(x) = divx

∫ 2π

0Φ(x, hr(τ)) [hq(τ)]⊥ψ′(τ) dτ, x ∈ IR2 \ D, (18.32)

i.e., we can interpret V ′r,qψ as a derivative of a single-layer potential. Therefore, fromTheorem 7.30 it follows that V ′r,qψ is continuous up to the boundary with boundaryvalues

(V ′r,qψ)(hr(t)) = −∫ 2π

0grady Φ(hr(t), hr(τ)) · [hq(τ)]⊥ψ′(τ) dτ

− ψ′(t)2 |h′r(t)|

h′r(t) · hq(t), t ∈ [0, 2π].

Proceeding analogously for the gradient of the double-layer potential, i.e., from The-orem 7.32, we find that

hq(t) · grad(Vrψ)(hr(t)) =∫ 2π

0grady Φ(hr(t), hr(τ)) · [hq(t)]⊥ψ′(τ) dτ

+ψ′(t)

2 |h′r(t)|h′r(t) · hq(t), t ∈ [0, 2π].

Adding the last two equations and performing an analogous partial integration inthe integral representing L′r,qψ now yields (18.31). For the latter partial integrationone has to take proper care of the singularity at t = τ. �

Note that (18.31) can be interpreted as formal differentiation of the jump relation2Vrψ = (I + Lr)ψ with respect to r. Analogously, the boundary condition (18.33)in the following theorem can be obtained by formally differentiating the boundarycondition (18.2) with respect to r.

Theorem 18.9. The boundary to data operator F : r → u|∂B is Frechet differen-tiable from U ⊂ C2

2π into C(∂B). The Frechet derivative is given by

F′(r; q) = w|∂B,


where w ∈ C2(IR2 \ D) ∩ C(IR2 \ D) denotes the uniquely determined boundedharmonic function satisfying the Dirichlet boundary condition

w = −ν ·(hq ◦ h−1

r

) ∂

∂ν{Φ(· , ζ) + u} on ∂D. (18.33)

Proof. In view of Theorem 6.25, we write the solution to (18.1)–(18.2) in the form

u = −2Vr(I + Lr)−1 fr in IR2 \ D,

where fr(t) := Φ(hr(t), ζ). We note that differentiability of r → Lr implies differ-entiability of r → (I + Lr)−1 with the derivative given by −(I + Lr)−1L′r,q(I + Lr)−1.Then, using the chain rule for the Frechet derivative, we obtain

u′ = −2V ′r,q(I + Lr)−1 fr + 2Vr(I + Lr)

−1L′r,q(I + Lr)−1 fr − 2Vr(I + Lr)

−1br,q

in IR2 \ D, where br,q(t) = hq(t) · gradΦ(hr(t), ζ) represents the Frechet derivativeof the boundary values with respect to the boundary, i.e., of r → fr . From this, it isobvious that u′ is bounded and harmonic. To derive the boundary condition (18.33)we use the jump relation 2Vr(I + Lr)−1 = I on ∂D for the double-layer potential.Now (18.33) follows, since (18.31) implies that

−2V ′r,q(I + Lr)−1 fr = −L′r,q(I + Lr)

−1 fr − hq · (grad u ◦ hr).

This completes the proof, provided we justify the application of (18.31) by showingthat (I + Lr)−1 fr ∈ C1,α

2π .From the proofs of Theorems 7.5 and 7.6 we know that the double-layer integral

operator K maps C(∂D) boundedly into C0,α(∂D) and that it maps C0,α(∂D) bound-edly into C1,α(∂D). Then, from Φ(· , ζ)|∂D ∈ C1,α(∂D), it follows that the solution ψof the integral equation ψ + Kψ = 2Φ(· , ζ) is in C1,α(∂D). �

The double-layer potential operator K from the proof of Theorem 6.25 in themodified form (18.7) has a continuous kernel (in two dimensions). Therefore it iscompact from L2(∂D) into itself and (I + K)−1 : L2(∂D) → L2(∂D) exists and isbounded. Denote by V : L2(∂D) → L2(∂B) the operator that takes densities on ∂Donto the trace on ∂B of their double-layer potential (and has the parameterized formVr as used above). Then we can express

F′(r; q) = −2V(I + K)−1

(ν ·

(hq ◦ h−1

r

) ∂

∂ν{Φ(· , ζ) + u}

). (18.34)

From this representation we observe that F′(r; ·) can be extended as a bounded lin-ear operator from L2[0, 2π] into L2(∂B). For this extension we have the followingtheorem.

Theorem 18.10. The linear operator F′(r; ·) : L2[0, 2π] → L2(∂B) is injective andhas dense range.

18.4 Iterative Methods 383

Proof. Relabeling the curves in Theorem 18.2, we observe that V is injective andhas dense range. Assume that F′(r; q) = 0. Then from (18.34) and the injectivity ofV and (I + K)−1 we conclude that

ν ·(hq ◦ h−1

r

) ∂∂ν{Φ(· , ζ) + u} = 0 on ∂D.

This in turn implies ν ◦ hr · hq = 0 since the normal derivative ∂ {Φ(· , ζ) + u} /∂νcannot vanish on open subsets of ∂D as a consequence of Holmgren’s Theorem 6.7and the boundary condition Φ(· , ζ) + u = 0 on ∂D. Elementary calculations yield

ν ◦ hr · hq =rq√

r2 + [r′]2(18.35)

and therefore (ν ◦ hr) · hq = 0 implies that q = 0.The above arguments also imply that the set

{(ν ◦ hr) · hq

∂

∂ν{Φ(· , ζ) + u} ◦ hr : q ∈ L2[0, 2π]

}

is dense in L2[0, 2π]. Then using the dense range of V and the surjectivity of (I+K)−1

from (18.34) we can conclude that the range of F′(r; ·) is dense. �

For the extension of the above analysis on Frechet differentiability to the Helm-holtz equation, we refer to [32, 194].

18.4 Iterative Methods

The Theorems 18.9 and 18.10 serve as theoretical foundation for the applicationof regularized Newton methods and related iteration schemes for the approximatesolution of (18.5). In this method, given u|∂B, the nonlinear equation

F(r) = u|∂B

is replaced by its linearization

F(r) + F′(r; q) = u|∂B, (18.36)

which has to solved for q in order to improve an approximate boundary given by rinto a new approximation given by r = r + q. As usual, Newton’s method consistsof iterating this procedure. From the form (18.34) of the Frechet derivative it canbe seen that the linear operator q → F′(r; q) is compact from L2[0, 2π] into L2(∂B).Therefore, for the solution of the linear equation (18.36) a regularization has to beincorporated, such as Tikhonov regularization. Injectivity and dense range of F′(r; ·)as prerequisites for regularization schemes are guaranteed by Theorem 18.10.


For practical computations q is taken from a finite-dimensional subspace Un ofL2[0, 2π] with dimension n and equation (18.36) is approximately solved by collo-cating it at m points x1, . . . , xm ∈ ∂B. Then writing

q =n∑

k=1

akqk

where q1, . . . , qn denotes a basis of Un, one has to solve the linear system

n∑k=1

ak(F′(r; qk)

)(x j) = u(x j) − (

F(r))(x j), j = 1, . . . ,m, (18.37)

for the real coefficients a1, . . . , an. In general, i.e., when m > n, the system (18.37)is overdetermined and has to be solved approximately by a least squares method.In addition, since we have to stabilize the ill-posed linearized equation (18.36), wereplace (18.37) by the least squares problem of minimizing the penalized defect

m∑j=1

∣∣∣∣∣∣∣n∑

k=1

ak(F′(r; qk)

)(x j) − u(x j) +

(F(r)

)(x j)

∣∣∣∣∣∣∣2

+ αn∑

k=1

a2k (18.38)

with some regularization parameter α > 0, that is, we employ a Tikhonov regulariza-tion in the spirit of the Levenberg–Marquardt algorithm (see [173]). Assuming thatthe basis functions q1, . . . , qn are orthonormal in L2[0, 2π], for example trigonomet-ric monomials, the penalty term in (18.38) corresponds to L2 penalization. However,numerical evidence strongly suggests to replace the L2 penalization by a Sobolevpenalization, i.e., by considering F′(r; ·)) as an operator from Hp[0, 2π] into L2(∂B)for p = 1 or p = 2 (see also Section 16.5).

In order to compute the right-hand sides of the linear system (18.37), in eachiteration step the direct Dirichlet problem for the boundary ∂D given by the radialfunction r has to be solved for the evaluation of

(F(r)

)(x j). For this, we can nu-

merically solve the integral equation of the second kind from Theorem 6.25 viathe Nystrom method of Section 12.2 using the composite trapezoidal rule. Then thevalues at the collocation points can be obtained via numerical quadrature with thecomposite trapezoidal rule for the modified double-layer potential of Theorem 6.25.For the evaluation of the normal derivative entering the boundary condition (18.33)for the Frechet derivatives one can use the approximation for the normal derivativeof the double-layer potential as described in Section 13.5. To compute the matrixentries

(F′(r; qk)

)(x j) we need to solve n additional direct Dirichlet problems for

the same boundary ∂D and different boundary values given by (18.33) for the basisfunctions q = q j, j = 1, . . . , n, that is, we have to solve the linear system set upalready for the evaluation of

(F(r)

)(x j) again for n different right-hand sides.

For numerical examples for related inverse Dirichlet problems for the Helmholtzequation we refer to [96, 124, 141, 144] in the two-dimensional case and to [51, 88]in the three-dimensional case.


In closing our analysis on Newton iterations for the boundary to data operator Fwe note as main advantages, which also hold in more general situations, that this ap-proach is conceptually simple and, as numerical examples indicate, leads to highlyaccurate reconstructions with reasonable stability against errors in the data. On theother hand, it should be noted that for the numerical implementation an efficient for-ward solver is needed for the solution of the corresponding direct boundary valueproblem for each iteration step. Furthermore, good a priori information is requiredin order to be able to choose an initial guess that ensures numerical convergence.In addition, on the theoretical side, although some progress has been made throughthe work of Hohage [97] and Potthast [195] the convergence of regularized New-ton iterations for the operator F has not been completely settled. At the time this isbeing written it remains an open question whether the general convergence resultson Newton type methods such as the Levenberg–Marquardt algorithm or the iter-atively regularized Gauss–Newton iterations as available in the literature [22, 115]are applicable to inverse boundary value problems.

For the following discussion on modified Newton type iterations with reducedcomputational costs, for variety and in order to avoid technical difficulties due tothe boundedness condition for the exterior Dirichlet problem in two dimensions, weconsider the Neumann boundary condition (18.3). We also replace the potential ap-proach by the direct approach based on Theorem 7.37, i.e, on Green’s integral the-orem and Green’s representation formula. Adding Green’s representation formula(6.12) applied to the solution u of the boundary value problem and Green’s secondintegral theorem applied to Φ(· , ζ) and observing the boundary condition (18.3) weobtain that

v(x) = Φ(x, ζ) +∫∂Dv(y)

∂Φ(x, y)∂ν(y)

ds(y), x ∈ IR2 \ D, x � ζ, (18.39)

for the total field v := u + Φ(· , ζ). From this, letting x tend to the boundary, by thejump relation for the double-layer potential we arrive at the integral equation of thesecond kind

ϕ(x) − 2∫∂Dϕ(y)

∂Φ(x, y)∂ν(y)

ds(y) = 2Φ(x, ζ), x ∈ ∂D, (18.40)

for the unknown boundary values ϕ := v|∂D. Given ∂D, by Theorem 6.21 and theRiesz theory the integral equation (18.40) is uniquely solvable. Evaluating (18.39)on ∂B we obtain that

∫∂Dϕ(y)

∂Φ(x, y)∂ν(y)

ds(y) = u(x), x ∈ ∂B. (18.41)

In terms of our inverse problem, we now can interpret (18.40) and (18.41) as asystem of two integral equations for the unknown boundary ∂D and the unknownbounday values ϕ of the total field v on ∂D. For the sequel it is convenient to call(18.41) the data equation since it contains the given trace of u on ∂B for the inverseproblem and (18.40) the field equation since it represents the boundary condition.


Both equations are linear with respect to ϕ and nonlinear with respect to ∂D. Equa-tion (18.41) is severely ill-posed (see Sections 16.6 and 18.2) whereas (18.40) iswell-posed.

Obviously there are three options for an iterative solution of (18.40) and (18.41).In a first method, given an approximation for the boundary ∂D one can solve thewell-posed integral equation of the second kind (18.40) for ϕ. Then, keeping ϕ fixed,equation (18.41) is linearized with respect to ∂D to update the boundary approxi-mation. In a second approach, one also can solve the system (18.40) and (18.41)simultaneously for ∂D and ϕ by Newton iterations, i.e., by linearizing both equa-tions with respect to both unknowns. Whereas in the first method the burden ofthe ill-posedness and nonlinearity is put on one equation, in a third method a moreeven distribution of the difficulties is obtained by reversing the roles of (18.40) and(18.41), i.e., by solving the severely ill-posed equation (18.41) for ϕ via regulariza-tion and then linearizing (18.40) to obtain the boundary update.

For a more detailed description of these ideas, using the parameterization (18.11)for starlike ∂D and recalling the mapping hr from (18.29), we introduce the param-eterized operators A1, A2 : C2

2π × L2[0, 2π]→ L2[0, 2π] by

A1(r, ψ)(t) :=∫ 2π

02 [h′r(τ)]⊥ · grady Φ(hr(t), hr(τ))ψ(τ) dτ

and

A2(r, ψ)(t) :=∫ 2π

0[h′r(τ)]⊥ · grady Φ(η(t), hr(τ))ψ(τ) dτ

for t ∈ [0, 2π], where we parametrized

∂B = {η(t) : t ∈ [0, 2π]}.Of course, the operators A1 and A2 are a renaming of the operators Lr (without theadded constant) and Vr from Theorems 18.7 and 18.8. Then the equations (18.40)and (18.41) can be written in the operator form

ψ − A1(r, ψ) = 2ui◦ hr (18.42)

andA2(r, ψ) = u ◦ η (18.43)

for ψ = ϕ ◦ hr, where we have set ui := Φ(· , ζ) indicating the incident field. Thelinearization of these equations requires the Frechet derivatives of the operators A1

and A2 with respect to r which are given by Theorems 18.7 and 18.8, i.e.,

(A′1(r, ψ; q))(t) = 2∫ 2π

0[h′r(τ)]⊥ ·Φ′′y (hr(t), hr(τ)) [hq(τ) − hq(t)]ψ(τ) d(τ)

+2∫ 2π

0[h′q(τ)]⊥ · grady Φ(hr(t), hr(τ))ψ(τ) d(τ)

(18.44)


and

(A′2(r, ψ; q))(t) =∫ 2π

0[h′r(τ)]⊥ ·Φ′′y (η(t), hr(τ)) hq(τ)ψ(τ) d(τ)

+

∫ 2π

0[h′q(τ)]⊥ · grady Φ(η(t), hr(τ))ψ(τ) d(τ)

(18.45)

for t ∈ [0, 2π]. Note that as opposed to the Frechet derivative of the boundary to dataoperator F as given in Theorem 18.9 the derivatives of A1 and A2 are given in anexplicit form as integral operators which offers computational advantages.

Now, given an approximation to the boundary parameterization r, the field equa-tion (18.42) can be solved for the density ψ. Then, keeping ψ fixed, linearizing thedata equation (18.43) with respect to r leads to the linear equation

A′2(r, ψ; q) = u ◦ η − A2(r, ψ) (18.46)

for q to update the radial function r via r + q using the data u ◦ η. This procedurecan be iterated. For fixed r and ψ the operator A′2(r, ψ; ·) has a smooth kernel andtherefore is severely ill-posed. This requires stabilization, for example via Tikhonovregularization. Injectivity of the linear operator A′2(r, ψ; ·) as the main prerequisitefor regularization schemes is guaranteed by the following theorem.

Theorem 18.11. Assume that r and ψ satisfy the field equation (18.42). Then thelinear operator A′2(r, ψ; ·) : L2[0, 2π]→ L2[0, 2π] is injective.

Proof. Analogous to the proof of Theorem 18.9 it can be seen that ψ ∈ C1,α2π . Assume

that A′2(r, ψ; q) = 0. In view of (18.35) we can rewrite hq = qν ◦ hr with some scalarfunction q such that q = 0 if and only if q = 0 (see also the concept of parallelcurves on p. 85). Then we have

divx

{Φ(x, hr(τ)) [hq(τ)]⊥

}= q(τ)

∂

∂τΦ(x, hr(τ))

and from (18.32) we obtain that

(A′2(r, ψ; q)(t) =∫ 2π

0q(τ)ψ′(τ)

∂

∂τΦ(η(τ), hr(τ)) dτ, t ∈ [0, 2π].

Now for the bounded harmonic function w in IR2 \ D defined by

w(x) :=∫ 2π

0q(τ)ψ′(τ)

∂

∂τΦ(x, hr(τ)) dτ, x ∈ IR2 \ D,

by the uniqueness for the exterior Dirichlet problem and the analyticity of harmonicfunctions from A′2(r, ψ; q) = 0, that is, from w = 0 on ∂B we can conclude thatw = 0 in IR2 \ D. Since the set of functions {Φ(x, ·) : x ∈ IR2 \ D} is complete inL2(∂D), from w = 0 in IR2 \ D we deduce that the product qψ′ has weak derivative


identically equal to zero. Therefore qψ′ = c in [0, 2π] for some constant c and sincethe derivative of the 2π-periodic function ψ must have at least one zero, it followsthat c = 0, that is, qψ′ = 0 in [0, 2π].

Now assume that ψ is constant on some open subinterval of [0, 2π], that is, v isconstant on an open subset Γ of ∂D. Since ∂v/∂ν = 0 on ∂D for the total field v,from Holmgren’s Theorem 6.7 we would have that v is constant in all of D whichcontradicts its singularity at the source point ζ. Hence we have q = 0 and conse-quently q = 0 which ends the proof. �

This approach for solving an inverse boundary value problem has been proposedby Johansson and Sleeman [111] first for an inverse obstacle scattering problem, i.e.,an inverse boundary value problem for the Helmholtz equation. It can be related tothe Newton iterations for the boundary to data operator F. From the derivation of(18.42) and (18.43) we have that

F(r) = A2

(r, [I − A1(r, ·)]−1(2ui◦ hr

)). (18.47)

By the product and chain rule this implies Frechet differentiability of the boundaryto data map F also in the case of the Neumann boundary condition (18.3) with thederivative given by

F′(r; q) = A′2(r, [I − A1(r, ·)]−1(2ui◦ hr

); q

)

+A2

(r, [I − A1(r, ·)]−1A′1

(r, [I − A1(r, ·)]−1(2ui◦ hr

); q

))

+A2

(r, [I − A1(r, ·)]−1(2(grad ui◦ hr) · hq

)).

(18.48)

Hence, the iteration scheme (18.46) can be interpreted as Newton iterations for Fwith the derivative of F approximated by the first term in the representation (18.48).As to be expected from this close relation, the quality of the reconstructions via(18.46) can compete with those of Newton iterations for the boundary to data oper-ator F with the benefit of reduced computational costs.

Without proof we note that it is also possible to characterize the Frechet deriva-tive in the Neumann case analogous to Theorem 18.9 via F′(r; q) = w ◦ η, wherew ∈ C2(IR2 \ D) ∩ C(IR2 \ D) denotes the uniquely determined harmonic functionvanishing at infinity and satisfying the Neumann boundary condition

∂w

∂ν=

dds

(ν ·

(hq ◦ h−1

r

) dds{Φ(· , ζ) + u}

)on ∂D, (18.49)

where d/ds denotes differentiation with respect to arc length (see [93]). In particular,analogous to (18.34) this implies the representation

(F′(r; q)) ◦ η−1 = 2V(I − K)−1

[dds

(ν ·

(hq ◦ h−1

r

) dds{Φ(· , ζ) + u}

)](18.50)


and proceeding as in the proof of Theorem 18.10 this can be used to show that thelinear operator F′(r; ·) : L2[0, 2π]→ L2[0, 2π] is injective and has dense range alsoin the Neumann case.

Following ideas first developed by Kress and Rundell [149] for the inverseDirichlet problem that we introduced at the end of Section 18.2 on p. 376, a secondapproach for iteratively solving the system (18.42) and (18.43) consists in simulta-neously linearizing both equations with respect to both unknowns. In this case, givenapproximations r and ψ both for the boundary parameterization and the Dirichletboundary values, the system of linear equations

− A′1(r, ψ; q) − 2(grad ui◦ hr

) · hq + χ − A1(r, χ) = −ψ + A1(r, ψ) + 2ui◦ hr (18.51)

andA′2(r, ψ; q) + A2(r, χ) = −A2(r, ψ) + u ◦ η (18.52)

has to be solved for q and χ in order to obtain updates r + q for the boundaryparameterization and ψ+χ for the boundary values. This procedure again is iteratedand coincides with traditional Newton iterations for the system (18.42) and (18.43).It has been analyzed including numerical examples by Ivanyshyn and Kress [108]for the inverse Neumann problem and also for inverse obstacle scattering problems(see [107, 109]). Due to the smoothness of the kernels in the second equation, thesystem (18.51) and (18.52) is severely ill-posed and requires regularization withrespect to both unknowns.

The simultaneous iterations (18.51) and (18.52) again exhibit connections to theNewton iteration for (18.5) as expressed through the following theorem.

Theorem 18.12. For fixed r let ψ be the unique solution of the field equation

ψ − A1(r, ψ) = 2ui◦ hr. (18.53)

If q satisfies the linearized boundary to data equation

F′(r; q) = u ◦ η − F(r) (18.54)

then q andχ := [I − A1(r, ·)]−1

(A′1(r, ψ; q) + 2(grad ui◦ hr) · hq

)(18.55)

satisfy the linearized data and field equations (18.51) and (18.52). Conversely, if qand χ satisfy (18.51) and (18.52) then q satisfies (18.54).

Proof. If q satisfies (18.54), from (18.47) and the equation (18.53) for ψ we have

F′(r; q) = u ◦ η − A2(r, ψ).

In view of (18.55), the representation (18.48) of the derivative of F yields

F′(r, q) = A′2(r, ψ; q) + A2(r, χ)


and combining this with the previous equation establishes that (18.52) holds. Fromthe definition (18.55) of χ we observe

χ − A1(r, χ) − A′1(r, ψ; q) − 2(grad ui◦ hr) · hq = 0

Therefore, in view of (18.53) we also have that (18.51) is satisfied.Conversely, the first equation (18.51) implies that χ can be expressed by (18.55)

and inserting this into (18.52) leads to

A′2(r, ψ; q)+ A2(r, [I − A1(r, ·)]−1(A′1(r, ψ; q)+2(gradui◦ hr) · hq

)= −A2(r, ψ)+ u ◦ η

and via (18.48) this implies (18.54). �

Theorem 18.12 illustrates the difference between the iteration method based on(18.51) and (18.52) and the Newton iterations (18.54) for the boundary to data oper-ator F. In general when performing (18.51) and (18.52) in the sequence of updatesthe relation (18.53) between the approximations r and ψ for the parameterizationand the boundary values will not be satisfied. This observation also indicates a pos-sibility to use (18.51) and (18.52) for implementing the Newton scheme (18.54)numerically. It is only necessary to replace the update ψ+χ for the boundary valuesby 2[I − A1(r + q, ·)]−1(ui◦ (hr + hq)

), i.e., at the expense of throwing away χ and

solving the field equation for the updated boundary with representation r+q for newboundary values.

The third possibility, namely given an approximation r for the boundary pa-rameterization first solving a regularized version of the severely ill-posed equation(18.41) for ϕ and then linearizing (18.40) to obtain the boundary update has not yetbeen explored in the literature. However, the hybrid method of Section 18.2 can beviewed as a modification of this approach (see [110]).

18.5 Sampling Methods

For the remainder of this chapter we explicitly indicate the dependence of the solu-tion to (18.1)–(18.2) on the source point by considering the bounded solution to theLaplace equation

Δu(· , y) = 0 in IR2 \ D (18.56)

satisfying the Dirichlet boundary condition

u(· , y) = −Φ(· , y) on ∂D (18.57)

for y ∈ ∂B and investigate the inverse boundary problem of determining ∂D froma knowledge of u(x, y) for all x, y ∈ ∂B. Since by Theorem 18.1 the knowledge ofu(x, y) for all x ∈ ∂B and only one y ∈ ∂B already uniquely determines ∂D, theinverse problem with data for all x, y ∈ ∂B is overdetermined.

18.5 Sampling Methods 391

Our subsequent analysis is based on the integral operator U : L2(∂B) → L2(∂B)with kernel given by the data of the inverse problems, i.e., U is defined by

(Ug)(x) := −∫∂B

u(x, y)g(y) ds(y), x ∈ ∂B. (18.58)

From the well-posedness of the exterior Dirichlet problem (Theorem 6.30) we de-duce that the solution u of (18.56)–(18.57) is continuous on ∂B × ∂B. Hence, U hasa continuous kernel and therefore is compact. In addition, we introduce an operatorH : L2(∂B)→ L2(∂D) by the single-layer potential

(Hg)(x) :=∫∂BΦ(x, y)g(y) ds(y), x ∈ ∂D,

which also is compact. The adjoint H∗ : L2(∂D)→ L2(∂B) of H is given by

(H∗ϕ)(x) =∫∂DΦ(x, y)ϕ(y) ds(y), x ∈ ∂B.

By R : v|∂D → v|∂Bwe denote the operator that maps the boundary values on ∂D ofbounded harmonic functions v ∈ C(IR2 \D)∩C2(IR2\ D) onto their restriction on ∂B.From the discussion of the representation of the Frechet derivative (18.34) we knowalready that we can write R = 2V(I+ K)−1, where V represents the modified double-layer potential (6.41) on ∂D and K is the modified double-layer integral operator asused in Theorem 6.25, and that R can be extended as a bounded injective operatorR : L2(∂D)→ L2(∂B) with dense range. For the solution of (18.56)–(18.57) we thencan write

−u(· , y)|∂B = RΦ(· , y)|∂D

for all y ∈ ∂B. From this, multiplying by g ∈ L2(∂B) and integrating over ∂B, inview of the well-posedness of the exterior Dirichlet problem and the continuity of Rwe deduce that Ug = RHg. Hence we have the factorization

U = RH. (18.59)

Now, for a constant unit vector e we define

Ψ (x, z) := e · gradx Φ(x, z), z ∈ B, x ∈ IR2 \ {z}, (18.60)

and consider the integral equation of the first kind

Ugz = Ψ (· , z)|∂B (18.61)

for a parameter point z ∈ D. Assume that gz ∈ L2(∂B) is a solution of (18.61). Thenthe single-layer potential

vz(x) :=∫∂BΦ(x.y)gz(y) ds(y), x ∈ B,


with density gz is harmonic in B and as consequence of (18.59) satisfies

Rvz|∂D = RHgz = Ψ (· , z)|∂B.

From this, by the uniqueness for the exterior Dirichlet problem and analyticity weobtain that

vz = Ψ (· , z) on ∂D. (18.62)

From the latter we conclude that ‖gz‖L2(∂B) → ∞ if the point z approaches the bound-ary ∂D. Therefore, in principle, the boundary ∂D may be found by solving the inte-gral equation (18.61) for z taken from a sufficiently fine grid in B and determining∂D as the location of those points z where ‖gz‖L2(∂B) becomes large. However, ingeneral, the unique solution of the interior Dirichlet problem in D with boundarycondition (18.62) will have an extension as a harmonic function across the bound-ary ∂D only in special cases. Hence, the integral equation of the first kind (18.61),in general, will not have a solution. Nevertheless, a mathematical foundation of theabove method can be provided. We will denote it as the linear sampling methodsince it is based on ideas of the linear sampling method in inverse obstacle scatter-ing as first proposed by Colton and Kirsch [30].

However, before we discuss the linear sampling method any further, we first con-sider a factorization method motivated by Kirsch’s [125] factorization method frominverse obstacle scattering. To this end we need to introduce the square root of apositive definite compact operator.

Theorem 18.13. Let A : X → X be a compact and positive definite operator in aHilbert space X. Then there exists a uniquely defined compact and positive definiteoperator A1/2 : X → X such that A1/2A1/2 = A.

Proof. In terms of the spectral decomposition from Theorem 15.12 in the form(15.13) we define

A1/2ϕ :=∞∑

n=1

√λn (ϕ, ϕn)ϕn (18.63)

for ϕ ∈ X with the positive square roots of the positive eigenvalues λn of A withorthonormal eigenelements ϕn. Then A1/2 : X → X is compact by Theorem 2.22since for the partial sums A1/2

m ϕ :=∑m

n=1

√λn (ϕ, ϕn)ϕn we have norm convergence

‖A1/2m − A1/2‖ = √λm+1 → 0 as m→ ∞. From

(A1/2ϕ, ψ) =∞∑

n=1

√λn (ϕ, ϕn)(ϕn, ψ)

for all ϕ, ψ ∈ X we observe that A1/2 is positive definite. Clearly A1/2ϕn =√λn ϕn

for n ∈ IN and therefore, in view of the self-adjointness of A1/2, we obtain

A1/2A1/2ϕ =

∞∑n=1

√λn (A1/2ϕ, ϕn)ϕn =

∞∑n=1

λn (ϕ, ϕn)ϕn = Aϕ


for all ϕ ∈ X. From the fact that any operator B with the required properties ofa square root satisfies B2 = A it follows that the eigenvalues of B are given byBϕn =

√λn ϕn. Therefore by the spectral theorem B must be of the form (18.63). �

Theorem 18.14. Let A : X → X be a compact and positive definite operator and forα > 0 let ϕα denote the Tikhovov regularized solution for the equation Aϕ = f forf ∈ X. Then limα→0(ϕα, f ) exists if and only if f ∈ A1/2(X). If f = A1/2g for g ∈ Xthen

limα→0

(ϕα, f ) = ‖g‖2.

Proof. From the representation for ϕα given in the proof of Theorem 15.23 we obtainthat

(ϕα, f ) =∞∑

n=1

λn

α + λ2n|( f , ϕn)|2 (18.64)

in terms of a singular system (λn, ϕn, ϕn) of A. Now assume that f = A1/2g for someg ∈ X . Then ( f , ϕn) = (g, A1/2ϕn) =

√λn (g, ϕn) and consequently

(ϕα, f ) =∞∑

n=1

λ2n

α + λ2n|(g, ϕn)|2 →

∞∑n=1

|(g, ϕn)|2 = ‖g‖2

as α→ 0.Conversely, assume that limα→0(ϕα, f ) exists. Then from (18.64) we observe that

there exists a constant M > 0 such that

∞∑n=1

λn

α + λ2n|( f , ϕn)|2 ≤ M

for all α > 0. Passing to the limit α→ 0, this implies that

∞∑n=1

1λn|( f , ϕn)|2 ≤ M.

Hence,

g :=∞∑

n=1

1√λn

( f , ϕn)ϕn ∈ X

is well defined and A1/2g = f . �

Theorem 18.15. Let X and Z be Hilbert spaces and let A : X → X and C : Z → Xbe compact linear operators such that A = CC∗ with the adjoint C∗ : X → Z of C.Assume further that A is positive definite and C is injective. Then

A1/2(X) = C(Z).

Proof. Since the operator A is compact and positive definite, by the spectral Theorem15.12 there exists a complete orthonormal system ϕn ∈ X of eigenelements of A with


positive eigenvalues λn, i.e.,

Aϕn = λnϕn, n = 1, 2, . . . .

Then the elements

gn :=1√λn

C∗ϕn, n = 1, 2, . . . , (18.65)

form a complete orthonormal system for Z. The orthonormality of the gn is an im-mediate consequence of the orthonormality of the ϕn and the property A = CC∗. Toestablish completeness, let g ∈ Z satisfy (g, gn) = 0, n = 1, 2, . . . . Then

(Cg, ϕn) = (g,C∗ϕn) =√λn (g, gn) = 0, n = 1, 2, . . . .

Hence, by Theorem 1.28, the completeness of the ϕn implies that Cϕ = 0. From thiswe obtain ϕ = 0, since C is injective. Therefore, again by Theorem 1.28, the gn arecomplete. This together with

Cgn =1√λn

Aϕn =√λn ϕn, n = 1, 2, . . . ,

implies that (√λn, gn, ϕn) is a singular system for the compact operator C. Together

with the singular system (√λn, ϕn, ϕn) of the operator A1/2, Picard’s Theorem 15.18

now implies that A1/2 (X) = C (Z). �

We want to apply Theorems 18.13 and 18.15 to the operator U and there-fore we need to establish that it is positive definite together with the existenceof an appropriate factorization. To this end we recall the single-layer operatorS : L2(∂D)→ L2(∂D) given by

(Sϕ)(x) :=∫∂DΦ(x, y)ϕ(y) ds(y), x ∈ ∂D,

where, for convenience, we have dropped a factor of 2.

Theorem 18.16. The operators U, R, and S are related through

U = RSR∗, (18.66)

where R∗ : L2(∂B)→ L2(∂D) is the adjoint of R.

Proof. By definition of the operators we have H∗ = RS , whence H = SR∗ follows.Inserting this into (18.59) completes the proof of (18.66). �

Theorem 18.17. Assume that the diameter of D is less than one. Then the operatorsS and U are positive definite.

Proof. Since its kernel is real-valued and symmetric, the operator S is self-adjoint.Consider the single-layer potential v with density ϕ ∈ L2(∂D) and assume that ϕ has


mean value zero. Then

v(x) = O

(1|x|

), grad v(x) = O

(1|x|

), |x| → ∞,

and, using the jump relations for single-layer potentials with L2 densities, an appli-cation of Green’s second integral theorem in D and IR2 \ D yields

(Sϕ, ϕ) =∫∂Dϕ Sϕ ds =

∫∂Dv

{∂v−∂ν− ∂v+∂ν

}ds =

∫IR2| grad v|2dx

whence (Sϕ, ϕ) ≥ 0 follows. For arbitrary ϕ ∈ L2(∂D) we write

ϕ = ψ0

∫∂Dϕ ds + ϕ

where ψ0 is the natural charge from Theorem 6.21 with∫∂Dψ0 ds = 1. Then∫

∂Dϕ ds = 0 and analogous to ϕ we decompose v into the sum

v = v0

∫∂Dϕ ds + v

with the single-layer potentials v0 and v with densities ψ0 and ϕ, respectively. ByTheorem 6.21 for the potential v0 we have that v0 = c on ∂D with some constantc and this, in particular, implies that (Sψ0, ϕ) = 0 = (Sϕ, ψ0). From

∫∂Dψ0 ds = 1

we observe that v0(x) → −∞ as |x| → ∞ and therefore, by the maximum-minimumprinciple for harmonic functions it follows that v0(x) ≤ c for all x ∈ IR2 \ D. Thisimplies that ψ0 = −∂v+/∂ν > 0 and therefore c = Sψ0 > 0, since by assumptionthe diameter of D is less than one and therefore ln |x − y| < 0 for all x, y ∈ ∂D withx � y. Consequently

(Sϕ, ϕ) = (Sψ0, ψ0)∣∣∣∣∣∫∂Dϕ ds

∣∣∣∣∣2

+ (Sϕ, ϕ) = c∣∣∣∣∣∫∂Dϕ ds

∣∣∣∣∣2

+ (Sϕ, ϕ) ≥ 0.

Equality implies that grad v = 0 in IR2 und∫∂Dϕ ds = 0 whence ϕ = 0 follows by

the jump relations. Therefore S is positive definite.From (18.66) we obtain that U is self-adjoint and that (Ug, g) = (SR∗g,R∗g) ≥ 0

for all g ∈ L2(∂B). By the positive definiteness of S equality holds in this inequalityonly if R∗g = 0. As already noted the operator R has dense range and therefore R∗is injective. Hence g = 0 and therefore U is positive definite. �

As first proposed by Hahner [84] and analyzed for the three-dimensional case,in the factorization method for solving our inverse problem the integral equation(18.61) is now replaced by

U1/2gz = Ψ (· , z)|∂B. (18.67)

Our aim is to show that this equation is solvable for gz in L2(∂B) if and only if z ∈ D.


Since under the assumption of Theorem 18.17 the single-layer operator S is com-pact and positive definite, by Theorem 18.13 there exists a uniquely defined compactand positive definite operator S 1/2 : L2(∂D)→ L2(∂D) satisfying S 1/2S 1/2 = S . Us-ing the boundedness of S from H−1/2(∂D) into H1/2(∂D) from Theorem 8.25 andthe duality pairing between H−1/2(∂D) and H1/2(∂D) we can estimate

∥∥∥S 1/2ϕ∥∥∥2

L2 = |(Sϕ, ϕ)| ≤ ‖Sϕ‖H1/2‖ϕ‖H−1/2 ≤ c‖ϕ‖2H−1/2

for all ϕ ∈ L2(∂D) and some positive constant c. Hence, by the denseness of L2(∂D)in H−1/2(∂D) it follows that S 1/2 is a bounded operator from H−1/2(∂D) into L2(∂D).Furthermore, by Corollary 2.12 we can estimate

∥∥∥S 1/2ϕ∥∥∥

H1/2 = sup‖ψ‖H−1/2=1

∣∣∣(S 1/2ϕ, ψ)∣∣∣ = sup

‖ψ‖H−1/2=1

∣∣∣(ϕ, S 1/2ψ)∣∣∣ ≤ c‖ϕ‖L2

for all ϕ ∈ L2(∂D). Therefore, finally, S 1/2 is also a bounded operator fromL2(∂D) into H1/2(∂D). Under the assumption that the diameter of ∂D is less thanone S : H−1/2(∂D) → H1/2(∂D) is an isomorphism (see p. 169). Hence, foreach f ∈ H1/2(∂D) there exists ψ ∈ H−1/2(∂D) such that Sψ = f . Then forϕ := S 1/2ψ ∈ L2(∂D) we have f = S 1/2ϕ, that is, S 1/2 : L2(∂D) → H1/2(∂D)also is an isomorphism.

Theorem 18.18. Assume that the diameter of D is less than one. Then

U1/2(L2(∂B)

)= R

(H1/2(∂D)

).

Proof. From Theorem 18.15 applied to the operators U : L2(∂B) → L2(∂B) andRS 1/2 : L2(∂D)→ L2(∂B) we obtain that that

U1/2(L2(∂B)

)= RS 1/2

(L2(∂D)

)

and the statement of the theorem follows from S 1/2(L2(∂D)

)= H1/2(∂D). �

Lemma 18.19. Ψ (·, z)|∂B ∈ R(H1/2(∂D)

)if and only if z ∈ D.

Proof. If z ∈ D then clearly Ψ (·, z) is a bounded harmonic function in IR2 \ D that iscontinously differentiable on ∂D. Therefore Ψ (·, z)|∂D ∈ H1/2(∂D) and Ψ (·, z)|∂B =

RΨ (·, z)|∂D.Conversely, let z � D and assume there exists f ∈ H1/2(∂D) such that R f =

Ψ (·, z)|∂B. Then, by the uniqueness for the exterior Dirichlet problem and analyt-icity, the solution v to the exterior Dirichlet problem with boundary trace v|∂D = fmust coincide with Ψ (·, z) in (IR2 \ D)\{z}. If z ∈ IR2 \ D this contradicts the analytic-ity of v. If z ∈ ∂D from the boundary condition it follows that Ψ (·, z)|∂D ∈ H1/2(∂D)which is a contradiction to Ψ (·, z) � H1(D) for z ∈ ∂D. �

Putting Theorem 18.18 and Lemma 18.19 together we arrive at the followingfinal characterization of the domain D.


Corollary 18.20. Assume that the diameter of D is less than one. Then z ∈ D if andonly if

U1/2gz = Ψ (·, z)|∂B

is solvable in L2(∂B).

Corollary 18.20 can be used for a reconstruction with the aid of a singular system(λn, gn, gn) of the operator U. Then, by Picard’s Theorem 15.18 as used in the proofof Theorem 18.15 we have that z ∈ D if and only if

∞∑n=1

|(gn, Ψ (· , z)|∂B)|2|λn| < ∞. (18.68)

At first glance Corollary 18.20 seems to imply that the nonlinear inverse problemhas been completely replaced by a linear problem. However, determining a singularsystem of U is nonlinear and there is still a nonlinear problem involved for findingthose points z where (18.68) is satisfied. Of course an obvious way to approximatelysolve this nonlinear problem is by truncating the series (18.68) through a finite sumfor z on a grid in B and approximately determining ∂D as the location of those pointsz where this sum is large.

We also note that the norm ‖gz‖L2(∂B) of the solution to (18.67) tends to infinityas z approaches ∂D. Assume to the contrary that ‖gzn‖L2(∂B) remains bounded for asequence (zn) in D with zn → z ∈ ∂D for n → ∞. Then, by Theorem 16.3, withoutloss of generality we may assume weak convergence gzn ⇀ gz ∈ L2(∂B) as n → ∞.The compactness of U1/2 implies that (see Problem 16.5)

U1/2gz = limn→∞U1/2gzn = lim

n→∞Ψ (· , zn)|∂B = Ψ (· , z)|∂B

i.e., we have a contradiction.We emphasize that in the derivation of Corollary 18.20 the unknown domain is

not required to be connected, i.e., it may consist of a finite number of componentsthat does not need to be known in advance. Furthermore, for the application of thefactorization method it is not necessary to know the boundary condition. Using theabove tools, in particular Theorem 18.15, it can be proven that Corollary 18.20 isalso valid for the Neumann boundary condition (18.3) (see [147]).

From a historical point of view, it is notable that the factorization method was firstdeveloped by Kirsch [125] in inverse obstacle scattering although this case requiresmuch deeper tools from functional analysis due to the fact that the far field operatoras counterpart of our data operator U no longer is self-adjoint. For the factorizationmethod in this more general setting we refer to [127] and for the application of thefactorization method to impedance tomography we refer to [86, 87].

In concluding this section, we briefly return to the linear sampling method.

Corollary 18.21. Assume that the diameter of D is less than one. For z ∈ D denoteby gz the solution of the equation (18.67) of the factorization method and for α > 0and z ∈ B let gαz denote the Tikhonov regularized solution of the equation (18.61) of


the linear sampling method, i.e., the solution of

αgαz + U∗Ugαz = U∗Ψ (·, z)|∂B,

and define the single-layer potential

vgαz (x) :=∫∂BΦ(x, y)gαz (y) ds, x ∈ B,

with density gαz . If z ∈ D then

limα→0

e · grad vgαz (z) = ‖gz‖2L2(∂B),

and if z � D then limα→0 e · grad vgαz (z) = ∞.Proof. In view of (18.60) we have e · grad vgαz (z) =

(gαz , Ψ (·, z)

)L2(∂B) and the state-

ment follows from Theorem 18.14 and Corollary 18.20. �

As pointed out above, the norm ‖gz‖L2(∂B) of the solution to (18.67) tends to in-finity as z→ ∂D. Therefore, by Corollary 18.21 also the limit limα→0 |e · grad vgαz (z)|tends to infinity when z approaches the boundary, i.e., the main feature from themotivation for the linear sampling method on p. 392 is verified.

Problems

18.1. Show that the Dirichlet problem for the Laplace equation in the doubly connected domainB \ D (in the geometric setting of this chapter) is uniquely solvable (see Problem 6.3).Hint: Find the solution in the form

u(x) =∫∂BϕB(y)

∂Φ(x, y)∂ν(y)

ds(y) +∫∂DϕD(y)

{∂Φ(x, y)∂ν(y)

+ Φ(x, y)

}ds(y), x ∈ B \ D.

18.2. Show that one pair of Cauchy data on ∂B of a harmonic function with zero trace on ∂Duniquely determines D (in the geometric setting of this chapter).

18.3. Analogous to Example 16.30, determine a singular value decomposition of the operator Adefined by (18.8).

18.4. For the kernel of the Frechet derivative of the double-layer operator in Theorem 18.7 showthat

k′(t, τ; r, q) = − 2π

[h′r(τ)]⊥ · {hr(t) − hr(τ)}|hr(t) − hr(τ)|4 {hr(t) − hr(τ)} · {hq(t) − hq(τ)}

+1π

[h′q(τ)]⊥ · {hr(t) − hr(τ)} + [h′r(τ)]⊥ · {hq(t) − hq(τ)}|hr(t) − hr(τ)|2

for t � τ. Use this representation to show that k′(· , · ; r, q) is continuous and to verify the inequality(18.30).

18.5. Determine a singular system for the operator U defined by (18.58) for the case where D andB are concentric disks.

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gleichung im IR3. Dissertation, Gottingen 1990.

Index

Abel’s integral equation, 2adjoint operator, 46Arzela–Ascoli theorem, 10

Baire theorem, 186Banach open mapping theorem, 188Banach space, 8Banach–Steinhaus theorem, 198best approximation, 13bijective, 2bilinear form, 45

non-degenerate, 45boundary element method, 255bounded linear functionals, 20bounded operator, 17bounded set, 7

totally, 9Bubnov–Galerkin method, 267

Cea’s lemma, 244Calderon projection, 131Cauchy data, 83Cauchy integral, 108Cauchy integral operator, 113Cauchy kernel, 118Cauchy principal value, 109Cauchy sequence, 8Cauchy–Schwarz inequality, 12Chebyshev polynomial, 215closed ball, 7closed set, 7closure, 7codimension, 70collocation method, 249compact, 8

relatively, 10sequentially, 9

compact operator, 25complete orthonormal system, 15complete set, 8completion, 8condition number, 280conjugate gradient method, 278constraint optimization, 329continuous

uniformly, 5continuous extension, 32continuous function, 5continuous operator, 5contraction, 23convergence, 5

mean square, 6norm, 19pointwise, 19, 185uniform, 6weak, 323

convex set, 14

defect correction iteration, 284dense set, 7direct sum, 35Dirichlet integral, 159Dirichlet problem, 83

weak, 160Dirichlet-to-Neumann operator, 130discrepancy, 334

principle, 303, 316, 336distance, 5domain of class Cn, 30dot product, 30double-layer heat potential, 175double-layer potential, 87dual least squares method, 359dual operator, 47

R. Kress, Linear Integral Equations, Applied Mathematical Sciences 82,DOI 10.1007/978-1-4614-9593-2, © Springer Science+Business Media New York 2014

409

410 Index

dual space, 20, 147dual system, 45

canonical, 46duality pairing, 148

eigenelement, 40eigenvalue, 40

multiplicity, 307equicontinuous, 10equivalent norm, 6

factorization method, 392fast multipole method, 293finite ascent, 37finite defect, 70finite descent, 38Fourier series, 15, 141Frechet derivative, 379Fredholm alternative, 55Fredholm integral equations, 1Fredholm operator, 72Fredholm theorems, 53function, 2

continuous, 5open, 190

function spaceC[a, b], 5Ck

2π, 145Cn(D), 76Cn(D), 76C0,α(G), 104C1,α(G), 108H1(D), 153H1[a, b], 338Hp(Γ), 151Hp[0, 2π], 142L2[a, b], 8

fundamental solutionheat equation, 174Helmholtz equation, 233Laplace equation, 76

Galerkin equation, 267Galerkin method, 267Gauss quadratures, 221Gauss trapezoidal product rule, 224Green’s formula, 77Green’s theorem, 76

Holder continuous, 103Holder space C0,α(G), 104Hahn–Banach theorem, 20harmonic function, 75hat functions, 203

heat equation, 171heat potential, 174Helmholtz equation, 233Hilbert inversion formula, 115Hilbert kernel, 115Hilbert scale, 149Hilbert space, 12Hilbert–Schmidt operator, 32Holmgren’s theorem, 79holomorphic

sectionally, 108hypersingular operator, 128

ill-posed problem, 298mildly, 312severely, 312

imbedding operator, 105improperly posed problem, 298index, 70, 116injective, 2inner product, 12integral equations

of the first kind, 1of the second kind, 1

integral operator, 22integro-differential equation, 341interpolation, 201

piecewise linear, 203trigonometric, 204

interpolation operator, 202interpolation space, 149inverse operator, 2inverse problem, 366involution, 48isometric, 8isomorphic, 8, 19isomorphism, 19iterated operator, 24

Jordan measurable, 22jump relation, 87

kernel, 22degenerate, 201weakly singular, 29, 31

Krylov subspaces, 278

Lagrange basis, 202Landweber–Fridman iteration, 319Laplace equation, 75Lax’s theorem, 51Lax–Milgram theorem, 269least squares solution, 270Legendre polynomial, 221

Index 411

lemmaCea, 244Riemann–Lebesgue, 321Riesz, 27

limit, 5linear operator, 17linear sampling method, 392Lippmann–Schwinger equation, 273Lipschitz domain, 169logarithmic double-layer potential, 87logarithmic single-layer potential, 87Lyapunov boundary, 96

mapping, 2maximum norm, 5maximum-minimum principle, 80, 172mean square norm, 6mean value theorem, 80method of fundamental solutions, 348minimum norm solution, 334minimum-maximum principle, 308moment collocation, 361moment solution, 358multigrid iteration, 288

full, 290

natural charge, 92Neumann problem, 84

weak, 161Neumann series, 23, 193Noether theorems, 122norm, 5

equivalent, 6maximum, 5mean square, 6stronger, 16weaker, 16

norm convergence, 19normal solvability, 69normal vector, 30normed space, 5nullspace, 34numerical integration operator, 224Nystrom method, 224

open ball, 7open set, 7operator, 2

adjoint, 46bijective, 2bounded, 17collectively compact, 190compact, 25continuous, 5

Dirichlet-to-Neumann, 130Fredholm, 72Hilbert–Schmidt, 32hypersingular, 128imbedding, 105injective, 2inverse, 2linear, 17nonnegative, 308positive definite, 267projection, 39self-adjoint, 129, 267, 304surjective, 2

operator equationof the first kind, 27, 297of the second kind, 23, 27

operator norm, 17orthogonal, 13

complement, 13system, 13

orthogonal projection, 242orthonormal basis, 306orthonormal system, 13

complete, 15

panel clustering methods, 293parallel surfaces, 84parametric representation, 30

regular, 30Parseval equality, 15Petrov–Galerkin method, 267Picard theorem, 311piecewise linear interpolation, 203pointwise convergence, 19Poisson integral, 174, 321polynomial

Chebyshev, 215Legendre, 221

pre-Hilbert space, 12product integration method, 230projection method, 242projection operator, 39, 241properly posed problem, 298

quadrature, 219convergent, 222interpolatory, 220points, 220weights, 220

quasi-solution, 329

range, 2Rayleigh–Ritz method, 267reduced wave equation, 233

412 Index

regular value, 40regularization

method, 301parameter, 301scheme, 301strategy, 302

regularizer, 63equivalent, 64left, 63right, 63

relatively compact, 10reproducing kernel, 360residual, 283residual correction, 284resolvent, 40resolvent set, 40Riemann problem, 116Riemann–Lebesgue lemma, 321Riesz lemma, 27Riesz number, 35Riesz theorem, 34, 49

scalar product, 12Schauder theory, 56self-adjoint operator, 129semi-norm, 104sequentially compact, 9series, 16sesquilinear form, 48

positive definite, 49symmetric, 49

Simpson’s rule, 221single-layer heat potential, 175single-layer potential, 87singular integral operator, 118singular system, 309singular value, 309

decomposition, 309Sobolev space

H1(D), 153H1[a, b], 338Hp(Γ), 151Hp[0, 2π], 142

Sobolev–Slobodeckij norm, 145Sokhotski–Plemelj theorem, 112space

Banach, 8Hilbert, 12normed, 5pre-Hilbert, 12Sobolev, 142

span, 15spectral cut-off, 316spectral method, 204

spectral radius, 40spectral theorem, 305spectrum, 40splines, 203Steklov theorem, 222successive approximations, 24, 195superconvergence, 278surface element, 30surface patch, 30surjective, 2Szego theorem, 222

tangent vector, 30theorem

Arzela–Ascoli, 10Baire, 186Banach open mapping, 188Banach–Steinhaus, 198Fredholm, 53Green, 76Hahn–Banach, 20Lax, 51Lax–Milgram, 269Noether, 122Picard, 311Riesz, 34, 49Sokhotski–Plemelj, 112Steklov, 222Szego, 222Weierstrass, 7

Tikhonov functional, 325Tikhonov regularization, 316totally bounded, 9trace operator, 154trapezoidal rule, 220trial and error, 303triangle inequality, 5trigonometric interpolation, 204two-grid iteration, 284

uniform boundedness principle, 186uniformly continuous, 5unisolvent, 201

Volterra integral equations, 41, 196

wavelets, 293weak derivative, 338weakly convergent, 323weakly singular kernel, 29Weierstrass approximation theorem, 7weight function, 220well-posed problem, 298

Education

Linear integral equations -rainer kress