Nonlinear Elliptic Equations of the Second Order

Qing Han

GRADUATE STUDIESIN MATHEMATICS 171

American Mathematical Society

Nonlinear Elliptic Equations of the Second Order

Qing Han

American Mathematical SocietyProvidence, Rhode Island

GRADUATE STUDIES IN MATHEMATICS 171

https://doi.org/10.1090//gsm/171

EDITORIAL COMMITTEE

Dan AbramovichDaniel S. Freed

Rafe Mazzeo (Chair)Gigliola Staffilani

2010 Mathematics Subject Classification. Primary 35J60, 35J25, 35J93, 35J96.

For additional information and updates on this book, visitwww.ams.org/bookpages/gsm-171

Library of Congress Cataloging-in-Publication Data

Names: Han, Qing, 1964–Title: Nonlinear elliptic equations of the second order / Qing Han.Description: Providence, Rhode Island : American Mathematical Society, [2016] | Series: Gradu-

ate studies in mathematics; volume 171 | Includes bibliographical references and index.Identifiers: LCCN 2015043419 | ISBN 9781470426071 (alk. paper)Subjects: LCSH: Differential equations, Elliptic. | Differential equations, Nonlinear. | AMS:

Partial differential equations – Elliptic equations and systems – Nonlinear elliptic equations.msc | Partial differential equations – Elliptic equations and systems – Boundary value problemsfor second-order elliptic equations. msc | Partial differential equations – Elliptic equations andsystems – Quasilinear elliptic equations with mean curvature operator. msc | Partial differentialequations – Elliptic equations and systems – Elliptic Monge-Ampere equations. msc

Classification: LCC QA377 .H31825 2016 | DDC 515/.3533–dc23 LC record available at http://lccn.loc.gov/2015043419

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10 9 8 7 6 5 4 3 2 1 21 20 19 18 17 16

To Yansu, Raymond, and Tommy

Contents

Preface vii

Introduction 1

Chapter 1. Linear Elliptic Equations 7

§1.1. The Maximum Principle 8

§1.2. Krylov-Safonov’s Harnack Inequality 23

§1.3. The Schauder Theory 42

Part 1. Quasilinear Elliptic Equations

Chapter 2. Quasilinear Uniformly Elliptic Equations 51

§2.1. Basic Properties 52

§2.2. Interior C1-Estimates 55

§2.3. Global C1-Estimates 58

§2.4. Interior C1,α-Estimates 61

§2.5. Global C1,α-Estimates 68

§2.6. Dirichlet Problems 73

Chapter 3. Mean Curvature Equations 79

§3.1. Principal Curvatures 80

§3.2. Global Estimates 87

§3.3. Interior Gradient Estimates 100

§3.4. Dirichlet Problems 105

v

vi Contents

Chapter 4. Minimal Surface Equations 115

§4.1. Integral Formulas 116

§4.2. Differential Identities 127

§4.3. Interior Gradient Estimates 136

§4.4. Interior Curvature Estimates 141

§4.5. Differential Identities: An Alternative Approach 151

Part 2. Fully Nonlinear Elliptic Equations

Chapter 5. Fully Nonlinear Uniformly Elliptic Equations 163

§5.1. Basic Properties 164

§5.2. Interior C2-Estimates 172

§5.3. Global C2-Estimates 194

§5.4. Interior C2,α-Estimates 200

§5.5. Global C2,α-Estimates 208

§5.6. Dirichlet Problems 213

Chapter 6. Monge-Ampere Equations 219

§6.1. Basic Properties 219

§6.2. Global C2-Estimates 223

§6.3. Interior C2-Estimates 236

§6.4. The Bernstein Problem 241

Chapter 7. Complex Monge-Ampere Equations 253

§7.1. Basic Properties 253

§7.2. Global C2-Estimates 258

Chapter 8. Generalized Solutions of Monge-Ampere Equations 277

§8.1. Monge-Ampere Measures 278

§8.2. Dirichlet Problems 300

§8.3. Global Holder Estimates 313

§8.4. Interior C1,α-Regularity 325

§8.5. Interior C2,α-Regularity 340

Bibliography 355

Index 365

Preface

The theory of nonlinear elliptic partial differential equations of the second or-der has flourished in the past half-century. The pioneering work of de Giorgiin 1957 opened the door to the study of general quasilinear elliptic differ-ential equations. Since then, the nonlinear elliptic differential equation hasbecome a diverse subject and has found applications in science and engi-neering. In mathematics, the development of elliptic differential equationshas influenced the development of the Riemannian geometry and complexgeometry. Meanwhile, the study of elliptic differential equations in a geo-metric setting has provided interesting new questions with fresh insights toold problems.

This book is written for those who have completed their study of thelinear elliptic differential equations and intend to explore the fascinating fieldof nonlinear elliptic differential equations. It covers two classes of nonlinearelliptic differential equations, quasilinear and fully nonlinear, and focuseson two important nonlinear elliptic differential equations closely related togeometry, the mean curvature equation and the Monge-Ampere equation.

This book presents a detailed discussion of the Dirichlet problems forquasilinear and fully nonlinear elliptic differential equations of the secondorder: quasilinear uniformly elliptic equations in arbitrary domains, meancurvature equations in domains with nonnegative boundary mean curvature,fully nonlinear uniformly elliptic equations in arbitrary domains, and Monge-Ampere equations in uniformly convex domains. Global solutions of theseequations are also characterized. The choice of topics is influenced by mypersonal taste. Some topics may be viewed by others as too advanced fora graduate textbook. Among those topics are the curvature estimates forminimal surface equations, the complex Monge-Ampere equation, and the

vii

viii Preface

generalized solutions of the (real) Monge-Ampere equations. Inclusion ofthese topics reflects their importance and their connections to many of themost active current research areas.

There is an inevitable overlap with the successful monograph by Gilbargand Trudinger. This book, designed as a textbook, is more focused on basicmaterials and techniques. Many results in this book are presented in specialforms. For example, the quasilinear and fully nonlinear uniformly ellipticdifferential equations studied in this book are not in their most generalform. The study of these equations serves as a prerequisite to the study ofthe mean curvature equation and the Monge-Ampere equation, respectively.More notably, our discussion of the Monge-Ampere equations is confinedto the pure Monge-Ampere equations, instead of the Monge-Ampere typeequations.

This book is based on one-semester courses I taught at Peking Univer-sity in the spring of 2011 and at the University of Notre Dame in the fallof 2011. Part of it was presented in the Special Lecture Series at PekingUniversity in the summer of 2007, in the Summer School in Mathematics atthe University of Science and Technology of China in the summer of 2008,and in a graduate course at Beijing International Center of MathematicalResearch in the spring of 2010.

During the writing of the book, I benefitted greatly from commentsand suggestions of many friends, colleagues, and students in my classes.Chuanqiang Chen, Xumin Jiang, Weiming Shen, and Yue Wang read themanuscript at various stages. Chuanqiang Chen and Jingang Xiong helpedwrite Chapter 8. Bo Guan, Marcus Khuri, Xinan Ma, and Yu Yuan providedvaluable suggestions on the arrangement of the book.

It is with pleasure that I record here my gratitude to my thesis advisor,Fanghua Lin, who guided me into the fascinating world of elliptic differentialequations more than twenty years ago.

I am grateful to Arlene O’Sean, my editor at the American MathematicalSociety, for reading the manuscript and guiding the effort to turn it into abook. Last but not least, I thank Sergei Gelfand at the AMS for his help inbringing the book to press.

The research related to this book was partially supported by grants fromthe National Science Foundation.

Qing Han

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Index

Aleksandrov’s maximum principle, 27

auxiliary functions, 14

barrier functions

for linear equations, 15Bernstein methods, 55, 59, 94, 100, 176,

179, 183, 186Bernstein theorems

minimal surface equations, 144Monge-Ampere equations, 250

boundary C2-estimatesfor complex Monge-Ampere

equations, 261for fully nonlinear equations, 195

for Monge-Ampere equations, 224boundary gradient estimates

for linear equations, 18for mean curvature equations, 89

for quasilinear equations, 58boundary Holder estimates

for linear equations, 19boundary Holder estimates of normal

derivativesfor linear equations, 38

Calderon-Zygmund decomposition, 30

Codazzi equations, 132comparison principles

for complex Monge-Ampereequations, 257

for fully nonlinear equations, 169for generalized solutions, 301

for linear equations, 10, 13

for Monge-Ampere equations, 222for quasilinear equations, 52

complex Monge-Ampere equations, 256boundary C2-estimates, 261

comparison principles, 257global C2-estimates, 273global gradient estimates, 259

contact sets, 23convex domains, 279

strictly convex domains, 310convex functions, 278

locally convex functions, 278

strictly convex functions, 278uniformly convex functions, 220

curvature estimates for minimal surfaceequations

integral estimates, 143pointwise estimates, 145

defining functions, 223, 258distance functions, 84

divergence, 116domains, 3

doubling condition, 322dyadic cubes, 30

Euclidean norms, 3existence for Dirichlet problems

for complex Monge-Ampereequations, 275

for fully nonlinear equations, 214, 215for generalized solutions, 308, 313

for linear equations, 44, 46

365

366 Index

for mean curvature equations, 105,107

for Monge-Ampere equations, 235for quasilinear equations, 73, 75

exterior sphere condition, 15

first fundamental forms, 80fully nonlinear equations

boundary C2-estimates, 195comparison principles, 169concavity, 165global C2-estimates, 199global C2,α-estimates, 208global gradient estimates, 194interior C2-estimates, 179, 186interior C1,α-estimates, 174interior C2,α-estimates, 200, 204interior gradient estimates, 176, 183Liouville theorems, 167, 176, 203uniform ellipticity, 165

Gauss curvatures, 81Gauss equations, 132generalized solutions, 296

comparison principles, 301Dirichlet problems in convex

domains, 308Dirichlet problems in strictly convex

domains, 313global L∞-estimates, 320global Holder estimates, 313, 314interior C1,α-estimates, 339interior C2,α-estimates, 353strict convexity, 333

global C2-estimatesfor complex Monge-Ampere

equations, 273for fully nonlinear equations, 199for Monge-Ampere equations, 233

global C1,α-estimatesfor quasilinear equations, 68

global C2,α-estimatesfor fully nonlinear equations, 208for quasilinear equations, 72

global L∞-estimatesfor generalized solutions, 320for linear equations, 14for mean curvature equations, 87for quasilinear equations, 58

global estimates on modulus ofcontinuity

for linear equations, 16

for mean curvature equations, 93global gradient estimates

for complex Monge-Ampereequations, 259

for fully nonlinear equations, 194for mean curvature equations, 94for Monge-Ampere equations, 223for quasilinear equations, 59

global Holder estimatesfor generalized solutions, 313, 314

gradients, 3

Harnack inequalities, 35Hausdorff distance, 325Hessian matrices, 3Hopf lemma, 10

integration by parts, 117interior C2-estimates

for fully nonlinear equations, 179, 186for Monge-Ampere equations, 237

interior C1,α-estimatesfor fully nonlinear equations, 174for generalized solutions, 339for quasilinear equations, 61

interior C2,α-estimatesfor fully nonlinear equations, 200, 204for generalized solutions, 353for quasilinear equations, 67

interior gradient estimatesfor fully nonlinear equations, 176, 183for mean curvature equations, 100for minimal surface equations, 137for quasilinear equations, 55

interior Holder estimatesfor linear equations, 37

interior sphere condition, 12

John lemma, 242, 316

Laplace-Beltrami operators, 117, 130Legendre transforms, 287level sets, 245Levi-Civita connections, 129linear equations

Aleksandrov’s maximum principle, 27barrier functions, 15boundary gradient estimates, 18boundary Holder estimates, 19boundary Holder estimates of normal

derivatives, 38boundary Schauder estimates, 45comparison principles, 10

Index 367

global L∞-estimates, 14global estimates on modulus of

continuity, 16global Schauder estimates, 46Harnack inequalities, 35interior L∞-estimates, 28interior Holder estimates, 37interior Schauder estimates, 43Liouville theorems, 38strong maximum principles, 12subsolutions, 8supersolutions, 8uniform ellipticity, 8weak Harnack inequalities, 31weak maximum principles, 8

Liouville theoremsfully nonlinear equations, 167, 176,

203linear equations, 38minimal surface equations, 104, 139,

140quasilinear equations, 53, 67

locally convex functions, 278

maximum principlesstrong maximum principles, 12weak maximum principles, 8

mean curvature equations, 83boundary gradient estimates, 89global L∞-estimates, 87global estimates on modulus of

continuity, 93global gradient estimates, 94interior gradient estimates, 100

mean curvatures, 81mean-value inequality, 121method of continuity, 73, 105, 213, 214,

235minimal surface equations, 83

Bernstein theorems, 144integral curvature estimates, 143interior gradient estimates, 137Liouville theorems, 104, 139, 140pointwise curvature estimates, 145

mollifiers, 285Monge-Ampere equations, 221

Bernstein theorems, 250boundary C2-estimates, 224comparison principles, 222generalized solutions, 296global C2-estimates, 233global gradient estimates, 223

interior C2-estimates, 237Monge-Ampere measures, 292monotonicity identity, 118multi-indices, 3

normal mappings, 279normalization, 244, 318

pluri-subharmonic functions, 254strictly pluri-subharmonic functions,

254principal curvatures, 80

quasilinear equationsboundary gradient estimates, 58comparison principles, 52global C1,α-estimates, 68global C2,α-estimates, 72global L∞-estimates, 58global gradient estimates, 59interior C1,α-estimates, 61interior C2,α-estimates, 67interior gradient estimates, 55Liouville theorems, 53, 67uniform ellipticity, 52

Riemann curvature tensors, 130

Schauder theoryboundary Schauder estimates, 45global Schauder estimates, 46interior Schauder estimates, 43

second fundamental forms, 80, 131sections, 298Simons identity, 133Sobolev inequality, 124strictly convex domains, 310strictly convex functions, 278strictly pluri-subharmonic functions,

254strongly pseudo-convex domains, 258subsolutions, 8

upper bounds, 28supersolutions, 8

weak Harnack inequalities, 31supporting functions, 278

tangential gradients, 116

uniformly convex domains, 223uniformly convex functions, 220uniformly elliptic equations

fully nonlinear equations, 165linear equations, 8

368 Index

quasilinear equations, 52uniqueness for Dirichlet problems

for fully nonlinear equations, 170for linear equations, 10for quasilinear equations, 53

Vitali covering lemma, 123

weak Harnack inequalities, 31

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Nonlinear elliptic differential equations are a diverse subject with important applications to the physical and social sciences and engi-neering. They also arise naturally in geometry. In particular, much of the progress in the area in the twentieth century was driven by geometric applications, from the Bernstein problem to the exis-tence of Kähler–Einstein metrics.

This book, designed as a textbook, provides a detailed discussion of the Dirichlet problems for quasilinear and fully nonlinear elliptic differential equations of the second order with an emphasis on mean curvature equa-tions and on Monge–Ampère equations. It gives a user-friendly introduction to the theory of nonlinear elliptic equations with special attention given to basic results and the most important techniques. Rather than presenting the topics in their full generality, the book aims at providing self-contained, clear, and “elementary” proofs for results in important special cases. This book will serve as a valuable resource for graduate students or anyone interested in this subject.

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