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Jindrich Necas Center for Mathematical ModelingLecture notes
Volume 4
Topics inmathematical
modelingVolume edited by M. Benes and E. Feireisl
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Jindrich Necas Center for Mathematical ModelingLecture notes
Volume 4
Editorial board
Michal BenesPavel DrabekEduard FeireislMiloslav FeistauerJosef MalekJan MalySarka NecasovaJirı NeustupaAntonın NovotnyKumbakonam R. RajagopalHans-Georg RoosTomas RoubıcekDaniel SevcovicVladimır Sverak
Managing editors
Petr KaplickyVıt Prusa
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Jindrich Necas Center for Mathematical ModelingLecture notes
Topics in mathematical modeling
Masato KimuraFaculty of MathematicsKyushu UniversityHakozaki 6-10-1, Higashi-ku, Fukuoka 812-8581Japan
Philippe LaurencotInstitut de Mathematiques de Toulouse, CNRS UMR 5219Universite de ToulouseFâ31062 Toulouse cedex 9France
Shigetoshi YazakiFaculty of EngineeringUniversity of Miyazaki1-1 Gakuen Kibanadai Nishi, Miyazaki 889-2192Japan
Volume edited by M. Benes and E. Feireisl
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2000 Mathematics Subject Classification. 35-06, 49Q10, 53C44, 35B40, 34A34
Key words and phrases. mathematical modeling, shape derivative, movinginterfaces, diffusive HamiltonâJacobi eqautions, crystalline curvature flow
equations
Abstract. The volume provides a record of lectures given by the visitors of the Jindrich NecasCenter for Mathematical Modeling in academic year 2007/2008.
All rights reserved, no part of this publication may be reproduced or transmitted in any formor by any means, electronic, mechanical, photocopying or otherwise, without the prior writtenpermission of the publisher.
c© Jindrich Necas Center for Mathematical Modeling, 2008c© MATFYZPRESS Publishing House of the Faculty of Mathematics and Physics
Charles University in Prague, 2008
ISBN 978-80-7378-060-9
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Preface
This volume of the Lecture Notes contains texts prepared by Masato Kimura,Philippe Laurencot and Shigetoshi Yazaki. They were long term visiting scientistsat the Necas Center for Mathematical Modeling in the years 2007 and 2008, andtheir topics are related each to other.
The Center creates a fruitful and productive environment for its members andfor its visitors. The visitors are invited to prepare mini-courses focused on theirresearch field. The texts covering the contents of mini-courses is published in theLecture Notes of the Necas Center for Mathematical Modeling.
First part of the volume of the Lecture Notes covers the lecture series of MasatoKimura on dynamics of hyperplanes in R
n. The text discusses aspects of formu-lation for the problems with moving interfaces including the shape derivatives ofenergy functionals. The author then derives the motion laws known in this domainby means of differential and variational calculus on hypersurfaces. Finally, thegradient-flow approach is used to put known examples of the hyperplane dynamicsinto a general framework.
Second part of the volume contains the lecture notes for the course of PhilippeLaurencot. This course was devoted to the aspects of diffusive HamiltonâJacobiequations which play a key role in several recently studied domains of applicationincluding the dynamics of hyperplanes, phase transitions and computer image pro-cessing. The text presents results related to the long-term behavior of the solutionof diffusive HamiltonâJacobi equations as well as results for the solution extinctionin finite time.
Third part of the volume is devoted to the lecture notes for the course ofShigetoshi Yazaki, which analyzed the motion of closed planar curves by crystallinecurvature with preservation of area. The text explains key issues related to the givendomain and provides insight into the physical application. The author discusses theextent of numerical solution of the given problem as well.
The presented three lecture notes are suitable for the students wishing to learnrecent advances in the given field of applied mathematics as well as they can serve asreviews for general mathematical audience. They also can be seen as vivid examplesof a cooperation established within the Necas Center for Mathematical Modeling.
October 2008 M. BenesE. Feireisl
vii
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Contents
Preface vii
Part 1. Shape derivative of minimum potential energy: abstracttheory and applications
Masato Kimura 1
Preface 5
Chapter 1. Shape derivative of minimum potential energy: abstract theoryand applications 7
1. Introduction 72. Multilinear map and Frechet derivative 103. Minimization problems 124. Banach contraction mapping theorem 155. Implicit function theorem 176. Parameter variation formulas 197. Lipschitz deformation of domains 238. Potential energy in deformed domains 289. Applications 30
Bibliography 37
Part 2. Geometry of hypersurfaces and moving hypersurfaces in Rm
for the study of moving boundary problemsMasato Kimura 39
Preface 43
Chapter 1. Preliminaries 451. Notation 452. Plane curves 463. Parametric representation 474. Graph representation and principal curvatures 50
Chapter 2. Differential calculus on hypersurfaces 531. Differential operators on Î 532. Weingarten map and principal curvatures 54
Chapter 3. Signed distance function 59
ix
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x CONTENTS
1. Signed distance function in general 592. Signed distance function for hypersurface 60
Chapter 4. Curvilinear coordinates 651. Differential and integral formulas in N Δ(Î) 65
Chapter 5. Moving hypersurfaces 691. Normal time derivatives 692. Signed distance function for moving hypersurface 723. Time derivatives of geometric quantities 73
Chapter 6. Variational formulas 751. Transport identities 752. Transport identities for curvatures 78
Chapter 7. Gradient structure and moving boundary problems 811. General gradient flow of hypersurfaces 812. Prescribed normal velocity motion 823. Mean curvature flow 824. Anisotropic mean curvature flow 835. Gaussian curvature flow 836. Willmore flow 847. Volume preserving mean curvature flow 848. Surface diffusion flow 859. HeleâShaw moving boundary problem 86
Bibliography 89
Appendix A. 911. Adjugate matrix 912. Jacobiâs formula 92
Part 3. Large time behaviour for diffusive HamiltonâJacobiequations
Philippe Laurencot 95
Chapter 1. Introduction 99
Chapter 2. Well-posedness and smoothing effects 1031. Gradient estimates 1052. Time derivative estimates 1083. Hessian estimates 1114. Existence 1145. Uniqueness 116Bibliographical notes 117
Chapter 3. Extinction in finite time 1191. An integral condition for extinction 1212. A pointwise condition for extinction 1233. Non-extinction 124
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CONTENTS xi
4. A lower bound near the extinction time 126Bibliographical notes 127
Chapter 4. Temporal decay estimates for integrable initial data: Ï = 1 1291. Decay rates 1312. Limit values of âuâ1 1343. Improved decay rates: q â (1, qâ) 137Bibliographical notes 139
Chapter 5. Temporal growth estimates for integrable initial data: Ï = â1 1411. Limit values of âuâ1 and âuââ 1422. Growth rates 149Bibliographical notes 150
Chapter 6. Convergence to self-similarity 1511. The diffusion-dominated case: Ï = 1 1512. The reaction-dominated case: Ï = â1 154Bibliographical notes 158
Bibliography 159
Appendix A. Self-similar large time behaviour 1631. Convergence to self-similarity for the heat equation 1632. Convergence to self-similarity for HamiltonâJacobi equations 164
Part 4. An area-preserving crystalline curvature flow equationShigetoshi Yazaki 169
Preface 173
Chapter 1. An area-preserving crystalline curvature flow equation:introduction to mathematical aspects, numerical computations,and a modeling perspective 175
1. Introduction 1752. Plane curve 1753. Moving plane curve 1784. Curvature flow equations 1795. Anisotropy 1816. The Frank diagram and the Wulff shape 1837. Crystalline energy 1878. Crystalline curvature flow equations 1899. An area-preserving motion by crystalline curvature 19510. Scenario of the proof of Theorem 1.50 19611. Numerical scheme 19812. Towards modeling the formation of negative ice crystals or vapor
figures produced by freezing of internal melt figures 202
Bibliography 207
Index 211
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xii CONTENTS
Appendix A. 2131. Strange examples 2132. A non-concave curve 2133. Anisotropic inequalityâproof of Lemma 1.53â 213
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Part 1
Shape derivative of minimum
potential energy: abstract theory
and applications
Masato Kimura
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2000 Mathematics Subject Classification. 49-01, 49Q10
Key words and phrases. shape derivative, fracture mechanics, lipschitzdeformation on domains
Abstract. The text provides an elementary introduction to mathematicalfoundation of shape derivative of potential energy and provides several ap-plications of the theory to some elliptic problems. Among many importantapplications the text focus on the energy release rate in fracture mechanics.
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Contents
Preface 5
Chapter 1. Shape derivative of minimum potential energy: abstract theoryand applications 7
1. Introduction 72. Multilinear map and Frechet derivative 103. Minimization problems 124. Banach contraction mapping theorem 155. Implicit function theorem 176. Parameter variation formulas 197. Lipschitz deformation of domains 238. Potential energy in deformed domains 289. Applications 30
Bibliography 37
3
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Preface
The principal aim of this short lecture note is to present a mathematical foun-dation of shape derivatives of potential energies with applications to some ellipticproblems. Among many important applications, in particular, we will focus on theenergy release rate in fracture mechanics. I hope that this work will contribute tofurther mathematical understanding of many applications of the shape derivativeincluding fracture mechanics.
This note was originally prepared for a series of lectures at Graduate Schoolof Mathematics, Kyushu University in 2005-2006. I also had occasions to give alecture at Department of Mathematics, Faculty of Nuclear Sciences and PhysicalEngineering, Czech Technical University in Prague in March 2006, and an intensivelecture at Graduate School of Informatics, Kyoto University in July 2006. I stayedat Czech Technical University in Prague for one year from April 2007 as a visitingresearcher of the Necas Center for Mathematical Modeling LC06052 financed byMSMT. During the stay, I again had a chance to present a part of this work inthe workshop âMatematika na Vysokych Skolach, Variacnı Principy v Matematicea Fyziceâ held at Herbertov in September 2007.
Based on the above lectures, this lecture note was completed during my stayin Prague with a support of NCMM. I would like to express my sincere thanksto all participants in the above lectures and to the organizers who gave me suchopportunities, in particular, Professor Yuusuke Iso and Dr. Masayoshi Kubo ofKyoto University. I also would like to be grateful to Professor Michal Benes andhis colleagues in the Mathematical Modelling Group of Faculty of Nuclear Sciencesand Physical Engineering, Czech Technical University in Prague, for their kindhospitalities and the stimulating research environment during my stay in Prague.
The mathematical results in this work have been obtained in a joint work[13, 14] with Dr. Isao Wakano of Kyoto University. I wish to acknowledge hiscooperation in this research. Professor Kohji Ohtsuka of Hiroshima Kokusai GakuinUniversity introduced me to his preceding work in this research area with fruitfuldiscussions. I would like to appreciate his kind encouragement during the research.
Masato KimuraPrague, February 2008
5
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CHAPTER 1
Shape derivative of minimum potential energy:
abstract theory and applications
1. Introduction
The shape derivative (or shape variation) is one of the fundamental techniquesin the theories of shape optimization, shape sensitivity analysis, and many freeboundary problems. The importance of these research fields has been greatly in-creasing for the last few decades, because of their many applications to numericalsimulations of industrial problems. The expansion of such applications of the shapederivative has been also supplying interesting and challenging problems to mathe-maticians.
We present a mathematical foundation of shape derivatives of potential ener-gies related to some elliptic problems. A central application which we keep in mindis the energy release rate in a cracked domain, which plays an important role infracture mechanics. The application of the shape derivative to the energy releaserate was originally studied in a series of works by Ohtsuka [16, 17, 18], and math-ematical justification of the existence and derivation of some formulas in a generalgeometric situation was carried out there. To the works by Ohtsuka, we will give analternative aspect with a new mathematical framework, which enables us to haverefined results.
We treat a class of shape derivatives, which includes the energy release rate,as a variational problem with a parameter in Banach spaces. A new parametervariational principle and the classical implicit function theorem in Banach spaceswill play key roles in our abstract theory. Similar mathematical approaches tothe shape derivative for various problems are found in Sokolowski and Zolesio [22]and [15] etc.
We briefly give an explanation of the energy release rate in fracture mechanics.Scientific investigation to understand crack evolution process in elastic body wasoriginated by Griffith [9] and has been studied from various viewpoints in engineer-ing, physics and mathematics since then. Griffithâs idea in the fracture mechanics iseven now the fundamental theory in modeling and analysis of the crack behaviour.We here make reference to only very few extended studies from mathematical pointof view, Cherepanov [4], Rice [20], Ohtsuka [16], [17], [18], Ohtsuka-Khludnev [19],and Francfort-Marigo [6]. For more complete list of crack problems and fracturemechanics, please see the references in the above papers.
The energy release rate G is a central concept in Griffithâs theory and its variousextended theories such as [6]. According to such theories, we treat crack evolutionsin brittle materials with linear elasticity under a quasi-static situation, in which
7
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8 1. SHAPE DERIVATIVE OF MINIMUM POTENTIAL ENERGY
applied boundary loading is supposed to change slowly and any inertial effect canbe ignored. The elastic energy at a fixed moment is supposed to be given byminimization of an elastostatic potential energy. According to the Griffithâs theory,the surface energy which is required in the crack evolution is supplied by relaxationof the potential energy along crack growth.
Roughly speaking, the energy release rate is defined as follows. Let 멉 bea bounded domain in R
n (n â„ 2), which corresponds to the un-cracked materialunder consideration. We assume that a crack ÎŁ exists in Ωâ, where ÎŁ is a closureof an n â 1 dimensional hypersurface. The cracked elastic body is represented byΩâ \ÎŁ. We consider a virtual crack extension ÎŁ(t) with parameter t â [0, T ), whereÎŁ = ÎŁ(0) â ÎŁ(t1) â ÎŁ(t2) (0 â€â t1 â€â t2 < T ).
Under the quasi-static assumption, the elastic potential energy E(t) in Ω(t) :=Ωâ \ ÎŁ(t) is given by
E(t) := minv
â«
Ωâ\ÎŁ(t)
W (x, v(x),âv(x))dx, (1.1)
where v is a possible displacement field in Ωâ\ÎŁ(t) with a given boundary conditionand W (x, v(x),âv(x)) represents a potential energy density including a body force.A given displacement field is imposed on a part of âΩâ, and no boundary conditionis imposed for the admissible displacement fields on the other part of âΩ(t) includingboth sides of ÎŁ(t). In other words, the normal stress free condition is imposed forthe minimizer on âΩ(t) implicitly.
The energy release rate G at t = 0 along the virtual crack extension ÎŁ(t)0â€t<T
is given by
G := limtâ+0
E(0) â E(t)
|ÎŁ(t) \ ÎŁ| . (1.2)
Since E(t) †E(0), G â„ 0 follows if the limit exists. The Griffithâs criterion forthe brittle crack extension is given by G â„ Gc, where Gc is an energy required tocreate a new crack per unit (nâ 1)-dimensional volume (i.e., length in 2d and areain 3d) and it is a constant depending on the material property and the position.
Cherepanov [4] and Rice [20] studied the so-called J-integral for straight crackin two dimensional linear elasticity, which is a path-independent integral expressionsof the energy release rate. Since these works, theoretical and practical studies ofcrack evolutions have been much developed by means of such useful mathematicalexpression of G in two dimensional case.
While most of these mathematically rigorous results have been restricted to twodimensional linear elasticity (and often only for straight cracks), Ohtsuka [16]â[18]and Ohtsuka-Khludnev [19] developed a mathematical formulation of the energyrelease rate for general curved cracks in multi-dimensional linear or semi-linearelliptic systems. They proved existence of the energy release rate, and obtained itsexpression by a domain integral and by generalized J-integral.
Based on the idea in [17], we shall give a new mathematical framework forshape derivative of potential energy including the energy release rate. Adoptingdomain perturbation Ï of Lipschitz class, the shape derivative is treated in anabstract parameter variation problem in Banach spaces, where Ï is considered as aparameter belonging to Lipschitz class. Instead of estimating the limit (1.2) directlyas in [17] and [19], we treat it by means of the Frechet derivative.
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1. INTRODUCTION 9
In our approach, the shape derivative of minimum potential energy is derivedas a Frechet derivative in a Banach space within an abstract parameter variationformulas and it is given in a domain integral expression. The key tools in theabstract parameter variation setting are the implicit function theorem and theLaxâMilgram theorem.
The organization of this lecture note is as follows. In Sections 2â5, some fun-damental results from nonlinear functional analysis are systematically described,particularly on the Frechet derivative and the implicit function theorem. Theirproofs are also given in most cases and some relatively simple proofs are left tothe readers. One of the primary sources of these sections is the introduction of thebook [11]. For more details of the nonlinear functional analysis, see [3] etc.
Based on the tools prepared in the previous sections, several abstract param-eter variation formulas are established in Section 6. A framework of Lipschitz de-formation of domains, which includes crack extensions, is established in Section 7,Minimization problems with a general potential energy in deformed domains arestudied in Section 8 as applications of the abstract parameter variation formulasof Section 6. Quadratic energy functionals corresponding to second order linearelliptic equations are treated in Section 9.
We remark that the theorems obtained in Sections 8â9 include the resultsin [17] and [19] under a weaker assumption for regularity of domain perturba-tion. In [19], they assumed that the domain perturbation t â Ï(t) belongs toC2([0, T ], W 2,â(Rn)n) and derived the domain integral expression of G, whereaswe prove it in Theorem 1.56 under a weaker assumption Ï â C1([0, T ], W 1,â(Rn)n).
In this note, we use the following notation. For an integer n â N, Rn denotes
the n-dimensional Euclidean space over R. Each point x â Rn is expressed by a
column vector x = (x1, · · · , xn)T, where T stands for the transpose of the vector ormatrix.
For a real-valued function f = f(x), its gradient is given by a column vector
âf(x) :=
(âf
âx1(x), · · · ,
âf
âxn(x)
)T
=
âfâx1
(x)...
âfâxn
(x)
,
and its Hessian matrix is denoted by
â2f := âT(âf) = â(âTf) =
â2fâx2
1
· · · â2fâx1âxn
......
â2fâxnâx1
· · · â2fâx2
n
â R
nĂn,
where we note that â2 does not mean the Laplacian operator in this lecture note.We denote the Laplacian of f by
âf := divâf = trâ2f =
nâ
i=1
â2f
âx2i
.
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10 1. SHAPE DERIVATIVE OF MINIMUM POTENTIAL ENERGY
2. Multilinear map and Frechet derivative
Our mathematical treatment for the shape derivative of the potential energyis based on infinite dimensional differential calculus, i.e. the Frechet derivative inBanach spaces. Several fundamental facts on multilinear maps and the Frechet de-rivative in Banach spaces are collected in this section. For the readersâ convenience,their definitions and basic theorems are stated with some exercises.
Definition 1.1 (Multilinear map). We suppose that Xk (k = 1, · · · , n, n â N),X and Y are real Banach spaces.
(1) A map g is called a multilinear map fromân
k=1 Xk to Y , if g :ân
k=1 Xk âY and the map [xk 7â g(x1, · · · , xn)] is linear from Xk to Y for eachk = 1, · · · , n. In the case that X1 = · · · = Xn = X , g is called a n-linearmap from X to Y .1
(2) A multilinear map g is called bounded, if there exists M > 0 such that
âg(x1, · · · , xn)âY †Mâx1âX1· · · âxnâXn
(â(x1, · · · , xn) â
nâ
k=1
Xk
).
The set of bounded multilinear maps g :ân
k=1 Xk â Y is denoted byB(X1, · · · , Xn; Y ). It will be shown (see Exercise 1.2) thatB(X1, · · · , Xn; Y ) is a Banach space with the norm
âgâB(X1,··· ,Xn;Y ) := supxkâXk
âg(x1, · · · , xn)âY
âx1âX1· · · âxnâXn
(g â B(X1, · · · , Xn; Y )).
The set of bounded n-linear maps from X to Y also becomes a Banachspace and it is denoted by Bn(X, Y ).
(3) An n-linear map g from X to Y is called symmetric, if g(x1, · · · , xn) isinvariant under all permutations of x1, · · · , xn. We write Bsym
n (X, Y ) :=g â Bn(X, Y ); g : symmetric.
Exercise 1.2. Prove that B(X1, · · · , Xn; Y ) is a Banach space by using thenatural identification B(X1, · · · , Xn; Y ) âŒ= B(X1, B(X2, · · · , Xn; Y )).
Similar to the case of linear operators (1-linear maps), the boundedness of ann-linear map is equivalent to its continuity as shown in the next proposition.
Proposition 1.3. Let Xk (k = 1, · · · , n, n â N), X and Y be Banach spaces.Then the following three conditions are equivalent.
(1) g is a continuous multilinear map fromân
k=1 Xk to Y .(2) g is a multilinear map from
ânk=1 Xk to Y , and is continuous at the origin
ofân
k=1 Xk.(3) g â B(X1, · · · , Xn; Y ).
Proof. 1.â2. This is trivial.2.â3. If not, for m â N, there exist xm
k â Xk such thatâg(xm
1 , · · · , xmn )âY â„ mnâxm
1 âX1· · · âxm
n âXn . Let us define Οmk := (mâxm
k âXk)â1xm
k .Then âg(Οm
1 , · · · , Οmn )âY â„ 1. Since Οm
k â 0 in Xk as m â â, taking the limit as
1âMultilinearâ and ân-linearâ are usually called, linear (n = 1), bilinear (n = 2), trilinear(n = 3), etc.
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2. MULTILINEAR MAP AND FRECHET DERIVATIVE 11
m â â, it follows that âg(0, · · · , 0)âY â„ 1 from the continuity of g at the origin.This contradicts the fact g(0, · · · , 0) = 0.3.â1. This is shown by a recursive argument with respect to n = 1, 2, · · · .
Some basic properties and typical examples of the n-linear map will be foundin the following exercises.
Exercise 1.4. If n â„ 2, prove that Bsymn (X, Y ) is a closed subspace of Bn(X, Y ).
Exercise 1.5. Let us consider a bounded bilinear map f0 â B(X, Y ; Z) andbounded multilinear maps g1 â B(X1, · · · , Xn; X) and g2 â B(Y1, · · · , Ym; Y ), andwe define
f(x1, · · · , xn, y1, · · · , ym) := f0(g1(x1, · · · , xn), g2(y1, · · · , ym)).
Then prove that f â B(X1, · · · , Xn, Y1, · · · , Ym; Z) and
âfâB(X1,··· ,Xn,Y1,··· ,Ym;Z) †âf0âB(X,Y ;Z)âg1âB(X1,··· ,XnY )âg2âB(Y1,··· ,Ym;Y ).
Exercise 1.6. Let Ω be a bounded domain in Rn and let p, q, r â [1,â]. For
u â Lp(Ω) and v â Lq(Ω), we define g(u, v)(x) := u(x)v(x) (x â Ω). Prove thatg â B(Lp(Ω), Lq(Ω); Lr(Ω)) for r †”(p, q), where
”(p, q) =
pq
p + q(p, q â [1,â)),
min p, q (p = â or q = â).
Exercise 1.7. Let Ω be a bounded domain in Rn and let k, m â NâȘ 0. For
(u1, · · · , uk) â Cm(Ω)k, we define
g(u1, · · · , uk)(x) := u1(x) · · ·uk(x) (x â Ω).
Prove that g â Bsymk (Cm(Ω), Cm(Ω)).
Let us define the Frechet derivative in Banach spaces together with the Gateauxderivative.
Definition 1.8 (Frechet derivative). Let X , Y be real Banach spaces, and letU be an open set in X . A mapping f : U â Y is Frechet differentiable at x0 â Uif there is A â B(X, Y ) such that
âf(x0 + h) â f(x0) â AhâY = o(âhâX) as âhâX â 0.
In this case, we write A = f âČ(x0) or fx(x0), and f âČ(x0) is called the Frechet de-rivative of f at x0. If f is Frechet differentiable at any x â U and the mappingU â x 7â f âČ(x) â B(X, Y ) is continuous, we say f â C1(U, Y ).
If f âČ â C1(U, B(X, Y )) then f âČâČ(x) = (f âČ)âČ(x) â B(X, B(X, Y )) = B2(X, Y ).If f âČâČ â C0(U, B2(X, Y )), we say f â C2(U, Y ). In the same manner, we candefine Cn(U, Y ) if the nth Frechet derivative f (n) â C0(U, Bn(X, Y )), and defineCâ(U, Y ) := â©nâNCn(U, Y ). In case that Y = R, we simply write Cn(U) =Cn(U, R).
Definition 1.9 (Gateaux derivative). Under the condition of Definition 1.8, forx0 â U and for (h1, h2, · · · , hn) â Xn (n â N), if there is O an open neighbourhood
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12 1. SHAPE DERIVATIVE OF MINIMUM POTENTIAL ENERGY
of the origin of Rn and [(t1, · · · , tn) 7â f(x0 + t1h1 + · · ·+ tnhn)] â Cn(O, Y ), then
we define nth Gateaux derivative at x0 as
dnf(x0, h1, h2, · · · , hn) :=ân
ât1 · · · âtnf(x0 + t1h1 + · · · + tnhn)
âŁâŁâŁâŁt1=···=tn=0
.
It is not difficult to show the Gateaux differentiability from the Frechet differ-entiability.
Exercise 1.10. For f â Cn(U, Y ) (n â„ 2), prove that there exists any nthGateaux derivative dnf(x, h1, h2 · · · , hn) for all x â U and it is given by
dnf(x, h1, h2, · · · , hn) = f (n)(x)[h1, · · · , hn], (1.3)
and that f (n)(x) â Bsymn (X, Y ) for x â U .
On the other hand, a converse proposition is also valid. A proof of the nexttheorem is found in § 2.1E of [3], for example.
Theorem 1.11. Let X, Y be real Banach spaces and U be an open set in X.For n â N, a mapping f : U â Y belongs to Cn(U, Y ), if and only if the nth Gateauxderivative dnf(x, h1, h2, · · · , hn) exists for all x â U and (h1, · · · , hn) â Xn and
[(h1, · · · , hn) 7â dnf(x, h1, h2, · · · , hn)] â Bn(X, Y ) (âx â U),
and dnf â C0(U, Bn(X, Y )). Furthermore, (1.3) holds in this case.
We conclude this section with some examples of Frechet derivatives in exercisesbelow. These examples will help the readers to understand the notion of higherFrechet and Gateaux derivatives.
Exercise 1.12. Let X, Y be real Banach spaces. For g â Bn(X, Y ) (n â N),we define f(x) := g(x, · · · , x) â Y (x â X). Prove that f â Câ(X, Y ) and f (k) = 0for k â„ n + 1. Moreover, in the case that g â Bsym
n (X, Y ), find the kth derivativef (k) â Bk(X, Y ) for k = 1, · · · , n.
Exercise 1.13. Let Ω be a bounded domain in Rn and let Ï â Ck(R) (k â
N âȘ 0). For u â Cm(Ω) (m â N âȘ 0), we define f(u)(x) := Ï(u(x)) (x â Ω).Prove that f â Ckâm(Cm(Ω), Cm(Ω)) if k â„ m.
Exercise 1.14. Let X be a real infinite dimensional Hilbert space and letekâk=1 â X be an orthonormal system. We define
J(u) :=ââ
k=1
a2k(kâ2 â a2
k), ak := (u, ek)X (u â U := w â X ; âwâX < 1).
Prove that J â Câ(U) and calculate its Frechet derivatives.
3. Minimization problems
This section is devoted to abstract minimization problems in Banach spaces inconnection with the LaxâMilgram theorem. We start our discussion with definitionsof global and local minimizers.
Definition 1.15. Let (S, d) be a metric space and J be a real valued functionaldefined on S.
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3. MINIMIZATION PROBLEMS 13
(1) u0 â S is called a global minimizer of J in S, if J(u0) †J(u) for all u â S.(2) u0 â S is called a local minimizer of J , if there exists an open neighbour-
hood O â S of u0 such that u0 is a global minimizer of J in O.(3) A local minimizer u0 of J is called isolated, if there exists Ï > 0 and
J(u0) < J(u) for all u â S with 0 < d(u, u0) < Ï.
Fundamental variational principles in Banach spaces are stated as follows. Itsproof will be left for the readersâ exercise.
Proposition 1.16. Let X be a Banach space and U be an open subset of X.We consider a functional J : U â R.
(1) If J â C1(U) and u0 â U is a local minimizer of J , we have J âČ(u0) = 0 âX âČ.
(2) If J â C2(U) and u0 â U is a local minimizer of J , we have J âČâČ(u0)[w, w] â„0 for all w â X.
(3) Suppose that J â C2(U) and J âČâČ(u)[w, w] â„ 0 for all w â X and u â U .If u0 â U with J âČ(u0) = 0 â X âČ and U is star-shaped 2 with respect to u0
then u0 is a global minimizer of J . Moreover if J âČâČ(u0)[w, w] = 0 impliesw = 0, then u0 is a unique global minimizer.
(4) Let J â C1(U) and u0 â U with J âČ(u0) = 0 â X âČ. If J âČâČ(u0) â B2(X, R)exists and if there exists α > 0 and J âČâČ(u0)[w, w] ℠αâwâ2
X for all w â X,then u0 is an isolated local minimizer of J .
Exercise 1.17. Prove Proposition 1.16.
Contrary to analogy with finite dimensional cases, the notion of the Gateauxderivative is not sufficient to describe the variational principles in infinite dimen-sional spaces like Proposition 1.16. Validity of the Frechet derivative is apparentalso from the next example.
Exercise 1.18. Let X be a Banach space and U be an open subset of X. Weassume that a functional J : U â R and u0 â U satisfy the condition;
For each v â X, t = 0 is a local minimizer of f(t) := J(u0 + tv) in R.
(1) Prove that J âČ(u0) = 0 â X âČ if J â C1(U).(2) Prove that J âČâČ(u0)[w, w] â„ 0 for all w â X if J â C2(U).(3) Is u0 a local minimizer of J? (Hint: Study the functional in Exercise 1.14
with u0 = 0.)
A unique existence of a global minimizer of a convex functional is obtained ifwe assume the coercivity condition (1.4) below.
Theorem 1.19. Let X be a real reflexive Banach space and let V be a closedsubspace of X. Assume that a real valued functional J â C2(X) satisfies the coer-civity condition:
J âČâČ(w)[v, v] ℠αâvâ2X (âv â V, âw â X), (1.4)
where α is a positive constant. Then the functional J admits a unique global min-imizer over V (g) := v â X ; v â g â V for arbitrary g â X. Furthermore, u is
2U is called star-shaped with respect to u0, if (1 â s)u0 + su â U for all u â U and s â [0, 1].
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14 1. SHAPE DERIVATIVE OF MINIMUM POTENTIAL ENERGY
the minimizer of J in V (g) if and only if u â V (g) satisfies
J âČ(u)[v] = 0 (âv â V ). (1.5)
Proof. For arbitrary v, w â V (g), we define f(t) := J((1 â t)w + tv),(0 †t †1). Then we have
f âČ(t) = J âČ((1 â t)w + tv)[v â w],
f âČâČ(t) = J âČâČ((1 â t)w + tv)[v â w, v â w] ℠αâv â wâ2X ,
and
J(v) â J(w) =
â« 1
0
(f âČ(0) +
â« s
0
f âČâČ(t)dt
)ds â„ J âČ(w)[v â w] +
α
2âv â wâ2
X . (1.6)
Putting w = g, we have
J(v) â J(g) â„ ââJ âČ(g)âXâČâv â gâX +α
2âv â gâ2
X .
Solving this quadratic inequality, we have
J(v) â„ J(g) â âJ âČ(g)â2XâČ
2α, (1.7)
âvâX †âgâX +1
α
(2âJ âČ(g)âXâČ +
â2α(|J(g)| + |J(v)|)
). (1.8)
From (1.7), the functional J is bounded from below. Let un â V (g) be aminimizing sequence which attains the infimum of J in V (g). From (1.8), unis bounded in X and there exists a subsequence un which converges weakly inX.3 The subsequence is again denoted by un and the weak limit is denoted by u.Since V (g) is weakly closed 4, u â V (g) follows.
From (1.6) with v = un and w = u,
J(un) â„ J(u) + J âČ(u)[un â u],
follows, and taking limit as n â â,
infV (g)
J = limnââ
J(un) â„ J(u) + J âČ(w)[u â u] = J(u).
Thus, u is a global minimizer of J in V (g). Since (1.5) holds, from (1.6), we have
J(v) â J(u) ℠α
2âv â uâ2
X (âv â V (g)), (1.9)
and the uniqueness of the minimizer follows. Conversely, if (1.5) holds, from (1.9),it follows that u is the minimizer of J .
Theorem 1.19 is related to the well-known LaxâMilgram theorem (see [21], forexample).
3A bounded closed convex subset of a reflexive Banach space is sequentially compact withrespect to the weak topology. See [24] § V.2 Theorem 1.
4A convex subset of a Banach space is strongly closed if and only if it is weakly closed. SeeMazurâs theorem [24] § V.1 Theorem 2.
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4. BANACH CONTRACTION MAPPING THEOREM 15
Theorem 1.20 (LaxâMilgram theorem). For a reflexive Banach space X, letA â B(X, X âČ) = B(X, B(X, R)) âŒ= B2(X, R) be selfadjoint, i.e., AâČ = A. Weassume the coercivity condition:
âα > 0 s.t. XâČăAw, wăX ℠αâwâ2X (âw â X).
Then A becomes a linear topological isomorphism from X to X âČ, i.e., A is a bijectivebounded linear operator from X to X âČ and Aâ1 â B(X âČ, X).
Exercise 1.21. In case that X is a real Hilbert space, prove the LaxâMilgramtheorem, applying Theorem 1.19 to the functional J(u) := 1
2XâČăAu, uăX âXâČ ăf, uăX
for arbitrary fixed f â X âČ.
Theorem 1.19 and the LaxâMilgram theorem have many important applicationsin elliptic partial differential equations. This is an example of a mixed boundaryvalue problem.
Exercise 1.22. Let Ω be a bounded domain in Rn with a Lipschitz boundary
Î. We suppose that ÎD is a relatively open subportion of Î, and ÎN := Î \ ÎD withÎD 6= â . For fixed f â L2(Ω) and gN â L2(ÎN), we consider
J(v, a) :=
â«
Ω
(1
2a(x)|âv(x)|2 â f(x)v(x)
)dx â
â«
ÎN
a(x)gN(x)v(x)dÏ
(v â H1(Ω), a â C0(Ω)).
For g â H1(Ω), we define closed affine subspaces of H1(Ω)
V (g) := v â H1(Ω); v = g on ÎD, V := V (0).
(1) Prove that J â Câ(H1(Ω) Ă C0(Ω)).(2) We define L â B(V, V âČ) by V âČăLv, zăV := â2
XJ(w, a)[v, z] (v, z â V ). Showthat L does not depend on w â H1(Ω) and write down the operator L asa differential operator formally.
(3) We fix a â C0(Ω) with a(x) ℠α > 0 (x â Ω) and fix g â H1(Ω). ApplyingTheorem 1.19 to J(·, a), prove the unique existence of a global minimizerof J(·, a) over V (g), and obtain the Euler equation (1.5) for it.
(4) Under the assumption of sufficient regularity for the minimizer and theother functions, show that the Euler equation (1.5) gives the followingboundary value problem of an elliptic partial differential equation5:
âdiv (a(x)âu) = f in Ω
u = g on ÎD
âu
âÎœ= gN on ÎN.
(1.10)
4. Banach contraction mapping theorem
We discuss the continuity of fixed points of contraction mappings with respectto a parameter in an abstract metric space. The results in this section will be usedfor a proof of the implicit function theorem in Section 5.
5The unique minimizer of J is called a weak (or H1) solution of the mixed boundary valueproblem (1.10).
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16 1. SHAPE DERIVATIVE OF MINIMUM POTENTIAL ENERGY
Theorem 1.23 (Banach contraction mapping theorem). Let (S, d) be a com-plete metric space. We suppose that T : S â S be a contraction, i.e. there isΞ â (0, 1) such that
d(T (x), T (y)) †Ξd(x, y) (âx, y â S).
Then there uniquely exists a â S such that T (a) = a. Moreover we have
d(a, T n(b)) †Ξnd(a, b) (âb â S, n â N). (1.11)
Proof. For arbitrary b â S and m, n â N (n < m), since
d(T m(b), T n(b)) â€mâ1â
k=n
d(T k+1(b), T k(b)) â€mâ1â
k=n
Ξkd(T (b), b) †Ξn
1 â Ξd(T (b), b),
T n(b)n is a Cauchy sequence in S. The limit of T n(b) is denoted by a â S. Thena = T (a) follows from
d(a, T (a)) †d(a, T n(b)) + d(T n(b), T (a))
†d(a, T n(b)) + Ξd(T nâ1(b), a) â 0 as n â â.
The inequality (1.11) is shown as
d(a, T n(b)) = d(T n(a), T n(b)) †Ξnd(a, b) (âb â S, n â N),
and the uniqueness of the fixed point a immediately follows from it.
We consider a family of âuniformâ contraction mappings. We find that, if it iscontinuous with respect to a parameter, the continuity of the fixed points follows.
Theorem 1.24. Let (S, d) be a complete metric space and let (Î, l) be a metricspace. We suppose that T : S Ă Î â S be a uniform contraction, i.e. there isΞ â (0, 1) such that
d(T (x, λ), T (y, λ)) †Ξd(x, y) (âx, y â S, âλ â Î). (1.12)
Then the unique fixed point g(λ) â S, T (g(λ), λ) = g(λ) exists for all λ â Î. IfT (x, ·) â C0(Î, S) for all x â S, then g â C0(Î, S).
Proof. We fix λ0 â Î. For arbitrary λ â Î, we have
d(g(λ), g(λ0)) = d(T (g(λ), λ), T (g(λ0), λ0))
†d(T (g(λ), λ), T (g(λ0), λ)) + d(T (g(λ0), λ), T (g(λ0), λ0)).
Applying (1.12) to the first term, we have
d(g(λ), g(λ0)) â€1
1 â Ξd(T (g(λ0), λ), T (g(λ0), λ0)), (1.13)
and g(λ) converges to g(λ0)) as λ â λ0 from the continuity of T .
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5. IMPLICIT FUNCTION THEOREM 17
5. Implicit function theorem
In this section, we prove the implicit function theorem in Banach spaces. Wefirst present some results on the differentiability of inverse operators and fixed pointsof contraction mappings.
Theorem 1.25. Let X, Y , Z be real Banach spaces, and let U be an open setin X and A â Ck(U, B(Y, Z)) for some k â N âȘ 0. We suppose that, for x â U ,there exists Î(x) â B(Z, Y ) such that Î(x) = [A(x)]â1. Then Î â Ck(U, B(Z, Y ))follows, and if k â„ 1 we have
ÎâČ(x)Ο = âÎ(x)(AâČ(x)Ο)Î(x) â B(Z, Y ) (âx â U, âΟ â X).
Proof. First, we assume k = 0. For arbitrary x0, x â U , we define Q :=âÎ(x)âÎ(x0)âB(Z,Y ) and Q0 := âÎ(x0)âB(Z,Y ). Since âÎ(x)âB(Z,Y ) †Q+Q0, forarbitrary z â Z, we have
â(Î(x) â Î(x0))zâY = âÎ(x)(A(x0) â A(x))Î(x0)zâY
†(Q + Q0)âA(x0) â A(x)âB(Y,Z)Q0âzâZ,
Hence, Q †(Q + Q0)Q0âA(x0)âA(x)âB(Y,Z) follows. Let x0 â U be fixed and let
âxâ x0âX be sufficiently small so that âA(x)âA(x0)âB(Y,Z) †(2Q0)â1. Then we
have1
2Q †Q2
0âA(x) â A(x0)âB(Y,Z) â 0 as x â x0 in X.
Hence, Î is continuous at x0.Next, we assume k = 1. We fix x0 â U and define R(x) := A(x) â A(x0) â
AâČ(x0)[x â x0] for x â U . Since R(x) = o(âx â x0âX) as x â x0, we have
Î(x) â Î(x0) + Î(x)(AâČ(x0)[x â x0])Î(x0)
= (Î(x) â Î(x0))(AâČ(x0)[x â x0])Î(x0) â Î(x)R(x)Î(x0)
= o(âx â x0âX) as x â x0.
It follows that there exists ÎâČ(x0) â B(X, B(Z, Y )), and furthermore we haveÎâČ(x0)Ο = âÎ(x0)(A
âČ(x0)Ο)Î(x0) for Ο â X . Since ÎâČ â C0(U, B(X, B(Z, Y ))),we have proved Î â C1(U, B(Z, Y )).
If k = 2, since Î â C1 and AâČ â C1, we have ÎâČ â C1 and Î â C2 follows. Fork = 3, 4, · · · , the assertion follows recursively.
Theorem 1.26 ([11]). Let X, Y be real Banach spaces, and let U , V be opensets in X and Y , respectively. We assume that T â Ck(U Ă V , V ) and V â y 7âT (x, y) â V is a contraction on V uniformly in x â U , i.e., there exists Ξ â (0, 1)such that
âT (x, y1) â T (x, y2)âY †Ξây1 â y2âY (âx â U, ây1, y2 â V ).
For x â U , let f(x) â V be the unique fixed point of T (x, ·) in V , i.e. T (x, f(x)) =f(x). Then f â Ck(U, Y ) follows.
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18 1. SHAPE DERIVATIVE OF MINIMUM POTENTIAL ENERGY
Proof. If k = 0, the assertion follows from Theorem 1.12. In the case k = 1,we define K(x) := Ty(x, f(x)) â B(Y ). Since âK(x)âB(Y ) †Ξ < 1, there exists
(I â K(x))â1 =ââ
m=0
K(x)m â B(Y ), â(I â K(x))â1âB(Y ) â€1
1 â Ξ.
For fixed x0 â U , we define α(x) := f(x) â f(x0) (x â U) and
A0 := (I â K(x0))â1Tx(x0, f(x0)) â B(X, Y ),
R(x) := (I â K(x0))α(x) â Tx(x0, f(x0))[x â x0] (x â U).
Then we have
f(x) â f(x0) = α(x) = A0(x â x0) + (I â K(x0))â1R(x) (x â U). (1.14)
Since
α(x) = T (x, f(x0) + α(x)) â T (x0, f(x0))
= Tx(x0, f(x0))[x â x0] + Ty(x0, f(x0))[α(x)] + R(x),
it follows that R(x) = o(âx â x0âX + âα(x)âY ) as âx â x0âX â 0. From (1.14),we also have α(x) = O(âxâ x0âX) as âxâ x0âX â 0. From these estimates, thereexists the Frechet derivative of f at x0 and f âČ(x0) = A0. Hence, we have proved
f âČ(x) = (I â K(x))â1Tx(x, f(x)) â B(X, Y ) (x â U).
From Theorem 1.25, (I âK(·))â1 â C0(U, B(Y )) holds, and we have f â C1(U, Y ).For the case k â„ 2, we can prove f â Ck(U, Y ) by a recursive argument for
k = 1, 2, · · · .
Corollary 1.27. Under the condition of Theorem 1.26, we have
âf(x) â f(x0)âY †1
1 â ΞâT (x, f(x0)) â f(x0)âY (âx0, x â U0).
Proof. This immediately follows from the inequality (1.13).
The implicit function theorem in Banach spaces is stated as follows.
Theorem 1.28 (Implicit function theorem). Let X, Y , Z be real Banach spacesand U , V be open sets in X and Y , respectively. We suppose that F : U Ă V â Zand (x0, y0) â U Ă V satisfy the conditions;
(1) F (x0, y0) = 0.(2) F â C0(U Ă V, Z).(3) F (x, ·) â C1(V, Z) for x â U and Fy is continuous at (x, y) = (x0, y0).(4) (Fy(x0, y0))
â1 â B(Z, Y ).
Then there exist a convex open neighbourhood of (x0, y0), U0 Ă V0 â U Ă V andf â C0(U0, V0), such that, for (x, y) â U0 ĂV0, F (x, y) = 0 if and only if y = f(x).Moreover, if F â Ck(U Ă V, Z) (k â N), then f â Ck(U0, V0).
Proof. Let us define
T (x, y) := y â Fy(x0, y0)â1F (x, y) â Y (x, y) â U Ă V.
Then F (x, y) = 0 is equivalent to y = T (x, y).
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6. PARAMETER VARIATION FORMULAS 19
For Ï > 0, we define U(Ï) := x â U ; âx â x0âX < Ï and V (Ï) := y âV ; ây â y0âY < Ï. We fix Ï0 > 0 such that y â Y ; ây â y0âY †Ï0 â V . SinceFy is continuous at (x0, y0), for arbitrary Δ > 0, there exist ÎŽ â (0, Ï0] such that
âFy(x, y) â Fy(x0, y0)âB(Y,Z) < Δ (â(x, y) â U(ÎŽ) Ă V (ÎŽ)).
For x â U(ÎŽ) and y1, y2 â V (ÎŽ), from the equality
T (x, y1)âT (x, y2) = Fy(x0, y0)â1
â« 1
0
Fy(x0, y0)âFy(x, y1+s(y2ây1))[y1ây2]ds,
we have
âT (x, y1) â T (x, y2)âY †LΔây1 â y2âY ,
where L := âFy(x0, y0)â1âB(Z,Y ).
Since F â C0(U Ă V, Z), for arbitrary Δ0 > 0, there exists ÎŽ0 â (0, ÎŽ] such that,for x â U(ÎŽ0), we have âF (x, y0)âZ < Δ0. For (x, y) â U(ÎŽ0) Ă V (ÎŽ), we have
âT (x, y)ây0âY = âT (x, y)âT (x, y0)âFy(x0, y0)â1F (x, y0)âY < LΔâyây0âY +LΔ0.
Hence, let Δ := (2L)â1 and we choose ÎŽ â (0, Ï0] as above and fix it, and, forΔ0 := ÎŽ(4L)â1, let us choose ÎŽ0 â (0, ÎŽ] as above. Then, for U0 := U(ÎŽ0) andV0 := V (ÎŽ/2), we have T (x, y) â V0 for (x, y) â U0 Ă V0 and
âT (x, y1) â T (x, y2)âY †1
2ây1 â y2âY (âx â U0,
ây1, y2 â V0).
Since T â C0(U0 ĂV0, V0), from Theorem 1.26, there exists f â C0(U0, V0) and theassertion follows.
In the case that F â Ck(U Ă V, Z) (k â N), T â Ck(U0 Ă V0, V0) holds andf â Ck(U0, V0) also follows from Theorem 1.26.
Corollary 1.29. Under the condition of the proof of Theorem 1.28, we have
âf(x) â f(x0)âY †2âFy(x0, y0)â1âB(Z,Y )âF (x, y0)âZ (âx â U0).
Proof. This immediately follows from Corollary 1.27.
6. Parameter variation formulas
We establish a theory of parameter variation of abstract potential energy inBanach spaces in this section. The implicit function theorem and the Lax-Milgramtheorem play essential roles in the arguments. The obtained results will providea fundamental framework for our analysis of shape derivative of elliptic potentialenergies.
As in the next theorem, we consider a family of minimizers u(”) of a functionalJ(u, ”) in a Banach space X , where ” is a parameter in another Banach space M .Our aim is to study the Frechet derivatives of u(”) and Jâ(”) = minu J(u, ”) withrespect to the parameter ” â M . A simple but basic observation in our analysis isthe relation (1.15) below.
Theorem 1.30. Let X and M be real Banach spaces. For U0 â X and an opensubset O0 â M , we consider a real valued functional J : U0 ĂO0 â R and a mapu : O0 â U0. We define Jâ(”) := J(u(”), ”) for ” â O0. We suppose the followingconditions.
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20 1. SHAPE DERIVATIVE OF MINIMUM POTENTIAL ENERGY
(1) J â C0(U0 Ă O0), J(w, ·) â C1(O0) for w â U0, and âMJ â C0(U0 ĂO0, M
âČ).(2) u â C0(O0, X) and u(”) is a global minimizer of J(·, ”) in U0 for each
” â O0.
Then we have Jâ â C1(O0) and
J âČâ(”) = âMJ(u(”), ”) (” â O0). (1.15)
Remark 1.31. If U0 is an open set and J â C1(U0,O0) and u â C1(O0, X),the formula (1.15) is easily obtained as follows;
J âČâ(”) = D”[J(u(”), ”)] = âXJ(u(”), ”)[uâČ(”)] + âMJ(u(”), ”) = âMJ(u(”), ”),
where the D” denotes the Frechet differential operator with respect to ” â M andthe last equality follows from âXJ(u(”), ”) = 0 â X âČ.
Proof of Theorem 1.30. We fix ”0 â O0 and we define u0 := u(”0) and
r(”) := Jâ(”) â Jâ(”0) â âMJ(u0, ”0)[” â ”0] (” â O0).
Since u(”) is a global minimizer and u â C0(O0, X), if ” is close to ”0, we have
r(”) †J(u0, ”) â J(u0, ”0) â âMJ(u0, ”0)[” â ”0] = o(â” â ”0âM ),
and
r(”) â„ J(u(”), ”) â J(u(”), ”0) â âMJ(u0, ”0)[” â ”0]
=
â« 1
0
âMJ(u(”), ”0 + s(” â ”0)) â âMJ(u0, ”0)[” â ”0]ds
= o(â” â ”0âM ).
It follows that r(”) = o(â”â ”0âM ), and we have (1.15) and J âČâ â C0(O0, M
âČ).
Theorem 1.30 is extended to the following Ck-version.
Corollary 1.32. Under the condition of Theorem 1.30, we assume that U0
is open. Let k â N. If âMJ â Ck(U0 Ă O0, MâČ) and u â Ck(O0, X), then Jâ â
Ck+1(O0).
Proof. This immediately follows from the formula (1.15).
In the above theorem, we have assumed the existence and the regularity of theminimizer u(”), whereas we can derive them from the LaxâMilgram theorem if weassume the coercivity of the functional J .
Theorem 1.33. Let X and M be real Banach spaces and U and O be opensubsets of X and M , respectively. We consider a real valued functional J : UĂO âR and fix ”0 â O. We assume
(1) J(·, ”) â C2(U) for ” â O and âXJ â C0(U ĂO, X âČ).(2) u0 â U satisfies âXJ(u0, ”0) = 0.(3) â2
XJ is continuous at (w, ”) = (u0, ”0).(4) There exists α > 0 such that â2
XJ(u0, ”0)[w, w] ℠αâwâ2X for w â X.
Then there exist a convex open neighbourhood of (u0, ”0), U0 Ă O0 â U Ă O andu â C0(O0,U0), such that, for ” â O0, the following three conditions are equivalent.
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6. PARAMETER VARIATION FORMULAS 21
(a) w â U0 is a local minimizer of J(·, ”).(b) w â U0 satisfies âXJ(w, ”) = 0.(c) w = u(”).
In this case, u(”) is a global minimizer of J(·, ”) on U0.
Proof. We define a map F := âXJ from U Ă O to X âČ and apply Theo-rem 1.28 at (w, ”) = (u0, ”0). From the LaxâMilgram theorem (Theorem 1.20),âXF (u0, ”0) = â2
XJ(u0, ”0) becomes a linear topological isomorphism from X toX âČ. Then, from the implicit function theorem (Theorem 1.28), there exist a convexopen neighbourhood of (u0, ”0), U0 ĂO0 â U Ă O and u â C0(O0, U0), such that,for ” â O0, w â U0 satisfies âXJ(w, ”) = 0 if and only if w = u(”).
From the continuity of â2XJ at (u0, ”0), without loss of generality, (after re-
placing U0 and O0 with smaller ones if we need) we can assume that
â2XJ(v, ”)[w, w] ℠α
2âwâ2
X (âw â X, â(v, ”) â U0 ĂO0). (1.16)
For ” â O0, if w â U0 is a local minimizer of J(·, ”) in U0, the âXJ(w, ”) = 0follows. Conversely, if w â U0 satisfies âXJ(w, ”) = 0, w is a local minimizer in U0
from the condition (1.16). It also follows from (1.16) that u(”) is a global minimizerof J(·, ”) on U0 (Proposition 1.16 (3)).
Higher regularity of u(”) also follows from the implicit function theorem.
Theorem 1.34. Under the condition of Theorem 1.33, we additionally assumethat âXJ â Ck(U ĂO, X âČ) for some k â N. Then u â Ck(O0,U0).
Proof. The assertion follows from the implicit function theorem (Theorem 1.28).
Under the condition of Theorem 1.33, we define
Jâ(”) := J(u(”), ”) (” â O0).
As a consequence of Theorem 1.34, a sufficient condition for Jâ â C1(O0) is J âC1(UĂO) and âXJ â C1(UĂO, X âČ). Due to Theorem 1.30, however, the conditionâXJ â C1(U ĂO, X âČ) is not necessary as shown in the next theorem.
Theorem 1.35. Under the condition of Theorem 1.33, we additionally assumethat J â Ck(U ĂO) for some k â N, then Jâ â Ck(O0) and it satisfies (1.15).
Proof. From Theorem 1.30, Jâ â C1(O0) and (1.15) immediately follows.Since u â Ckâ1(O0, X) follows from Theorem 1.34, Jâ â Ck(O0) is obtained fromthe formula (1.15).
In Theorem 1.35 with k = 1, we have derived Jâ â C1(O0) without assumingany differentiability of u(”) more than u â C0(O0, X) (this is from Theorem 1.33).Actually, we have a Holder regularity of u.
Proposition 1.36. Under the condition of Theorem 1.33, we additionally as-sume that J â C1(U ĂO), then we have
âu(”) â u0âX = o(â” â ”0â1/2
M
)as â” â ”0âM â 0.
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22 1. SHAPE DERIVATIVE OF MINIMUM POTENTIAL ENERGY
Proof. Let Ï0 > 0 with v â X ; âv â u0âX †Ï0 â U0. For h â X withâhâX = 1, ” â O0 and Ï â (0, Ï0], we have
J(u0 + Ïh, ”) = J(u0, ”) + ÏâXJ(u0, ”)[h] + Ï2
â« 1
0
(1 â s)â2XJ(u0 + sÏh, ”)[h, h]ds.
âXJ(u0, ”)[h] = âXJ(u0, ”)[h] â âXJ(u0, ”0)[h]
=
Ïâ1(J(u0 + Ïh, ”) â J(u0, ”)) â Ï
â« 1
0
(1 â s)â2XJ(u0 + sÏh, ”)[h, h]ds
â
Ïâ1(J(u0 + Ïh, ”0) â J(u0, ”0)) â Ï
â« 1
0
(1 â s)â2XJ(u0 + sÏh, ”0)[h, h]ds
= Ïâ1
â« 1
0
âMJ(u0 + Ïh, ”0 + t(” â ”0)) â âMJ(u0, ”0 + t(” â ”0))[” â ”0]dt
â Ï
â« 1
0
(1 â s)â2XJ(u0 + sÏh, ”) â â2
XJ(u0 + sÏh, ”0)[h, h]ds
Choosing Ï := â” â ”0â1/2M , we have
ââXJ(u0, ”)âXâČ â€ 2Ï(Ï)Ï (” â O0, â” â ”0âM †Ï20),
where, for r > 0,
S(r) :=(w, λ) â X Ă M ; âw â u0âX †r, âλ â ”0âM †r2
,
Ï(r) := sup(w,λ)âS(r)
ââMJ(w, λ) â âMJ(u0, ”0)âM âČ
+ sup(w,λ)âS(r)
ââ2XJ(w, λ) â â2
XJ(u0, ”0)âB2(X,R).
We remark that Ï(r) â 0 as r â +0. Hence, from Corollary 1.29, we haveâu(”) â u0âX †2αâ1ââXJ(u0, ”)âXâČ â€ 4αâ1Ï(Ï)Ï = o(Ï).
Under the conditions of Theorem 1.33, â2XJ(u(”), ”) can be regarded as a linear
topological isomorphism from X to X âČ from the LaxâMilgram theorem. Therefore,we can define Î(”) â B(X âČ, X) which satisfies
â2XJ(u(”), ”)[Î(”)h, w] = h[w] (âw â X, âh â X âČ).
Theorem 1.37. Under the condition of Theorem 1.34 with k = 1,
uâČ(”) = âÎ(”)h0(”) (” â O0), (1.17)
holds, where h0(”) := âMâXJ(u(”), ”) â B(M, X âČ).
Proof. Differentiating âXJ(u(”), ”) = 0 â X âČ by ”, we have
â2XJ(u(”), ”)[uâČ(”)] + âMâXJ(u(”), ”) = 0 â B(M, X âČ).
This is equivalent to (1.17) from the LaxâMilgram theorem.
The second Frechet derivative of Jâ is given by the next formula.
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7. LIPSCHITZ DEFORMATION OF DOMAINS 23
Theorem 1.38. Under the condition of Theorem 1.33, we additionally assumethat J â C2(U ĂO) then Jâ â C2(O0) and it satisfies
J âČâČâ (”)[”1, ”2] = â2
MJ(u(”), ”)[”1, ”2] â XăÎ(”)h0(”)[”1], h0(”)[”2]ăXâČ
(” â O0, ”1, ”2 â M).
Proof. Differentiating the formula (1.15) by ” and substituting (1.17), weobtain the formula.
7. Lipschitz deformation of domains
A systematic investigation of domain deformation with Lipschitz domain map-pings is carried out in this section. The domain mapping method is one of theimportant techniques of the shape sensitivity analysis and optimal shape designtheory.
We consider a domain deformation with Lipschitz transform Ï : Ω â Ï(Ω),where Ω is a bounded domain in R
n (n â N) and Ï is Rn-valued W 1,â function. It
is known that a function in W 1,â is Lipschitz in the following sense. For a functionu : Ω â R
k, we define
|u|Lip,Ω := supx,yâΩ,x 6=y
|u(x) â u(y)||x â y| .
If |u|Lip,Ω < â, u is called uniformly Lipschitz continuous on Ω.6
Proposition 1.39. Let Ω be a domain in Rn. For u â W 1,â(Ω), there is
u â C0(Ω) such that u(x) = u(x) for a.e. x â Ω, in other words, we can regardW 1,â(Ω) â C0(Ω). If Ω is convex, W 1,â(Ω) = C0,1(Ω) as a subset of C0(Ω).Moreover, we have
ââuâLâ(Ω) = |u|Lip,Ω (u â W 1,â(Ω) â© C0(Ω)).
Exercise 1.40. Prove Proposition 1.39. (Hint: W 1,â â C0,1 is shown byusing the Friedrichsâ mollifier. The converse is obtained by direct calculation of thedistributional derivative as a limit of a finite difference.)
In the following arguments, we fix a bounded convex domain Ω0 â Rn and we
identify W 1,â(Ω0, Rn) with C0,1(Ω0, R
n). We denote by Ï0 the identity map onR
n, i.e., Ï0(x) = x (x â Rn).
Proposition 1.41. Suppose that Ï â W 1,â(Ω0, Rn) satisfies |ÏâÏ0|Lip,Ω0
< 1.Then Ï is a bi-Lipschitz transform from Ω0 to Ï(Ω0), i.e. Ï is bijective from Ω0
onto an open set and Ï and Ïâ1 are both uniformly Lipschitz continuous.
Proof. Let ” := ÏâÏ0 and Ξ := |”|Lip,Ω0â (0, 1). First, we show that Ï(Ω0)
is open. We fix arbitrary y0 â Ï(Ω0) with y0 = Ï(x0), x0 â Ω0. Let ÎŽ > 0 such
that BÎŽ(x0) â Ω0, where BÎŽ(x0) := x â Rn; |x â x0| < ÎŽ. For y â B(1âΞ)ÎŽ(y0),
we show that y â Ï(Ω0). It is easily checked that T (Ο) := y â ”(Ο) is a uniform
contraction on BÎŽ(x0). From the contraction mapping theorem (Theorem 1.23),
6For a Lipschitz domain Ω â Rn and α â (0, 1], we define C0,α(Ω) := u â C0(Ω) â©
Lâ(Ω); âK > 0, |u(x) â u(y)| †K|x â y|α (âx, y â Ω). u â C0,α(Ω) is called an uniform
α-Holder function if α â (0, 1), and u â C0,1(Ω) is called a uniform Lipschitz function.
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24 1. SHAPE DERIVATIVE OF MINIMUM POTENTIAL ENERGY
there is a fixed point x = T (x) = yâ”(x) in BÎŽ(x0), that is y = Ï(x). Hence, Ï(Ω0)is an open set. Since |Ï(x1)âÏ(x2)| â„ |x1 âx2|â |”(x1)â”(x2)| â„ (1â Ξ)|x1âx2|for x1, x2 â Ω0, it follows that Ï is injective and Ïâ1 satisfies uniform Lipschitzcondition on Ï(Ω0).
For an arbitrary open set Ω with Ω â Ω0, we define
O(Ω) :=Ï â W 1,â(Ω0, R
n); |Ï â Ï0|Lip,Ω0< 1, Ï(Ω) â Ω0
. (1.18)
We note that O(Ω) is an open subset of W 1,â(Ω0, Rn). For Ï â O(Ω), we denote
by Ω(Ï) the deformed domain, i.e., Ω(Ï) := Ï(Ω), hereafter.A virtual crack extension in a cracked elastic body for the energy release rate
(1.2) is treated as follows. Let Ωâ be an un-cracked elastic body in Rn with Ωâ â Ω0,
and let ÎŁ â Ωâ be an initial crack. We assume that ÎŁ is a closure of an (n â 1)-dimensional smooth hypersurface. Then, we consider the energy release rate of acracked domain Ω := Ωâ \ ÎŁ along a virtual crack extension ÎŁ(t) parametrized by0 †t < T , where ÎŁ(t) is closed and satisfies
ÎŁ = ÎŁ(0) â ÎŁ(t1) â ÎŁ(t2) (0 â€â t1 â€â t2 < T ).
We suppose that the virtual crack extension is expressed by a parametrizeddomain mappings Ï(t) â O(Ω) with Ï(0) = Ï0, Ï(t)(ÎŁ) = ÎŁ(t), Ω(Ï(t)) = Ωâ\ÎŁ(t),and Ï(t)(x) = x in a neighbourhood of âΩâ. Without loss of generality, we choosea parameter t by t = Hnâ1(ÎŁ(t) \ÎŁ), where Hnâ1 denotes the (nâ 1)-dimensionalHausdorff measure.
If the elastic potential energy in the deformed domain Ω(Ï) is denoted byEâ(Ï), the energy release rate G along ÎŁ(t)0â€t<T at t = 0 is expressed as
G = EâČâ(Ï0)[Ï(0)],
where EâČâ is the Frechet derivative of Eâ with respect to Ï and Ï is the t derivative
of Ï. Namely, the calculation and the mathematical justification of the energyrelease rate G are reduced to those of the Frechet derivative of Eâ.
Example 1.42. Let 멉 be a bounded domain in R2. We define
ÎŁ(t) :=(x1, 0)T â R
2; c0 †x1 †c1 + t
(0 †t †T ),
and assume that ÎŁ(T ) â Ωâ. We consider a virtual crack extension ÎŁ(t)0â€t<T
in a cracked domain Ω := Ωâ \ ÎŁ(0). We suppose that Ω0 be a bounded convexdomain with Ωâ â Ω0 and that a cut off function q â W 1,â(Ω0) with supp(q) âx â Ωâ; x1 > c0 and q(x) = 1 at x = (c1, 0)T. Then we define
Ï(t)(x) := x + t q(x)
(10
)(x â Ω0, t â„ 0).
If |Ï(t) â Ï0|Lip,Ω0= t |q|Lip,Ω0
< 1, then Ï(t) is a bi-Lipschitz transform from Ωâ
onto itself and satisfies Ï(t)(ÎŁ(0)) = ÎŁ(t) and Ω(Ï(t)) = Ωâ \ ÎŁ(t).
Example 1.43. Let Ω = ΩâČ Ă (0, 1) be a bounded cylindrical domain in Rn,
where ΩâČ is a bounded Lipschitz domain in Rnâ1. For fixed h â W 1,â(ΩâČ), we
consider a domain perturbation of Ω:
Ω(t) := (xâČ, xn) â ΩâČ Ă R; t h(xâČ) < xn < 1 ,
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7. LIPSCHITZ DEFORMATION OF DOMAINS 25
for small |t| << 1. We fix a bounded convex domain Ωâ with Ω â Ωâ. Then,this Lipschitz boundary deformation is expressed by Ï(t) â O(Ω) as follows. Letq â W 1,â(Ω0) be a cut off function with supp(q) â (xâČ, xn) â Ω0; xn < 1 andq(xâČ, 0) = 1 for xâČ â ΩâČ. Then we define
Ï(t)(x) :=
(xâČ
xn + tq(x)h(xâČ)
)(x = (xâČ, xn)T â Ω0, t â„ 0).
If |Ï(t)âÏ0|Lip,Ω0= t |qh|Lip,Ω0
< 1, then Ï(t) is a bi-Lipschitz transform from Ωâ
onto itself and satisfies Ω(Ï(t)) = Ω(t).
For each deformation Ï â O(Ω), we define a pushforward operator Ïâ whichtransforms a function v on Ω to a function Ïâv := v Ïâ1 on Ω(Ï), if Ï satisfiesProposition 1.41. We define
âÏT(x) :=
(âÏj
âxi(x)
)
iâ,jâ
â RnĂn, (x â Ω0),
A(Ï) :=(âÏT
)â1 â Lâ(Ω0, RnĂn), Îș(Ï) := detâÏT â Lâ(Ω0, R).
These Jacobi matrices and Jacobian appear in the pullback of differentiation andintegration on Ω(Ï) to Ω. For a function v on Ω, we have
[â(Ïâv)] Ï = A(Ï)âv a.e in Ω (v â W 1,1(Ω)),â«
Ω(Ï)
(Ïâv)(y)dy =
â«
Ω
v(x)Îș(Ï)(x)dx (v â L1(Ω)).
These equalities are well known in the case Ï â C1. However, for Ï â W 1,â = C0,1,these are not so trivial. See [5] and [25] etc. for details.
Exercise 1.44. Under the condition of Proposition 1.41, prove that
ess- infΩ0
Îș(Ï) > 0.
Frechet derivatives of Îș(Ï) and A(Ï) with respect to Ï are obtained as follows.
Theorem 1.45. Let Ω be an open subset of Ω0.
(1) It holds that Îș â Câ(W 1,â(Ω, Rn), Lâ(Ω)), and the (n + 1)-th Frechetderivative of Îș vanishes, i.e., Îș(n+1) = 0 â Bn+1(W
1,â(Ω, Rn), Lâ(Ω)).In particular, we have
ÎșâČ(Ï0)[”] = div ” for ” â W 1,â(Ω, Rn).
(2) We define O0(Ω) := Ï â W 1,â(Ω, Rn); ess- infΩ Îș(Ï) > 0, which is anopen subset of W 1,â(Ω, Rn). Then A â Câ(O0(Ω), Lâ(Ω, RnĂn)) holdsand, in particular, we have the formula:
AâČ(Ï0)[”] = ââ”T for ” â W 1,â(Ω, Rn).
Proof. Since the determinant is a polynomial of degree n, the propositionÎș â Câ(W 1,â(Ω, Rn), Lâ(Ω)) is clear. For fixed ” â W 1,â(Ω, Rn), we define
mij(t) := ÎŽij + tâ”j
âxiâ Lâ(Ω) (i, j = 1, · · · , n, t â R),
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26 1. SHAPE DERIVATIVE OF MINIMUM POTENTIAL ENERGY
where ÎŽij is Kroneckerâs delta. Then we have
ÎșâČ(Ï0)[”] =d
dt
âŁâŁâŁâŁt=0
Îș(Ï0 + t”)
=d
dt
âŁâŁâŁâŁt=0
det (mij(t)) =d
dt
âŁâŁâŁâŁt=0
(â
ÏâSn
sgn(Ï)m1Ï(1)(t) · · ·mnÏ(n)(t)
)
=d
dt
âŁâŁâŁâŁt=0
(m11(t) · · ·mnn(t)) =
nâ
i=1
m11(0) · · ·mâČii(0) · · ·mnn(0) = div”.
Let the (i, j) component of A(Ï) be denoted by aij(Ï) â Lâ(Ω). Then wehave aij(Ï) = αij(Ï)/Îș(Ï), where αij(Ï) is the (i, j) cofactor of âÏT, which is
a polynomial of âÏk
âxlof degree n â 1. Since ess- infΩ Îș(Ï) > 0 for Ï â O0(Ω),
aij â Câ(O0(Ω), Lâ(Ω)) follows. For fixed ” â W 1,â(Ω, Rn), differentiating theidentity
A(Ï0 + t”)(I + tâ”T) = I (I: identity matrix of degree n),
by t â R at t = 0, we have
AâČ(Ï0)[”] + A(Ï0)â”T = O.
Since A(Ï0) = I, we have AâČ(Ï0)[”] = ââ”T.
We conclude this section with three basic propositions for the pushforwardoperator Ïâ.
Proposition 1.46. Under the condition of Proposition 1.41, for p â [1,â],Ïâ is a linear topological isomorphism from Lp(Ω) onto Lp(Ω(Ï)), and a lineartopological isomorphism from W 1,p(Ω) onto W 1,p(Ω(Ï)). More precisely, we have
m(Ï)âvâLp(Ω) †âÏâvâLp(Ω(Ï)) †M(Ï)nâvâLp(Ω) (v â Lp(Ω), 1 †p †â),
m(Ï)
M(Ï)ââvâLp(Ω) †ââ(Ïâv)âLp(Ω(Ï)) â€
M(Ï)nâ1
m(Ï)ââvâLp(Ω)
(v â W 1,p(Ω), 1 †p †â),
where
m(Ï) := min(1, ess- inf
ΩÎș(Ï)
),
M(Ï) := max
(1, n max
1â€i,jâ€n
â„â„â„â„âÏj
âxi
â„â„â„â„Lâ(Ω)
).
Proof. Since we have
m(Ï) †Îș(Ï)(x) †M(Ï)n a.e. x â Ω,
the estimates for Lp norms are obtained. We also have
|âv(x)| †M(Ï)|â(Ïâv)(Ï(x))| a.e. x â Ω,
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7. LIPSCHITZ DEFORMATION OF DOMAINS 27
from the equality âv = (âÏT)â(Ïâv).7 Note that αij is given by a cofactor ofâÏT and is estimated as |αij | †M(Ï)nâ1 a.e. in Ω. Hence, we have
|â(Ïâv)(Ï(x))| †M(Ï)nâ1
m(Ï)|âv(x)| a.e. x â Ω.
From these estimates, we obtain the Lp estimates for the gradients.
Proposition 1.47. Let k â N âȘ 0, l â 0, 1, and p â [1,â]. We supposethat f â W k+l,p(Ω0) if p â [1,â), and f â Ck+l(Ω0) if p = â. Then the mapping[Ï 7â f Ï] belongs to Ck(O(Ω), W l,p(Ω)).
Proof. We fix a domain Ω1 with a smooth boundary such that Ω â Ω1 âΩ1 â Ω0. Let ηΔ be a Friedrichsâ mollifier and define fΔ := ηΔ â f , where f isthe zero extension of f to R
n. Then fΔ â Câ(Rn) and fΔ tends to f strongly inW k+l,p(Ω1) as Δ â +0.
First, we prove the theorem for k = 0. We fix Ï1 â O(Ω), and suppose thatÏ â O(Ω) satisfies âÏâÏ1âW 1,â(Ω0) â€ Ï for sufficiently small fixed Ï > 0. Without
loss of generality, we assume that Ω(Ï1) â Ω1 and Ω(Ï) â Ω1. We have
âf Ï1 â f ÏâW l,p(Ω) †âf Ï1 â fΔ Ï1âW l,p(Ω)
+ âfΔ Ï1 â fΔ ÏâW l,p(Ω) + âfΔ Ï â f ÏâW l,p(Ω).
The last term (and the first term as its special case) is estimated as
âfΔ Ï â f ÏâW l,p(Ω) †C(Ï, p)âfΔ â fâW l,p(Ω(Ï)) †C(Ï, p)âfΔ â fâW l,p(Ω1),
where C(Ï, p) is a positive constant depending only on Ï and p. For the secondterm, if p â [1,â), we have the estimates
âfΔ Ï1 â fΔ ÏâpLp(Ω) =
â«
Ω
|fΔ(Ï1(x)) â fΔ(Ï(x))|pdx
â€â«
Ω
(ââfΔâLâ(Ω0)âÏ1 â ÏâLâ(Ω0)
)pdx = |Ω|ââfΔâp
Lâ(Ω0)âÏ1 â ÏâpLâ(Ω0),
and
ââ(fΔ Ï1 â fΔ Ï)âpLp(Ω) =
â«
Ω
âŁâŁ[âÏT
1 (x)]âfΔ(Ï1(x)) â [âÏT(x)]âfΔ(Ï(x))âŁâŁp dx
â€â«
Ω
âŁâŁ[âÏT
1 (x)] (âfΔ(Ï1(x)) ââfΔ(Ï(x)))âŁâŁ
+âŁâŁ([âÏT
1 (x)] â [âÏT(x)])âfΔ(Ï(x))
âŁâŁpdx
†C(Ï, Δ, p)âÏ1 â ÏâpW 1,â(Ω0),
where C(Ï, Δ, p) is a positive constant depending only on Ï, Δ and p. For the casep = â, similarly we have
âfΔ Ï1 â fΔ ÏâW 1,â(Ω) †CâČ(Ï, Δ)âÏ1 â ÏâW 1,â(Ω0).
7For x â Rn and n Ă n matrix A = (aij ), |Ax| †(
P
i,j a2ij)
1/2|x| holds, where |x| =
(Pn
i=1x2
i )1/2.
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28 1. SHAPE DERIVATIVE OF MINIMUM POTENTIAL ENERGY
From these estimates, it follows that the mapping [Ï 7â f Ï] is continuous atÏ = Ï1. Since Ï1 is arbitrary in O(Ω), it belongs to C0(O(Ω), W l,p(Ω)).
Let us proceed to the case k = 1. For an arbitrary ” â W 1,â(Ω0), we considerthe Gateaux derivative d
dt |t=0f (Ï + t”) in W l,p(Ω). We define
F (t) := f (Ï + t”) â W l,p(Ω), FΔ(t) := fΔ (Ï + t”) â W l,p(Ω).
Then it is easy to see that there exists a > 0 such that
limΔâ+0
âFΔ â FâC1([âa,a],W l,p(Ω)) = 0,
and that
d
dtf (Ï+ t”) = lim
Δâ+0
d
dtfΔ (Ï+ t”) = lim
Δâ+0(âfΔ (Ï+ t”)) ·” = (âf (Ï+ t”)) ·”.
Hence [Ï 7â f Ï] is Gateaux differentiable and the derivative is given by (âf Ï) ·” â W l,p(Ω). Since [Ï 7â âf Ï] â C0(O(Ω), W l,p(Ω)), in other words, we have[Ï 7â âf Ï] â C0(O(Ω), B(W 1,â(Ω0), W
l,p(Ω))). From Theorem 1.11, it followsthat [Ï 7â f Ï] belongs to C1(O(Ω), W l,p(Ω)).
For the case k â„ 2, we can prove the assertion in the same way.
Proposition 1.48. We assume ” â W 1,â(Ω0, Rn) with supp(”) â Ω.
(1) If |”|Lip,Ω0< 1, then Ï = Ï0 + ” is a bi-Lipschitz transform from Ω onto
itself.(2) For t â R with |t”|Lip,Ω0
< 1, we define a bi-Lipschitz transform Ï(t) =Ï0 + t” from Ω to itself. Let l â 0, 1 and p â [1,â]. Suppose thatf â W l,p(Ω) if p â [1,â), and f â Cl(Ω) â© W l,â(Ω) if p = â. ThenÏ(t)âf â f strongly in W l,p(Ω) as t â 0.
Proof. From Proposition 1.41, the claim 1 is clear. For the claim 2, let us fixt0 > 0 with |t0”|Lip,Ω0
< 1. Then, from Proposition 1.46, there exist C > 0 suchthat the following inequalities hold for |t| †t0,
âÏ(t)âf â fâW l,p(Ω) = âÏ(t)â(f â f Ï(t))âW l,p(Ω) †Câf â f Ï(t)âW l,p(Ω).
Since [Ï 7â f Ï] â C0(O(Ω), W l,p(Ω)) from Proposition 1.47, we obtain
âf â f Ï(t)âW l,p(Ω) = âf Ï0 â f Ï(t)âW l,p(Ω) â 0,
as t â 0.
8. Potential energy in deformed domains
Let Ω0 be a fixed bounded convex open set of Rn (n ℠2). We consider an
open set Ω whose closure is contained in Ω0. We suppose that v â H1(Ω) be ascalar valued function which describes a physical state in Ω â R
n. For such v(x),we introduce the following energy functional:
E(v, Ω) :=
â«
Ω
W (x, v(x),âv(x))dx,
where
W (Ο, η, ζ) â R for (Ο, η, ζ) â Ω0 Ă R Ă Rn,
âtopics-in-mathematical-modelingâ â 2008/12/5 â 8:30 â page 29 â #41
8. POTENTIAL ENERGY IN DEFORMED DOMAINS 29
is a given energy density function. We assume some suitable regularity condi-tions and boundedness of its derivatives in the following argument. For sim-plicity, the partial derivatives of W with respect to Ο, η and ζ will be denotedby âΟW = (âW
âΟ1, · · · , âW
âΟn)T, Wη = âW
âη , and âζW = (âWâζ1
, · · · , âWâζn
)T, respec-
tively. Moreover, for v â H1(Ω), we often write W (v(x)) = W (x, v(x),âv(x)),âΟW (v(x)) = âΟW (x, v(x),âv(x)), etc.
We consider the following minimization problem.
Problem 1.49. Let V be a closed subspace of H1(Ω) with H10 (Ω) â V â H1(Ω),
and let V (g) := v â H1(Ω); vâ g â V for g â H1(Ω). For given g â H1(Ω), finda local minimizer u of E(·, Ω) in V (g), i.e. u â V (g) and there exists Ï > 0 suchthat
E(u, Ω) †E(w, Ω) (âw â V (g) with âw â uâH1(Ω) < Ï). (1.19)
If u is a local minimizer, under suitable regularity conditions for W , formallywe obtain the variation formulas;
â«
Ω
Wη(u(x))v(x) + âζW (u(x)) · âv(x) dx = 0 (âv â V ), (1.20)
âdiv [âζW (u(x))] + Wη(u(x)) = 0 in Ω
For fixed Ω and V â H1(Ω) as Problem 1.49, we consider a family of min-imization problems which is parametrized by Ï â O(Ω), where O(Ω) is definedby (1.18).
We define
V (Ï, g) := Ïâ(V (g)) = v â H1(Ω(Ï)); Ïâ1â (v) â g â V
(Ï â O(Ω), g â H1(Ω)).
Problem 1.50. For given Ï â O(Ω) and g â H1(Ω), find a local minimizeru(Ï) of E(·, Ω(Ï)) in V (Ï, g), i.e. u(Ï) â V (Ï, g) and there exists Ï > 0 such that
E(u(Ï), Ω(Ï)) †E(w, Ω(Ï)) (âw â V (Ï, g) with âw â u(Ï)âH1(Ω(Ï)) < Ï).(1.21)
In many mathematical models of actual (quasi-)static systems, the potentialenergy in a deformed domain should be a local minimum. So, we define
Eâ(Ï) := E(u(Ï), Ω(Ï)), (1.22)
for local minimizers u(Ï).We note that v â V is equivalent to Ïâ(v + g) â V (Ï, g). Using the formulas
in § 7, we haveE(Ïâ(v + g), Ω(Ï)) = E(v, Ï) (v â V ),
where
E(v, Ï) := E(v + g, Ï) (v â H1(Ω)),
E(v, Ï) :=
â«
Ω
W (Ï(x), v(x), [A(Ï)(x)]âv(x)) Îș(Ï)(x)dx (v â H1(Ω)).
We define v(Ï) := Ïâ1â (u(Ï))âg â V . Then u(Ï) is a local minimizer of E(·, Ω(Ï))
in V (Ï, g), if and only if v(Ï) is a local minimizer of E(·, Ï) in V . The followingtheorem is a direct consequence of Theorem 1.30.
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30 1. SHAPE DERIVATIVE OF MINIMUM POTENTIAL ENERGY
Theorem 1.51. Suppose that E â C1(H1(Ω) ĂO(Ω)). Let U0 â V and let O0
be an open subset of O(Ω) with Ï0 â O0. If v â C0(O0, V ) and v(Ï) is a globalminimizer of E(·, Ï) in U0, then we have Eâ â C1(O0) and
EâČâ(Ï) = EÏ(v(Ï), Ï),
where Eâ is defined by (1.22) with u(Ï) = Ïâ(v(Ï) + g).
We remark that
EâČâ(Ï0) = EÏ(v(Ï0), Ï0) = EÏ(u(Ï0), Ï0).
Under the suitable regularity conditions for W (Ο, η, ζ), we have the following for-mula. For ” â W 1,â(Ω0, R
n),
EÏ(u, Ï0)[”]
=d
dt
â«
Ω
W (x + t”(x), u(x), [A(Ï0 + t”)(x)]âu(x)) Îș(Ï0 + t”)(x)dx
âŁâŁâŁâŁt=0
=
â«
Ω
(âΟW (u) · ” â (âζW (u))T(â”T)âu + W (u)div”
)dx. (1.23)
By means of the above domain mappings, we obtain another type of variationformula for Problem 1.49, so called âinterior variationâ which is different from (1.20).
Theorem 1.52. Suppose that E â C1(H1(Ω)ĂO(Ω)). If u is a local minimizerof E(·, Ω) in V (g) as in Problem 1.49, then we have
EÏ(u, Ï0)[”] = 0 (” â W 1,â(Ω), supp(”) â Ω).
Proof. For ” â W 1,â(Ω) with supp(”) â Ω, we define Ï(t) = Ï0±t” for t â„ 0.From Proposition 1.46 and 1.48, if |t”|Lip,Ω0
< 1, the corresponding pushforwardoperator Ï(t)â is a linear topological isomorphism from H1(Ω) onto itself, and
limtâ0
âÏ(t)âu â uâH1(Ω) = 0.
Since
E(Ï(t)âu, Ω) â„ E(u, Ω) = E(Ï(0)âu, Ω),
for 0 †t << 1, we obtain
± EÏ(u, Ï0)[”] =d
dtE(u, Ï(t))
âŁâŁâŁâŁt=0
=d
dtE(Ï(t)âu, Ω)
âŁâŁâŁâŁt=0
â„ 0.
Hence, we have EÏ(u, Ï0)[”] = 0.
9. Applications
Based on the discussions in previous sections, we study the shape derivative ofquadratic elliptic potential energies which correspond to scalar valued linear ellipticproblems of second order. However, our framework is applicable even to vectorvalued elliptic systems such as linear elasticity problems and to some semilinearproblems as shown in [17].
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9. APPLICATIONS 31
Example 1.53. We consider the following potential energy.
W (Ο, η, ζ) =1
2
(ζTB(Ο)ζ + b(Ο)η2
)â f(Ο)η, (1.24)
E(v, Ω) =
â«
Ω
1
2
((âv)TB(x)âv + b(x)v2
)â f(x)v
dx,
where k â NâȘ0 and B(Ο) is an nĂn symmetric matrix (which is denoted by RnĂnsym )
and it satisfies B = (bij) â Ck(Ω0, RnĂnsym ) and âÎČ0 > 0 such that ζTB(Ο)ζ â„ ÎČ0|ζ|2
(âΟ â Ω0,âζ â R
n), and
b â Ck(Ω0, R), f â W k,2(Ω0, R), (1.25)
with the condition b(Ο) â„ 0 for Ο â Ω0. We remark that
âΟW (Ο, η, ζ) =1
2
nâ
i,j=1
âbij(Ο)ζiζj + âb(Ο)η2
ââf(Ο)η,
Wη(Ο, η, ζ) = b(Ο)η â f(Ο),
âζW (Ο, η, ζ) = B(Ο)ζ.
We suppose that ÎD is a nonempty Lipschitz portion of âΩ and that a boundedtrace operator Îł0 : H1(Ω) â L2(ÎD) is defined and
V := v â H1(Ω); Îł0(v) = 0 6= 0, (1.26)
holds.8 Then the minimization problem 1.21 corresponds to the following linearelliptic boundary value problem.
âdiv(B(x)âu) + b(x)u = f(x) in Ω(Ï),
u = g on ÎD,
(Bâu) · Îœ = 0 on âΩ \ ÎD,
where Îœ is a unit normal vector on âΩ.From the Poincare inequality and the LaxâMilgram theorem, there exists a
unique global minimizer u(Ï) â V (g, Ï) for Ï â O(Ω).
The Poisson equation is a special case of Example 1.53.
Example 1.54. We consider the following potential energy which correspondsto the Poisson equation.
W (Ο, η, ζ) =1
2|ζ|2 â f(Ο)η,
E(v, Ω) =
â«
Ω
(1
2|âv|2 â f(x)v
)dx,
where k â N âȘ 0 and f â W k,2(Ω0, R). This is a special case of Example 1.53.We remark that
âΟW (Ο, η, ζ) = ââf(Ο)η, Wη(Ο, η, ζ) = âf(Ο), âζW (Ο, η, ζ) = ζ.
8Under the condition (1.26), from the well known Poincare inequality, the coercivity of thisfunctional follows. We remark that, if b(Ο) > 0 for Ο â Ω0, we do not need the condition (1.26).
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32 1. SHAPE DERIVATIVE OF MINIMUM POTENTIAL ENERGY
Under the same boundary condition, the minimization problem 1.21 correspondsto the following boundary value problem of the Poisson equation:
ââu = f(x) in Ω(Ï),
u = g on ÎD,
âu
âÎœ= 0 on âΩ \ ÎD.
Lemma 1.55. For the potential energy of Example 1.53, E â Ck(H1(Ω)ĂO(Ω))and (1.23) holds.
Proof. In the case of Example 1.53, E becomes
E(v, Ï)
=
â«
Ω
1
2(A(Ï)âv)
TB(Ï(x)) (A(Ï)âv) +
1
2b(Ï(x))v2 â f(Ï(x))v
Îș(Ï)dx
=
â«
Ω
1
2Κ1(v, Ï)(x) +
1
2Κ2(v, Ï)(x) â Κ3(v, Ï)(x)
Îș(Ï)(x)dx
where Κ1(v, Ï) := (A(Ï)âv)T (B Ï) (A(Ï)âv) , Κ2(v, Ï) := (bÏ)v2 , Κ3(v, Ï) :=(f Ï)v. Under the assumptions, from Theorem 1.45 and Proposition 1.47, we have
[(v, Ï) 7â A(Ï)âv] â Câ(H1(Ω) ĂO(Ω), L2(Ω)n),
[Ï 7â B Ï] â Ck(O(Ω), Lâ(Ω, RnĂn)),
[(w, B) 7â wTBw] â Câ(L2(Ω)n Ă Lâ(Ω, RnĂn), L1(Ω)).
From these regularities, it follows that Κ1 â Ck(H1(Ω) ĂO(Ω), L1(Ω)). Similarly,from the following regularities
[Ï 7â b Ï] â Ck(O(Ω), Lâ(Ω)),
[(b, v) 7â bv2] â Câ(Lâ(Ω) Ă H1(Ω), L1(Ω)),
[Ï 7â f Ï] â Ck(O(Ω), L2(Ω))
[(f, v) 7â fv] â Câ(L2(Ω) Ă H1(Ω), L1(Ω)),
Κ2 and Κ3 also belong to Ck(H1(Ω)ĂO(Ω), L1(Ω)). Since Îș â Câ(O(Ω), Lâ(Ω))
from Theorem 1.45, we conclude that E â Ck(H1(Ω) ĂO(Ω)).
We obtain the following theorem from Lemma 1.55 and Theorem 1.35.
Theorem 1.56. For the potential energy of Example 1.53, Eâ â Ck(O(Ω))holds, and if k â„ 1 we have
EâČâ(Ï0)[”] =
â«
Ω
(âΟW (u) · ” â (âζW (u))T(â”T)âu + W (u)div”
)dx
(” â W 1,â(Ω0, Rn)),
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9. APPLICATIONS 33
where u is the global minimizer of E(·, Ω) in V (g). In particular, in the case ofExample 1.54, we have
EâČâ(Ï0)[”] =
â«
Ω
â(âf · ”)u â (âTu)(â”T)(âu) +
(1
2|âu|2 â fu
)div”
dx
(” â W 1,â(Ω0, Rn)).
We consider the case of Example 1.53. The global minimizer u belongs toH1(Ω), but not to H2(Ω) in general. The regularity loss is caused by geometricalsingularities such as a crack tip and a corner of the domain boundary, or by anincompatible point of a mixed boundary condition.
Theorem 1.57. We consider Example 1.53 with k = 1. Let G be a nonemptyopen set in R
n with a Lipschitz boundary. If there exists a sequence of subdomainsΩll in which the GaussâGreen formula holds and
Ω1 â Ω2 â · · ·Ω with
ââ
l=1
Ωl = Ω,
and if the global minimizer u belongs to H2(Ωl ⩠G) for each l, then we have
EâČâ(Ï0)[”] = lim
lââ
â«
âΩl
W (u) ” · Îœ â (âζW (u) · Îœ) (âu · ”) dHnâ1x , (1.27)
for ” â W 1,â(Ω0, Rn) with supp(”) â G.
Proof. We define Ωl := Ωl ⩠G. Under the conditions, we have
âdiv [âζW (u)] + Wη(u) = 0 in L2(Ωl),
which is equivalent to
âdiv(Bâu) + bu = f in L2(Ωl).
From the following equalities:9â«
eΩl
W (u(x))div”(x)dx =
â«
âeΩl
W (u(x))”(x) · ÎœdHnâ1x â
â«
eΩl
âx[W (u(x))] · ”(x)dx,
âx[W (u(x))] = âΟW (u(x)) + Wη(u(x))âu(x) + [â2u(x)]âζW (u(x)) in Ωl,
â«
eΩl
Wη(u(x))âu(x) · ”(x)dx =
â«
eΩl
div [âζW (u)]âu(x) · ”(x)dx
=
â«
âeΩl
(âζW (u) · Îœ) (âu(x) · ”(x)) dHnâ1x â
â«
eΩl
âT
ζ W (u)â (âu(x) · ”(x)) dx
=
â«
âΩl
(âζW (u) · Îœ) (âu(x) · ”(x)) dHnâ1x
ââ«
Ωl
âT
ζ W (u)(â2u(x))”(x) + (â”T)âu
dx,
9In this note, âu(x) denotes the gradient of u as a column vector and â2u(x) denotes theHessian matrix.
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34 1. SHAPE DERIVATIVE OF MINIMUM POTENTIAL ENERGY
â«
eΩl
âx[W (u(x))] · ”(x)dx
=
â«
Ωl
âΟW (u(x)) + [â2u(x)]âζW (u(x))
· ”(x)dx
=
â«
âΩl
(âζW (u) · Îœ) (âu(x) · ”(x)) dHnâ1x
ââ«
Ωl
âT
ζ W (u)(â2u(x))”(x) + (â”T)âu
dx
=
â«
âΩl
(âζW (u) · Îœ) (âu(x) · ”(x)) dHnâ1x
+
â«
Ωl
âΟW (u) · ”dx ââT
ζ W (u)(â”T)âu
dx,
we obtain
EâČâ(Ï0)[”] =
â«
Ω
(âΟW (u) · ” â (âζW (u))T(â”T)âu + W (u)div”
)dx
=
â«
Ω\Ωl
(âΟW (u) · ” â (âζW (u))T(â”T)âu + W (u)div”
)dx
+
â«
Ωl
(âΟW (u) · ” â (âζW (u))T(â”T)âu
)dx
+
â«
âΩl
W (u) ” · Îœ dHnâ1x â
â«
Ωl
âx[W (u)] · ”dx
=
â«
Ω\Ωl
(âΟW (u) · ” â (âζW (u))T(â”T)âu + W (u)div”
)dx
+
â«
âΩl
W (u) ” · Îœ â (âζW (u) · Îœ) (âu · ”) dHnâ1x ,
where Îœ denotes the outward unit normal of âΩl. The first term tends to 0 if l â â.Hence, we have the formula (1.27).
Corollary 1.58. In particular, for the case of Example 1.54, we have
EâČâ(Ï0)[”] = lim
lââ
â«
âΩl
(1
2|âu|2 â fu
)” · Îœ â (âu · Îœ) (âu · ”)
dHnâ1
x . (1.28)
In the absence of singularity, Corollary 1.58 has the following form.
Theorem 1.59. Let Ω be a bounded domain with C2-boundary. Under thecondition of Example 1.54, we assume that ÎD = âΩ with g ⥠0. Then the followingformula holds.
EâČâ(Ï0)[”] = â1
2
â«
âΩ
|âu|2” · Îœ dHnâ1x .
Proof. From the regularity theorem of elliptic boundary value problems, wehave u â H2(Ω). So, we can choose Ωl = Ω. We remark that
âu · ” = (âu · Îœ)(” · Îœ), |âu · Îœ| = |âu| on âΩ,
âtopics-in-mathematical-modelingâ â 2008/12/5 â 8:30 â page 35 â #47
9. APPLICATIONS 35
since u and its tangential derivatives vanish on the boundary. Hence we have
EâČâ(Ï0)[”] =
â«
âΩ
(1
2|âu|2 â fu
)” · Îœ â (âu · Îœ) (âu · ”)
dHnâ1
x
=
â«
âΩ
(1
2|âu|2
)” · Îœ â
(|âu|2
)” · Μ
dHnâ1
x = â1
2
â«
âΩ
|âu|2 ” · Îœ dHnâ1x .
This theorem can be generalized as follows.
Theorem 1.60. Under the condition of Example 1.54, we assume that ” âW 1,â(Ω0, R
n) satisfies
supp(”) â© supp( g |ÎD) = â , supp(”) â© ÎD â© âΩ \ ÎD = â ,
âG : an open set of Rn s.t. G â supp(”) and âΩ â© G is C2-class.
Then the following formula holds.
EâČâ(Ï0)[”] = â1
2
â«
ÎD
|âu|2” · Îœ dHnâ1x â
â«
âΩ\ÎD
f u ” · Îœ dHnâ1x .
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âtopics-in-mathematical-modelingâ â 2008/12/5 â 8:30 â page 37 â #49
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[21] M. Schechter, Principles of Functional Analysis. second edition, GraduateStudies in Mathematics, Vol.36, American Mathematical Society, Providence,RI, (2002).
[22] J. Sokolowski and J.-P. Zolesio, Introduction to Shape Optimization. ShapeSensitivity Analysis. Springer Series in Computational Mathematics, Vol.16,Springer-Verlag, Berlin, (1992).
[23] I. Wakano, Analysis for stress intensity factors in two dimensional elasticity.Proc. Japan Acad., Vol.73, Ser.A, No.5 (1997), 86-88.
[24] K. Yosida, Functional Analysis. sixth edition, Springer-Verlag, Berlin-NewYork (1980).
[25] W. P. Ziemer, Weakly Differentiable Functions, Sobolev Spaces and Functionsof Bounded Variation. Graduate Texts in Mathematics, 120, Springer-Verlag,New York (1989).
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Part 2
Geometry of hypersurfaces and
moving hypersurfaces in Rm for the
study of moving boundary
problems
Masato Kimura
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2000 Mathematics Subject Classification. 14Q10, 35R35, 53C44
Key words and phrases. hypersurfaces, moving hypersurfaces, gradient flows
Abstract. The text provides a record of an intensive lecture course on themoving boundary problems. Keeping the applications to moving boundaryproblems in R2 or R3 in mind, the text systematically constructs the math-ematical foundations of hypersurfaces and moving hypersurfaces in Rm form â„ 2. Main basic concepts treated in the text are differential and inte-gral formulas for hypersurfaces, geometric quantities and the signed distancefunctions for moving hypersurfaces, the variational formulas and the transportidentities, and the gradient structure of moving boundary problems.
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Contents
Preface 43
Chapter 1. Preliminaries 451. Notation 452. Plane curves 463. Parametric representation 474. Graph representation and principal curvatures 50
Chapter 2. Differential calculus on hypersurfaces 531. Differential operators on Î 532. Weingarten map and principal curvatures 54
Chapter 3. Signed distance function 591. Signed distance function in general 592. Signed distance function for hypersurface 60
Chapter 4. Curvilinear coordinates 651. Differential and integral formulas in N Δ(Î) 65
Chapter 5. Moving hypersurfaces 691. Normal time derivatives 692. Signed distance function for moving hypersurface 723. Time derivatives of geometric quantities 73
Chapter 6. Variational formulas 751. Transport identities 752. Transport identities for curvatures 78
Chapter 7. Gradient structure and moving boundary problems 811. General gradient flow of hypersurfaces 812. Prescribed normal velocity motion 823. Mean curvature flow 824. Anisotropic mean curvature flow 835. Gaussian curvature flow 836. Willmore flow 847. Volume preserving mean curvature flow 848. Surface diffusion flow 859. HeleâShaw moving boundary problem 86
Bibliography 89
41
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42 CONTENTS
Appendix A. 911. Adjugate matrix 912. Jacobiâs formula 92
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Preface
From the end of March 2007, I had an occasion to spend one year at De-partment of Mathematics, Faculty of Nuclear Sciences and Physical Engineering,Czech Technical University in Prague as a visiting researcher of the Necas Centerfor Mathematical Modeling. One of the common research subjects in the Mathe-matical Modelling Group of Professor Michal Benes in Czech Technical University,where I visited, is the moving boundary problems related to physics, chemistry,biology and engineering. This is a note for my intensive lectures on the movingboundary problems which I gave at Czech Technical University in November 2007.
The principal object of my intensive lectures was to construct a basic mathe-matical framework for the moving boundary problems and to derive several usefulformulas without assuming any knowledge on the differential geometry. Most ofthe topics in this note are basically included in the course of the classical differ-ential geometry. However, the standard description of the differential geometry isnot always suitable for the applied mathematicians who intend to study movingboundary problems.
Keeping the applications to moving boundary problems in R2 or R
3 in mind,we systematically construct the mathematical foundations of hypersurfaces andmoving hypersurfaces in R
m for m â„ 2. Main basic concepts treated in this noteare differential and integral formulas for hypersurfaces, geometric quantities and thesigned distance functions for moving hypersurfaces, the variational formulas and thetransport identities, and the gradient structure of moving boundary problems. Bythe word âgeometricâ, we just mean that the quantity or the operator does notdepend on the choice of the Cartesian coordinate in R
m.On the other hand, we had to omit many important topics, such as the con-
struction of the partition of unity, the hight function and the domain mappingmethod, the Sobolev spaces on hypersurfaces, the well-posedness of some movingboundary problems, etc. We are not able to touch on these issues except for someexamples and exercises without proofs.
Finally, I would like to thank Professor Michal Benes and his colleagues for giv-ing me this opportunity and for attending the lectures. Especially, I am grateful toMr.Tomas Oberhuber who carefully read the first manuscript and gave me severaluseful comments on this work.
Masato KimuraFukuoka, July 2008
43
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CHAPTER 1
Preliminaries
In this chapter, we collect our notation in this lecture note and several basicfacts on the geometries of hypersurfaces in the standard style of geometrical treat-ment. We start from the well-known Frenetâs formula for plane curves. In generaldimension, we introduce the standard local parametric representation of a hyper-surface and define the surface integral on the hypersurface. In the last section, wegive a definition of the principal curvatures based on the graph representation ofthe hypersurface in general dimension.
1. Notation
We use the following notation throughout this note. For an integer m â N
with m â„ 2, Rm denotes the m-dimensional Euclidean space over R. Each point
x â Rm is expressed by a column vector x = (x1, · · · , xm)T, where T stands for the
transpose of the vector or matrix.For two column vectors a = (a1, · · · , am)T â R
m and b = (b1, · · · , bm)T â Rm,
their inner product is denoted by
a · b := aT b =
mâ
i=1
aibi,
and |x| :=â
x · x. For two square matrices A = (aij) â RmĂm and B = (bij) â
RmĂm, their componentwise inner product is also denoted by
A : B := tr(ATB
)=
mâ
i=1
mâ
j=1
aijbij ,
where tr stands for the trace of a square matrix.For a real-valued function f = f(x), its gradient is given by a column vector
âf(x) = âxf(x) :=
(âf
âx1(x), · · · ,
âf
âxm(x)
)T
=
âfâx1
(x)...
âfâxm
(x)
,
and its transpose is denoted by âT
âTf(x) =
(âf
âx1(x), · · · ,
âf
âxm(x)
).
For a (column) vector-valued function h(x) = (h1(x), · · · , hm(x))T â Rm, its trans-
pose is denoted by hT(x) = h(x)T = (h1(x), · · · , hm(x)). The above gradient
45
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46 1. PRELIMINARIES
operator acts on h as follows.
âhT(x) := (âh1(x), · · · , âhm(x)) =
âh1
âx1(x) · · · âhm
âx1(x)
......
âh1
âxm(x) · · · âhm
âxm(x)
â R
mĂm.
âTh(x) := (âhT(x))T =
âTh1(x)...
âThm(x)
=
âh1
âx1(x) · · · âh1
âxm(x)
......
âhm
âx1(x) · · · âhm
âxm(x)
â R
mĂm.
We note that âT does not mean the divergence operator in this paper. The diver-gence operator is denoted by
divxh(x) = divh(x) :=mâ
i=1
âhi
âxi(x) = tr(âhT) = tr(âTh).
This div acts on a column vector.For a real-valued function f = f(x), its Hessian matrix is denoted by
â2f := âT(âf) = â(âTf) =
â2fâx2
1
· · · â2fâx1âxm
......
â2fâxmâx1
· · · â2fâx2
m
â R
mĂm,
where we note that â2 does not mean the Laplacian operator in this paper. Wedenote the Laplacian of f by
âf := divâf = trâ2f =
mâ
i=1
â2f
âx2i
.
2. Plane curves
We start from the plane curve. Let Î be a C2-class curve in R2 parametrized
by a length parameter s â I, where I is an interval:
Î = Îł(s) = (Îł1(s), Îł2(s))T â R
2; s â I, Îł â C2(I, R2), |ÎłâČ(s)| = 1.
The tangential and normal unit vectors are given by
Ï (s) := ÎłâČ(s), Îœ(s) := (âÎłâČ2(s), Îł
âČ1(s))
T.
Exercise 1.1. Prove that Ï âČ(s) â Îœ(s) and Îœ âČ(s) â Ï (s).
The signed curvature Îș(s) is defined by the formulas
Ï âČ(s) = Îș(s)Îœ(s), Îœ âČ(s) = âÎș(s)Ï (s).
These are known as the Frenet formula for plane curves. We draw the readersâattention to the above sign convention of Îœ and Îș, which we use throughout thisnote.
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3. PARAMETRIC REPRESENTATION 47
Example 1.2. We consider a circle of radius r > 0: Î = x â R2; |x| = r.
Then we have the following relations:
Îł(s) =(r cos
(s
r
), r sin
(s
r
))T
(0 †s < 2Ïr),
Ï (s) =(â sin
(s
r
), cos
(s
r
))T
, Îœ(s) = â(cos(s
r
), sin
(s
r
))T
Ï âČ(s) =
(â1
rcos(s
r
), â1
rsin(s
r
))T
=1
rÎœ(s), Îș(s) =
1
r.
Exercise 1.3. For Î given by a graph η = u(Ο) (Ο â I1 â R) in Ο-η plane withu â C2(I1) and ds
dΟ > 0, prove that
Îș =uâČâČ(Ο)
(1 + |uâČ(Ο)|2) 32
.
Exercise 1.4. For Î = Ï(Ο) = (Ï1(Ο), Ï2(Ο))T â R
2; Ο â I2 â R withÏ â C2(I2) and ds
dΟ > 0, prove that
Îș =ÏâČ
1(Ο)ÏâČâČ2 (Ο) â ÏâČâČ
1 (Ο)ÏâČ2(Ο)
|ÏâČ(Ο)|3 .
3. Parametric representation
Let m â N and m â„ 2. We consider (mâ 1)-dimensional hypersurfaces embed-ded in R
m.
Definition 1.5. Let k â N. A subset Î â Rm is called a Ck-class hypersurface
in Rm, if, for each x â Î, there exist,
O : a nonempty bounded domain in Rm,
U : a nonempty bounded domain in Rmâ1,
Ï â Ck(U , Rm)(1.1)
such that x â O and they satisfy the conditions:
Ï : U â Î â©O is bijective, and rank(âT
Ο Ï(Ο)) = m â 1 (âΟ â U). (1.2)
The triplet (O, U , Ï) gives a kind of local embedding Ck-coordinate of the hyper-surface Î. We define
Ck(Î) := (O, U , Ï); (O, U , Ï) satisfies (1.1) and (1.2) .
If a subset (O”, U”, Ï”)”âM â Ck(Î) satisfies Î â âȘ”âMO”, then we call
(O”, U”, Ï”)”âM a local embedding Ck-coordinate system of Î. Moreover, ifÎ is compact, we can choose its finite subset (O”j , U”j , Ï”j
)j=1,2,··· ,J such that
it is a local Ck-embedding coordinate system of Î, too.
Exercise 1.6. Let k â N. For a Ck-class hypersurface Î in Rm, suppose that
(O, U , Ï) and (O, U , Ï) are two triplets belonging to Ck(Î) with ÎŁ := Îâ©Oâ©O 6= â .Then, prove that Ïâ1 Ï â Ck(Ïâ1(ÎŁ), Ïâ1(ÎŁ)).
To represent a hypersurface, there are mainly four ways;
(i) parametric representation by local coordinate (Definition 1.5),
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48 1. PRELIMINARIES
(ii) local graph representation (next section),(iii) level set approach,(iv) domain mapping method.
Although the domain mapping method is important and widely used, we omitit in this note except for Section 1.
The level set approach enables us to treat a hypersurface globally in a fixedorthogonal coordinate system. It is useful especially to represent moving hypersur-faces.
Exercise 1.7. Let O be an open set in Rm, and let w â Ck(O). Then, for a
fixed c â R, we define a (c-)level set Î of w by
Î := x â O; w(x) = c.Prove that Î is a Ck-class hypersurface in R
m provided Î 6= â and âw(x) 6= 0 forall x â Î. (Conversely, any Ck-class hypersurface can be represented as a level setof a Ck-class function. An example of such functions is given by the signed distancefunction (Section 2).)
Definition 1.8. Let k â N and let l be an integer with 0 †l †k. For aCk-class hypersurface Î in R
m and a function f defined on Î, f is called of Cl-classif f Ï â Cl(U) for any (O, U , Ï) â Ck(Î) . The set of all Cl-class functions on Îis denoted by Cl(Î).
Exercise 1.9. Let k â N and let l be an integer with 0 †l †k. Let(O”, U”, Ï”)”âM be a local embedding Ck-coordinate system of Î. Suppose that
a function f defined on Î satisfies f Ï” â Cl(U”) for all ” â M . Then, prove
that f â Cl(Î).
Let (O, U , Ï) â Ck(Î). Then, for each Ο â U ,âÏâΟ1
(Ο), · · · ,âÏ
âΟmâ1(Ο) are m â 1
independent Rm vectors, which are tangent to Î at x = Ï(Ο) â Î. We denote by
Tx(Î) the tangent space at x = Ï(Ο) â Î:
Tx(Î) := span
âšâÏ
âΟ1(Ο), · · · ,
âÏ
âΟmâ1(Ο)
â©.
This is an m â 1 dimensional subspace of Rm.
Exercise 1.10. Prove that Tx(Î) does not depend on the choice of local Ck-embedding coordinate of Î which covers x.
We define the following functions of Ο â U :
G(Ο) :=(âΟÏT(Ο)
) (âT
Ο Ï(Ο))â R
(mâ1)Ă(mâ1)sym ,
i.e., G(Ο) = (gij(Ο)), gij =âÏ
âΟi· âÏ
âΟj,
g(Ο) := detG(Ο).
Ïi(Ο) := (Ï1(Ο), · · · , Ïiâ1(Ο), Ïi+1(Ο), · · · , Ïm(Ο))T â R
mâ1,
α(Ο) := (α1(Ο), · · · , αm(Ο))T â Rm, αi(Ο) := (â1)m+i det
(âT
Ο Ïi(Ο)).
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3. PARAMETRIC REPRESENTATION 49
Theorem 1.11. Under the above conditions, for Ο â U , we have α(Ο) âTÏ(Ο)(Î)â„ and
det(âT
Ο Ï(Ο), α(Ο))
= g(Ο) = |α(Ο)|2 > 0.
Proof. Since rank(âT
Ο Ï(Ο)) = m â 1, at least one of αi is not zero, i.e.,
α(Ο) 6= 0. From the equality
α · âÏ
âΟj=
mâ
i=1
αiâÏi
âΟj=
mâ
i=1
(â1)m+i det(âT
Ο Ïi) âÏi
âΟj
= det
(âT
Ο Ï(Ο),âÏ
âΟj
)= 0,
for each j = 1, · · · , m â 1, we obtain α(Ο) â TÏ(Ο)(Î)â„.
We define J(Ο) :=(âT
Ο Ï(Ο), α(Ο)). Then we have
detJ =
mâ
i=1
(â1)m+i det(âT
Ο Ïi)αi =
mâ
i=1
|αi|2 = |α|2,
and
(detJ)2 = det(JTJ) = det
((âΟÏ
T
αT
)(âT
Ο Ï, α))
= det
(G 00T |α|2
)= (detG)|α|2.
Hence, we obtain the assertions.
Corollary 1.12. We can define a unit normal vector field Îœ â Ckâ1(Î â©O, Rm) by
Μ(x) :=α(Ο)
|α(Ο)| (x = Ï(Ο) â Î â© O),
and we have âg(Ο) = det
(âT
Ο Ï(Ο), Îœ(Ï(Ο))).
We remark that the tangent space is expressed as
Tx(Î) = y â Rm; y · Îœ(x) = 0.
Definition 1.13. A Ck-class hypersurface Î in Rm is called oriented, if there
exists a (single-valued) unit normal vector field on Î, i.e., Îœ â Ckâ1(Î, Rm) withÎœ(x) â Tx(Î)â„ and |Îœ(x)| = 1 (x â Î).
Under the conditions of Exercise 1.7, the unit normal vector field is expressedas
Îœ(x) =âw(x)
|âw(x)| or â âw(x)
|âw(x)| (x â Î),
and Î becomes an oriented hypersurface (see Lemma 2.1).We define the surface integral on Î as follows. Let k â N and let Î be a Ck-class
hypersurface in Rm. For a function f defined on Î, we consider the surface integral:
â«
Î
f dHmâ1 =
â«
Î
f(x) dHmâ1x ,
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50 1. PRELIMINARIES
where Hmâ1x is the mâ1 dimensional Hausdorff measure with respect to x (see [5],
[20] etc.), provided f is Hmâ1-integrable. In particular, if supp(f) â (Î â© O) andf Ï â L1(U) for (O, U , Ï) â Ck(Î), we have
â«
Î
f(x) dHmâ1x =
â«
U
f(Ï(Ο))â
g(Ο) dΟ.
4. Graph representation and principal curvatures
Let Î be a Ck-class hypersurface in Rm (k â N). We fix an arbitrary point
on Î and assume that it is the origin 0 â Î â Rm without loss of generality. We
choose a suitable orthogonal coordinate (Ο1, Ο2, · · · , Οmâ1, η)T of Rm such that the
tangent space and the unit normal vector at 0 â Î are given by
T0(Î) =(Ο, 0)T â R
m; Ο = (Ο1, · · · , Οmâ1)T â R
mâ1
,
Îœ(0) = (0, · · · , 0, 1)T â Rm.
In a neighborhood of the origin, from the implicit function theorem, Î is locallyexpressed by a graph η = u(Ο) (Ο â U), where U is a bounded domain of R
mâ1
with 0 â U , and
u â Ck(U), âΟu(0) = 0 â Rmâ1.
The unit normal vector of Î in the neighborhood of the origin is given by
Îœ(x) =1â
1 + |âΟu(Ο)|2(
ââΟu(Ο)1
) (x =
(Ο
u(Ο)
)â Î
). (1.3)
Proposition 1.14. Let Î be a bounded Ck-class hypersurface (k â„ 1). Suppose
that Î is contained in a Ck-class hypersurface Î. Then, for f â Cl(Î) (0 †l †k),
there exists an open set O â Rm with Î â O and f â Cl(O) such that f |Î = f .
Proof. First, we assume that
supp(f) â ÎâČ := (Ο, u(Ο))T â Rm; Ο â U (1.4)
in the above local graph representation. Then we can construct f by f(Ο, η) :=f(Ο, u(Ο)) in a neighborhood of ÎâČ. For general f , applying a partition of unity, we
decompose f =âN
j=1 fj, where each fj satisfies the condition (1.4), and we define
f :=âN
j=1 fj.
We assume that k â„ 2 hereafter, and denote by â2Ο u the Hessian matrix of u(Ο)
â2Ο u(Ο) =
(â2u
âΟiâΟj(Ο)
)
i,j=1,··· ,mâ1
â R(mâ1)Ă(mâ1)sym .
Definition 1.15 (principal curvatures and directions). Let the eigenvaluesand the eigenvectors of the symmetric matrix â2
Ο u(0) be denoted by Își â R and
eâČi â R
mâ1 (i = 1, · · · , m â 1) with
â2Ο u(0)eâČ
i = ÎșieâČi, eâČ
i · eâČj = ÎŽij .
Then Își (i = 1, · · · , m â 1) are called principal curvatures of Î at 0 â Î, andei := (eâČ
i, 0)T â T0(Î) â Rm is called a principal direction with respect to Își.
The sum of the principal curvatures is denoted by Îș :=âmâ1
i=1 Își, and call Îș mean
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4. GRAPH REPRESENTATION AND PRINCIPAL CURVATURES 51
curvature in stead of Îș/(mâ1) in this lecture note. The Gauss-Kronecker curvatureis also denoted by Îșg := Î mâ1
i=1 Își.
The meaning of the principal curvatures is clarified in the next proposition.
Proposition 1.16. Under the conditions of Definition 1.15, for Ï = (Ï âČ, 0)T âT0(Î) with Ï âČ â R
mâ1 and |Ï | = |Ï âČ| = 1, we define a plane curve in Ï -η planewith a coordinate (Ï, η)T â R
2 by
ÎÏ :=(Ï, u(ÏÏ âČ))T â R
2; Ï â I
,
where I is an open interval with 0 â I. Let the curvature of ÎÏ at (Ï, η) = (0, 0)be denoted by ÎșÏ . Then, it is given as
ÎșÏ = (Ï âČ)T(â2
Ο u(0))
Ï âČ =
mâ1â
i=1
Își|Ï Â· ei|2.
In particular, Îșei = Își holds.
Proof. We obtain the assertions from the equality:
ÎșÏ =d2
dÏ2u(ÏÏ âČ)
âŁâŁâŁâŁÏ=0
= (Ï âČ)T(â2
Ο u(0))
Ï âČ.
The following lemma will be used in proving Theorem 2.10.
Lemma 1.17. Under the conditions of Definition 1.15, let y â C1(I, Î) withy(0) = 0 and yâČ(0) = ei, where I is an open interval with 0 â I. Then, we have
d
dsΜ(y(s))
âŁâŁâŁâŁs=0
= âÎșiei.
Proof. We denote by Μm(x) the m-th component of Μ(x). At x = (Ο, u(Ο))T,we have
Μ(x) = Μm(x)
(ââΟu(Ο)
1
).
We note that
d
dsÎœm(y(s))
âŁâŁâŁâŁs=0
= Μ(0) · d
dsΜ(y(s))
âŁâŁâŁâŁs=0
=1
2
d
ds|Μ(y(s))|2
âŁâŁâŁâŁs=0
= 0.
We define Ο(s) â Rmâ1 for small s such that y(s) = (Ο(s), u(Ο(s)))T. Since
dΟds (0) =
eâČi, we obtain
d
dsΜ(y(s))
âŁâŁâŁâŁs=0
=d
ds
(Îœm(y(s))
(ââΟu(Ο(s))
1
))âŁâŁâŁâŁs=0
= Îœm(0)
â d
dsâΟu(Ο(s))
âŁâŁâŁâŁs=0
0
= â( â2
Ο u(0)eâČi
0
)
= âÎșiei.
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CHAPTER 2
Differential calculus on hypersurfaces
In this chapter, we introduce some differential operators on hypersurfaces withtheir basic formulas and define the Weingarten map, which will play an essentialrole in our study of principal curvatures on the hypersurfaces.
Throughout this chapter, we suppose that Î is an oriented Ck-class hypersur-face in R
m with k â„ 1 and m â„ 2, and that O is an open set of Rm with Î â O.
1. Differential operators on Î
We derive several reduction formulas from the differential operators in Rm to
the ones on hypersurfaces in this section.
Lemma 2.1. If f â C1(O) and f |Î = 0, then âf(x) â Tx(Î)â„.
Proof. We assume that Î is locally parametrized as x = Ï(Ο) â Î â Rm
by Ο â Rmâ1. Since Tx(Î) = ăâÏ
âΟ1(Ο), · · · ,
âÏâΟmâ1
(Ο)ă for x = Ï(Ο), the assertion
follows from
0 =â
âΟif(Ï(Ο)) = (âf(x))T
âÏ
âΟi(Ο) (i = 1, · · · , m â 1).
Definition 2.2 (Gradient on Î). For f â C1(Î), we define
âÎf(x) := Î x âf(x) (x â Î),
where Î x := (I â Îœ(x)Îœ(x)T) is the orthogonal projection from Rm to Tx(Î),
and f â C1(O) is arbitrary C1-extension of f to an open neighborhood of Î with
f |Î = f . From Lemma 2.1, âÎf does not depend on the choice of f .
Definition 2.3 (Divergence on Î). For h â C1(Î, Rm), we define
divÎh := trâÎhT.
Definition 2.4 (LaplaceâBeltrami operator). Let k â„ 2. For f â C2(Î), wedefine
âÎf := divÎâÎf.
This âÎ is called the LaplaceâBeltrami operator on Î.
Differential rules for these differential operators on Î are collected in the fol-lowing three propositions.
Proposition 2.5. We suppose that f â Ck(Î) and h â Ck(Î, Rm).
(i) âÎf â Ckâ1(Î, Rm).
53
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54 2. DIFFERENTIAL CALCULUS ON HYPERSURFACES
(ii) divÎh â Ckâ1(Î).(iii) âÎf â Ckâ2(Î, R
m) (if k â„ 2).
Proposition 2.6 (Product rules). Suppose that f, g â C1(Î) and that h âC1(Î, R
m). Then we have the following formulas on Î.
(i) âÎ(fg) = (âÎf)g + (âÎg)f .(ii) divÎ(fh) = f divÎh + âÎf · h.(iii) âÎ(fg) = (âÎf) g + 2(âÎf) · (âÎg) + f (âÎg)
(if k â„ 2 and f, g â C2(Î)).
Proposition 2.7 (Chain rules). Suppose that f â C1(Î) and g â C1(R), andthat y â C1(R, R
m) with y(s) â Î for s â R. Then we have the following formulason Î.
(i)d
ds(f y(s)) = âT
Îf(y(s)) yâČ(s) (s â R).
(ii) âÎ(g f)(x) = (âÎf(x)) (gâČ f(x))) (x â Î).
Exercise 2.8. Prove the above three propositions.
The reduction formula for divÎ is given as follows.
Proposition 2.9. For h â C1(O, Rm), we have
divÎh = divh â Îœ · âh
âÎœon Î.
Proof. From the definition of divÎ, we obtain
divÎh = tr(I â ΜΜT)âhT
= divh â tr
Μ
âhT
âÎœ
= divh â Îœ · âh
âÎœ,
on Î.
2. Weingarten map and principal curvatures
We assume that k â„ 2 in this section.
Theorem 2.10. We define W â Ckâ2(Î, RmĂm) by
W (x) := ââT
ÎÎœ(x) x â Î.
Then, W (x) is symmetric and
W (x)ei = Îșiei (i = 1, · · · , m â 1)
W (x)Μ(x) = 0
holds, where Își and ei are the principal curvatures and the corresponding principaldirections at x â Î with ei · ej = ÎŽij.
Proof. We fix x â Î. The equality W (x)Îœ(x) = 0 is clear from the definitionof W . For each principal direction ei, there exists y â C1(I, Î) such that y(0) = x
and yâČ(0) = ei, where I is an open interval with 0 â I. Then, from Lemma 1.17,we have
Îșiei = â d
ds[Îœ(y(s))]|s=0 = â
(âT
ÎÎœ(y(0)))yâČ(0) = W (x)ei. (2.1)
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2. WEINGARTEN MAP AND PRINCIPAL CURVATURES 55
Since the eigenvectors of matrix W (x) consists of the orthonormal basis e1, · · · ,emâ1, Îœ(x), it follows that the matrix W (x) is symmetric.
Definition 2.11 (Weingarten map). We consider the linear mapping corre-sponding to the matrix W (x)
W (x) : Rm ââ Tx(Î) â R
m
â â
p 7ââ W (x)p =
mâ1â
i=1
Își(p · ei)ei
which does not depend on the choice of the orthogonal coordinate system. Wecall it the (extended) Weingarten map at x â Î. If we restrict the map W (x) tothe tangent space Tx(Î), this is usually called the Weingarten map or the secondfundamental tensor.
The following corollary is clear from the fact that Îș1(x), · · · , Îșmâ1(x), 0 arethe eigenvalues of W (x).
Corollary 2.12.
Îș(x) = tr W (x) = âdivÎÎœ(x) (x â Î),
Îșg(x) = det (W (x) + Îœ(x)Îœ(x)T) (x â Î).
The following proposition is useful to calculate the mean curvature.
Proposition 2.13. If an extension Μ of Μ satisfies
Îœ â C1(O, Rm), |Îœ(x)| = 1 (x â O), Îœ|Î = Îœ,
then Îș = âdivÎœ holds on Î.
Proof. We fix x â Î and Îœ = Îœ(x). For Ï â R, if |Ï| << 1, then
|Îœ(x â ÏÎœ)|2 = 1.
Differentiating this equality by Ï at Ï = 0, we obtain
0 =d
dÏ|Îœ(x â ÏÎœ)|2
âŁâŁâŁâŁÏ=0
= 2Îœ(x)TâTÎœ(x)(âÎœ)
= â2ÎœT
âÎœ
âÎœ(x).
Hence, from Proposition 2.9, we have
divÎœ(x) = divÎÎœ(x) + Îœ · âÎœ
âÎœ(x) = divÎÎœ(x) = âÎș(x).
Example 2.14. Under the conditions of Exercise 1.7, if k ℠2 and Μ is in thedirection from the domain w > c to w < c, then the mean curvature of the levelset Πis given as
Îș(x) = div
( âw(x)
|âw(x)|
)(x â Î = x â O; w(x) = c).
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56 2. DIFFERENTIAL CALCULUS ON HYPERSURFACES
Exercise 2.15. If Î is given by a graph η = u(Ο) with u â C2(Rmâ1) and mthcomponent of Îœ is positive, i.e., Îœm > 0, then prove that
Îș(x) = divΟ
(âΟ u(Ο)â
1 + |âΟ u(Ο)|2
)for x =
(Ο
u(Ο)
)â Î.
Proposition 2.16. For f â C2(O), we have
âÎf = âf + Îșâf
âÎœâ â2f
âÎœ2on Î.
Proof. At a point x â Î, we have
âf = div(âf) = divÎ(âf) + ÎœT(â2f)Îœ
= divÎ
(âÎf + Îœ
âf
âÎœ
)+
â2f
âÎœ2
= âÎf + (divÎÎœ)âf
âÎœ+ ÎœTâÎ
(âf
âÎœ
)+
â2f
âÎœ2
= âÎf â Îșâf
âÎœ+
â2f
âÎœ2.
The next theorem is an extension of the Frenetâs formula for plane curves toR
m.
Theorem 2.17. Let Îł be the identity map on Î, i.e., Îł(x) := x for x â Î.Then we have the following formulas:
âÎÎłT(x) = I â Îœ(x)ÎœT(x) = Î x(Î) (x â Î),
âÎÎł(x) = Îș(x)Îœ(x) (x â Î).
Proof. We extend γ to whole Rm as γ(x) = (γ1(x), · · · , γm(x))T := x. Then
we have
âÎÎłT = (I â ΜΜT)âÎłT = (I â ΜΜT) = I â (ΜΜ1, · · · , ΜΜm),
âÎÎłT = â(divÎ(ΜΜ1), · · · , divÎ(ΜΜm))
= â(divÎÎœ)ÎœT â ÎœT(âÎÎœT) = â(divÎÎœ)ÎœT = ÎșÎœT.
One of the most important tools in our analysis is the following GaussâGreenformula on Î.
Theorem 2.18 (GaussâGreen formula on Î). Let h â C1(Î, Rm) and f âC1(Î). We assume that supp(hf) is compact in Î. 1 Then we have
â«
Î
h · âÎfdHmâ1 = ââ«
Î
(divÎh + ÎșÎœ · h)fdHmâ1.
1If Î is compact, this condition is always satisfied.
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2. WEINGARTEN MAP AND PRINCIPAL CURVATURES 57
A proof of this theorem will be given in Section 1.
Corollary 2.19. For f â C1(Î) and g â C2(Î) with compact supp(fg), wehave â«
Î
âÎf · âÎg dHmâ1 = ââ«
Î
f âÎg dHmâ1.
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CHAPTER 3
Signed distance function
We study the signed distance function for a hypersurface in this chapter. Inthe first section, we show the Lipschitz property of the signed distance function fora general closed set. In the second section, we give various differential formulas forsmooth hypersurfaces. The signed distance function has many good properties andit is a useful mathematical tool in mathematical and numerical analysis for freeboundary problems (see [12, 14, 15, 16] etc.).
1. Signed distance function in general
We begin from the situation without any regularity assumption on Î.We suppose that there are open subsets Ω+ and Ω+ of R
m with Ω+ â©Î©â = â ,and we define a closed set Î := R
m \ (Ω+ âȘ Ωâ). Then we define
d(x) =
dist(x, Î) (x â Ω+),0 (x â Î),âdist(x, Î) (x â Ωâ),
where
dist(x, Î) := infyâÎ
|x â y|.
This d(x) is called the signed distance function for Î.
Exercise 3.1. Prove that, for a closed set Î â Rm and x â R
m, there existsx â Î such that
|x â x| = minyâÎ
|x â y| = dist(x, Î).
Theorem 3.2. The signed distance function is Lipschitz continuous with Lip-schitz constant 1, i.e.,
|d(x) â d(y)| †|x â y| (x, y â Rm).
Proof. We consider the following two cases depending on the value of d(x)d(y).If d(x)d(y) â„ 0, without loss of generality, we can assume that x and y are bothin Ω+ âȘ Î. In case that they are both in Ωâ âȘ Î, we can prove the assertion in thesame way. Let y â Î satisfy d(y) = |y â y|. Since d(x) = dist(x, Î), we have
d(x) â d(y) = d(x) â |y â y| †|x â y| â |y â y| †|x â y|.The other inequality d(y) â d(x) †|xâ y| is obtained in the same way. Hence weobtain |d(x) â d(y)| †|x â y|.
59
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60 3. SIGNED DISTANCE FUNCTION
If d(x)d(y) < 0, without loss of generality, we can assume that x â Ω+ andy â Ωâ. Since Ω+ and Ωâ are open sets, there exists Ξ â (0, 1) such that z :=(1 â Ξ)x + Ξy â Î. Then we have
|d(x) â d(y)| = d(x) â d(y) = dist(x, Î) + dist(y, Î) †|x â z|+ |z â y| = |xâ y|.
Corollary 3.3. The signed distance function d is differentiable a.e. in Rm,
and its gradient âd coincides with the distribution sense gradient of d. Moreover,d â W 1,â(Rm) and it satisfies
|âd(x)| †1 a.e. in Rm.
Proof. This corollary follows from the Rademacherâs Theorem (see [5] etc.).
Let d be an another signed distance function for a closed set Î = Rm\(Ω+âȘΩâ).
We assume that Î and Î are both compact (bounded and closed). The Hausdorff
distance between two compact sets Î and Î is defined by
distH(Î, Î) := max
(maxxâÎ
dist(x, Î), maxxâÎ
dist(x, Î)
).
Theorem 3.4. Under the assumptions, we have distH(Î, Î) †âdâ dâLâ(Rm).
Proof. For x â Î, since d(x) = 0, we obtain
dist(x, Î) = |d(x)| = |d(x) â d(x)| †âd â dâLâ(Rm),
and
maxxâÎ
dist(x, Î) †âd â dâLâ(Rm).
In the same way, we obtain the other inequality.
2. Signed distance function for hypersurface
In this section, we assume that Î is an oriented mâ1 dimensional hypersurfaceembedded in R
m (m â„ 2) of Ck-class (k â N, k â„ 2). Then, Îœ â Ckâ1(Î, Rm)follows. We define
X(y, r) := y â rÎœ(y) â Rm (y â Î, r â R), (3.1)
N Δ(Î0) := X(y, r); y â Î0, |r| < Δ (Î0 â Î, Δ > 0).
N Δ±(Î0) := X(y, r); y â Î0, 0 < ±r < Δ (Î0 â Î, Δ > 0).
From Theorem 2.10 and 2.17, we have
âT
ÎX(y, r) = I â Îœ(y)ÎœT(y) + rW (y) (y â Î, r â R). (3.2)
Lemma 3.5. We fix a point xâ â Î and denote by Își (i = 1, · · · , m â 1) theprincipal curvatures of Î at xâ. We define
Îș+ := max1â€iâ€mâ1
Își, Îșâ := min1â€iâ€mâ1
Își,
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2. SIGNED DISTANCE FUNCTION FOR HYPERSURFACE 61
Δââ :=
â 1
Îș+if Îș+ > 0
ââ if Îș+ †0,Δâ+ :=
â 1
Îșâif Îșâ < 0
+â if Îșâ â„ 0.
For any Δââ <â Δâ <â Δ+ < Δâ+, there exists a bounded domain Oâ â Rm with
xâ â Oâ such that X is Ckâ1-diffeomorphism from (Îâ©Oâ)Ă [Δâ, Δ+] to its imagein R
m.
Proof. For r â [Δâ, Δ+], 1 + rÎși > 0 (i = 1, · · · , m â 1) holds. It follows thatrank(I + rW (xâ)) = m â 1 and I + rW (xâ) is bijective from Txâ(Î) onto itself.
We consider a local embedding coordinate (O, U , Ï) â Ck(Î) with xâ â O,and we define Οâ := Ïâ1(xâ) â U â R
mâ1. Let us define a new coordinate
Ο = (Ο, r)T = (Ο1, · · · , Οmâ1, r)T â U Ă R â R
m. We define
Ί(Ο, r) := X(Ï(Ο), r) ((Ο, r) â U Ă R). (3.3)
Then its Jacobian matrix is given by
âT
ΟΊ(Ο) =
(âT
Ο Ί,âΊ
âr
)=(âT
ÎX(y, r)âT
Ο Ï(Ο), âÎœ(y)),
where y = Ï(Ο) for Ο â U . From (3.2), we obtain
âT
ÎX(y, r)âT
Ο Ï(Ο) =(I â Îœ(y)ÎœT(y) + rW (y)
)âT
Ο Ï(Ο)
= (I + rW (y))âT
Ο Ï(Ο),
and
âT
ΟΊ(Ο) =
((I + rW (y))âT
Ο Ï(Ο), âÎœ(y)). (3.4)
Since ââΟi
Ï(Οâ)i=1,··· ,mâ1 is a basis of Txâ(Î), if r â [Δâ, Δ+], so is the set of
column vectors of (I + rW (xâ))âT
Ο Ï(Οâ), too. Hence, from
âT
ΟΊ(Οâ, Οm) =
((I + ΟmW (xâ))âT
Ο Ï(Οâ), âÎœ(xâ)),
it follows that detâT
ΟΊ(Οâ, Οm) 6= 0.
We choose a bounded domain Uâ â U such that Οâ â Uâ and that Ί becomes aCkâ1-diffeomorphism from UâĂ[Δâ, Δ+] onto its image. There also exists a boundeddomain Oâ â R
m such that Î â©Oâ = Ï(Uâ). Then, it satisfies the assertion.
Proposition 3.6. We suppose that Î0 is a bounded hypersurface and it satisfiesÎ0 â Î. Then there exists Δ0 > 0 such that X is Ckâ1-diffeomorphism fromÎ0 Ă [âΔ0, Δ0] onto N Δ0(Î0).
Exercise 3.7. Prove Proposition 3.6.
In the following argument, for simple description, we suppose the followingcondition:
âΔ > 0 s.t. X is a Ckâ1-diffeomorphism from Î Ă (âΔ, Δ) onto N Δ(Î). (3.5)
Then, we can define
(Îł(x), d(x)) := Xâ1(x) (x â N Δ(Î)). (3.6)
We remark that Îł(x) â Î stands for the perpendicular foot on Î from x â N Δ(Î).The function d(x) is none other than the signed distance function for Î.
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62 3. SIGNED DISTANCE FUNCTION
For simplicity, we often write the perpendicular foot from x â N Δ(Î) as x :=Îł(x) â Î. Moreover, for a function f defined on Î, we define an extension of f by
f(x) := f(x) (x â N Δ(Î)).
We remark thatx = x â d(x)Îœ(x) (x â N Δ(Î)). (3.7)
We have the following theorem.
Theorem 3.8. Under the above conditions, Îł â Ckâ1(N Δ(Î), Rm) and d âCk(N Δ(Î)) hold. Furthermore they satisfy the following formulas for x â N Δ(Î),
âd(x) = âÎœ(x), Îł(x) = x â d(x)âd(x),
â2d(x) =(I + d(x)W
)â1W , âd(x) =
mâ1â
i=1
Își(x)
1 + d(x)Își(x),
âTÎł(x) =(I + d(x)W
)â1Î x = Î x
(I + d(x)W
)â1, (3.8)
where x := Îł(x) and W := W (x).
Proof. From the inverse function theorem, it follows that Îł and d belong toCkâ1-class.
We differentiate the equation:
r = d(X(y, r)), (3.9)
by y â Î, then we have
0T = âTd(X(y, r))âT
ÎX(y, r). (3.10)
The principal direction at x â Î corresponding to Își(x) is denoted by ei for i =1, · · · , m â 1. Substituting y = x and r = d(x) to (3.10) and multiplying it to ei,we obtain
0 = âTd(x)âT
ÎX(x, d(x))ei
= âTd(x)(I â Îœ(x)ÎœT(x) + d(x)W )ei = âTd(x)(1 + d(x)Își(x))ei.
Since 1+ d(x)Își(x) > 0, we have âd(x) · ei = 0 for i = 1, · · · , mâ 1. On the otherhand, differentiating (3.9) by r and substituting y = x and r = d(x), we obtain
1 = âTd(X(y, r))âX
âr(y, r)|y=x,r=d(x) = ââTd(x)Îœ(x).
Thus we obtain âd(x) = âÎœ(x) = âÎœ(Îł(x)). Since Îœ and Îł both belong to Ckâ1-class, âd â Ckâ1(N Δ(Î), Rm) and hence d â Ck(N Δ(Î)) follows. The equationÎł(x) = x â d(x)âd(x) also follows from (3.7).
Applying the gradient operator âT to (3.7), we have
I = âTÎł(x) â Îœ(x)âTd(x) â d(x)âT
ÎÎœ(x)âTÎł(x)
= (I + d(x)W )âTÎł(x) + Îœ(x)ÎœT(x).
Since
det(I + d(x)W ) =
mâ1â
i=1
(1 + d(x)Își(x)) > 0,
we obtainâTÎł(x) =
(I + d(x)W
)â1Î x.
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2. SIGNED DISTANCE FUNCTION FOR HYPERSURFACE 63
We remark that (I + d(x)W )â1 and Î x are commutative because Î xW = W =WÎ x. Hence finally, we arrive at
â2d(x) = âT(âd(x)) = ââT(Îœ(Îł(x))) = ââT
ÎÎœ(x)âTÎł(x)
= W(I + d(x)W
)â1Î x =
(I + d(x)W
)â1WÎ x
=(I + d(x)W
)â1W .
The Laplacian âd(x) = trâ2d(x) is easily calculated from this formula.
Theorem 3.9. If f â Ckâ1(Î) then f â Ckâ1(N Δ(Î)) and
âf(x) = (I + d(x)W (x))â1âÎf(x) (x â N Δ(Î)),
in particular,âf(x) = âÎf(x) (x â Î). (3.11)
Moreover, if k â„ 3, then the equation
â2f(x) = âÎ(âT
Îf)(x) + Îœ(x)âT
Îf(x)W (x) (x â Î),
holds. In particular,âf(x) = âÎf(x) (x â Î).
Proof. Since f = f Îł, f â Ckâ1(N Δ(Î)) is clear. From Theorem 3.8, wehave
âf(x) = âÎłT(x)âÎf(x)
= (I + d(x)W (x))â1Î xâÎf(x)
= (I + d(x)W (x))â1âÎf(x).
The latter part is shown as follows. To the equation
âTÎf(x) =
(I + d(x)W (x))âf (x)
T
= âTf(x)(I + d(x)W (x)) (x â N Δ(Î)),
we multiply â at x â Î, then, from (3.11), we obtain
âÎ(âT
Îf)(x) = â(âT
Îf)(x)
= â(âTf(x) + d(x)âTf(x) W (x))
= â2f(x) + âd(x)(âTf(x) W (x)
)+ d(x)â
(âTf(x) W (x))
)
= â2f(x) â Îœ(x)âT
Îf(x)W (x),
and
âf(x) = trâ2f(x) = tr(âÎ(âT
Îf)(x) + Îœ(x) (W (x)âÎf(x))T)
= divÎâÎf(x) + Îœ(x) · (W (x)âÎf(x)) = âÎf(x).
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CHAPTER 4
Curvilinear coordinates
In this chapter, we study various differential and integral formulas in a curvi-linear coordinate of N Δ(Î). A proof of the GaussâGreenâs theorem on Î is alsogiven.
1. Differential and integral formulas in N Δ(Î)
Let Î be a compact Ck-class hypersurface in Rm with k â„ 2. We suppose the
condition (3.5) in this section.For r â (âΔ, Δ), we define
Îr := x â N Δ(Î); d(x) = r.Then, we have
N Δ(Î) =â
|r|<Δ
Îr .
By dr, Îœr, Îșr and Wr , we denote the signed distance function, the unit normalvector, the mean curvature and the Weingarten map, respectively, for Îr. Then wehave the following proposition.
Proposition 4.1. Under the above conditions, for r â (âΔ, Δ), Îr is a Ck-classhypersurface and
dr(x) = d(x) â r (x â N Δ(Î))
Îœr(x) = Îœ(x) (x â Îr)
Îșr(x) =mâ1â
i=1
Își(x)
1 + rÎși(x)(x â Îr)
Wr(x) = (I + rW )â1W (x â Îr)
Proof. The equality dr(x) = d(x)â r is clear, and the other assertions followfrom Theorem 3.8.
Theorem 4.2. For f â C0(Îr), we haveâ«
Îr
f(x) dHmâ1x =
â«
Î
f(X(y, r)) det(I + rW (y)) dHmâ1y . (4.1)
Proof. We fix a local coordinate (O, U , Ï) â Ck(Î). First, we assume that
supp(f X(·, r)) â (Î â© O). (4.2)
65
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66 4. CURVILINEAR COORDINATES
Then, for Ί defined by (3.3), we have
âT
Ο Ί(Ο, r) = (I + rW (y))âT
Ο Ï(Ο).
Similar to Section 3, we define
Gr(Ο) :=(âΟΊ
T(Ο, r)) (
âT
Ο Ί(Ο, r))
=(âΟÏ
T(Ο))(I + rW (y))2
(âT
Ο Ï(Ο)),
gr(Ο) := detGr(Ο) = (det (I + rW (y)))2detG(Ο),
âgr(Ο) = det (I + rW (y))
âg(Ο).
Hence, the surface integral on Îr is given by (4.1) under the condition (4.2). For
general f , applying a partition of unity, we decompose f =âN
j=1 fj , where each fj
satisfies the condition (4.2).
Theorem 4.3. For x â N Δ(Î), we define
y := Îł(x) â Î, r := d(x) â (âΔ, Δ).
Then, we have
dx = det(I + rW (y))dHmâ1y dr in N Δ(Î).
i.e.
â«
N Δ(Î)
f(x)dx =
⫠Δ
âΔ
â«
Î
f(X(y, r)) det(I + rW (y)) dHmâ1y dr,
for f â C0(N Δ(Î)). In other words, we haveâ«
N Δ(Î)
f(x)dx =
⫠Δ
âΔ
â«
Îr
f(x) dHmâ1x dr .
Proof. From (3.4) and Corollary 1.12, the Jacobian becomes
detâT
ΟΊ(Ο, r) = â det (I + rW (y))
âg(Ο).
This gives the formulas of the theorem.
From this theorem, we immediately have the following corollary.
Corollary 4.4. For f â C0(N Δ(Î)),
limrâ0
1
2r
â«
N r(Î)
f(x) dx =
â«
Î
f(x) dHmâ1x ,
holds.
We shall prove the GaussâGreen theorem on Î by using Corollary 4.4.
Proof of Theorem 2.18. We decompose the vector field h = h0 + h1 asfollows.
h0(x) := Î xh(x), h1(x) := h(x) â h0(x) (x â Î).
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1. DIFFERENTIAL AND INTEGRAL FORMULAS IN N Δ(Î) 67
We have â«
Î
h · âÎf dHmâ1 =
â«
Î
h0 · âf dHmâ1
= limrâ0
1
2r
â«
N r(Î)
h0 · âf dx
= â limrâ0
1
2r
â«
N r(Î)
(div h0) f dx
= ââ«
Î
(div h0) f dHmâ1
= ââ«
Î
(divÎh0) f dHmâ1
The term divÎh0 is computed as follows:
divÎh0 = divÎ
((I â ΜΜT)h
)
= divÎh â (divÎÎœ)ÎœTh â ÎœTâÎ(ÎœTh)
= divÎh â ÎșÎœTh.
Substituting this to the above equation, we obtain the GaussâGreenâs theoremon Î.
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CHAPTER 5
Moving hypersurfaces
We extend our formulation to moving hypersurfaces, i.e., time dependent hy-persurfaces. In the first section, we introduce a new geometric quantity the normalvelocity and a new derivative the normal time derivative. We give several formu-las for the signed distance function in Section 2, and for the normal derivatives ofseveral geometric quantities in Section 3.
1. Normal time derivatives
Let us consider a time dependent hypersurface Î(t) (t â I), where I is aninterval. We assume that each Î(t) is a nonempty oriented (m â 1)-dimensionalhypersurface in R
m. We define
M =â
tâI
(Î(t) Ă t) â Rm+1, (5.1)
and we call M a moving hypersurface in Rm. The geometric quantities of Î(t) such
as Îœ, Îș and W etc. are denoted by Îœ(x, t), Îș(x, t) and W (x, t) etc. for x â Î(t).In this note, we assume that the moving hypersurface M has the following
regularity.
Definition 5.1 (C2,1-class moving hypersurface). An oriented moving hyper-surface M of the form (5.1) is called of C2,1-class, if and only if M is a C1-classm-dimensional hypersurface in R
m Ă R and Îœ â C1(M, Rm).
Proposition 5.2. Let M be an oriented moving hypersurface of the form (5.1).Then, M belongs to C2,1-class if and only if M is locally represented by a graphη = u(Ο, t) (Ο â U â R
mâ1, I0 â I) in a suitable Cartesian coordinate (Ο, η) ofR
m withu â C1(U Ă I0), âΟu â C1(U Ă I0, R
mâ1). (5.2)
More precisely, for any (x, t) â M, there exist a suitable Cartesian coordinate (Ο, η)of R
m, and a bounded domain U â Rmâ1, an open subinterval I0 â I, a function
u satisfying (5.2), and a bounded domain Q â Rm Ă R, such that (x, t) â O and
x â Î(t); (x, t) â Q = (Ο, u(Ο, t)); Ο â U (t â I0).
Exercise 5.3. Prove Proposition 5.2.
In the following arguments, we always assume that M is of C2,1 class. We cally a C1-trajectory on M if y â C1(I0, R
m) with y(t) â Î(t) for t â I0, where I0 isa subinterval of I.
Definition 5.4 (Normal velocity). We define a normal velocity at (x, t) â Mby v(x, t) := yâČ(t) · Îœ(x, t), where y is a C1-trajectory on M through (x, t).
69
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70 5. MOVING HYPERSURFACES
Theorem 5.5. The normal velocity v â C0(M) is well-defined. Moreover, itsatisfies v(·, t) â C1(Î(t)) and âÎv â C0(M, Rm).
Proof. Let y(t) be a C1-trajectory on M. Without loss of generality, wesuppose that y(t) â M â© Q for t â I0 under the condition of Proposition 5.2.Then, there exists ζ â C1(I0,U) such that y(t) = (ζ(t), u(ζ(t), t))T for t â I0.From (1.3) and the equality
yâČ(t) =
(ζâČ(t)
âΟu(ζ(t), t) · ζâČ(t) + ut(ζ(t), t)
), (5.3)
we obtain
v(y(t), t) = Îœ(x, t) · yâČ(t) =ut(ζ(t), t)â
1 + |âΟu(ζ(t), t)|2.
For fixed t â I0 and x = (Ο, η)T = y(t) â Î(t), since ζ(t) = Ο â U , it follows that
v(x, t) =ut(Ο, t)â
1 + |âΟu(Ο, t)|2,
and the right hand side does not depend on the choice of the C1-trajectory. More-over, the assertions on the regularity follow from this expression.
Proposition 5.6. Under the condition of Proposition 5.2, we have
v(x, t) =ut(Ο, t)â
1 + |âΟu(Ο, t)|2(x = (Ο, u(Ο, t))T â Î(t)), (5.4)
for Ο â U and t â I0.
Definition 5.7. A C1-trajectory y(t) (t â I0) on M is called a normal tra-jectory on M, if and only if yâČ(t) â Ty(t)(Î(t))â„ for t â I0.
We remark that yâČ(t) = v(y(t), t)Îœ(y(t), t) holds for any normal trajectoryy(t).
Proposition 5.8. For (x0, t0) â M, there uniquely exists a normal trajectoryon M through (x0, t0).
Proof. Under the conditions of the proof of Theorem 5.5, y(t) is a normaltrajectory on M through (x0, t0) if and only if
yâČ(t) = v(y(t), t)Îœ(y(t), t)
y(t0) = x0.(5.5)
Substituting (1.3), (5.3) and (5.4), we obtain
ζâČ(t) = â ut(ζ(t), t)
1 + |âΟu(ζ(t), t)|2 âΟu(ζ(t), t),
ζ(t0) = Ο0,
(5.6)
where x0 = (Ο0, u(Ο0, t0)). The system of ordinary differential equations (5.6) isequivalent to (5.5) and permits a unique solution since the right hand side of thefirst equation of (5.6) satisfies the local Lipschitz condition with respect to theζ-variable.
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1. NORMAL TIME DERIVATIVES 71
Definition 5.9 (Normal time derivative). Let f â C1(M, Rk). For (x0, t0) âM and the normal trajectory y(t) on M through (x0, t0), we define
Dtf(x0, t0) :=d
dt[f(y(t), t)]
âŁâŁâŁâŁt=t0
,
and Dtf is called the normal time derivative of f on M.
We denote the identity map on M by Îł(x, t) := x ((x, t) â M)). Then wehave
DtÎł = vÎœ on M.
We also remark that even if a function f = f(x) does not depend on the timevariable t, the normal time derivative of f may not be zero. In general, for f(x, t) =f(x), we have
Dtf(x, t) =d
dtf(y(t)) = v(x, t) (âTf(x))Îœ(x, t) (x = y(t) â Î(t)).
Proposition 5.10. Let Q be an open neighbourhood of M in Rm+1. For
f â C1(Q),
Dtf(x, t) = ft(x, t) + v(x, t)âf
âÎœ(x, t) ((x, t) â M),
holds, where âfâÎœ denotes the partial derivative with respect to x in the direction
Μ(x, t).
Proof. Let y(t) be a normal trajectory on M. Then, at x = y(t) â Î(t), wehave
Dtf(x, t) =d
dtf(y(t), t) = âTf(x, t)yâČ(t) + ft(x, t)
= v(x, t)âTf(x, t)Îœ(x, t) + ft(x, t).
Proposition 5.11. For f â C1(M) and a C1-trajectory y(t) on M, we obtain
d
dtf(y(t), t) = Dtf(x, t) + âT
Î(t)f(x, t)yâČ(t) (x = y(t) â Î(t)).
Proof. Let f â C1(Q) be an extension of f . Then, for x = y(t) â Î(t), weobtain
d
dtf(y(t), t) =
d
dtf(y(t), t)
= âTf(x, t)yâČ(t) + ft(x, t)
=
(âÎ(t)f(x, t) + Îœ(x, t)
âf
âÎœ(x, t)
)T
yâČ(t) + ft(x, t)
= âT
Î(t)f(x, t)yâČ(t) +
(âf
âÎœ(x, t)ÎœT(x, t)yâČ(t) + ft(x, t)
)
= âT
Î(t)f(x, t)yâČ(t) + Dtf(x, t).
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72 5. MOVING HYPERSURFACES
2. Signed distance function for moving hypersurface
Let M be a C2,1-class moving hypersurface of the form (5.1). For simplicity, weassume that there exists Δ > 0 and Î(t) satisfies the condition (3.5) for each t â I.The signed distance function and the perpendicular foot and other quantities forÎ(t) are denoted by d(x, t) and Îł(x, t), etc. We define
N Δ(M) := (x, t) â Rm Ă R; x â N Δ(Î(t)), t â I.
Lemma 5.12. Under the above condition, Îł â C1(N Δ(M), Rm) and d â C1(N Δ(M))hold.
Exercise 5.13. Using the implicit function theorem, prove Lemma 5.12.
Theorem 5.14. Under the above condition, âd â C1(N Δ(M), Rm) and Îł âC1(N Δ(M), Rm) hold, and, for (x, t) â N Δ(M), we have
dt(x, t) = v(x, t),
âdt(x, t) = (I + d(x, t)W (x, t))â1âÎ(t)v(x, t),
Îłt(x, t) = v(x, t)Îœ(x, t) â d(x, t)(I + d(x, t)W (x, t))â1âÎ(t)v(x, t),
where x = Îł(x, t) â Î(t). Furthermore, if W â C1(M, RmĂm), then â2d âC1(N Δ(M), RmĂm) holds and, for (x, t) â M, we obtain
â2dt(x, t) = âÎ(t)(âT
Î(t)v)(x, t) + Îœ(x, t)âT
Î(t)v(x, t)W (x, t),
âdt(x, t) = âÎ(t)v(x, t).
Proof. Since
âd(x, t) = âÎœ(Îł(x, t), t) ((x, t) â N Δ(M)),
from Lemma 5.12, we obtain âd â C1(N Δ(M), Rm).Differentiating the equality
x = Îł(x, t) + d(x, t)âd(x, t) ((x, t) â N Δ(M)),
by t, then we obtain
0 = Îłt(x, t) + dt(x, t)âd(x, t) + d(x, t)âdt(x, t). (5.7)
Taking an inner product with âd(x, t), we get
dt(x, t) = dt(x, t)|âd(x, t)|2= ââd(x, t) · (Îłt(x, t) + d(x, t)âdt(x, t))
= Îœ(Îł(x, t), t) · Îłt(x, t) â d(x, t)âd(x, t) · âdt(x, t)
= v(Îł(x, t), t) â d(x, t)1
2
â
ât|âd(x, t)|2
= v(Îł(x, t), t).
Hence, dt = v holds and the expression of âdt follows from Theorem 3.9. The timederivative of Îł is obtained from (5.7).
In the case W â C1(M, RmĂm), from the expression
â2d(x, t) =(I + d(x, t)W
)â1W ((x, t) â N Δ(M)),
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3. TIME DERIVATIVES OF GEOMETRIC QUANTITIES 73
where W = W (Îł(x, t), t), â2d â C1(N Δ(M), RmĂm) holds. The last two equalitiesalso follow from dt = v and Theorem 3.9.
3. Time derivatives of geometric quantities
The normal time derivatives of the geometric quantities of a moving hypersur-face M are given as follows.
Theorem 5.15. If M belongs to C2,1-class,
DtÎœ = ââÎ(t)v,
holds on M. Furthermore, if W â C1(M, RmĂm), then the following equalitieshold on M:
DtW = vW 2 + âÎ(t)(âT
Î(t)v) + Îœ(âT
Î(t)v)
W,
DtÎș = v
mâ1â
i=1
Îș2i + âÎ(t)v,
DtÎșg = vÎșÎșg + adj(W + ΜΜT) : âÎ(t)(âT
Î(t)v).
Proof. Let y(t) be a normal trajectory on M. Then, at x = y(t) â Î(t), wehave
DtΜ(x, t) =d
dt(Îœ(y(t), t)) = â d
dt(âd(y(t), t))
= ââ2d(x, t)yâČ(t) ââdt(x, t)
= âW (x, t)Îœ(x, t)v(x, t) ââv(x, t)
= ââv(x, t) = ââÎ(t)v(x, t).
The second equality is similarly calculated as follows.
DtW (x, t) =d
dt
(â2d(y(t), t)
)= v(x, t)
â
âÎœâ2d(x, t) + â2dt(x, t).
From the equality
â
âÎœâ2d(x, t) =
d
dr
[â2d(x + rÎœ(x, t), t)
]âŁâŁâŁâŁr=0
=d
dr
[(I â rW (x, t))â1W (x, t)
]âŁâŁâŁâŁr=0
= W (x, t)2,
and Theorem 5.14, we obtain the expression of DtW . The normal time derivativeof the mean curvature is obtained from
DtÎș = Dt(trW ) = tr(DtW ).
For the Gauss-Kronecker curvature Îșg, from the Jacobiâs formula, we have
DtÎșg = Dt det(W + ΜΜT) = tr(ADt(W + ΜΜT)
),
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74 5. MOVING HYPERSURFACES
where A = adj(W +ΜΜT) denotes the adjugate matrix of W +ΜΜT (see Section 1).From Proposition A.1, it satisfies
AW = Îșg(I â ΜΜT), AÎœ = ÎșgÎœ, Aei =
â
j 6=i
Îșj
ei (i = 1, · · · , m â 1).
Since
Dt(W + ΜΜT)
= DtW + (DtΜ)ΜT + Μ(DtΜ)T
= vW 2 + âÎ(t)(âT
Î(t)v) + Îœ(âT
Î(t)v)
W â (âÎ(t)v)ÎœT â Îœ âT
Î(t)v
= vW 2 + âÎ(t)(âT
Î(t)v) â (âÎ(t)v)ÎœT + Îœ(âT
Î(t)v)
(W â I),
we obtain
DtÎșg
= tr(vÎșgW + A âÎ(t)(âT
Î(t)v) â A(âÎ(t)v)ÎœT + ÎșgÎœ(âT
Î(t)v)
(W â I))
= vÎșgÎș + tr(A âÎ(t)(âT
Î(t)v))â(AâÎ(t)v
)· Îœ + ÎșgÎœ ·
((W â I)âÎ(t)v
)
= vÎșÎșg + tr(A âÎ(t)(âT
Î(t)v))
= vÎșÎșg + A : âÎ(t)(âT
Î(t)v) .
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CHAPTER 6
Variational formulas
One of the most important applications of our formulation for moving hypersur-faces is to derive various variational formulas with respect to the shape deformationof hypersurfaces. In this chapter, we prove some basic transport identities, i.e., vari-ations of volume or surface integrals in the first section. A transport identity of thesymmetric polynomials of the principal curvatures is derived in the second section.This is an interesting extension of the Gauss-Bonnet theorem for two dimensionalsurfaces to multidimensional cases.
1. Transport identities
We suppose that M is an oriented C2,1-class moving hypersurface of the form(5.1) and that Î(t) is compact for each t â I.
Theorem 6.1. If f â C1(M), then
d
dt
â«
Î(t)
f(x, t) dHmâ1 =
â«
Î(t)
(Dtf â fÎșv) dHmâ1.
In particular, we have
d
dt|Î(t)| = â
â«
Î(t)
Îșv dHmâ1. (6.1)
Proof. Under the condition of Proposition 5.2, we define
Ï(Ο, t) := (Ο, u(Ο, t))T â Rm (Ο â U , t â I0).
We first suppose the condition:
supp(f(·, t)) â Ï(U , t) (t â I0).
Then, from Corollary 1.12, we haveâ«
Î(t)
f(x, t) dHmâ1 =
â«
U
f(Ï(Ο, t), t)â
g(Ο, t)dΟ,
whereA(Ο, t) :=
(âT
Ο Ï(Ο, t), Îœ(Ï(Ο, t), t)),
g(Ο, t) := (detA(Ο, t))2.
Without loss of generality, we assume thatâ
g(Ο, t) = detA(Ο, t) > 0.
From Proposition A.3, we have
â
ât
âg(Ο, t) =
â
âtdetA(Ο, t) = tr
(A(Ο, t)â1At(Ο, t)
)âg(Ο, t).
75
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76 6. VARIATIONAL FORMULAS
The last term is computed as follows. Since x â Î(t) and Ο â U are correspondingto each other by the map x = Ï(Ο, t), we can write it conversely as Ο = Ο(x, t),where
x = Ï(Ο(x, t), t) (x â Î(t)).
It follows that, for x = Ï(Ο, t),
A(Ο, t)â1 =
( âT
Î(t)Ο(x, t)
ÎœT(x, t)
),
and
tr(A(Ο, t)â1At(Ο, t)
)
= tr
âT
Î(t)Ο
ÎœT
(âT
Ο Ït(Ο, t),â
ât[Îœ(Ï(Ο, t), t)]
)
= tr
((âT
Î(t)Ο)(âT
Ο Ït(Ο, t)) ââ 1
2âât |Îœ|2
)
= tr((âT
Î(t)Ο)(âT
Ο Ït(Ο, t)))
= tr((âÎ(t)Ο
T)(âΟÏT
t (Ο, t)))
= trâÎ
(ÏT
t (Ο(x, t), t))
= divÎ(t),x (Ït(Ο(x, t), t)) .
Hence, applying Proposition 5.11 and the GaussâGreen formula (Theorem 2.18),we obtain
d
dt
â«
Î(t)
f(x, t) dHmâ1
=d
dt
â«
U
f(Ï(Ο, t), t)â
g(Ο, t)dΟ
=
â«
U
â
ât(f(Ï(Ο, t), t))
âg + f
â
ât
âg(Ο, t)
dΟ
=
â«
U
(Dtf + (âT
Î(t)f)Ït
)âg + f divÎ(t),x(Ït(Ο(x, t), t))
âg
dΟ
=
â«
Î(t)
Dtf + (âT
Î(t)f)Ït + f divÎ(t),x(Ït(Ο(x, t), t))
dHmâ1x
=
â«
Î(t)
(Dtf â fÎșv)dHmâ1.
For more general f , we can apply the above result with a partition of unityof M.
We additionally suppose that there exist a bounded open set Ωâ(t) and aunbounded open set Ω+(t) of R
m satisfying the following conditions:
Rm \ Î(t) = Ωâ(t) âȘ Ω+(t), Ωâ(t) ⩠Ω+(t) = â , N Δ
±(Î(t)) â Ω±(t),
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1. TRANSPORT IDENTITIES 77
for sufficiently small Δ > 0. We define
Q :=â
tâI
Ωâ(t) Ă t.
Then M is represented by the domain mapping method as follows.
Proposition 6.2. Under the above conditions, for a fixed t0 â I, there ex-ist a bounded domain Ω0 â R
m with a smooth boundary Î0 = âΩ0 and Ί âC1(I0, C
1(Ω0, Rm)) with an open subinterval I0 â I including t0, such that
Ωâ(t) = Ί(Ω0, t), Î(t) = Ί(Î0, t) (t â I0),
and that Ί(·, t) is a C1-diffeomorphism from Ω0 to Ωâ(t) for each t â I0.
We remark that Ί(·, t) represents the map from y â Ω0 â Rm to x â Ωâ(t) â
Rm as x = Ί(y, t), and that the condition Ί â C1(I, C1(Ω0, R
m)) stands for that
Ί(y, t), âyΊT(y, t),âΊ
ât(y, t), âyΊT
t (y, t) =â
âtâyΊT(y, t),
are all continuous with respect to (y, t) â Ω0 Ă I0.
Exercise 6.3. Prove Proposition 6.2.
Theorem 6.4. Under the condition of Proposition 6.2, if u â C1(Q), then wehave
d
dt
â«
멉(t)
u(x, t) dx =
â«
멉(t)
ut(x, t) dx ââ«
Î(t)
u(x, t)v(x, t) dHmâ1.
In particular, we have
d
dt|Ωâ(t)| = â
â«
Î(t)
v dHmâ1.
Proof. Without loss of generality, we assume that
detâyΊT(y, t) > 0 (y â Ω0).
We define Κ(·, t) := Ί(·, t)â1 â C1(Ωâ(t), Ω0). From the Jacobiâs formula (Propo-sition A.3), we have
â
ât(detâT
y Ί(y, t)) = (detâT
y Ί) tr[(âxΚT)(âyΊT
t )]
= (detâT
y Ί) trâx
[ΊT
t (Κ(x, t), t)]
= (detâT
yΊ(y, t)) divx [Ίt(Κ(x, t), t)] .
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78 6. VARIATIONAL FORMULAS
Hence, we obtain
d
dt
â«
멉(t)
u(x, t) dx
=d
dt
â«
Ω0
u(Ί(y, t), t) detâyΊT(y, t) dy
=
â«
Ω0
(ΊT
t (y, t)âxu(x, t) + ut(x, t))(detâyΊT(y, t))
+u(x, t) (detâT
yΊ(y, t)) divx [Ίt(Κ(x, t), t)]
dy
=
â«
멉(t)
ΊT
t (y, t)âxu(x, t) + ut(x, t) + u(x, t) divx [Ίt(Κ(x, t), t)]
dx
=
â«
멉(t)
ut(x, t) dx ââ«
Î(t)
u(x, t)Ίt(Κ(x, t), t) · Îœ(x, t) dHmâ1x .
Since v(x, t) = Ίt(Κ(x, t), t) · Μ(x, t), the transport identity follows.
2. Transport identities for curvatures
We again suppose that M is an oriented C2,1-class moving hypersurface of theform (5.1) and that Î(t) is compact for each t â I.
Theorem 6.5 (Transport identity for Îșg).
d
dt
â«
Î(t)
Îșg dHmâ1 = 0. (6.2)
This theorem indicates thatâ«Î Îșg dHmâ1 is a topological invariant. In R
2, itis well known that
â«
Î
Îș dHmâ1 = #(connected components of Î) Ă 2Ï.
In the three dimensional case, Theorem 6.5 follows from the famous Gauss-Bonnettheorem for two dimensional manifolds.
If Î(t) is a solution of the Gaussian curvature flow v = Îșg (Section 5), then wehave
d
dt|Ωâ(t)| = â
â«
Î(t)
vdHmâ1 = ââ«
Î(t)
ÎșgdHmâ1,
where the last integral does not depend on t. This indicates that the solution ofthe Gaussian curvature flow possesses the constant volume speed property.
A direct approach to prove Theorem 6.5 is as follows. From Theorem 6.1 andTheorem 5.15, we have
d
dt
â«
Î(t)
Îșg dHmâ1 =
â«
Î(t)
(DtÎșg â vÎșÎșg) dHmâ1
=
â«
Î(t)
(adj(W + ΜΜT) : âÎ(t)(âT
Î(t)v))
dHmâ1.
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2. TRANSPORT IDENTITIES FOR CURVATURES 79
Using the integration by parts on Î(t), it is possible to show that the last integralvanishes, but it is lengthy and complicated. So, we choose another way to proveTheorem 6.5 in a more general form.
For r â R, we define
Ar(x, t) := I + rW (x, t) ((x, t) â M).
Then we have
detAr =mâ1â
i=1
(1 + rÎși). (6.3)
Lemma 6.6.
d
dt|Îr(t)| = â
â«
Î(t)
(â
ârdetAr(x, t)
)v(x, t) dHmâ1
x .
Proof. For y â Îr(t), the perpendicular foot on Î(t) from y is denoted byx = Îł(y, t) â Î(t). From Proposition 4.1 and (6.3), we have
vr(y, t) =â
âtdr(y, t) =
â
ât(d(y, t) â r) = v(x, t).
and
Îșr(y, t) =mâ1â
i=1
Își
1 + rÎși=
mâ1â
i=1
â
ârlog(1 + rÎși)
=â
ârlog detAr =
1
det Ar
â
ârdetAr ,
where Își = Își(x, t) and Ar = Ar(x, t).
d
dt|Îr(t)| = = â
â«
Îr(t)
Îșrvr dHmâ1
= ââ«
Î(t)
1
detAr
(â
ârdetAr
)v det Ar dHmâ1
= ââ«
Î(t)
(â
ârdet Ar
)v dHmâ1.
Let Sl = Sl(Îș1, · · · , Îșmâ1) be the lth elementary symmetric polynomial ofÎș1, · · · , Îșmâ1, i.e.,
S0 = 1, S1 =
mâ1â
i=1
Își = Îș, S2 =â
i<j
ÎșiÎșj , · · · , Smâ1 =
mâ1â
i=1
Își = Îșg .
Then we have
detAr =
mâ1â
i=1
(1 + rÎși) =
mâ
l=0
Slrl,
where Sm := 0.
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80 6. VARIATIONAL FORMULAS
Theorem 6.7.d
dt
â«
Î(t)
Sl dHmâ1 = â(l + 1)
â«
Î(t)
Sl+1 v dHmâ1 (l = 0, · · · , m â 1).
Proof. Since
|Îr(t)| =
â«
Î(t)
detAr dHmâ1 =
mâ1â
l=0
rl
â«
Î(t)
Sl dHmâ1,
we haved
dt|Îr(t)| =
mâ1â
l=0
rl d
dt
â«
Î(t)
Sl dHmâ1. (6.4)
On the other hand, from Lemma 6.6 and the equality
â
ârdetAr =
mâ1â
l=0
(l + 1)Sl+1rl,
we also obtain
d
dt|Îr(t)| =
mâ1â
l=0
rl
(â(l + 1)
â«
Î(t)
Sl+1 v dHmâ1
). (6.5)
Comparing (6.4) and (6.5), we obtain the result.
Theorem 6.7 includes the two important formulas (6.1) with l = 0 and (6.2)with l = mâ1. This theorem enables us to comprehend them in a unified approach.
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CHAPTER 7
Gradient structure and moving boundary
problems
Many geometric moving boundary problems have their own gradient flow struc-tures. A gradient structure is defined by an energy functional and an inner productfor each hypersurface. We consider several examples of moving boundary problemsand give formal gradient structures for them. In this chapter, we suppose M is asufficiently smooth moving hypersurface with compact Î(t).
1. General gradient flow of hypersurfaces
We illustrate a general framework of moving boundary problems with a formalgradient structure. Let G be a set of hypersurfaces in R
m with âsomeâ regularityconditions and with a constraint R(Î) = 0. We suppose that Î(t) â G for t â I.Then formally we can write
d
dtR(Î(t)) = ăRâČ(Î(t)), v(·, t)ă ,
where the right hand side is linear with respect to the normal velocity v. This leadsto the constraint for the normal velocity:
v â TÎ(t)(G) := v ; ăRâČ(Î(t)), vă = 0.An typical example of the constraint is the volume preserving condition:
|Ωâ| = a0, R(Î) := |Ωâ| â a0 = 0, (7.1)
where a0 > 0 is a fixed constant. In this case, we have
d
dtR(Î(t)) = â
â«
Î(t)
v dHmâ1,
v â TÎ(G) :=
v ;
â«
Î
v dHmâ1 = 0
. (7.2)
We consider an energy:
E : G â Î 7â E(Î) â R.
Then the first variation of the energy ddtE(Î(t)) becomes a linear functional of
v â TÎ(G).To consider a gradient flow of E(Î), we need to identify the first variation of the
energy with a linear functional on TÎ(G). We introduce an inner product ăf, găÎ81
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82 7. GRADIENT STRUCTURE AND MOVING BOUNDARY PROBLEMS
defined on a linear space TÎ(G), by which we identify the variation of energy witha element of TÎ(G), namely,
d
dtE(Î(t)) = ăâE(Î(t)), văÎ(t) .
The expression of the first variation âE(Î) depends on the choice of the innerproduct. The most standard choice is the L2-inner product:
ăf, găÎ =
â«
Î
fg dHmâ1 .
The gradient flow of E(Î) with respect to ă·, ·ăÎ is formally given as
v = ââE(Î(t)).
An important property of the gradient flow is the energy decreasing property:
d
dtE(Î(t)) = ăâE(Î(t)), văÎ(t) = âăv, văÎ(t) †0.
We will show various examples of moving boundary problems with the gradientstructure in the following sections.
2. Prescribed normal velocity motion
Condition: f â Câ(Rm) is given.
Energy: E(Î) :=
â«
멉
f(x)dHm
Variation of energy:d
dtE(Î(t)) = â
â«
Î(t)
f(x)v dHmâ1
Inner product: ăf, găÎ :=
â«
Î
fg dHmâ1
Gradient flow: v = f(x)
We remark that, if f ⥠1, then E(Î) = |Ωâ| and this is the constant speed motionv = 1, which is deeply related to the eikonal equation (see [18] etc.).
3. Mean curvature flow
The mean curvature flow is one of the most important moving boundary prob-lems and is related to many applications. We refer [6] and [18] and reference therein.
Energy: E(Î) :=
â«
Î
dHmâ1
Variation of energy:d
dtE(Î(t)) = â
â«
Î(t)
Îșv dHmâ1
Inner product: ăf, găÎ :=
â«
Î
fg dHmâ1
Gradient flow: v = Îș
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5. GAUSSIAN CURVATURE FLOW 83
4. Anisotropic mean curvature flow
We also consider an anisotropic surface energy
E(Î) :=
â«
Î
α(Îœ(x)) dHmâ1,
where α â C2(Smâ1, R+) is a given anisotropic energy density, where Smâ1 :=x â R
m; |x| = 1. We extend α to the whole space as
α(x) := α(x/|x|) (x â Rm, x 6= 0).
The variation of the energy is given as
d
dtE(Î(t)) = â
â«
Î(t)
â2α(Îœ) : W + α(Îœ)Îș
v dHmâ1. (7.3)
Then we have the gradient flow as follows.
Inner product: ăf, găÎ :=
â«
Î
fg dHmâ1
Gradient flow: v = â2α(Îœ) : W + α(Îœ)Îș
In the case m = 2, we can write ÎČ(Ξ) := α(Îœ) with (Îœ = (cos Ξ, sin Ξ)T). Then thegradient flow becomes
v = (ÎČâČâČ(Ξ) + ÎČ(Ξ)) Îș. (7.4)
Exercise 7.1. Prove (7.3) and (7.4).
For more details about the anisotropic mean curvature flow, see [6] and thereferences therein.
5. Gaussian curvature flow
Let Î(t) be convex (i.e. Ωâ(t) is a convex domain). In the case m = 3, themoving boundary problem
v = Îșg,
is called the Gaussian curvature flow and is well studied (see [6], [19] and refer-ences therein). In particular, it possesses the constant volume speed property (seeSection 2).
We consider a more general form in Rm using the result of Theorem 6.7.
Energy: E(Î) :=1
l + 1
â«
Î
Sl(Îș1, · · · , Îșmâ1) dHmâ1
Variation of energy:d
dtE(Î(t)) = â
â«
Î(t)
Sl+1(Îș1, · · · , Îșmâ1) v dHmâ1
Inner product: ăf, găÎ :=
â«
Î
fg dHmâ1
Gradient flow: v = Sl+1(Îș1, · · · , Îșmâ1)
This is an extension of the mean curvature flow (l = 0) and the Gaussian curvatureflow (m = 3 and l = 2).
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84 7. GRADIENT STRUCTURE AND MOVING BOUNDARY PROBLEMS
6. Willmore flow
The following gradient flow is called the Willmore flow (see [1], [17] and thereferences therein).
Energy: E(Î) :=1
2
â«
Î
Îș2 dHmâ1
Variation of energy:d
dtE(Î(t)) =
â«
Î(t)
(âÎÎș + Îș
mâ1â
i=1
Îș2i â
1
2Îș3
)v dHmâ1
Inner product: ăf, găÎ :=
â«
Î
fg dHmâ1
Gradient flow: v = ââÎÎș â Îș
mâ1â
i=1
Îș2i +
1
2Îș3
Gradient flow (m=3): v = ââÎÎș â 1
2Îș3 + 2ÎșÎșg
7. Volume preserving mean curvature flow
We suppose the volume preserving condition (7.1) and the constraint on v(7.2) with the same energy and inner product as Section 3. Since we impose theconstraint, the inner product is
ăf, găÎ =
â«
Î
fg dHmâ1 for f, g â TÎ(G).
The variation of the energy is identified with an element of TÎ(G) as follows. Wedefine
ăÎșă :=1
|Î|
â«
Î
Îș dHmâ1.
Then we have
d
dtE(Î(t)) = â
â«
Î(t)
Îșv dHmâ1
=
â«
Î(t)
(ăÎșă â Îș)v dHmâ1
= ăăÎșă â Îș, v(·, t)ăÎ(t) .
Hence, we obtain
âE(Î) = ăÎșă â Îș.
The gradient flow under the volume preserving condition becomes as follows.
Gradient flow: v = Îș â ăÎșăThis is called the volume preserving mean curvature flow (area preserving meancurvature flow, if m = 2), which keeps the volume enclosed by Î(t) and decreasesthe surface area. See [11] etc., for details.
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8. SURFACE DIFFUSION FLOW 85
8. Surface diffusion flow
We again suppose the volume preserving condition (7.1) and the constraint onv (7.2) with the same energy as Section 3. But we consider another inner producton TÎ(G).
We consider a Sobolev space on Î:
H1(Î) := f : Î â R; f â L2(Î), |âÎf | â L2(Î),with the norm
âfâH1(Î) :=
(â«
Î
(|f |2 + |âÎf |2) dHmâ1
) 12
.
Its dual space is denoted by Hâ1(Î). For f â L2(Î), it is identified with an elementof Hâ1(Î) by
Hâ1(Î)ă f, g ăH1(Î) =
â«
Î
fg dHmâ1 (âg â H1(Î)).
We define the following Sobolev spaces with constraint.
H1(Î) := f â H1(Î);
â«
Î
f dHmâ1 = 0,
Hâ1(Î) := f â Hâ1(Î); H1(Î)ă 1, f ăHâ1(Î) = 0.Then it follows that the LaplaceâBeltrami operator can be extended to a topo-logical linear isomorphism from H1(Î) onto Hâ1(Î). We denote it by âÎ âB(H1(Î), Hâ1(Î)). Our inner product is defined as
ăf, găÎ := â H1(Î)
âšââ1
Î f, gâ©
Hâ1(Î)(f, g â TÎ(G) â Hâ1(Î)) .
This can be regarded as an inner product of Hâ1(Î).Then we can identify the variation of the energy in the following sense. Sinceâ«
ÎâÎfdHmâ1 = 0, we formally have
ââ1Î (âÎf) = f â ăfă (f â H1(Î)),
and
d
dtE(Î(t)) = â
â«
Î(t)
(Îș â ăÎșă) v dHmâ1
= ââ«
Î(t)
ââ1Î (âÎÎș) v dHmâ1
= ăâÎÎș, v(·, t)ăÎ(t) .
Hence, we obtain
âE(Î) = âÎÎș
The gradient flow with respect to the inner product becomes as follows.
Gradient flow: v = ââÎÎș
This is called the surface diffusion flow, which keeps the volume enclosed by Î(t)and decreases the surface area. See [2] etc., for details.
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86 7. GRADIENT STRUCTURE AND MOVING BOUNDARY PROBLEMS
9. HeleâShaw moving boundary problem
HeleâShaw flow is two-dimensional slow flow of viscous fluid between two par-allel horizontal plates with a thin gap. It was studied by H. S. HeleâShaw [9]experimentally at first, and some mathematical models have been studied in sev-eral papers [3], [4], [10], [13], etc.
A typical mathematical model for the moving boundary problem of the HeleâShaw flow with the surface tension effect is as follows.
âp(x, t) = 0 (x â Ωâ(t)),
p(x, t) = Îș (x â Î(t)),
v(x, t) = âÎœ(x, t) · âp(x, t) (x â Î(t)),
with a given initial boundary Î(0), where p(x, t) is a pressure field of the fluid inΩâ(t).
This HeleâShaw moving boundary problem also has a gradient structure. Let
H12 (Î) be the trace space on Î from H1(Ωâ). Then it is well known that, for
q â H12 (Î), there exists u = u(q) â H1(Ωâ) with
âu = 0 in Ωâ, u|Î = q.
We define the Dirichlet-to-Neumann map:
ÎÎq := ââu(q)
âÎœ, ÎÎ â B(H
12 (Î), Hâ 1
2 (Î)),
where Hâ 12 (Î) := (H
12 (Î))âČ. We define quotient spaces
H12 (Î) :=
f â H
12 (Î);
â«
Î
f dHmâ1 = 0
,
Hâ 12 (Î) :=
f â Hâ 1
2 (Î);H
12 (Î)
ă 1, f ăHâ
12 (Î)
= 0
.
Then ÎÎ := ÎÎ|H
12 (Î)
belongs to B(H12 (Î), Hâ 1
2 (Î)) and is bijective. We define
U(f) := u(Îâ1Î f). Their inner products are defined by
(f, g)H
12 (Î)
:=
â«
멉
âu(f) · âu(g)dx,
(f, g)Hâ
12 (Î)
:=
â«
멉
âU(f) · âU(g)dx,
We remark that
(f, g)H
12 (Î)
=
â«
Î
(ÎÎf) g dHmâ1 (if ÎÎf â L2(Î)),
(f, g)Hâ
12 (Î)
=
â«
Î
(Îâ1Î f) g dHmâ1 (if g â L2(Î)).
We again suppose the volume preserving condition (7.1) and the constraint on
v (7.2) with the same energy as Section 3. We adopt the inner product of Hâ 12 (Î):
ăf, găÎ := (f, g)Hâ
12 (Î)
.
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9. HELEâSHAW MOVING BOUNDARY PROBLEM 87
Then we can identify the variation of the energy in the following sense. Sinceâ«Î(ÎÎf) dHmâ1 = 0, we formally have
Îâ1Î (ÎÎf) = f â ăfă (f â H
12 (Î)),
andd
dtE(Î(t)) = â
â«
Î(t)
(Îș â ăÎșă) v dHmâ1
= ââ«
Î(t)
Îâ1Î (ÎÎÎș) v dHmâ1
= âăÎÎÎș, v(·, t)ăÎ(t) .
Hence, we obtain
âE(Î) = âÎÎÎș =âu(Îș)
âÎœ.
The gradient flow with respect to the inner product becomes as follows.
Gradient flow: v = ââu(Îș)
âÎœThis is nothing but the HeleâShaw moving boundary problem. It also keeps thevolume enclosed by Î(t) and decreases the surface area. Physically, these propertiescorrespond to the volume conservation law and the surface tension effect. See theabove references for details.
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Bibliography
[1] K. Deckelnick and G. Dziuk, Error estimates for the Willmore flow of graphs,Interfaces and Free Boundaries Vol.8 (2006), pp.21-46.
[2] C. M. Elliott and H. Garcke, Existence results for diffusive surface motionlaws, Adv. Math. Sci. Appl. Vol.7 (1997), pp.467-490.
[3] J. Escher and G. Simonett; Classical solutions of multidimensional Hele-Shawmodels, SIAM J. Math. Anal. Vol.28 (1997), pp.1028-1047.
[4] J. Escher and G. Simonett; Classical solutions for Hele-Shaw models withsurface tension, Adv. Diff. Eq. Vol.2 (1997), pp.619-642.
[5] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Func-tions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL (1992).
[6] Y. Giga, Surface Evolution Equations, A Level Set Approach, Monographs inMathematics Vol.99, Birkhauser Verlag, Basel (2006).
[7] D. Gilberg and N. S. Trudinger, Elliptic Partial Differential Equations of Sec-ond Order, second edition, Springer-Verlag (1983).
[8] M. E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane,Oxford Mathematical Monographs, The Clarendon Press, Oxford (1993).
[9] H. S. Hele-Shaw, The flow of water, Nature Vol.58 (1898), pp.34-36.[10] S. D. Howison; Fingering in Hele-Shaw cells, J. Fluid Mech. Vol.167 (1986),
pp.439-453.[11] G. Huisken, The volume preserving mean curvature flow, J. Reine Angew.
Math. Vol.382 (1987), pp.35-48.[12] M. Kimura, Numerical analysis of moving boundary problems using the bound-
ary tracking method, Japan J. Indust. Appl. Math. Vol.14 (1997), pp.373-398.[13] M. Kimura, Time local existence of a moving boundary of the Hele-Shaw flow
with suction, Euro. J. Appl. Math. Vol.10 (2000), pp.581-605.[14] M. Kimura, Shape derivative and linearization of moving boundary prob-
lems, GAKUTO International Series, Mathematical Sciences and ApplicationsVol.13 (2000), pp.167-179.
[15] M. Kimura, Numerical analysis of moving boundary problems, Sugaku Expo-sitions, Vol.15 (2002), pp.71-88.
[16] M. Kimura and H. Notsu, A level set method using the signed distance func-tion, Japan J. Indust. Appl. Math. Vol.19 (2002), pp.415-446.
[17] T. Oberhuber, Finite difference scheme for the Willmore flow of graphs, Ky-bernetika Vol.43, No. 6 (2007), pp.855-867.
[18] J. A. Sethian, Level Set Methods and Fast Marching Methods, Evolving In-terfaces in Computational Geometry, Fluid Mechanics, Computer Vision, andMaterials Science, second edition, Cambridge Monographs on Applied and
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90 Bibliography
Computational Mathematics, Vol.3, Cambridge University Press, Cambridge(1999).
[19] T. K. Ushijima and H. Yagisita, Approximation of the Gauss curvature flow bya three-dimensional crystalline motion, Proceedings of Czech-Japanese Semi-nar in Applied Mathematics 2005, COE Lect. Note Vol.3, Kyushu Univ. (2006),pp.139-145.
[20] W. P. Ziemer, Weakly Differentiable Functions, Sobolev Spaces and Functionsof Bounded Variation, Graduate Texts in Mathematics, Vol.120, Springer-Verlag, New York (1989).
âtopics-in-mathematical-modelingâ â 2008/12/5 â 8:30 â page 91 â #103
APPENDIX A
1. Adjugate matrix
For A â RmĂm, we denote by adjA â R
mĂm the transpose of the cofactormatrix of A. It is called the adjugate of A. It is also called âadjointâ, but the
adjoint matrix of A often means the conjugate transpose Aâ := AT
. To avoid theambiguity, we adopt the word âadjugateâ for adjA. It is well known that
A (adjA) = (adjA)A = (detA) I,
and, for invertible A, the inverse matrix is given by
Aâ1 =1
detAadjA .
We remark the next proposition.
Proposition A.1. Let A be diagonalizable. Namely, there exists an invertiblematrix P such that
A = Pâ1
λ1 O. . .
O λm
P,
where λ1, · · · , λm are the eigenvalues of A. Then, the adjugate of A is representedin the form:
adjA = Pâ1
”1 O. . .
O ”m
P,
where”i :=
â
1â€jâ€m, j 6=i
λj (i = 1, · · · , m).
Proof. The assertion is clear if detA = λ1 · · ·λm 6= 0. For general A, sincedet(A + ΔI) = (λ1 + Δ) · · · (λm + Δ) 6= 0 for 0 < Δ << 1, we have
adj(A + ΔI) = Pâ1
”1(Δ) O. . .
O ”m(Δ)
P,
where”i(Δ) :=
â
1â€jâ€m, j 6=i
(λj + Δ) (i = 1, · · · , m).
Taking the limit Δ â +0, we obtain the assertion.
91
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92 A
2. Jacobiâs formula
Lemma A.2. Let A = (a1, · · · , am), B = (b1, · · · , bm) â RmĂm. Then we
have
tr (adjA)B =
mâ
k=1
det (a1, · · · , akâ1, bk , ak+1, · · · , am) .
Proof. It is sufficient to prove the assertion of the lemma in the case detA 6= 0.We remark that adjA = (detA)Aâ1. The jth column vector of the identity matrixI â R
mĂm is denoted by ÎŽj = (ÎŽ1j , · · · , ÎŽmj)T â R
m. We have
mâ
k=1
det (a1, · · · , akâ1, bk , ak+1, · · · , am)
=mâ
k=1
(detA) det(Aâ1(a1, · · · , akâ1, bk , ak+1, · · · , am)
)
= (det A)
mâ
k=1
det(ÎŽ1, · · · , ÎŽkâ1, A
â1bk , ÎŽk+1, · · · , ÎŽm
)
= (det A)mâ
k=1
(the kth component of Aâ1bk )
= (det A) tr (Aâ1B ).
For general A, we can derive the formula in the similar way to the proof of Propo-sition A.1.
Proposition A.3 (Jacobiâs formula). We suppose that A â C1(I, RmĂm).Then we have
d
dtdetA(t) = tr (adjA(t))AâČ(t) (t â I).
In particular, if detA(t) 6= 0 then
d
dtdetA(t) = detA(t) tr
(A(t)â1AâČ(t)
)(t â I).
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2. JACOBIâS FORMULA 93
Proof. Let aj(t) = (a1j(t), · · · , amj(t))T â R
m be the jth column vector of
the matrix A(t) â RmĂm. The t derivative is denoted by âČ = d
dt . We have
d
dtdetA(t)
=d
dt
â
ÏâSm
sgn(Ï) aÏ(1)1 · · · aÏ(m)m
=â
ÏâSm
sgn(Ï)
(mâ
k=1
aÏ(1)1 · · · aÏ(kâ1)kâ1 aâČÏ(k)k aÏ(k+1)k+1 · · · aÏ(m)m
)
=
mâ
k=1
(â
ÏâSm
sgn(Ï) aÏ(1)1 · · · aÏ(kâ1)kâ1 aâČÏ(k)k aÏ(k+1)k+1 · · · aÏ(m)m
)
=
mâ
k=1
det (a1, · · · , akâ1, aâČk , ak+1, · · · , am)
= tr (adjA(t))AâČ(t) .
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Part 3
Large time behaviour for diffusive
HamiltonâJacobi equations
Philippe Laurencot
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2000 Mathematics Subject Classification. Primary 35B40, 35B33, 35K15, 35K55,49L25
Key words and phrases. diffusive HamiltonâJacobi equation, large time behaviour,gradient estimates, extinction, convergence to self-similarity
Abstract. The large time behaviour of non-negative and integrable solutionsto the semilinear diffusive HamiltonâJacobi equation âtu â âu ± |âu|q = 0in (0,â) Ă RN is surveyed in these lecture notes, the HamiltonâJacobi termacting either as an absorption or a source term. Temporal decay estimates areestablished for the solution, its gradient and time derivative as well as one-sidedestimates for its Hessian matrix. Next, according to the values of the param-eters q > 0 and N â„ 1, the time evolution is shown to be either dominatedby the diffusion term, the reaction term, or governed by a balance betweenboth terms. The occurrence of finite time extinction is also investigated forq â (0, 1).
Acknowledgement. These notes grew out from lectures which I gave at Fu-dan University, Shangai, in November 2006 and at the Necas Center for Math-ematical Modeling, Praha, in April 2008. I thank Professors Songmu Zhengand Eduard Feireisl for their kind invitations as well as both institutions fortheir hospitality and support. I also thank Saıd Benachour and the referee foruseful comments on the manuscript.
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Contents
Chapter 1. Introduction 99
Chapter 2. Well-posedness and smoothing effects 1031. Gradient estimates 1051.1. Gradient estimates: Ï = 1 and q > 1 1061.2. Gradient estimates: q > 1 1071.3. Gradient estimates: q = 1 1071.4. Gradient estimates: q â (0, 1) 1082. Time derivative estimates 1083. Hessian estimates 1114. Existence 1145. Uniqueness 116Bibliographical notes 117
Chapter 3. Extinction in finite time 1191. An integral condition for extinction 1212. A pointwise condition for extinction 1233. Non-extinction 1244. A lower bound near the extinction time 126Bibliographical notes 127
Chapter 4. Temporal decay estimates for integrable initial data: Ï = 1 1291. Decay rates 1311.1. Decay rates: q = N/(N + 1) 1321.2. Decay rates: q = 1 1332. Limit values of âuâ1 1343. Improved decay rates: q â (1, qâ) 137Bibliographical notes 139
Chapter 5. Temporal growth estimates for integrable initial data: Ï = â1 1411. Limit values of âuâ1 and âuââ 1421.1. The case q â„ 2 1421.2. The case q â (qâ, 2) 1441.3. The case q â [1, qâ] 1472. Growth rates 149Bibliographical notes 150
Chapter 6. Convergence to self-similarity 1511. The diffusion-dominated case: Ï = 1 151
97
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98 CONTENTS
2. The reaction-dominated case: Ï = â1 154Bibliographical notes 158
Bibliography 159
Appendix A. Self-similar large time behaviour 1631. Convergence to self-similarity for the heat equation 1632. Convergence to self-similarity for HamiltonâJacobi equations 164
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CHAPTER 1
Introduction
The aim of these lectures is to present several qualitative properties of non-negative solutions to the Cauchy problem for a semilinear parabolic equation in-volving a nonlinearity depending on the gradient of the solution which might becalled a diffusive (or viscous) HamiltonâJacobi equation and reads
âtu â âu + Ï |âu|q = 0 , (t, x) â Qâ := (0,â) Ă RN , (1.1)
u(0) = u0 , x â RN , (1.2)
where u = u(t, x) is a function of the time t â„ 0 and the space variable x â RN ,
N â„ 1, and the parameters Ï and q are such that
q â (0,â) and Ï â â1, 1 . (1.3)
The equation (1.1) features two mechanisms acting on the space variable, thelinear diffusion âu and the nonlinearity Ï |âu|q. Taken apart, these two mecha-nisms have different properties and there is thus a competition between them in(1.1). The main issue to be addressed in these notes is whether one of these twoterms dominates the dynamics at large times. To start the discussion on this mat-ter, it is obvious that the sign of Ï is of utmost importance: indeed, if Ï = 1, thenonlinear term |âu|q is non-negative and thus acts as an absorption term whichenhances the effect of the linear diffusion. While, if Ï = â1, â|âu|q is non-positiveand is thus a source term which opposes to the diffusive effect. Observe also that,in contrast to the semilinear parabolic equation
âtv â âv ± vp = 0 , (t, x) â Qâ ,
which has been thoroughly studied, the nonlinearity in (1.1) is only effective in thespatial regions where u is steep and not flat. In particular, constants are solutionsto (1.1).
In order to determine which of these mechanisms governs the large time dy-namics, let us first describe briefly some salient features of their own and restrictourselves to non-negative integrable and bounded continuous initial data: we thusassume that
u0 â L1(RN ) â© BC(RN ) , u0 â„ 0, u0 6⥠0 . (1.4)
On the one hand, it is well-known that the linear heat equation
âtc â âc = 0 , (t, x) â Qâ , (1.5)
99
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100 1. INTRODUCTION
with initial condition c(0) = u0 has a unique smooth solution c : t 7ââ etâu0 whichis given by
c(t, x) :=(etâu0
)(x) = (g(t) â u0) (x) =
1
tN/2
â«
RN
G
(x â y
t1/2
)u0(y) dy (1.6)
with the notation
g(t, x) :=1
tN/2G( x
t1/2
)and G(x) :=
1
(4Ï)N/2exp
(â|x|2
4
)(1.7)
for (t, x) â Qâ. An important property enjoyed by non-negative and integrablesolutions to (1.5) is the invariance through time evolution of the L1-norm, that is
â„â„etâu0
â„â„1
= âu0â1 for every t â„ 0 . (1.8)
On the other hand, the HamiltonâJacobi equation
âth + Ï |âh|q = 0 , (t, x) â Qâ , (1.9)
does not have smooth solutions and might even have several weak solutions, adrawback which is also met in the study of scalar conservation laws. Still, there is aselection principle which allows to single out one solution among the weak solutions,the so-called viscosity solution. We refer the reader to [4, 6, 25, 27] for the precisedefinition and several properties of viscosity solutions. In particular, (1.9) has aunique viscosity solution with initial condition h(0) = u0. Moreover, if q â„ 1, it isgiven by the Hopf-Lax-Oleinik representation formula
Ï h(t, x) := infyâRN
Ï u0(y) +
q â 1
qq/(qâ1)|x â y|q/(qâ1) tâ1/(qâ1)
(1.10)
if q > 1 and
Ï h(t, x) := inf|yâx|â€t
Ï u0(y) if q = 1 , (1.11)
for (t, x) â Qâ. According to (1.10) and (1.11) the HamiltonâJacobi equation (1.9)enjoys the following invariance property
infxâRN
Ï h(t, x) = infxâRN
Ï u0(x) for every t â„ 0 . (1.12)
A natural guess is then that
âu(t)â1 and infxâRN
Ï u(t, x)
are two relevant quantities for the large time dynamics.Let us first look at the case Ï = 1. In that case, u(t) turns out to be integrable
for each t â„ 0 so that
infxâRN
u(t, x) = infxâRN
u0(x) = 0 ,
and this quantity does not seem to provide any useful information on the large timebehaviour. Concerning the L1-norm of u it formally follows from (1.1) that
âu(t)â1 = âu0â1 ââ« t
0
â«
RN
|âu(s, x)|q dxds , t â„ 0 , (1.13)
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1. INTRODUCTION 101
from which we readily deduce that t 7ââ âu(t)â1 is non-increasing and
I1(â) := limtââ
âu(t)â1 = âu0â1 ââ« â
0
â«
RN
|âu(s, x)|q dxds â„ 0 .
The question is then whether I1(â) > 0 or I1(â) = 0. In the latter case it meansthat the damping of the nonlinearity is sufficiently strong so as to drive the L1-norm of u(t) to zero as t â â. The nonlinear term is thus expected to play anon-negligible role for large times.
In the same vein, u is a subsolution to the linear heat equation and we inferfrom the comparison principle that
âu(t)ââ â€â„â„etâu0
â„â„â
†C tâN/2 , t â (0,â) .
As the nonlinear term |âu|q acts as an absorption term in that case it enhancesthe dissipation due to the diffusion and the main issue is to figure out whether thisadditional dissipative mechanism speeds up the convergence to zero of the Lâ-normof u(t) as t â â.
When Ï = â1, the nonlinear term |âu|q is a source term and thus slows downor even impedes the dissipation of the diffusion. Indeed, it formally follows from(1.1) by integration that
âu(t)â1 = âu0â1 +
â« t
0
â«
RN
|âu(s, x)|q dxds , t â„ 0 .
Consequently t 7ââ âu(t)â1 and
t 7ââ infxâRN
âu(t, x) = ââu(t)ââ
are non-decreasing. Setting
I1(â) := limtââ
âu(t)â1 â [âu0â1,â] and Iâ(â) := limtââ
âu(t)ââ â [0, âu0ââ] ,
the questions here are whether I1(â) is finite or not and whether Iâ(â) is positiveor zero. While we expect the diffusion to be dominant for large times if I1(â) < âand the nonlinearity to be dominant for large times if Iâ(â) > 0, we shall seebelow that the situation is more complicated than for Ï = 1.
We finally collect some elementary properties of (1.1).
â If u is a non-negative solution to (1.1), (1.2) then
u(t, x) :=1 â Ï
2âu0ââ + Ï u(t, x) , (t, x) â [0,â) Ă R
N , (1.14)
is a non-negative solution to
âtu â âu + |âu|q = 0 , (t, x) â Qâ , (1.15)
u(0) = (1 â Ï)âu0ââ
2+ Ï u0 , x â R
N . (1.16)
Of course, u = u if Ï = 1.â The equation (1.1) is an autonomous equation, that is, if u is a solution
to (1.1), (1.2), and t0 > 0, then u(t, x) := u(t + t0, x) solves (1.1) withinitial condition u(t0).
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102 1. INTRODUCTION
â If u is a solution to (1.1), (1.2) and λ > 0, then the function uλ definedby
uλ(t, x) := λ2âq u(λ2(qâ1)t, λqâ1x
), (t, x) â [0,â) Ă R
N
is also a solution to (1.1) with initial condition uλ(0).
Finally there are two cases for which (1.1) can be transformed to a linearequation:
â if q = 2, the celebrated HopfâCole transformation reduces (1.1) to thelinear heat equation. Indeed, introducing v = eâÏu, we have v(t) =etâeâÏu0 for t â„ 0.
â If q = 1, N = 1 and u0 is non-increasing on (0,â) and non-decreasingon (ââ, 0), then so is u(t) for each t â„ 0 and u actually solves the linearequation
âtu â â2xu â Ï sign(x) âxu = 0 , (t, x) â [0,â) Ă R .
Notations. In the following, C and Ci, i â„ 1, denote positive constants that onlydepend on N and q and may vary from place to place. The dependence of theseconstants upon additional parameters is indicated explicitly.
For p â [1,â] and w â Lp(RN ), âwâp denotes the norm of w in Lp(RN ).We also define the space BC(RN ) of bounded and continuous functions in R
N byBC(RN ) := C(RN ) â© Lâ(RN ) while BUC(RN ) denotes the space of bounded anduniformly continuous functions in R
N .Given a real number r â R, we define its positive part r+ by r+ := max r, 0.
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CHAPTER 2
Well-posedness and smoothing effects
We begin with the well-posedness of (1.1), (1.2) in BC(RN ).
Theorem 2.1. [32] Let u0 be a non-negative function in BC(RN ) and Ï ââ1, 1. Then the initial-value problem (1.1), (1.2) has a unique non-negativeclassical solution
u â BC([0,â) Ă RN ) â© C1,2(Qâ) .
In addition, t 7ââ âu(t)ââ is a non-increasing function of time and
0 †u(t, x) †âu0ââ , (t, x) â [0,â) Ă RN . (2.1)
We also have
u(t) = etâu0 â Ï
â« t
0
e(tâs)â|âu(s)|q ds , t â„ 0 . (2.2)
Recall that t 7ââ etâu0 denotes the solution to the linear heat equation (1.5) withinitial condition u0 and is given by (1.6), (1.7). Furthermore, if u0 â W 1,â(RN ),then
ââu(t)ââ †ââu0ââ , t â„ 0 . (2.3)
While the uniqueness stated in Theorem 2.1 is a consequence of the comparisonprinciple [32] (see Section 5 below), the proof of the existence of a solution to (1.1),(1.2) requires more ingredients among which the following gradient estimates arecrucial.
Theorem 2.2. [11, 32] Let u0 be a non-negative function in BC(RN ),Ï â â1, 1, and denote by u the corresponding classical solution to (1.1), (1.2).Then âu(t) â Lâ(RN ) and
ââu(t)ââ †C1 âu(s)ââ (t â s)â1/2 if q > 0 , (2.4)
ââu(t)ââ †C2 âu(s)â1/qâ (t â s)â1/q if q > 0 , q 6= 1 , (2.5)
for t > s â„ 0.Furthermore, if Ï = 1 and q > 1, then
â„â„â„âu(qâ1)/q(t)â„â„â„â
†C3 âu(s)â(qâ1)/qâ (t â s)â1/2 , (2.6)
â„â„â„âu(qâ1)/q(t)â„â„â„â
†C4 tâ1/q (2.7)
for every t > s â„ 0.
103
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104 2. WELL-POSEDNESS AND SMOOTHING EFFECTS
As a consequence of Theorem 2.2 there is a smoothing effect from Lâ(RN ) toW 1,â(RN ) for solutions to (1.1), (1.2). But Theorem 2.2 also provides quantitativeinformation about the temporal decay of ââu(t)ââ which will be used in the studyof the large time behaviour. In fact, non-negative solutions to the linear heatequation (1.5) also satisfy (2.4) and (2.6) while (2.5) and (2.7) are also enjoyedby non-negative (viscosity) solutions to (1.9). Thus, in that particular case, non-negative solutions to (1.1), (1.2) inherit the smoothing effects from both termsacting on the space variable.
The proof of the gradient estimates (2.4)â(2.7) relies on a modified Bernsteintechnique [21]: introducing a new unknown function v defined by v := fâ1(u) forsome strictly monotone function f to be determined, one looks for time-dependentsupersolutions to the nonlinear parabolic equation satisfied by w := |âv|2 and thenapply the comparison principle.
Smoothing effects for the time derivative are also available for both (1.5) and(1.9) and it turns out that this property is still shared by (1.1).
Theorem 2.3. [31] Let u0 be a non-negative function in BC(RN ), Ï â â1, 1,and denote by u the corresponding classical solution to (1.1), (1.2). Then, fori â 1, 2, x â R
N and t > 0, we have
âC6 Ï0(t) â€ Ï âtu(t, x) †C5 Ïi(t) , (t, x) â Qâ , (2.8)
with
Ï0(t) :=
âu0ââ tâ1 if q 6= 1 ,
âu0ââ(tâ1 + tâ1/2
)if q = 1 ,
Ï1(t) :=
âu0â1/(2qâ1)
â tâ(q+1)/(2qâ1) if q â„ 2 ,
âu0ââ tâ1 if q â (0, 2) ,
Ï2(t) :=
âu0ââ tâ1 if q â„ 2 ,
âu0â1/(2qâ1)â tâ(q+1)/(2qâ1) if q â (1, 2) ,
âu0ââ tâ3/2 (1 + log (1 + t)) if q = 1 ,
âu0â(2âq)/qâ tâ2/q if q â (0, 1) .
The proof of Theorem 2.3 also relies on the comparison principle applied to theequation satisfied by (ÎŽ âtu â A)/Ï(âu) for some suitable choices of ÎŽ â â1, 1,A â R and Ï â C2(RN ; (0,â)).
The situation is slightly different for the Hessian matrix: indeed, while it alsoenjoys a smoothing effect for (1.5), this is no longer true for (1.9). It is then not clearwhether a smoothing effect is available for (1.1). Still, non-negative solutions to(1.9) become instantaneously semiconcave if Ï = 1 [27, Chapter 3.3] and semiconvexif Ï = â1 and non-negative solutions to (1.1) also enjoy this property if q â (1, 2).
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1. GRADIENT ESTIMATES 105
Theorem 2.4. [9] Let u0 be a non-negative function in BC(RN ), Ï â â1, 1,and denote by u the corresponding classical solution to (1.1), (1.2). If q â (1, 2],then the Hessian matrix D2u = (âiâju)1â€i,jâ€N satisfies
Ï D2u(t, x) †C7 âu0â(2âq)/qâ tâ2/q IN (2.9)
for (t, x) â Qâ, that is,
ÏNâ
i,j=1
(âiâju) (t, x) Οi Οj †C7 âu0â(2âq)/qâ |Ο|2 tâ2/q
for each Ο â RN . Here IN denotes the N Ă N identity matrix.
Furthermore, if u0 â W 2,â(RN ),
Ï D2u(t, x) †âD2u0ââ IN . (2.10)
The proof of Theorem 2.4 still relies on the comparison principle. For q = 2,the estimate (2.9) follows from the analysis of Hamilton [34] (since, if f is a non-negative solution to the linear heat equation âtf = âf , the function â log f solves(1.1) with q = 2). It is also established in [41, Lemma 5.1], still for q = 2.
The estimate (2.9) may also be seen as an extension to a multidimensionalsetting of a weak form of the Oleinik gradient estimate for scalar conservation laws.Indeed, if N = 1 and U = âxu, then U is a solution to âtU â â2
xU + Ï âx (|U |q) = 0in (0,â) Ă R. The estimate (2.9) then reads
Ï âxU †C âu0â(2âq)/qâ tâ2/q
for t > 0, respectively, and we thus recover the results of [29, 37] in that case.When q = 1 and Ï = â1 it is rather log u that becomes instantaneously semi-
convex, a property reminiscent from the Aronson-Benilan one-sided inequality forthe Laplacian of log
(etâu0
)[3].
Proposition 2.5. Let u0 be a non-negative function in BC(RN ) and denote byu the corresponding classical solution to (1.1), (1.2). If Ï = â1 and q = 1, thenthe matrix D2 log u(t, x)+ (1/2t) IN is a non-negative matrix for each (t, x) â Qâ,that is, for each Ο â R
N ,
Nâ
i,j=1
(âiâj log u) (t, x) Οi Οj +|Ο|22t
â„ 0 . (2.11)
The remainder of this chapter is devoted to the proof of the above results: sincethe nonlinearity and the initial condition need not have the required regularity toapply the comparison principle, we give a formal proof of Theorems 2.2, 2.3, 2.4 andProposition 2.5 in Sections 1, 2 and 3. We sketch rigorous proofs in Section 4 (usingapproximation arguments) and complete the proof of Theorem 2.1 in Section 5.
1. Gradient estimates
The proof of the gradient estimates listed in Theorem 2.2 relies on a modifica-tion of the Bernstein technique [21]. The cornerstone of such a technique is to finda strictly monotone function f such that the equation satisfied by w := |âv|2 withv := fâ1(u) has a supersolution which is a sole function of time.
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106 2. WELL-POSEDNESS AND SMOOTHING EFFECTS
Let us thus consider a strictly monotone function f and put
v := fâ1(u) and w := |âv|2 . (2.12)
It follows from (1.1) that v solves
âtv â âv â(
f âČâČ
f âČ
)(v) |âv|2 +
1
f âČ(v)F((f âČ(v))2 |âv|2
)= 0 ,
with F (r) := Ï rq/2 for r â„ 0. Next
âtw = 2
Nâ
i=1
âiv ââiv + 2
(f âČâČ
f âČ
)(v) âv · âw
+ 2
(f âČâČ
f âČ
)âČ
(v) w2 + 2
(f âČâČ
f âČ2
)(v) F
((f âČ(v))2 w
)w
â 2 f âČ(v) F âČ((f âČ(v))2 w
)âv · âw â 4 f âČâČ(v) F âČ
((f âČ(v))2 w
)w2 .
Using the inequality
âw â„ 2Nâ
i=1
âiv ââiv ,
we obtain
Lw â 2
(f âČâČ
f âČ
)âČ
(v) w2 + 2
(f âČâČ
f âČ2
)(v) Î
((f âČ(v))2 w
)w †0 , (2.13)
where
Lz := âtz â âz + 2
f âČ(v) F âČ
((f âČ(v))2 w
)â(
f âČâČ
f âČ
)(v)
âv · âz , (2.14)
and
Î(r) := 2 r F âČ(r) â F (r) = Ï (q â 1) rq/2 , r â [0,â) . (2.15)
At this point, assuming that u0 â W 1,â(RN ) and choosing f(r) = r for r â„ 0,we find that L(ââu0ââ) = 0 which, together with (2.13) and the comparisonprinciple, gives (2.3).
After this preparation we are in a position to establish (2.4)â(2.7) by a suitablechoice of the function f .
1.1. Gradient estimates: Ï = 1 and q > 1. In that case, we choosef(r) = rq/(qâ1) for r â„ 0 and notice that
â2
(f âČâČ
f âČ
)âČ
(v) =2
q â 1
1
v2and
(f âČâČ
f âČ2
)(v) =
1
qf(v). (2.16)
On the one hand, since f and Î are non-negative and
â2
(f âČâČ
f âČ
)âČ
(v) â„ 2
q â 1âu0ââ2(qâ1)/q
â
by (2.1) it follows from (2.13) that
Lw +2
q â 1âu0ââ2(qâ1)/q
â w2 †0 . (2.17)
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1. GRADIENT ESTIMATES 107
The function
Y1(t) :=(q â 1) âu0â2(qâ1)/q
â
2t, t â (0,â) ,
is clearly a supersolution to (2.17) and we deduce from the comparison principlethat w(t, x) †Y1(t) for (t, x) â Qâ. We have thus proved (2.6) for s = 0 and usethe autonomous property of (1.1) to obtain (2.6) for t > s â„ 0.
We next turn to the proof of (2.7). For that purpose we infer from (2.13), (2.15)and (2.16) that
Lw + 2
(q
q â 1
)qâ1
w(q+2)/2 †0 . (2.18)
The function
Y2(t) :=
(q â 1
q
)2
(q â 1)â2/q tâ2/q , t â (0,â) ,
is a supersolution to (2.18) and (2.7) follows by the comparison principle.
1.2. Gradient estimates: q > 1. We put
u :=1 â Ï
2âu0ââ + Ï u
and notice that u solves (1.15), (1.16) and satisfies 0 †u(t, x) †âu0ââ for all(t, x) â Qâ. Since
âu(t, x) =Ï q
q â 1u1/q âu(qâ1)/q(t, x)
and u enjoys the properties (2.6) and (2.7) as shown in the previous section 1.1, weinfer from (2.6) and (2.7) that
ââu(t)ââ †q
q â 1âu0â1/q
â
â„â„â„âu(qâ1)/q(t)â„â„â„â
â€
q C3
q â 1âu0ââ tâ1/2 ,
q C4
q â 1âu0â1/q
â tâ1/q ,
hence (2.4) and (2.5).
1.3. Gradient estimates: q = 1. In that case, the function Î defined in
(2.15) is equal to zero and we choose f(r) = âu0ââ â r2 for r â[0, âu0â1/2
â
]. Then
â2
(f âČâČ
f âČ
)âČ
(v) =2
v2â„ 2
âu0ââand (2.13) reads
Lw +2
âu0ââw2 †0 . (2.19)
The function t 7ââ âu0ââ/(2t) is clearly a supersolution to (2.19) and the compar-
ison principle ensures that |âv(t, x)| †âu0â1/2â (2t)â1/2 for (t, x) â Qâ. Finally,
|âu(t, x)| †2 v(t, x) |âv(t, x)| â€â
2 (âu0ââ â u(t, x))1/2 âu0â1/2â tâ1/2
â€â
2 âu0ââ tâ1/2 ,
hence (2.4) for q = 1.
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108 2. WELL-POSEDNESS AND SMOOTHING EFFECTS
1.4. Gradient estimates: q â (0, 1). We choose
f(r) =1 + Ï
2âu0ââ â Ï r2 for r â
[0,
2 âu0â1/2â
1 + Ï
),
so that
â2
(f âČâČ
f âČ
)âČ
(v) =2
v2â„ 2
âu0ââ,
2
(f âČâČ
f âČ2
)(v) Î
((f âČ(v))2 w
)w = 2q (1 â q) vqâ2 w(q+2)/2
â„ 2q (1 â q) âu0â(qâ2)/2â w(q+2)/2 .
We then deduce from (2.13) that
Lw +2
âu0ââw2 †0 , (2.20)
Lw + 2q (1 â q) âu0â(qâ2)/2â w(q+2)/2 †0 . (2.21)
On the one hand we proceed as in the previous section to show that (2.4) followsfrom (2.20). On the other hand, the function
Y3(t) :=
(21âq âu0â(2âq)/2
â
q(1 â q)
)2/q
tâ2/q , t > 0 ,
is a supersolution to (2.21). By the comparison principle, we have w(t, x) †Y3(t)for (t, x) â Qâ from which (2.5) readily follows.
2. Time derivative estimates
For Ο â RN , we put F (Ο) := |Ο|q and consider a positive function Ï â C2(RN )
and two real numbers ÎŽ â â1, 1 and A â R. Introducing
w :=1
Ï(âu)(ÎŽ âtu â A) , (2.22)
we infer from (1.1) that
âtw = â (âÏ)(âu)
Ï(âu)· [ââu â ÏâF (âu)] w
+1
Ï(âu)[âÏ(âu) w â Ï (âF )(âu) · âÏ(âu) w] .
Observing that
(âÏ)(âu) · âF (âu) = âÏ(âu) · (âF )(âu) ,
we obtain
âtw =w
Ï(âu)
âÏ(âu) â
â
i,j
(âiÏ)(âu) â2j âiu
+ b · âw + âw ,
where
b := 2âÏ(âu)
Ï(âu)â Ï (âF )(âu) .
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2. TIME DERIVATIVE ESTIMATES 109
Since
âÏ(âu) =â
i,j
(âiÏ)(âu) â2j âiu +
â
i,j,k
(âiâkÏ)(âu) âjâku âjâiu ,
we end up with
âtw â b · âw â âw =w
Ï(âu)
â
i,j,k
(âiâk)(Ï)(âu) âjâku âjâiu , (2.23)
and we are left to find appropriate functions Ï for which the quadratic form on theright-hand side of (2.23) will be negative.
Let Ί â C2(R) be a convex and even function such that Ί(0) = 0 and ΊâČ(s) â„ 0for s â„ 0. For Δ â (0,â) and Ο â R
N , |Ο| †ââu0ââ, we put
Ï(Ο) := Δ +N
4
(Ί(N1/2 ââu0ââ
)â
Nâ
i=1
Ί(Οi)
), (2.24)
and note that Ï(Ο) ℠Δ and (âiâkÏ)(Ο) = 0 if i 6= k and (â2i Ï)(Ο) = âN ΊâČâČ(Οi)/4.
Then (2.23) reads
âtw â b · âw â âw +N
4Ï(âu)
â
i,j
ΊâČâČ(âiu) (âiâju)2 w = 0 . (2.25)
Before going further, let us recall that, by the Cauchy-Schwarz inequality,
Nâ
i,j
(âiâju)2 â„ Nâ
i
(â2i u)2 â„ (âu)2 . (2.26)
(a) Ί(s) = s2/”, ” > 0 . Then ΊâČâČ(s) = 2/” and
(âu)2 = (Ï(âu) w + A + ÎŽÏ F (âu))2
.
We then infer from (2.25) that
L w := âtw â b · âw â âw +Ï(âu)
2”w3 +
A + ÎŽÏ F (âu)
”w2
+1
2”Ï(âu)
N
â
i,j
(âiâju)2 â (âu)2 + (A + ÎŽÏ F (âu))2
w = 0 .
We first take Δ = ”, ÎŽ = âÏ and A = F (ââu0ââ) = ââu0âqâ. Then Ï(âu) ℠”
and it follows from (2.3) and (2.26) that L(tâ1/2
)â„ 0. The comparison principle
then entails that w(t, x) †tâ1/2 for (t, x) â Qâ, hence
Ï âtu(t, x) â„ âââu0âqâ â Ï(âu)
t1/2â„ âââu0âq
â â(
” +N2 ââu0â2
â
4”
)tâ1/2 .
Choosing ” = ââu0ââ, we conclude that
Ï âtu(t, x) â„ âââu0âqâ â C ââu0ââ tâ1/2
for (t, x) â Qâ. The equation (1.1) being autonomous, we infer from the previousinequality that
Ï âtu(t, x) â„ âââu(t/2)âqâ â
â2 C ââu(t/2)ââ tâ1/2 ,
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110 2. WELL-POSEDNESS AND SMOOTHING EFFECTS
and we use (2.4) and (2.5) to obtain that
Ï âtu(t, x) â„ âC âu0ââ tâ1 , (t, x) â Qâ , (2.27)
if q 6= 1 and
Ï âtu(t, x) â„ âC âu0ââ(tâ1/2 + tâ1
), (t, x) â Qâ , (2.28)
if q = 1.We next take Δ = ”, ÎŽ = Ï and A = 0. Then Ï(âu) ℠” and it follows from
(2.3) and (2.26) that L(tâ1/2
)â„ 0. The comparison principle then entails that
w(t, x) †tâ1/2 for (t, x) â Qâ, hence
Ï âtu(t, x) †Ï(âu)
t1/2â€(
” +N2 ââu0â2
â
4”
)tâ1/2 .
Choosing ” = ââu0ââ, we conclude that
Ï âtu(t, x) †C ââu0ââ tâ1/2
for (t, x) â Qâ and we proceed as before (with the help of (2.4) and (2.5)) toconclude that
Ï âtu(t, x) †C âu0ââ tâ1 , (t, x) â Qâ . (2.29)
(b) ΊâČâČ(s) = 1/(A + |s|q) . In that case, ΊâČâČ is non-increasing. Consequently,
ΊâČâČ(âiu) ℠ΊâČâČ(|âu|) and
(âu)2 = (Ï(âu) w + A + ÎŽÏ F (âu))2
= 4 Ï(âu) (A + ÎŽÏ F (âu)) w + (Ï(âu) w â A â ÎŽÏ F (âu))2
.
We then infer from (2.25) that
L w := âtw â b · âw â âw + ΊâČâČ(|âu|) (A + ÎŽÏ F (âu)) w2
+ΊâČâČ(|âu|)4Ï(âu)
Nâ
i,j
ΊâČâČ(âiu)
ΊâČâČ(|âu|) (âiâju)2 â (âu)2
w
+ΊâČâČ(|âu|)4Ï(âu)
[Ï(âu) w â A â ÎŽÏ F (âu)]2
w = 0 .
Taking ÎŽ = Ï yields that ΊâČâČ(|âu|) (A + ÎŽÏ F (âu)) = 1 and thus L(tâ1)â„ 0. We
then deduce from the comparison principle that
Ï âtu(t, x) †A +
(Δ +
N
4Ί(N1/2ââu0ââ
))tâ1
for (t, x) â Qâ. Letting Δ â 0 in the above inequality, we end up with
Ï âtu(t, x) †A +N
4Ί(N1/2ââu0ââ
)tâ1 , (t, x) â Qâ . (2.30)
If q â (0, 1), we may take A = 0 to obtain that Ί(s) = s2âq/((2 â q)(1 â q))and infer from (2.30) that
Ï âtu(t, x) †C ââu0â2âqâ tâ1 , (t, x) â Qâ .
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3. HESSIAN ESTIMATES 111
Using once more the fact that (1.1) is an autonomous equation and (2.5) we finallyobtain
Ï âtu(t, x) †C âu0â(2âq)/qâ tâ2/q , (t, x) â Qâ . (2.31)
If q = 1 we have
Ί(s) =
â« s
0
s â z
A + zdz †s log
(1 +
s
A
), s â„ 0 ,
which, together with (2.30), yields
Ï âtu(t, x) †A + C ââu0ââ log
(1 +
ââu0ââA
)tâ1 , (t, x) â Qâ .
The choice A = ââu0ââ log (2 + t) tâ1 then gives
Ï âtu(t, x) †C ââu0ââ1 + log (1 + t)
t, (t, x) â Qâ .
Owing to (2.4), we may proceed as before to conclude that
Ï âtu(t, x) †C âu0ââ1 + log (1 + t)
t3/2, (t, x) â Qâ . (2.32)
Finally, if q > 1, we have
Ί(s) =
â« s
0
s â z
A + zqdz †Aâ(qâ1)/q s
â« â
0
dz
1 + zq, s â„ 0 .
Choosing A = ââu0âq/(2qâ1)â tâq/(2qâ1) we infer from (2.30) that
Ï âtu(t, x) †C ââu0âq/(2qâ1)â tâq/(2qâ1) , (t, x) â Qâ .
Recalling (2.5) we may proceed as before to conclude that
Ï âtu(t, x) †C âu0â1/(2qâ1)â tâ(q+1)/(2qâ1) , (t, x) â Qâ , (2.33)
and thus complete the proof of Theorem 2.3.
3. Hessian estimates
Let us first handle the case q â (1, 2].
Proof of Theorem 2.4. For 1 †i, j †N , we put wij := Ï âiâju. It followsfrom (1.1) that
âtwij â âwij = âÏq âi
(|âu|qâ2
(Nâ
k=1
âku wjk
))
= âq |âu|qâ2Nâ
k=1
wik wjk â Ïq |âu|qâ2Nâ
k=1
âku âiwjk
â q (q â 2) |âu|qâ4
(Nâ
k=1
âku wik
) (Nâ
k=1
âku wjk
).(2.34)
Consider now Ο â RN \ 0 and set
w :=
Nâ
i=1
Nâ
j=1
wij Οi Οj .
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112 2. WELL-POSEDNESS AND SMOOTHING EFFECTS
Multiplying (2.34) by Οi Οj and summing up the resulting identities yield
âtw â âw = âq |âu|qâ2Nâ
k=1
(Nâ
i=1
wik Οi
)2
â Ïq |âu|qâ2 âu · âw
â q (q â 2) |âu|qâ4
Nâ
i=1
Nâ
j=1
âju wij Οi
2
. (2.35)
Thanks to the following inequalities
|âu|qâ4
Nâ
i=1
Nâ
j=1
âju wij Οi
2
†|âu|qâ4Nâ
j=1
|âju|2Nâ
j=1
(Nâ
i=1
wij Οi
)2
†|âu|qâ2Nâ
k=1
(Nâ
i=1
wik Οi
)2
,
and
w2 †|Ο|2Nâ
k=1
(Nâ
i=1
wik Οi
)2
,
and since q †2, the right-hand side of (2.35) can be bounded from above. We thusobtain
âtw â âw †âq (q â 1) |âu|qâ2Nâ
k=1
(Nâ
i=1
wik Οi
)2
â Ïq |âu|qâ2 âu · âw
†âÏq |âu|qâ2 âu · âw â q (q â 1) |âu|qâ2
|Ο|2 w2 .
Consequently,
Lw †0 in RN Ă (0,â) , (2.36)
where the parabolic differential operator L is given by
Lz := zt â âz + Ïq |âu|qâ2 âu · âz +q (q â 1) |âu|qâ2
|Ο|2 z2 .
On the one hand, since q â (1, 2] and |âu(t, x)| †ââu0ââ, it is straightforwardto check that
W1(t) :=
(1
âw(0)ââ+
q (q â 1) t
|Ο|2 ââu0â2âqâ
)â1
, t > 0 ,
satisfies LW1 â„ 0 with W1(0) â„ w(0, x) for all x â RN . The comparison principle
then entails that w(t, x) †W1(t) for (t, x) â Qâ, from which we conclude that
w(t, x) †âw(0)ââ †âD2u0ââ |Ο|2 ,
whence (2.10). Observe that we also obtain that
w(t, x) †|Ο|2 ââu0â2âqâ
q (q â 1) tfor (t, x) â Qâ . (2.37)
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3. HESSIAN ESTIMATES 113
On the other hand, we infer from (2.5) that
W2(t) :=2 C2âq
2 |Ο|2q2 (q â 1)
âu0â(2âq)/qâ
t2/q, t > 0 ,
satisfies LW2 â„ 0 with W2(0) = â â„ w(0, x) for all x â RN . We then use again
the comparison principle as above and obtain (2.9).
For further use, we report the following weaker version of Theorem 2.4.
Corollary 2.6. Under the assumptions of Theorem 2.4 there is C9 such that
Ï âu(t, x) †C9 âu0â(2âq)/qâ
t2/q, (2.38)
for (t, x) â (0,â) Ă RN .
In addition, if u0 â W 2,â(RN ),
supxâRN
Ï âu(t, x) †supxâRN
Ï âu0(x) , t â„ 0 . (2.39)
Proof. Consider i â 1, . . . , N and define Οi = (Οij) â R
N by Οii = 1 and
Οij = 0 if j 6= i. We take Ο = Οi in (2.36) and obtain that L(Ï â2
i u) †0, that is,
ât
(Ï â2
i u)ââ
(Ï â2
i u)+Ïq |âu|qâ2 âu·â
(Ï â2
i u)+q (qâ1) |âu|qâ2
(Ï â2
i u)2 †0
in Qâ. Summing the above inequality over i â 1, . . . , N and recalling that
|âu|2 †N
Nâ
i=1
(â2
i u)2
,
we end up with
(Ï âu)t â â(Ï âu) + Ïq |âu|qâ2 âu · â (Ï âu) +q (q â 1) |âu|qâ2
N|Ï âu|2 †0
in Qâ. We next proceed as in the proof of Theorem 2.4 to complete the proof ofCorollary 2.6.
We next establish the log-semiconvexity of u in the particular case Ï = â1 andq = 1.
Proof of Proposition 2.5. We follow the approach of [34] and introducev := log u. We deduce from (1.1) and the positivity of u [32, Corollary 4.2] that vsolves
âtv â âv = |âv| + |âv|2 in Qâ . (2.40)
We next put wij := âiâjv for 1 †i, j †N , P := 1/|âv| and deduce from (2.40)that wij solves
âtwij = âwij + 2
Nâ
k=1
wik wjk + 2 âv · âwij + P
Nâ
k=1
wik wjk
+ P âv · âwij â P 3
(Nâ
k=1
wik âkv
) (Nâ
l=1
wjl âlv
)
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114 2. WELL-POSEDNESS AND SMOOTHING EFFECTS
Consider now Ο â RN \ 0 and put
w :=
Nâ
i,j=1
wij Οi Οj .
Then
âtw = âw + 2
Nâ
k=1
(Nâ
i=1
wik Οi
)2
+ 2 âv · âw + P
Nâ
k=1
(Nâ
i=1
wik Οi
)2
+ P âv · âw â P 3
Nâ
i,k=1
wik Οi âkv
2
.
By the Cauchy-Schwarz inequality, we have
Nâ
i,k=1
wik Οi âkv
2
†|âv|2Nâ
k=1
(Nâ
i=1
wik Οi
)2
.
Therefore, we may estimate from below the last term of the right-hand side of theequation satisfied by w and obtain
âtw â„ âw + (2 + P ) âv · âw + 2Nâ
k=1
(Nâ
i=1
wik Οi
)2
.
Using once more the Cauchy-Schwarz inequality, we realize that
w2 †|Ο|2Nâ
k=1
(Nâ
i=1
wik Οi
)2
,
whence
âtw â„ âw + (2 + P ) âv · âw +2 w2
|Ο|2 .
The comparison principle then ensures that
w(t, x) â„ â|Ο|22t
, (t, x) â Qâ ,
which gives the result.
4. Existence
Two different approaches have been used to investigate the well-posedness of(1.1), (1.2), one relying on the comparison principle and the previous gradient esti-mates [11, 32] and the other one on the integral formulation (2.2) and the smooth-ing properties of the heat semigroup (etâ)tâ„0 [20]. Both methods actually allow tohandle different classes of initial data but in different ranges of the parameter q.
We sketch the first approach which is actually a compactness method and is thusbased on a sequence of approximations to (1.1), (1.2). In view of the transformation(1.14), we only need to consider the case Ï = 1. We thus assume that Ï = 1 and
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4. EXISTENCE 115
consider a non-negative function u0 â BC(RN ). There is a sequence of functions(u0,k)kâ„1 such that, for each integer k â„ 1, u0,k â BCâ(RN ),
0 †u0,k(x) †u0,k+1(x) †u0(x) , x â RN , (2.41)
and (u0,k) converges uniformly towards u0 on compact subsets of RN . In addition,
if u0 â W 1,â(RN ) we may assume that
ââu0,kââ â€(
1 +K1
k
)ââu0ââ , (2.42)
for some constant K1 > 0 depending only on the approximation process. Next,since Ο 7ââ |Ο|q, Ο â R
N , is not regular enough for small values of q, we set
FΔ(r) :=(Δ2 + r
)q/2 â Δq , r â„ 0 , (2.43)
for Δ â (0, 1). Then the Cauchy problem
âtuk,Δ â âuk,Δ + FΔ
(|âuk,Δ|2
)= 0 , (t, x) â Qâ , (2.44)
uk,Δ(0) = u0,k + Δ , x â RN , (2.45)
has a unique classical solution uk,Δ â C(3+α)/2,3+α([0,â)ĂRN ) for some α â (0, 1)
[38]. Observing that Δ and âu0ââ + Δ are solutions to (2.44) with Δ †uk,Δ(0, x) â€âu0ââ + Δ, the comparison principle warrants that
Δ †uk,Δ(t, x) †âu0ââ + Δ , (t, x) â [0,â) Ă RN .
Moreover, arguing as in Section 1, we obtain similar gradient estimates for uk,Δ
which does not depend on k â„ 1 and have a mild dependence on Δ â (0, 1). More
precisely, introducing vk,Δ := fâ1 (uk,Δ) and wk,Δ := |âvk,Δ|2 for some strictlymonotone function f to be specified, we proceed along the lines of Section 1 toestablish that:
âą if q > 1 (with f(r) = rq/(qâ1)),
âŁâŁâŁâu(qâ1)/qk,Δ (t, x)
âŁâŁâŁ â€(
q â 1
2
)1/2
(âu0ââ + Δ)(qâ1)/q
tâ1/2 , t > 0 ,
âą if q > 2 (with f(r) = rq/(qâ1)),
âŁâŁâŁâu(qâ1)/qk,Δ (t, x)
âŁâŁâŁ â€(
2
qK2(η)+
K3(η)
K2(η)Δ(qâ1)/2
)1/q
tâ1/q , t â(0, Δ(1âq)/2
),
for every η â (0, q â 1) with
K2(η) :=2(q â 1 â η)
q
(q
q â 1
)q
and K3(η) :=4
q
(q â 2
η
)(qâ2)/2
,
âą if q â (1, 2] (with f(r) = rq/(qâ1)),
âŁâŁâŁâu(qâ1)/qk,Δ (t, x)
âŁâŁâŁ †q â 1
q
(1
q â 1+ Δ(qâ1)/2
)1/q
tâ1/q , t â(0, Δ(1âq)/2
),
âą if q â (0, 1] (with f(r) = âu0ââ + Δ + Δq/2 â r2),
|âuk,Δ(t, x)| â€(2 + 2Δq/4
)1/2 (âu0ââ + Δq/2
)tâ1/2 , t â
(0, Δâq/4
),
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116 2. WELL-POSEDNESS AND SMOOTHING EFFECTS
âą if q â (0, 1) (with f(r) = âu0ââ + Δ + Δq/2 â r2),
|âuk,Δ(t, x)| â€(
2 + q Δq/4
2q q(1 â q)
)1/q (âu0ââ + Δ + Δq/2
)1/q
tâ1/q , t â(0, Δâq/4
).
In addition, we have
ââuk,Δ(t)ââ †ââu0,kââ , t â„ 0 .
Combining these estimates with classical parabolic regularity results we firstlet Δ â 0 and then k â â to obtain the existence part of Theorem 2.1, togetherwith Theorem 2.2. The proofs of Theorems 2.3, 2.4 and Proposition 2.5 are donein a similar way.
5. Uniqueness
So far, given u0 â BC(RN ), we have constructed a solution u to (1.1), (1.2) en-joying the properties stated in Theorems 2.2, 2.3, 2.4 and Proposition 2.5. To provethe uniqueness assertion of Theorem 2.1, we follow the proof of [32, Theorem 4]and show that any other solution to (1.1), (1.2) in the sense of Theorem 2.1 coin-cides with the solution u already obtained in the previous section. More precisely,assume that
v â BC([0,â) Ă RN ) â© C1,2(Qâ)
is such that
ÎŽ âtv â âv + Ï |âv|q â„ 0 , (t, x) â Qâ , (2.46)
and
ÎŽ v(0, x) â„ ÎŽ u0(x) , x â RN , (2.47)
for some ÎŽ â â1, +1.Then we claim that
ÎŽ v(t, x) â„ ÎŽ u(t, x) , (t, x) â [0,â) Ă RN . (2.48)
Indeed, fix T > 0, Δ â (0, 1) and put α := min 1/q, 1 and
AΔ :=
2q
[N T (qâ1)/q + q
(C2 âu0â1/q
â + Δ T 1/q)qâ1
]Δ if q > 1 ,
2 (Δq + N Δ) if q â (0, 1] .
Introducing
z(t, x) := ÎŽ (u â v)(t, x) â AΔ tα â Δ(1 + |x|2
)1/2, (t, x) â [0, T )Ă R
N ,
we infer from the boundedness of u and v and (2.47) that there is RΔ > 0 such thatz(t, x) †âΔ < 0 for t â [0, T ] and |x| â„ RΔ and also for (t, x) â 0 Ă R
N . Conse-quently, if z is positive somewhere in [0, T ]ĂR
N it must have a positive maximumat some point (t0, x0) â (0, T ] Ă |x| †RΔ. This implies that âz(t0, x0) = 0,âz(t0, x0) †0 and âtz(t0, x0) â„ 0 so that
âtz(t0, x0) â âz(t0, x0) â„ 0 . (2.49)
Now, by (1.1) and (2.46) we have
âtz(t0, x0) â âz(t0, x0) â€âŁâŁâŁ|âv(t0, x0)|q â |âu(t0, x0)|q
âŁâŁâŁâ α AΔ tαâ10 + N Δ
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BIBLIOGRAPHICAL NOTES 117
and we use the relation âz(t0, x0) = 0 to compute âv(t0, x0) and obtain
âtz(t0, x0) â âz(t0, x0) â€âŁâŁâŁâŁâŁ
âŁâŁâŁâŁâŁâu(t0, x0) â ΎΔx0
(1 + |x0|2)1/2
âŁâŁâŁâŁâŁ
q
â |âu(t0, x0)|qâŁâŁâŁâŁâŁ
â α AΔ tαâ10 + N Δ .
Either q > 1 and it follows from the mean value theorem, (2.5) and the choice ofAΔ that
âtz(t0, x0) â âz(t0, x0) †q
(|âu(t0, x0)| +
Δ |x0|(1 + |x0|2)1/2
)qâ1Δ |x0|
(1 + |x0|2)1/2
âAΔ
qtâ(qâ1)/q0 + N Δ
†q Δ(C2 âu0â1/q
â tâ1/q0 + Δ
)qâ1
â AΔ
qtâ(qâ1)/q0 + N Δ
†âAΔ
2qt뱉10 < 0 ,
which contradicts (2.49). Or q â (0, 1] and we deduce from the Holder continuityof r 7â rq and the choice of AΔ that
âtz(t0, x0) â âz(t0, x0) â€(
Δ |x0|(1 + |x0|2)1/2
)q
â AΔ + N Δ †âAΔ
2< 0 ,
again contradicting (2.49). Therefore, z cannot take positive values in [0, T ]Ă RN
and thus
Ύ u(t, x) †Ύ v(t, x) + AΔ tα + Δ(1 + |x|2
)1/2, (t, x) â [0, T )Ă R
N .
Since AΔ â 0 as Δ â 0 we may pass to the limit as Δ â 0 in the previous inequalityand conclude that (2.48) holds true.
The uniqueness statement of Theorem 2.1 then readily follows.
Bibliographical notes
The gradient estimates (2.5) and (2.7) (with Ï = 1) are also true for non-negative viscosity solutions to the non-diffusive Hamiton-Jacobi equation (1.9): thisfact is proved in [42] by a different method, still using the comparison principle.No such gradient estimate seems to be available for q = 1 as a consequence of thelack of strict convexity (or concavity) of |âu|.
An alternative approach to study the well-posedness of (1.1), (1.2) in Lebesguespaces Lp(RN ), p â„ 1, is employed in [20]. It relies on the formulation (2.2) of(1.1), (1.2) which is the starting point of a fixed point procedure. This approach isrestricted to q â [1, 2) but provides the existence and uniqueness of a weak solution(which is classical for positive times) for u0 â Lr(RN ) whenever r â„ 1 and
r >N(q â 1)
2 â qor r =
N(q â 1)
2 â q> 1 .
When q = 1, previous existence and uniqueness results were obtained in [15, 16,17, 18].
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CHAPTER 3
Extinction in finite time
Let us first recall that, if X is a vector space, a solution f : [0,â) â X toan evolution problem enjoys the extinction in finite time property if there existsTâ > 0 such that
f(t) 6= 0 for t â [0, Tâ) ,
f(t) ⥠0 for t â„ Tâ .
A typical example of an evolution problem which exhibits the previous propertyis the ordinary differential equation f âČ(t) + f(t)Îł = 0 with f(0) = f0 > 0 andÎł â (0, 1). Then, for t â„ 0,
f(t) = (1 â Îł)1/(1âÎł) (Tâ â t)1/(1âÎł)+ with Tâ :=
f1âÎł0
1 â Îł.
Extinction in finite time also shows up for non-negative solutions to the fast diffusionequation âtv = âvm in R
N if m â (0, (N â2)+/N) and to the p-Laplacian equationâtv = div
(|âv|pâ2 âv
)in R
N if p â (0, 2N/(N +1)) (see, e.g., [22, 35]). Noticingthat the exponent below which extinction in finite time takes place is smaller thanone in the previous examples, it is rather natural to wonder whether extinctionphenomena could also occur for non-negative solutions to (1.1), (1.2) when q â(0, 1). It is however obvious that extinction in finite time cannot take place ifÏ = â1: indeed, in that case, we have u(t) â„ etâu0 for each t â„ 0 by the comparisonprinciple, which readily implies that u(t, x) > 0 for (t, x) â Qâ if u0 6⥠0.
Thus, throughout this chapter we assume that
Ï = 1 and q â (0, 1) . (3.1)
There are two different conditions on the initial condition u0 which warrant thatthe corresponding classical solution u to (1.1), (1.2) vanishes after a finite time: thefirst one requires some integrability at infinity for u0 while the second involves apointwise bound.
Theorem 3.1. Consider a non-negative function u0 â BC(RN ) and denote byu the corresponding classical solution to (1.1), (1.2). If u0 â L1(RN ; |x|m dx) forsome m â„ 0 and q â (0, (m + N)/(m + N + 1)), then there exists Tâ â [0,â) suchthat
u(t, x) = 0 for (t, x) â [Tâ,â) Ă RN .
Theorem 3.2. [14] Let u0 be a non-negative function in BC(RN ) satisfying
lim sup|x|ââ
|x|q/(1âq) u0(x) < â . (3.2)
119
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120 3. EXTINCTION IN FINITE TIME
Denoting by u the corresponding classical solution to (1.1), (1.2), there existsTâ â [0,â) such that
u(t, x) = 0 for (t, x) â [Tâ,â) Ă RN .
Let us first point out that the sets of initial data involved in Theorems 3.1and 3.2 have a non-empty intersection but are different. Indeed, both results applyfor compactly supported initial data but, if m â„ 0 and q â (0, (m+N)/(m+N +1)),a function u0 satisfying (3.2) need not be in L1(RN ; |x|m dx). Next, as we shall seebelow, the proofs of Theorems 3.1 and 3.2 rely on different arguments : indeed thelatter is achieved by constructing suitable supersolutions while the former followsfrom a differential inequality involving the Lâ-norm of u.
The next issue to be considered is the optimality of the algebraic growth con-dition (3.2). In that direction, we report the following result:
Theorem 3.3. [14] Consider u0 â BC(RN ) satisfying
u0(x) > 0, x â RN , (3.3)
lim|x|ââ
|x|q/(1âq) u0(x) = â . (3.4)
Denoting by u the corresponding classical solution to (1.1), (1.2), we have u(t, x) >0 for every (t, x) â Qâ.
It turns out that Theorem 3.3 also guarantees the optimality of the exponent(m + N)/(m + N + 1) in Theorem 3.1: indeed, if m â„ 0 and q â ((m + N)/(m +N +1), 1), there is at least a non-negative function u0 â L1(RN ; |x|m dx) such thatthe corresponding classical solution u to (1.1), (1.2) satisfies u(t, x) > 0 for every(t, x) â Qâ (see Corollary 3.10 below).
Remark 3.4. An explicit computation shows that, if T > 0 and K â [0,â),the function z defined by
z(t, x) := (T â t)1/(1âq)+
[K(1âq)/q +
(1 â q)(qâ1)/q
q|x|]âq/(1âq)
for (t, x) â Qâ, is a solution to the HamiltonâJacobi equation âtz + |âz|q = 0 inQâ. Observe that z(t, x) > 0 for every (t, x) â [0, T ) Ă R
N and z(T ) ⥠0. Thesupersolutions and subsolutions we will construct in the proofs of Theorems 3.2and 3.3 are somehow related to this function.
We finally establish a lower bound for the Lâ-norm of u near the extinctiontime.
Proposition 3.5. Let u0 be a non-negative function in BC(RN ) with compactsupport and u denote the corresponding classical solution to (1.1), (1.2). There isa positive real number Îș depending only on N , q and |supp u0| such that
âu(t)ââ â„ Îș (Tâ â t)1/(1âq)
, t â [0, Tâ] , (3.5)
where Tâ â (0,â) is the extinction time of u. Here, |supp u0| denotes the (Lebesgue)measure of supp u0.
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1. AN INTEGRAL CONDITION FOR EXTINCTION 121
1. An integral condition for extinction
Consider m â„ 0 and a non-negative function u0 â L1(RN ; |x|m dx) â© BC(RN ).We denote by u the corresponding classical solution to (1.1), (1.2) and first establishthat u(t) still belongs to L1(RN ; |x|m dx) for t > 0.
Lemma 3.6. There exists K1 > 0 depending only on m, N , q and u0 such thatâ«
RN
|x|m u(t, x) dx †K1 , t ℠0 . (3.6)
Proof. For x â RN and Δ > 0, we put bΔ(x) :=
(1 + Δ |x|2
)1/2. We first
multiply (1.1) by b1(x)m and integrate over RN : owing to the non-negativity of u
and Ï = 1, we obtain
d
dt
â«
RN
b1(x)m u(t, x) dx
†âm
â«
RN
b1(x)mâ2 x · âu(t, x) dx
â€â«
RN
Nm b1(x)mâ2 + m(m â 2) |x|2 b1(x)mâ4
u(t, x) dx
†C
â«
RN
b1(x)m u(t, x) dx ,
whence â«
RN
b1(x)m u(t, x) dx †C
â«
RN
b1(x)m u0(x) dx , t â [0, 1] . (3.7)
Next, for t ℠1, we multiply (1.1) by bΔ(x)m and integrate over RN to obtain
d
dt
â«
RN
bΔ(x)m u(t, x) dx +
â«
RN
bΔ(x)m |âu(t, x)|q dx
†mΔ
â«
RN
bΔ(x)mâ2 |x| |âu(t, x)| dx
†mΔ1/2 ââu(t)â1âqâ
â«
RN
bΔ(x)m |âu(t, x)|q dx .
Since ââu(t)ââ †C2 âu0â1/qâ for t â„ 1 by (2.5), we end up with
d
dt
â«
RN
bΔ(x)m u(t, x) dx
+(1 â m C1âq
2 âu0â(1âq)/qâ Δ1/2
) â«
RN
bΔ(x)m |âu(t, x)|q dx †0
for t ℠1. Choosing Δ = 1/(m2 C
2(1âq)2 âu0â2(1âq)/q
â
), we conclude that
â«
RN
bΔ(x)m u(t, x) dx â€â«
RN
bΔ(x)m u(1, x) dx , t ℠1 . (3.8)
Lemma 3.6 now readily follows from (3.7) and (3.8) as bΔ(x) ℠Δm |x|m.
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122 3. EXTINCTION IN FINITE TIME
Another useful tool for the proof of Theorem 3.1 is the following interpolationinequality:
Lemma 3.7. If w â L1(RN ; |x|m dx) â© Lâ(RN ), then w â L1(RN ) and thereis a constant K2 > 0 depending only on N such that
âwâ1 †K2 âwâm/(m+N)â
(â«
RN
|x|m w(x) dx
)N/(m+N)
.
Proof. Consider w â L1(RN ; |x|m dx) â© Lâ(RN ) and R > 0. Then
âwâ1 =
â«
|x|â€R
|w(x)| dx +
â«
|x|>R
|w(x)| dx
†C RN âwââ +1
Rm
â«
RN
|x|m w(x) dx .
Choosing
R =
(â«
RN
|x|m w(x) dx
)1/(m+N)
âwââ1/(m+N)â
yields Lemma 3.7.
Proof of Theorem 3.1. Owing to (2.5), (3.6), Lemma 3.7 and the GagliardoâNirenberg inequality
âwââ †C ââwâN/(N+1)â âwâ1/(N+1)
1 for w â L1(RN ) â© W 1,â(RN ) , (3.9)
we have for t > s â„ 0
âu(t)âqâ †C ââu(t)âqN/(N+1)
â âu(t)âq/(N+1)1
†C (t â s)âN/(N+1) âu(s)âN/(N+1)â âu(t)âqm/((N+1)(N+m))
â
Ă(â«
RN
|x|m u(t, x) dx
)qN/((N+1)(N+m))
†C (t â s)âN/(N+1) âu(s)âN/(N+1)â âu(t)âqm/((N+1)(N+m))
â ,
whence
âu(t)âqâ †C (t â s)â(m+N)/(m+N+1) âu(s)â(m+N)/(m+N+1)
â .
We multiply the above inequality by 1/t and integrate with respect to t over (s,â)to deduce that
Z(s) :=
â« â
s
âu(t)âqâ
tdt †C sâ(m+N)/(m+N+1) âu(s)â(m+N)/(m+N+1)
â .
Since dZ/ds(s) = ââu(s)âqâ/s, the above inequality also reads
s1âq dZ
ds(s) + C Z(s)q(m+N+1)/(m+N) †0 , s ℠0 .
Introducing Z(s) := Z(s1/q
)for s â„ 0, we readily obtain
dZ
ds(s) + C Z(s)q(m+N+1)/(m+N) †0 , s ℠0 ,
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2. A POINTWISE CONDITION FOR EXTINCTION 123
whence Z(s) = 0 for s sufficiently large since q < (m+N)/(m+N+1). Consequentlyu(t) = 0 for t large enough.
2. A pointwise condition for extinction
We introduce the following positive real numbers
α =2 â q
2(1 â q)and ÎČ = α â 1 =
q
2(1 â q). (3.10)
A classical tool to establish extinction in finite time is to construct supersolutionsto (1.1) enjoying such a property and in fact to look for self-similar supersolutions.
Lemma 3.8. Consider T â (0,â) and B â„ 1 such that
Bq/2 ℠α + ÎČ + 4 α (ÎČ + 1)
min ÎČ, (2ÎČ)q . (3.11)
The function WB,T defined by
WB,T (t, x) = (T â t)α fB
(|x| (T â t)â1/2
), (t, x) â [0, T ]Ă R
N ,
fB(y) =(A + B y2
)âÎČ, y â R, with A =
ÎČ
αB,
is a supersolution to (1.1) in [0, T )Ă RN .
Proof. We denote by L the parabolic operator defined by
Lw = âtw â âw + |âw|q , (3.12)
and put
y0 =
(ÎČ
α B
)1/2
.
Let (t, x) â [0, T )Ă RN and compute LWB,T (t, x). Putting y = |x| (T â t)â1/2 we
obtainLWB,T (t, x) = (T â t)αâ1
(A + B y2
)âÎČâ1H(y),
where
H(y) := 2 ÎČ N B â α A + (2 ÎČ B)q yq(A + B y2
)(2âq)/2
â(α + ÎČ) B y2 â 4 ÎČ (ÎČ + 1)B2 y2
A + B y2
â„ ÎČ B + (2 ÎČ)q B(2+q)/2 y2 â (α + ÎČ) B y2 â 4 ÎČ (ÎČ + 1)B2 y2
A + B y2.
Now, on the one hand, we have for y â [0, y0]
H(y) â„ ÎČ B â (α + ÎČ) B y20 â 4 ÎČ (ÎČ + 1)
B2 y20
A + B y20
â„ ÎČ B â (α + ÎČ)ÎČ
αâ 4 ÎČ (ÎČ + 1)
B
1 + B.
As ÎČ < α and B â„ Bq/2 â„ 1 we further obtain, thanks to (3.11),
H(y) â„ ÎČ B â (α + ÎČ) â 4 ÎČ (ÎČ + 1) â„ 0.
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124 3. EXTINCTION IN FINITE TIME
On the other hand, there holds for y â„ y0
H(y) â„ (2 ÎČ)q B(2+q)/2 y2 â (α + ÎČ) B y2 â 4 ÎČ (ÎČ + 1)B2 y2
A + B y2
â„(
(2 ÎČ)q Bq/2 â (α + ÎČ) â 4 ÎČ (ÎČ + 1)B
A + B y20
)B y2
â„(
(2 ÎČ)q Bq/2 â (α + ÎČ) â 4 α (ÎČ + 1)B
1 + B
)B y2,
hence H(y) â„ 0 by (3.11). The proof of Lemma 3.8 is thus complete.
Proof of Theorem 3.2. Owing to (3.2) and the boundedness of u0 there isa constant C0 > 0 such that
u0(x) †C0 (1 + |x|)âq/(1âq), x â R
N .
Since (1 + |x|)2 â„ 1 + |x|2 for x â RN we further obtain that u0 satisfies
u0(x) †C0
(1 + |x|2
)âÎČ, x â R
N ,
the real number ÎČ being defined in (3.10). We next fix B â„ 1 such that (3.11) holdstrue and consider T > 0 such that
T â„ 1 and T â„ C1/α0 BÎČ/α. (3.13)
With this choice of the parameters T and B the function WB,T defined in Lemma 3.8is a supersolution to (1.1). In addition, α > ÎČ and we infer from (3.13) that, forx â R
N ,
WB,T (0, x) = T α BâÎČ
(ÎČ
α+
|x|2T
)âÎČ
â„ C0
(1 + |x|2
)âÎČ â„ u0(x) .
The comparison principle [32, Theorem 4] then ensures that u(t, x) †WB,T (t, x) for(t, x) â [0, T ]ĂR
N . Therefore, since u is non-negative and WB,T vanishes identicallyat time T , we conclude that u(T ) ⥠0, whence u(t, x) = 0 for (t, x) â [T,â)ĂR
N .
3. Non-extinction
The proof of Theorem 3.3 also relies on a comparison argument, but now withsuitable subsolutions.
Lemma 3.9. Let T , a and b be three positive real numbers and define
wa,b,T (t, x) = (T â t)1/(1âq)(a + b |x|2
)âÎČ, (t, x) â [0, T ]Ă R
N ,
the parameter ÎČ being defined in (3.10). If
Ï := bq/2 qq (1 â q)1âq â 1 < 0 and b †(1 â q) |Ï |N q (N + 2 â q)
a
T, (3.14)
then the function wa,b,T is a subsolution to (1.1) on [0, T ]Ă RN .
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3. NON-EXTINCTION 125
Proof. The parabolic operator L being defined by (3.12), we compute Lwa,b,T .Let (t, x) â [0, T ]Ă R
N and put y = a + b |x|2. Since b|x|2 †y, we have
Lwa,b,T (t, x) †(T â t)q/(1âq)
1 â qyâ2âq/(2â2q) F (y),
where
F (y) := Ï y2 + b q (T â t)
(a (2 â q)
1 â q+
(N â 2) â q (N â 1)
1 â qy
).
Now we have
F (y) â€ Ï y2 + b q (T â t)
(a (2 â q)
1 â q+
N
1 â qy
)
â€ Ï y2 +b N q T
1 â qy +
a b q (2 â q) T
1 â q=: G(y).
As GâČâČ = 2 Ï < 0 by (3.14) and y â„ a, we infer from (3.14) that
GâČ(y) †GâČ(a) = 2 Ï a +b N q T
1 â q†0 .
Consequently, G is a non-increasing function on [a,â) and (3.14) ensures that
G(y) †G(a) = a
(Ï a +
b q T
1 â q(N + 2 â q)
)†0
for y â [a,â). Therefore Lwa,b,T (t, x) †0 and the proof of Lemma 3.9 is complete.
Proof of Theorem 3.3. We fix b â (0, 1) such that bq/2 qq (1â q)1âq †1/2and T > 0. On the one hand it follows from (3.4) that there is RT > 0 such that
u0(x) â„(b Tâ2/q |x|2
)âq/(2â2q)
, |x| â„ RT . (3.15)
On the other hand the continuity of u0 and (3.3) ensure that
mT := min|x|â€RT
u0(x) > 0 .
We then define a(T ) by
a(T ) = T 2/q mâ(2â2q)/qT +
N q (N + 2 â q)
(1 â q) |Ï | b T, (3.16)
where Ï is defined in (3.14) and notice that(a(T ) Tâ2/q
)âq/(2â2q)
†mT . (3.17)
Owing to (3.16) and Lemma 3.9 the function wa(T ),b,T is a subsolution to (1.1) on
[0, T ] à RN and we infer from (3.15) and (3.17) that wa(T ),b,T (0, x) †u0(x) for
x â RN . The comparison principle [32, Theorem 4] then entails that
wa(T ),b,T (t, x) †u(t, x), (t, x) â [0, T ]Ă RN .
Therefore u(T/2, x) > 0 for each x â RN and, as T is arbitrary in (0,â), Theo-
rem 3.3 follows.
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126 3. EXTINCTION IN FINITE TIME
Corollary 3.10. Consider m â„ 0 and q â ((m + N)/(m + N + 1), 1). Thenthere is at least a non-negative function u0 â BC(RN ) â© L1(RN ; |x|m dx) suchthat the corresponding classical solution u to (1.1), (1.2) satisfies u(t, x) > 0 for(t, x) â [0,â) Ă R
N .
In other words, the assertion of Theorem 3.1 is false for q â ((m + N)/(m +N + 1), 1).
Proof. The condition on q implies that m + N < q/(1 â q). We may thenchoose Ï â (0,â) such that
m + N <q
1 â qâ Ï .
We next put
u0(x) = (1 + |x|)Ïâq/(1âq), x â R
N .
Thanks to the choice of Ï, u0 satisfies the assumptions of Theorem 3.3 and u0
belongs to L1(RN ; |x|m dx), which yields the expected result.
4. A lower bound near the extinction time
Since u0 is compactly supported, it clearly satisfies (3.2) and Theorem 3.2implies that Tâ < â. Introducing the positivity set
P(t) :=x â R
N : u(t, x) > 0
of u at time t â„ 0, we proceed as in [31, Theorem 9] to prove that
P(t) â
x â RN : d(x,P(0)) <
(âu0ââA0
)(1âq)/(2âq)
(3.18)
where A0 := (1 â q)(2âq)/(1âq) (N(1 â q) + q)â1/(1âq)/(2 â q). Indeed, con-sider x0 â R
N such that d(x0,P(0))(2âq)/(1âq) â„ âu0ââ/A0. Introducing S(x) :=A0 |x â x0|(2âq)/(1âq) for x â R
N , we have u0(x) = 0 †S(x) if x 6â P(0) and
u0(x) †âu0ââ †A0 d(x0,P(0))(2âq)/(1âq) †A0 |x â x0|(2âq)/(1âq) = S(x)
if x â P(0), the last inequality following from the choice of x0. Therefore,u0(x) †S(x) for x â R
N and S is actually a stationary solution to (1.1). Thecomparison principle then entails that u(t, x) †S(x) for (t, x) â [0,â) Ă R
N . Inparticular, u(t, x0) †S(x0) = 0 for t ℠0 which, together with the non-negativityof u, implies that u(t, x0) = 0 for t ℠0 and completes the proof of (3.18).
Consider next t â [0, Tâ) and s â [0, t). On the one hand, the gradient estimate(2.5), (3.18) and the GagliardoâNirenberg inequality (3.9) give
âu(t)ââ †C ââu(t)âN/(N+1)â âu(t)â1/(N+1)
1
†C âu(s)âN/q(N+1)â (t â s)âN/q(N+1) âu(t)â1/(N+1)
â |P(t)|1/(N+1)
†C(|P(0)|) âu(s)âN/q(N+1)â (t â s)âN/q(N+1) âu(t)â1/(N+1)
â .
As t < Tâ, we have âu(t)ââ > 0 and we deduce from the previous inequality that
(t â s) âu(t)âqâ †C(|P(0)|) âu(s)ââ , 0 †s < t < Tâ . (3.19)
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BIBLIOGRAPHICAL NOTES 127
Now, for T â (0, Tâ), we put
m(T ) := infsâ[0,T )
âu(s)ââ(T â s)1/(1âq)
> 0 .
We infer from (3.19) that, for s â [0, T ) and t â (s, T ),
(t â s)(T â t)q/(1âq)
(T â s)1/(1âq)
( âu(t)ââ(T â t)1/(1âq)
)q
=t â s
(T â s)1/(1âq)âu(t)âq
â
†C(|P(0)|) âu(s)ââ(T â s)1/(1âq)
.
Choosing t = (1 â q)T + qs â (s, T ) in the previous inequality leads us to
C(|P(0)|) m(T )q †âu(s)ââ(T â s)1/(1âq)
, s â [0, T ) .
Therefore, C(|P(0)|) m(T )q †m(T ) and the positivity of m(T ) allows us toconclude that m(T ) â„ C(|P(0)|) > 0. We have thus proved that âu(s)ââ â„C(|P(0)|) (T â s)1/(1âq) for s â [0, T ) and T â (0, Tâ) with a positive constantC(|P(0)|) which does not depend on T . We then let T â Tâ in the previousinequality to complete the proof of Proposition 3.5.
Bibliographical notes
Theorem 3.1 improves [13, Theorem 1] which only deals with the case m = 0.The proof given here is also simpler. The extinction in finite time of non-negativesolutions to (1.1), (1.2) with a compactly supported initial condition u0 is alsoestablished in [31, Corollary 9.1] by a comparison argument but with a differentsupersolution (travelling wave). Proposition 3.5 seems to be new and is a first steptowards a better understanding of the behaviour near the extinction time.
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CHAPTER 4
Temporal decay estimates for integrable initial
data: Ï = 1
Throughout this section we assume that
Ï = 1 and u0 is a non-negative function in L1(RN ) â© BC(RN ) (4.1)
and denote by u the corresponding classical solution to (1.1), (1.2). We then in-vestigate the time behaviour of the Lp-norms of u for p = 1 and p = â. As a firststep, we check that u(t) actually remains in L1(RN ) for t > 0.
Lemma 4.1. If q > 0, then u â C([0,â), L1(RN )) and t 7ââ âu(t)â1 is anon-increasing function of time with
I1(â) := limtââ
âu(t)â1 â [0, âu0â1] . (4.2)
In addition, |âu|q â L1(Qâ) andâ« â
0
â«
RN
|âu(t, x)|q dxdt †âu0â1 . (4.3)
We now turn to the time evolution of the Lp-norms of u for p = 1 and p = â.The main issue here is to figure out whether the additional dissipative mechanism|âu|q speeds up the convergence to zero in Lp(RN ) for p = â. For that purpose,we introduce the following positive real numbers:
qâ :=N + 2
N + 1, (4.4)
a :=(N + 1)(qâ â q)
2(q â 1)for q â (1, qâ)
b :=1
q(N + 1) â Nfor q >
N
N + 1,
(4.5)
and first study the behaviour of the Lâ-norm of u.
Proposition 4.2. [13]
(a): If q = N/(N + 1), then there is a constant C10 such that
âu(t)ââ †C10 âu0ââ expâC10 âu0ââ1/(N+1)
1 t
, t â (0,â) . (4.6)
(b): If q â (N/(N + 1), qâ], then there is a constant C10 such that
âu(t)ââ †C10 âu0âqb1 tâNb , t â (0,â) . (4.7)
129
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130 4. TEMPORAL DECAY ESTIMATES FOR INTEGRABLE INITIAL DATA: Ï = 1
(c): If q â„ qâ, then there is a constant C10 such that
âu(t)ââ †C10 âu0â1 tâN/2 , t â (0,â) . (4.8)
Observe that, for q â [N/(N + 1), qâ) the temporal decay estimates (4.6) and(4.7) are faster for large times than the one which follows from the heat equation bycomparison, thus showing the influence of the absorption term when q < qâ. Thedecay to zero is even faster for q â (0, N/(N + 1)) as u vanishes identically after afinite time by Theorem 3.1 (with m = 0). For q > qâ, the absorption term does notspeed up the convergence of âu(t)ââ to zero.
We next investigate the behaviour of the L1-norm of u.
Proposition 4.3. [11, 14, 19] We have
I1(â) > 0 ââ q > qâ , (4.9)
where the critical exponent qâ is given by (4.4).
Recalling that the L1-norm of non-negative solutions to the linear heat equationremains constant (and positive if u0 6⥠0) throughout time evolution, a consequenceof Proposition 4.3 is that the absorption is sufficiently strong for q â (0, qâ] so asto drive the L1-norm of u(t) to zero as t â â.
When q â (1, qâ), it is actually possible to obtain more precise information onthe convergence to zero of âu(t)â1.
Proposition 4.4. [2] If q â (1, qâ), there is C11 such that
âu(t)â1 †C11
(â«
|x|â„t1/2u0(x) dx + tâa
), t â„ 0 . (4.10)
As a consequence of Proposition 4.4 we realize that, if the initial condition u0
is such that
supR>0
R2a
â«
|x|â„R
u0(x) dx
< â , (4.11)
then âu(t)â1 †C tâa for t > 0. In addition, since (1.1) is autonomous, we inferfrom Proposition 4.2 (b) that
âu(t)ââ †2Nb C10
â„â„â„â„u(
t
2
)â„â„â„â„qb
1
tâNb †C tâ(2a+N)/2
for t > 0. It is however not possible for a solution to (1.1), (1.2) to decay to zeroin Lâ(RN ) at a faster algebraic rate as the following result shows:
Proposition 4.5. [13, 19] Assume that
(a): either q â (1, qâ] and there is α > a such that
âu(t)âLâ †C tâ(N/2)âα for t â„ 1,
(b): or q = qâ and there is Îł > N + 1 such that
âu(t)âLâ †C tâN/2 (log t)âÎł for t â„ 1,
then u ⥠0.
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1. DECAY RATES 131
Combining the outcome of Proposition 4.4 and Proposition 4.5 we concludethat, if q â (1, qâ), there are initial data for which âu(t)â1 decays as tâa butâu(t)â1 cannot decay at a faster algebraic rate. This is in sharp contrast with thecase q = 1 for which exponential decay rates are possible [15, 16]. Nevertheless,âu(t)â1 also cannot decay arbitrarily fast in that case since
âu(t)â1 â„ C tâ3/2 eât/4 , t â„ 1 ,
if N = 1 andâu(t)â1 â„ eâ(Ct1/3+t)/4 , t â„ 1 ,
if N â„ 2.
1. Decay rates
We first state an easy consequence of the comparison principle.
Lemma 4.6. There is a constant C such that, for q â (0,â) and t > 0,
âu(t)ââ †C âu0â1 tâN/2 , (4.12)
ââu(t)ââ †C âu0â1 tâ(N+1)/2 . (4.13)
Proof. Since Ï = 1, u is a subsolution to the linear heat equation and thecomparison principle ensures that u(t, x) â€
(etâu0
)(x) for (t, x) â [0,â) Ă R
N .Since u0 is non-negative, we infer from the temporal decay estimates for integrablesolutions to the linear heat equation that
âu(t)ââ â€â„â„etâu0
â„â„â
†C âu0â1 tâN/2
for t > 0. It next follows from (2.4) with s = t/2 and the previous inequality that
ââu(t)ââ †C1
â„â„â„â„u(
t
2
)â„â„â„â„â
(2
t
)1/2
†C âu0â1 tâ(N+1)/2
for t > 0.
We next use the gradient estimate (2.5) to obtain another decay rate.
Lemma 4.7. There is a constant C such that, for q â (N/(N + 1),â), q 6= 1,and t > 0,
âu(t)ââ †C âu0âqb1 tâNb , (4.14)
ââu(t)ââ †C âu0âb1 tâ(N+1)b . (4.15)
Proof. For t > 0, we infer from (2.5) (with s = t/2), Lemma 4.1 and theGagliardoâNirenberg inequality (3.9) that
âu(t)ââ †C âu(t)â1/(N+1)1 ââu(t)âN/(N+1)
â
†C âu0â1/(N+1)1
â„â„â„â„u(
t
2
)â„â„â„â„N/q(N+1)
â
(2
t
)N/q(N+1)
.
Multiplying both sides of the above inequality by tNb leads to
tNb âu(t)ââ †C âu0â1/(N+1)1
(t
2
)Nb â„â„â„â„u(
t
2
)â„â„â„â„â
N/q(N+1)
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132 4. TEMPORAL DECAY ESTIMATES FOR INTEGRABLE INITIAL DATA: Ï = 1
for t > 0. Introducing
Ï(T ) := suptâ[0,T ]
tNb âu(t)ââ
for T > 0, it follows from the previous inequality that, for t â [0, T ],
tNb âu(t)ââ †C âu0â1/(N+1)1 Ï(T )N/q(N+1) .
Consequently,
Ï(T ) †C âu0â1/(N+1)1 Ï(T )N/q(N+1) ,
Ï(T ) †C âu0âqb1 ,
and thus
âu(t)ââ †C âu0âqb1 tâNb
for t â [0, T ]. Since T is arbitrary, we have proved (4.14). Using again (2.5) (withs = t/2) and (4.14), we further obtain that
ââu(t)ââ †C2
â„â„â„â„u(
t
2
)â„â„â„â„1/q
â
(2
t
)1/q
†C âu0âb1 tâ(N+1)b
for t > 0.
The temporal decay rates stated in (b) and (c) of Proposition 4.2 for q â(N/(N + 1),â), q 6= 1, follow from Lemma 4.6 for q â„ qâ and Lemma 4.7 forq < qâ. We next turn to the remaining cases q = N/(N + 1) and q = 1.
1.1. Decay rates: q = N/(N + 1). We proceed as in the proof of Lemma 4.7and infer from (2.5), Lemma 4.1 and the GagliardoâNirenberg inequality (3.9) that,for t > s > 0,
âu(t)ââ †C âu(t)â1/(N+1)1 ââu(t)âN/(N+1)
â
†C âu0â1/(N+1)1 âu(s)âN/q(N+1)
â (t â s)âN/q(N+1)
†C âu0â1/(N+1)1 âu(s)ââ (t â s)â1 .
Let B be a positive real number to be specified later and assume that t > B.Choosing s = tâB > 0 and multiplying both sides of the above inequality by et/B
lead to
et/B âu(t)ââ †C e
Bâu0â1/(N+1)
1
e(tâB)/B âu(tâ B)ââ
for t > B. Introducing
Ï(T ) := suptâ[0,T ]
et/B âu(t)ââ
for T > B, it follows from the previous inequality that, for t â (B, T ],
et/B âu(t)ââ †C e
Bâu0â1/(N+1)
1 Ï(T ) ,
while, for t â [0, B],
et/B âu(t)ââ †e âu0ââ
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1. DECAY RATES 133
by (2.1). Therefore,
Ï(T ) †C e
Bâu0â1/(N+1)
1 Ï(T ) + e âu0ââ .
Choosing B = 2 C e âu0â1/(N+1)1 we end up with
Ï(T ) †2 e âu0ââ .
As T is arbitrary we conclude that
âu(t)ââ †2 e âu0ââ exp
t
2 C e âu0â1/(N+1)1
for t â„ 0 and the proof of (4.6) is complete.
1.2. Decay rates: q = 1. We give two proofs of Proposition 4.2 for q = 1, thefirst one relying on a Moser technique [13] and the second one on the L1-euclideanlogarithmic Sobolev inequality.
First proof of Proposition 4.2: q = 1. We employ a Moser technique asin [26, Section 4]. Consider r â„ 1, s2 â (0,â) and s1 â [0, s2). It follows from (1.1)after multiplication by r urâ1 and integration over (s1, s2) Ă R
N that
âu(s2)ârr +
â« s2
s1
â«
RN
|â (ur) (s, x)| dxds †âu(s1)ârr .
We next use the Sobolev inequality to obtain thatâ« s2
s1
âur(s)â1â ds †C âu(s1)ârr, (4.16)
where 1â = â if N = 1 and 1â = N/(N â 1) otherwise.Fix t â (0,â). If N = 1 we choose r = 1, s1 = 0 and s2 = t in (4.16) and
use the monotonicity of s 7ââ âu(s)ââ to obtain (4.7) for q = 1. Assume now thatN â„ 2. As u is non-negative s 7â âu(s)âm is a non-increasing function for eachm â [1,â] and we infer from (4.16) that
âu(s2)ârN/(Nâ1) †C1/r (s2 â s1)â1/r âu(s1)âr, 0 †s1 < s2. (4.17)
Introducing for k â„ 0 the sequences rk = (N/(N â 1))k
and tk = t(1â2â(k+1)), weproceed as in [26, Section 4] and write (4.17) with r = rk, s1 = tk and s2 = tk+1.Arguing by induction we eventually arrive at the following inequality:
âu(t)ârk†âu(tk)ârk
†Cαk 2ÎČk tâαk âu(t/2)â1 (4.18)
for k â„ 1 with
αk :=
kâ1â
n=0
(N â 1
N
)n
and ÎČk :=
kâ1â
n=0
(n + 2)
(N â 1
N
)n
, k â„ 0.
We may now let k â â in (4.18) to obtain (4.7) for q = 1.
We now give an alternative proof of Proposition 4.2 for q = 1 employing theL1-euclidean logarithmic Sobolev inequality [8, Theorem 2] which we recall now.
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134 4. TEMPORAL DECAY ESTIMATES FOR INTEGRABLE INITIAL DATA: Ï = 1
Theorem 4.8. For each w â W 1,1(RN ) and ” â (0,â), we haveâ«
RN
|w(x)| log
( |w(x)|âwâ1
)dx + N log
(”e
NL1
)âwâ1 †” ââwâ1 (4.19)
with L1 := Î((N + 2)/2)1/N/(NÏ1/2
).
Second proof of Proposition 4.2: q = 1. Fix t â (0,â) and let âC1([0, t)) be an increasing function such that (0) = 1 and (s) â â as s â t.Introducing F (s) := âu(s)â(s) for s â [0, t), we infer from (1.1) that
F âČ(s)
F (s)=
âČ(s)
(s)2
â«
RN
u(s, x)(s)
F (s)(s)log
(u(s, x)(s)
F (s)(s)
)dx
+
â«
RN
u(s, x)(s)â1
F (s)(s)(âu(s, x) â |âu(s, x)|) dx
†âČ(s)
(s)2
â«
RN
u(s, x)(s)
F (s)(s)log
(u(s, x)(s)
F (s)(s)
)dx
â 1
(s)
â«
RN
âŁâŁâu(s)(s, x)âŁâŁ
F (s)(s)dx .
It then follows from (4.19) with w = u(s) and ” = (s)/âČ(s) that
F âČ(s)
F (s)†N
âČ(s)
(s)2log
(NL1
e
âČ(s)
(s)
), s â [0, t) .
With the choice (s) = t/(t â s), we obtain
F âČ(s)
F (s)†âN
tlog
(e
NL1(t â s)
), s â [0, t) ,
which yields after integration
log F (s) †log F (0) â N
t
(log
(e
NL1
)s + (t â s) â (t â s) log (t â s)
)
+ N (1 â log t)
for s â [0, t). We then let s â t in the previous inequality and end up with
log âu(t)ââ †log âu0â1 + N log (NL1) â N log t ,
that is, (4.7) for q = 1 with C10 = (NL1)N
.
2. Limit values of âuâ1
Lemma 4.9. [19, Lemma 3.2] If q > 1 and u0 6⥠0 then âu(t)â1 > 0 for everyt â„ 0.
Proof. Since u is a classical solution to âtu â âu + A · âu = 0 in Qâ withA := |âu|qâ2 âu â C(Qâ; RN ), the assertion of Lemma 4.9 readily follows fromthe strong maximum principle.
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2. LIMIT VALUES OF âuâ1 135
Proof of Proposition 4.3: q > qâ. Combining (2.6) (with the choice s =t/2) and (4.12) yields
â„â„â„âu(qâ1)/q(t)â„â„â„
Lâ
†C âu0â(qâ1)/q1 tâ(q(N+1)âN)/(2q) , t > 0 . (4.20)
Consider now s â (0,â) and t â (s,â). Since âu = (q/(qâ1)) u1/q âu(qâ1)/q,it follows from (1.1) and (4.20) that
âu(t)â1 = âu(s)â1 â(
q
q â 1
)q â« t
s
â«
RN
âŁâŁâŁâu(qâ1)/q(Ï, x)âŁâŁâŁq
u(Ï, x) dxdÏ
â„ âu(s)â1 â C
â« t
s
Ïâ(q(N+1)âN)/2 âu(Ï)â1 dÏ .
Owing to the monotonicity of Ï 7ââ âu(Ï)â1, we further obtain
âu(t)â1 â„ âu(s)â1
(1 â C
â« t
s
Ïâ(q(N+1)âN)/2 dÏ
).
Since q > qâ the right-hand side of the above inequality has a finite limit as t â â.We may then let t â â and use (4.2) to obtain
I1(â) â„ âu(s)âL1
(1 â C sâ(N+1)(qâqâ)/2
), s > 0 .
Consequently, for s large enough, we have I1(â) > âu(s)â1/2 which is positive byLemma 4.9.
Proof of Proposition 4.3: q â [1, qâ]. It first follows from (4.3) that
Ï(t) :=
â« â
t
ââu(s)âqq ds ââ
tââ0 . (4.21)
We next consider a Câ-smooth function in RN such that 0 ††1 and
(x) = 0 if |x| †1 and (x) = 1 if |x| ℠2 .
For R > 0 and x â RN we put R(x) = (x/R). We multiply (1.1) by R(x) and
integrate over (t1, t2) Ă RN to obtain
â«
RN
u(t2, x) R(x) dx â€â«
RN
u(t1, x) R(x) dx â 1
R
â« t2
t1
â( x
R
)âu(s, x) dxds .
Using the Holder inequality and the properties of we deduce that
â«
|x|â„2R
u(t2, x) dx â€â«
|x|â„R
u(t1, x) dx+C R(NqâNâq)/q Ï(t1)1/q (t2ât1)
(qâ1)/q .
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136 4. TEMPORAL DECAY ESTIMATES FOR INTEGRABLE INITIAL DATA: Ï = 1
Combining the above inequality with (4.7) and Lemma 4.1 yields
âu(t2)â1 =
â«
|x|â€2R
u(t2, x) dx +
â«
|x|â„2R
u(t2, x) dx
†C RN âu(t2)ââ +
â«
|x|â„R
u(t1, x) dx
+ C R(NqâNâq)/q Ï(t1)1/q (t2 â t1)
(qâ1)/q
â€â«
|x|â„R
u(t1, x) dx + C RN (t2 â t1)âNb
+ C R(NqâNâq)/q Ï(t1)1/q (t2 â t1)
(qâ1)/q .
Choosing
R = R(t1, t2) := Ï(t1)1/(q+N) (t2 â t1)
(qNb+qâ1)/(q+N)
we are led to
âu(t2)â1 â€â«
|x|â„R(t1,t2)
u(t1, x) dx
+ C Ï(t1)N/(q+N) (t2 â t1)
âqNb(N+1)(qââq)/(q+N) .
Since qâ â„ q â„ 1 > N/(N + 1) (so that b > 0) we may let t2 â â in the previousinequality to conclude that I1(â) †0 if q â [1, qâ) and I1(â) †C Ï(t1)
N/(qâ+N) ifq = qâ. We have used here that R(t1, t2) â â as t2 â â and that u(t1) â L1(RN ).Owing to the non-negativity of I1(â), we readily obtain that I1(â) = 0 if q â [1, qâ).When q = qâ, we let t1 â â and use (4.21) to conclude that I1(â) = 0 also inthat case.
Proof of Proposition 4.3: q â [N/(N + 1), 1). As in the previous proof weconsider a Câ-smooth function in R
N such that 0 ††1 and
(x) = 0 if |x| †1 and (x) = 1 if |x| ℠2 .
For R > 0 and x â RN we put R(x) = (x/R). We multiply (1.1) by R(x) and
integrate over (1, t) Ă RN to obtain
â«
|x|â„2R
u(t, x) dx â€â«
|x|â„R
u(1, x) dx
+ââââ
R
â« t
1
ââu(s)â1âqâ
â«
RN
|âu(s)|q dxds .
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3. IMPROVED DECAY RATES: q â (1, qâ) 137
We then infer from (2.5) and (4.3) that
â«
|x|â„2R
u(t, x) dx â€â«
|x|â„R
u(1, x) dx
+C
R
â« â
1
sâ(1âq)/q
â«
RN
|âu(s)|q dxds
â€â«
|x|â„R
u(1, x) dx +C
R
â« â
1
â«
RN
|âu(s)|q dxds
â€â«
|x|â„R
u(1, x) dx +C
R.
Therefore
âu(t)â1 †C RNâ„â„etâu0
â„â„â
+
â«
|x|â„R
u(1, x) dx +C
R,
and the choice R = R(t) :=â„â„etâu0
â„â„â1/(N+1)
âyields
âu(t)â1 â€â«
|x|â„R(t)
u(1, x) dx +â„â„etâu0
â„â„1/(N+1)
â.
Letting t â â gives the result sinceâ„â„etâu0
â„â„â
â 0 as t â â.
Proof of Proposition 4.3: q â (0, N/(N + 1)). As u(t) vanishes identicallyafter a finite time by Theorem 3.1 (with m = 0), we clearly have I1(â) = 0.
3. Improved decay rates: q â (1, qâ)
Proof of Proposition 4.4. We fix T > 0. For R > 0 and t â [0, T ], wehave
âu(t)â1 =
â«
|x|â€2R
u(t, x) dx +
â«
|x|>2R
u(t, x) dx .
On the one hand, it follows from (4.7) that
â«
|x|â€2R
u(t, x) dx †C RN âu(t)ââ †C RN
â„â„â„â„u(
t
2
)â„â„â„â„qb
1
tâNb . (4.22)
On the other hand, consider a Câ-smooth function ζ in RN such that 0 †ζ †1
and
ζ(x) = 0 if |x| †1 and ζ(x) = 1 if |x| ℠2 .
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138 4. TEMPORAL DECAY ESTIMATES FOR INTEGRABLE INITIAL DATA: Ï = 1
For x â RN we put ζR(x) = ζ(x/R). It follows from (1.1) and the Holder and
Young inequalities that
d
dt
â«
RN
ζR(x)q/(qâ1) u(t, x) dx +
â«
RN
ζR(x)q/(qâ1) |âu(t, x)|q dx
= â q
q â 1
â«
RN
ζR(x)1/(qâ1) âζR(x) · âu(t, x) dx
†C R((Nâ1)qâN)/q
(â«
RN
ζR(x)q/(qâ1) |âu(t, x)|q dx
)1/q
†1
2
â«
RN
ζR(x)q/(qâ1) |âu(t, x)|q dx + C R((Nâ1)qâN)/(qâ1) ,
whence, after integration with respect to time,â«
RN
ζR(x)q/(qâ1) u(t, x) dx â€â«
RN
ζR(x)q/(qâ1) u0(x) dx + C t R((Nâ1)qâN)/(qâ1) .
Using the properties of ζ and since t â [0, T ], we end up withâ«
|x|>2R
u(t, x) dx â€â«
|x|â„R
u0(x) dx + C T R((Nâ1)qâN)/(qâ1) . (4.23)
Collecting (4.22) and (4.23), we obtain
âu(t)â1 â€â«
|x|â„R
u0(x) dx + C T R((Nâ1)qâN)/(qâ1) (4.24)
+ C RN
â„â„â„â„u(
t
2
)â„â„â„â„qb
1
tâNb .
Now, we put
z(t) := t1/(qâ1) Râ1/b(qâ1) âu(t)â1 , t â [0, T ]
Y (T, R) := T 1/(qâ1) Râ1/b(qâ1)
(â«
|x|â„R
u0(x) dx + C T R((Nâ1)qâN)/(qâ1)
).
Multiplying both sides of (4.24) by t1/(qâ1) Râ1/b(qâ1) we obtain for t â [0, T ]
z(t) †Y (T, R)
(t
T
)1/(qâ1)
+ C
(tb
R
)(1âNb(qâ1)âqb)/b(qâ1) z
(t
2
)qb
†Y (T, R) + C
z
(t
2
)qb
as Nb(q â 1) â 1 + qb = 0 by the definition (4.5) of b. Consequently,
sup[0,T ]
z †Y (T, R) + C
(sup[0,T ]
z)qb
.
Since q > 1, we have qb < 1 and infer from the Young inequality that
sup[0,T ]
z †Y (T, R) + C +1
2sup[0,T ]
z
sup[0,T ]
z †2 Y (T, R) + C .
âtopics-in-mathematical-modelingâ â 2008/12/5 â 8:30 â page 139 â #151
BIBLIOGRAPHICAL NOTES 139
In particular, z(T ) †2 Y (T, R) + C, that is,
âu(T )â1 †2
â«
|x|â„R
u0(x) dx+C T R((Nâ1)qâN)/(qâ1) +C R1/((qâ1)b) Tâ1/(qâ1) .
The choice R = T 1/2 then yields (4.10).
We finally show that, when q â (1, qâ], the Lâ-norm of u cannot decay to zeroat an arbitrary fast rate.
Proof of Proposition 4.5. It follows from (1.1) that, for t â„ s â„ 2,
âu(s)â1 â âu(t)â1 †C
â« t
s
â„â„â„â(u(qâ1)/q
)(Ï)â„â„â„
q
ââu(Ï)â1 dÏ .
We infer from (2.6) (with s = Ï/2) and the time monotonicity of the L1-norm of uthat
âu(s)â1 â âu(t)â1 †C âu(s)â1
â« t
s/2
âu(Ï)âqâ1â Ïâq/2 dÏ .
But it is easy to check that the assumptions (a) or (b) imply that Ï 7â âu(Ï)âqâ1â Ïâq/2
belongs to L1(1,â). Consequently we may find s0 large enough such that
âu(s0)â1 â âu(t)â1 †1
2âu(s0)â1 for t â„ s0 .
We then pass to the limit as t â â and deduce from (4.9) that âu(s0)â1 â€2 I1(â) = 0, whence u(s0) ⥠0. Applying Lemma 4.9 we conclude that u ⥠0.
Bibliographical notes
For q â (1, 2), an alternative proof of Lemma 4.9 relying on Lp-estimates maybe found in [2, Lemma 4.1] (and applies also if the Laplacian is replaced by adegenerate diffusion). Proposition 4.2 is proved in [13] but the argument givenhere is simpler. The alternative proof of Proposition 4.2 for q = 1 built upon theL1-euclidean logarithmic Sobolev inequality is also new. Proposition 4.3 is provedin [15, 16] when q = 1 and in [1] when q â (1, qâ) and q = 2. The general case q > 1is performed in [11, 19] and is extended to q â (0, 1] in [13, 14]. Next, if q â (1, qâ)and u0 fulfils (4.11), we have âu(t)â1 †C tâa for t > 0 as already mentioned. Thisproperty was previously established in [12] under the stronger condition
R(u0) := inf
R > 0 such that sup
|x|â„R
|x|(2âq)/(qâ1) u0(x)
†γq
< â ,
the proof combining (2.7) and the comparison with the supersolution
x 7ââ Îłq |x|â(2âq)/(qâ1) with Îłq :=((q â 1)(2(1 + a))1/(qâ1)
)/(2 â q) .
Proposition 4.4 is proved in [2] with a slightly different proof. It unfortunatelydoes not cover the critical exponent q = qâ. In that case, it has been shown in[30] that, for initial data decaying sufficiently rapidly as |x| â â, then âu(t)â1 â€C (log t)â(N+1) and âu(t)ââ †C (log t)â(N+1) tâN/2 for t â„ 1, the proof relyingon completely different arguments (invariant manifold). Note that, according to
âtopics-in-mathematical-modelingâ â 2008/12/5 â 8:30 â page 140 â #152
140 4. TEMPORAL DECAY ESTIMATES FOR INTEGRABLE INITIAL DATA: Ï = 1
Proposition 4.4, tN/2âu(t)ââ cannot decay at a faster logarithmic rate. Finally,Proposition 4.5 is actually a slight improvement of [19, Corollary 3.5].
âtopics-in-mathematical-modelingâ â 2008/12/5 â 8:30 â page 141 â #153
CHAPTER 5
Temporal growth estimates for integrable initial
data: Ï = â1
Throughout this section we assume that
Ï = â1 and u0 is a non-negative function in L1(RN ) â© W 1,â(RN ) (5.1)
and denote by u the corresponding classical solution to (1.1), (1.2). We then in-vestigate the time behaviour of the Lp-norms of u for p = 1 and p = â. As a firststep, we specify conditions ensuring that u(t) remains in L1(RN ) for t > 0.
Lemma 5.1. If q â„ 1, then u â C([0,â), L1(RN )) and t 7ââ âu(t)â1 is anon-decreasing function of time with
I1(â) := limtââ
âu(t)â1 â [âu0â1,â] . (5.2)
In addition,
âu(t)â1 = âu0â1 +
â« t
0
ââu(s)âqq ds , t â„ 0 . (5.3)
Remark 5.2. It is not known whether Lemma 5.1 remains valid for q â (0, 1)when Ï = â1.
When Ï = â1, the nonlinear term |âu|q is a source term and thus slows downor even impedes the dissipation of the diffusion. In that case, the counterpart ofProposition 4.3 (though less complete and more complicated) reads:
Proposition 5.3. [9, 31, 39] We have
I1(â) â [âu0â1,â) and Iâ(â) = 0 if q â„ 2 ,
I1(â) â [âu0â1,â] and Iâ(â) â [0, âu0ââ] if q â (qâ, 2) ,
I1(â) = â and Iâ(â) â (0, âu0ââ] if q â [1, qâ] ,
(5.4)
where qâ is defined in (4.4) and
Iâ(â) := limtââ
âu(t)ââ â [0, âu0ââ] . (5.5)
In addition, if Iâ(â) > 0 then I1(â) = â and
âu(t)â1 â„ C tN/q , t â„ 0 . (5.6)
Observe that, if q â [1, qâ], the strength of the source term leads to the un-boundedness of the L1-norm of u(t) as t â â. The behaviour of âu(t)â1 is muchmore complex when q ranges in (qâ, 2) since it is unbounded for large times only for
141
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142 5. TEMPORAL GROWTH ESTIMATES FOR INTEGRABLE INITIAL DATA: Ï = â1
some initial data. Let us also emphasize at this point that the converse of the laststatement of Proposition 5.3 is false, that is, there is at least one initial conditionu0 such that I1(â) = â and Iâ(â) = 0, see Proposition 5.6 below.
Additional properties are gathered in the next proposition.
Proposition 5.4. [31]
(a): If q â (qâ, 2) and Iâ(â) = 0, then
âu(t)ââ †C tâ(2âq)/(2(qâ1)) , t > 0 . (5.7)
(b): If q â (1, 2) and Iâ(â) > 0, then
0 †t(2âq)/q (âu(t)ââ â Iâ(â)) †C âu0â(2âq)/qâ , t > 0 . (5.8)
(c): If q = 1, then
0 †t1/2
log t(âu(t)ââ â Iâ(â)) †C âu0ââ , t â„ 2 . (5.9)
1. Limit values of âuâ1 and âuââThis section is devoted to the proof of Proposition 5.3 which requires to handle
separately the cases q â„ 2, q â (qâ, 2) and q â [1, qâ].We first check the last statement of Proposition 5.3. We thus assume that
Iâ(â) > 0 and fix t > 0. For k â„ 1, let xk â RN (depending possibly on t) be
such that âu(t)ââ â 1/k †u(t, xk). For R > 0, it follows from (2.5) (with s = 0)and the time monotonicity of âuââ that
âu(t)â1 â„â«
|xâxk|â€R
u(t, x) dx
â„â«
|xâxk|â€R
(u(t, xk) â |x â xk| ââu(t)ââ) dx
â„ C
(RN
N
(âu(t)ââ â 1
k
)â C1
RN+1
N + 1âu0â1/q
â tâ1/q
)
â„ C RN
(Iâ(â) â 1
kâ C1 R tâ1/q
).
Letting k â â and choosing R = R(t) :=(Iâ(â)t1/q
)/(2C1) gives the result.
1.1. The case q â„ 2. We put
A := âu0ââ ââu0âqâ2â and B := ââu0âqâ2
â , (5.10)
and consider t > 0 and an integer p â„ 1. We multiply (1.1) by up and integrateover (0, t) Ă R
N to obtain
p
â« t
0
â«
RN
upâ1 |âu|2 dxds =1
p + 1
â«
RN
(up+1
0 â up+1(t))
dx
+
â« t
0
â«
RN
up |âu|q dxds ,
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1. LIMIT VALUES OF âuâ1 AND âuââ 143
whence, thanks to (2.3),â« t
0
â«
RN
upâ1 |âu|2 dxds †âu0â1 âu0âpâ
p(p + 1)+
B
p
â« t
0
â«
RN
up |âu|2 dxds . (5.11)
We next claim thatâ« t
0
â«
RN
|âu|q dxds †âu0â1
kâ
i=1
Ai
(i + 1)!+
Bk+1
k!
â« t
0
â«
RN
uk |âu|2 dxds (5.12)
for every integer k â„ 1. Indeed, it follows from (2.3) and (5.11) with p = 1 thatâ« t
0
â«
RN
|âu|q dxds †B
â« t
0
â«
RN
|âu|2 dxds
†A
2âu0â1 + B2
â« t
0
â«
RN
u |âu|2 dxds
whence (5.12) for k = 1. We next proceed by induction and assume that (5.12)holds true for some k â„ 1. Inserting (5.11) with p = k + 1 in (5.12) yields
â« t
0
â«
RN
|âu|q dxds †âu0â1
kâ
i=1
Ai
(i + 1)!+
Bk+1
k!
âu0â1 âu0âk+1â
(k + 1)(k + 2)
+Bk+2
(k + 1)!
â« t
0
â«
RN
uk+1 |âu|2 dxds
†âu0â1
k+1â
i=1
Ai
(i + 1)!+
Bk+2
(k + 1)!
â« t
0
â«
RN
uk+1 |âu|2 dxds ,
whence (5.12) for k + 1 and the proof of (5.12) is complete.Finally, let p0 â„ 1 be the smallest integer satisfying p0 > A. Using (5.11) with
p = p0 + 1 and (2.3), we obtain that
(p0 + 1)
â« t
0
â«
RN
up0 |âu|2 dxds †âu0â1 âu0âp0+1â
p0 + 2+ A
â« t
0
â«
RN
up0 |âu|2 dxds ,
from which we deduce, since p0 + 1 â A â„ 1,â« t
0
â«
RN
up0 |âu|2 dxds †âu0â1 âu0âp0+1â
p0 + 2.
We insert the above estimate in (5.12) with k = p0 and end up with
â« t
0
â«
RN
|âu|q dxds †âu0â1
p0â
i=1
Ai
(i + 1)!+
Bp0+1
p0!
âu0â1 âu0âp0+1â
p0 + 2†âu0â1
p0+1â
i=1
Ai
i!.
Consequently, |âu|q â L1(Qâ) and we may integrate (1.1) over Qâ to concludethat
I1(â) †âu0â1
p0+1â
i=0
Ai
i!< â ,
and the proof of (5.4) for q â„ 2 is complete.
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144 5. TEMPORAL GROWTH ESTIMATES FOR INTEGRABLE INITIAL DATA: Ï = â1
1.2. The case q â (qâ, 2). In that case, the statement of Proposition 5.3 is anobvious consequence of Theorem 2.1 and Lemma 5.1.
Nevertheless, some additional information on I1(â) and Iâ(â) may be ob-tained when q â (qâ, 2). The first result in that direction is the following:
Proposition 5.5. Assume that q â (qâ, 2). There is Îł = Îł(N, q) > 0 such thatI1(â) < â (and thus Iâ(â) = 0 by Proposition 5.3) whenever
âu0â1 ââu0â(N+1)(qâqâ)â †γ . (5.13)
Proof. Owing to (5.3), it is clear that I1(â) < â whenever t 7ââ ââu(t)âqq â
L1(0,â). Such a property is certainly true if we know that
t 7ââ ââu(t)âkk â L1(0,â) for some k â [1, q] (5.14)
since ââuâqq †ââu0âqâk
â ââuâkk by (2.3). We proceed in two steps. Under a small-
ness assumption on u0, we obtain a first temporal decay estimate for ââu(t)âk. In asecond step, we improve this estimate so as to obtain the required time integrabilityof t 7ââ ââu(t)âk
k.Step 1. Fix k â (qâ, q] (close to qâ) such that
qâ < k < min
N
N â 1, 2
, p :=
N(k â 1)
2 â kâ (1, k) and
1
2(k â 1)â N
2k>
1
q.
(5.15)
Such a choice is always possible since the function k 7ââ N(k â 1)/(2â k) is equalto 1 for k = qâ and the function k 7ââ (1/(2(k â 1))) â (N/(2k)) is equal to 1/qâ
for k = qâ. Putting A(s, x) := |âu(s, x)|qâk, the integral identity (2.2) reads
u(t) = etâu0 +
â« t
0
e(tâs)âA(s)|âu(s)|k ds , t â„ 0 . (5.16)
Recall that âA(s)ââ †Aâ := ââu0âqâkâ in view of (2.3).
We next introduce
α :=N
2
(1
pâ 1
k
)+
1
2=
1
2(k â 1)â N
2k>
1
2
and
M(t) := supsâ(0,t)
sαââu(s)âk
for t > 0. Using a fixed point argument in a suitable space and the uniquenessassertion of Theorem 2.1, we proceed as in [20, 39] to establish that M(t) is finitefor each t > 0 and converges to zero as t â 0. It follows from (5.16) and the decayproperties of the heat semigroup that
ââu(t)âk †C tâα âu0âp + C
â« t
0
(t â s)â(k+N(kâ1))/(2k) Aâ
â„â„|âu(s)|kâ„â„
1ds
†C tâα âu0âp + C Aâ M(t)k
â« t
0
(t â s)â(k+N(kâ1))/(2k) sâkα ds .
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1. LIMIT VALUES OF âuâ1 AND âuââ 145
Consequently,
tα ââu(t)âk †C âu0âp + C Aâ M(t)k
â« 1
0
(1 â s)â(k+N(kâ1))/(2k) sâkα ds ,
and the last integral in the right-hand side of the previous inequality is finite thanksto the choice of k â (qâ, N/(N â 1)). Thus
M(t) †K1
(âu0âp + Aâ M(t)k
)for t > 0 . (5.17)
Now, assume that
âu0âkâ1p Aâ †2â(k+1) Kâk
1 . (5.18)
Since M(t) â 0 as t â 0 and M is non-decreasing, there is a maximal t0 â (0,â]such that M(t) †2K1 âu0âp for t â [0, t0). Assuming for contradiction that t0 < â,we have M(t0) = 2K1 âu0âp while (5.17) implies that
M(t0) †K1
(âu0âp + Aâ (2K1)
k âu0âkp
)†3K1
2âu0âp ,
hence a contradiction. Consequently, t0 = â and we deduce that
M(t) †2K1 âu0âp for all t > 0 (5.19)
whenever u0 fulfils (5.18). Observe that, since kα < 1, the estimate (5.19) does notguarantee (5.14). A better estimate will be obtained in the next step.Step 2. We combine (2.5) and (5.19) to obtain
ââu(t)âqq †ââu(t)âqâk
â ââu(t)âkk †C âu0â(qâk)/q
â tâ1âk(αâ(1/q))
for t â„ 1. Recalling that α > 1/q by (5.15), we conclude that t 7ââ ââu(t)âqq be-
longs to L1(0,â), whence I1(â) < â. Finally, owing to the GagliardoâNirenberginequality
âwâp †C ââwâ(N(pâ1))/(p(N+1))â âwâ(p+N)/(p(N+1))
1 , w â L1(RN )â©W 1,â(RN ) ,
we realize that
âu0âkâ1p Aâ = âu0âkâ1
p ââu0âqâkâ †C âu0â1/(N+1)
1 ââu0âqâqââ ,
so that choosing Îł sufficiently small in (5.13) implies (5.18).
Before identifying a class of initial data for which Iâ(â) > 0 (and thusI1(â) = â by Proposition 5.3) we report the following result.
Proposition 5.6. [20] Assume that q â (qâ, 2). There exists u0 â L1(RN ) â©BC2(RN ) such that I1(â) = â and Iâ(â) = 0. More precisely, this particularsolution to (1.1), (1.2) enjoys the following properties: for t â„ 0,
âu(t)â1 = (1+t)((N+1)(qâqâ))/(2(qâ1)) âu0â1 and âu(t)ââ =âu0ââ
(1 + t)(2âq)/(2(qâ1)).
Proposition 5.6 is established in [20, Theorem 3.5] by constructing a self-similarsolution to (1.1) of the form
(t, x) 7ââ tâ(2âq)/(2(qâ1)) Ws
(|x|tâ1/2
).
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146 5. TEMPORAL GROWTH ESTIMATES FOR INTEGRABLE INITIAL DATA: Ï = â1
The profile Ws then satisfies a nonlinear ordinary differential equation, which turnsout to have an integrable and smooth solution Ws if q â (qâ, 2). Taking u0 = Ws
gives the result.We end up this section with the following result:
Proposition 5.7. [9] There is a constant K = K(q) > 0 such that, if u0 âW 2,â(RN ) satisfies
âu0âââŁâŁâŁâŁ infyâRN
âu0âŁâŁâŁâŁâ(2âq)/q
> K , (5.20)
then Iâ(â) > 0.
Proof. Since u is a classical solution to (1.1), (1.2) with Ï = â1, it followsfrom (1.1) that
u(t, x) â„ u0(x) +
â« t
0
âu(s, x) ds â„ u0(x) +
â« t
0
infyâRN
âu(s, y) ds
for every x â RN and t â„ 0. Therefore,
âu(t)ââ â„ âu0ââ +
â« t
0
infyâRN
âu(s, y) ds ,
and we infer from (2.38) and (2.39) that
âu(t)ââ â„ âu0ââ + T infyâRN
âu0 â C âu0â(2âq)/qâ
â« t
T
sâ2/q ds
for T > 0 and t > T . Since q < 2, we may let t â â in the above inequality
and obtain with the choice T = âu0â(2âq)/2â
âŁâŁinfyâRN âu0âŁâŁâq/2
that there is aconstant K depending only on q such that
Iâ(â) â„ âu0ââ â Kq/2
âŁâŁâŁâŁ infyâRN
âu0âŁâŁâŁâŁ(2âq)/2
âu0â(2âq)/2â . (5.21)
Therefore, if âu0ââ > KâŁâŁinfyâRN âu0
âŁâŁ(2âq)/q, we readily conclude from (5.21)
that Iâ(â) > 0, whence Proposition 5.7.
Unfortunately, the conditions (5.13) and (5.20) do not involve the same quan-tities. Still, we can prove that if u0 fulfils
âu0ââ âD2u0ââ(2âq)/qâ > Î
(which clearly implies (5.20) since q < 2), the quantity âu0â1 ââu0â(N+1)(qâqâ)â
cannot be small. Indeed, there is a constant C depending only on q and N suchthat
(âu0ââ âD2u0ââ(2âq)/q
â
)q(N+1)/2
†Câu0â1 ââu0â(N+1)(qâqâ)â . (5.22)
To prove (5.22), we put B = âu0ââ âD2u0ââ(2âq)/qâ and note that the Gagliardoâ
Nirenberg inequalities
âu0ââ †C ââu0âN/(N+1)â âu0â1/(N+1)
1 ,
ââu0ââ †C âD2u0â(N+1)/(N+2)â âu0â1/(N+2)
1 ,
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1. LIMIT VALUES OF âuâ1 AND âuââ 147
imply that
ââu0â(2âq)(N+2)â †C âD2u0â(2âq)(N+1)
â âu0â2âq1
= C Bâq(N+1) âu0âq(N+1)â âu0â2âq
1
†C Bâq(N+1) ââu0âqNâ âu0â2
1 ,
whence the above claim.
1.3. The case q â [1, qâ]. We first show that I1(â) = â.
Lemma 5.8. Assume that q â [1, qâ]. Then I1(â) = â.
Proof. For T > 0 and t > T , we infer from (1.1) that
âu(t)â1 = âu(T )â1 +
â« t
T
ââu(s)âqq ds â„
â« t
T
ââu(s)âqq ds . (5.23)
Case 1: N â„ 2. Since 1 †q †qâ †2, it follows from (5.23), the pointwiseinequality u(s, x) â„
(esâu0
)(x) and the Sobolev inequality that
âu(t)â1 â„ C
â« t
T
âu(s)âqQ ds â„ C
â« t
T
â„â„esâu0
â„â„q
Qds
with Q := Nq/(N â q). Sinceâ„â„esâu0
â„â„Q
â„ âu0â1 âg(s)âQ ââ„â„esâu0 â âu0â1 g(s)
â„â„Q
= sâN(Qâ1)/(2Q) âu0â1 âGâQ ââ„â„esâu0 â âu0â1 g(s)
â„â„Q
â„ âu0â1 âGâQ
2s(Nâq(N+1))/(2q)
for s â„ T and T large enough by Proposition A.1 (the functions g and G beingdefined in (1.7)), we conclude that
âu(t)â1 â„ C(t((N+1)(qââq))/2 â T ((N+1)(qââq))/2
)if q â [1, qâ)
and
âu(t)â1 â„ C (log t â log T ) if q = qâ ,
which gives the expected result for N â„ 2.
Case 2: N = 1. It follows from (1.1), the time monotonicity of âuâ1 and theGagliardoâNirenberg inequality
âwâ2qâ1â †C ââxwâq
q âwâqâ11 , w â L1(R) â© W 1,q(R) ,
that, for t > T > 0,
âu(t)â1 â„ C
â« t
T
âu(s)â2qâ1â âu(s)ââ(qâ1)
1 ds â„ C âu(t)ââ(qâ1)1
â« t
T
âu(s)â2qâ1â ds .
Consequently,
âu(t)âq1 â„ C
â« t
T
âu(s)â2qâ1â ds ,
and we proceed as in the previous case to complete the proof.
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148 5. TEMPORAL GROWTH ESTIMATES FOR INTEGRABLE INITIAL DATA: Ï = â1
Having excluded the finiteness of I1(â), we turn to Iâ(â) and establish thefollowing lemma:
Lemma 5.9. [31] Assume that q â (1, 2) and Iâ(â) = 0. Then âu(t)ââ â€C tâ(2âq)/(2(qâ1)) for all t â„ 0.
Proof. infer from (1.1) that, for each T > t,
u(T, x) = u(t, x) +
â« T
t
âtu(s, x) ds â„ u(t, x) +
â« T
t
âu(s, x) ds .
Since (1.1) is autonomous, we infer from (2.9) that
ââu(s, x) †N C7
â„â„â„â„u(
t
2
)â„â„â„â„(2âq)/q
â
(s â t
2
)â2/q
for s â (t, T ) ,
so that
u(T, x) â„ u(t, x) â N C7
â„â„â„â„u(
t
2
)â„â„â„â„(2âq)/q
â
â« T
t
(s â t
2
)â2/q
ds
â„ u(t, x) â C
â„â„â„â„u(
t
2
)â„â„â„â„(2âq)/q
â
tâ(2âq)/q .
Letting T â â we obtain
u(t, x) †C
â„â„â„â„u(
t
2
)â„â„â„â„(2âq)/q
â
tâ(2âq)/q + Iâ(â)
= C
â„â„â„â„u(
t
2
)â„â„â„â„(2âq)/q
â
tâ(2âq)/q ,
whence
t(2âq)/(2qâ2) âu(t)ââ †C
((t
2
)(2âq)/(2qâ2) â„â„â„â„u(
t
2
)â„â„â„â„â
)(2âq)/q
.
Introducing
A(t) := supsâ[0,t]
s(2âq)/(2qâ2) âu(s)ââ
,
we deduce from the previous inequality that A(t) †C A(t)(2âq)/q, hence A(t) â€Cq/(2qâ2). This bound being valid for each t â„ 0, the proof of Lemma 5.9 iscomplete.
Proof of Proposition 5.3: q â [1, qâ]. Since u(t) â„ e(tâT )âu(T ) for T â„ 0and t > T by the comparison principle, we have
(t â T )N/2 âu(t)ââ â„ âu(T )â1 âGâââ (t â T )N/2
â„â„â„e(tâT )âu(T ) â âu(T )â1 g(t â T )â„â„â„â
,
and we infer from Proposition A.1 that
lim inftââ
tN/2 âu(t)ââ
â„ lim inf
tââ
(t â T )N/2 âu(t)ââ
â„ âu(T )â1 âGââ (5.24)
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2. GROWTH RATES 149
for every T > 0, the functions g and G being still defined by (1.7).We first consider the case q > 1 and assume for contradiction that Iâ(â) = 0.
Then Lemma 5.9 entails that
tN/2 âu(t)ââ †C t((N+1)(qâqâ))/(2(qâ1)) ,
from which we deduce that
lim suptââ
tN/2 âu(t)ââ
†C (5.25)
as q â [1, qâ]. Combining (5.24) and (5.25) yields that âu(T )â1 âGââ †C for everyT > 0 and thus contradicts Lemma 5.8. Therefore Iâ(â) > 0.
Next, if q = 1, we observe that (2.3) ensures that
âtu â âu â„ |âu|qâ
ââu0âqââ1â
in Qâ. Consequently, u := u/ââu0ââ satisfies âtuââu â„ |âu|qâ in Qâ. We theninfer from the comparison principle that u(t, x) â„ v(t, x) for (t, x) â [0,â) Ă R
N ,where v denotes the unique classical solution to âtv â âv = |âv|qâ in Qâ withinitial condition v(0) = u0/ââu0ââ. Therefore,
Iâ(â) = ââu0ââ limtââ
âu(t)ââ â„ ââu0ââ limtââ
âv(t)ââ > 0 ,
which completes the proof of Proposition 5.3.
2. Growth rates
First, the assertion (a) of Proposition 5.4 is nothing but Lemma 5.9. We thenturn to the proof of the assertions (b) and (c).
Proof of Proposition 5.4 (b). For (t, x) â Qâ, we infer from (1.1) and(2.38) that, for each T > t,
u(T, x) = u(t, x) +
â« T
t
âtu(s, x) ds â„ u(t, x) +
â« T
t
âu(s, x) ds
â„ u(t, x) â C9 âu0â(2âq)/qâ
â« T
t
sâ2/q ds
â„ u(t, x) â C âu0â(2âq)/qâ tâ(2âq)/q ,
âu(T )ââ â„ âu(t)ââ â C âu0â(2âq)/qâ tâ(2âq)/q .
Letting T â â gives (5.8).
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150 5. TEMPORAL GROWTH ESTIMATES FOR INTEGRABLE INITIAL DATA: Ï = â1
Proof of Proposition 5.4 (c). We proceed as in the previous proof but use(2.8) instead of (2.38) to obtain
u(T, x) = u(t, x) +
â« T
t
âtu(s, x) ds â„ u(t, x) â C5
â« T
t
Ï2(s) ds
â„ u(t, x) â C âu0âââ« T
t
sâ3/2 (1 + log (1 + s)) ds
â„ u(t, x) â C âu0ââ tâ1/2 (2 + log (1 + t)) ,
âu(T )ââ â„ âu(t)ââ â C âu0ââ tâ1/2 (2 + log (1 + t)) .
Letting T â â gives (5.9).
Bibliographical notes
Lemma 5.8 was first proved in [39]. The proof given here is slightly differentand is inspired by the proof of [31, Theorem 3]. The fact that Iâ(â) > 0 forq â [1, qâ] was first establish in [9] for sufficiently large initial data (fulfilling acondition similar to (5.20)). The proof for general initial data is due to Gilding[31, Theorem 3]. When q â (1, 2) and Iâ(â) > 0 the growth rate (5.6) seems tobe optimal, see Section 2. When q = 1 the growth rate (5.6) is also optimal in thesense that there is a large class of initial data (including compactly supported ones)such that âu(t)â1 †C tN for t â„ 1 [40]. Concerning self-similar solutions to (1.1)for Ï = â1, their existence is investigated in [20, 33].
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CHAPTER 6
Convergence to self-similarity
1. The diffusion-dominated case: Ï = 1
In this section, we shall prove that, when Ï = 1, the diffusion governs the largetime dynamics as soon as I1(â) is non-zero. More precisely, we have the followingresult which is similar to the one known for the linear heat equation (1.5) andrecalled in Section 1 below.
Theorem 6.1. Assume that Ï = 1 and q > qâ. Let u0 â L1(RN ) â© W 1,â(RN )be a non-negative function and denote by u the corresponding classical solution to(1.1), (1.2). Then
I1(â) := limtââ
âu(t)â1 â (0,â) , (6.1)
and, for every p â [1,â],
limtââ
tN(pâ1)/2pâu(t) â I1(â) g(t)âp = 0 , (6.2)
limtââ
tN(pâ1)/2p+(1/2)ââu(t) â I1(â) âg(t)âp = 0 , (6.3)
where the functions g and G are defined in (1.7).
The proof of Theorem 6.1 relies on the following properties of the inhomoge-neous heat equation.
Lemma 6.2. Assume that v is the solution to the Cauchy problem for the linearinhomogeneous heat equation
âtv â âv = f , (t, x) â Qâ ,
v(0) = v0 , x â RN ,
with v0 â L1(RN ) and f â L1(Qâ). Then
limtââ
âv(t) â Kâ g(t)â1 = 0 , (6.4)
where
Kâ := limtââ
â«
RN
v(t, x) dx =
â«
RN
v0(x) dx +
â« â
0
â«
RN
f(t, x) dx dt .
Assume further that there is p â [1,â] such that f(t) â Lp(RN ) for every t > 0and
limtââ
t1+(N/2)(1â1/p) âf(t)âp = 0 . (6.5)
Then
limtââ
t(N/2)(1â1/p) âv(t) â Kâ g(t)âp = 0 , (6.6)
151
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152 6. CONVERGENCE TO SELF-SIMILARITY
and
limtââ
t(N/2)(1â1/p)+1/2 ââv(t) â Kâ âg(t)âp = 0 . (6.7)
Proof. We first observe that the assumptions on v0 and f warrant that Kâ
is finite and put
K(T ) :=
â«
RN
v(T, x) dx =
â«
RN
v0(x) dx +
â« T
0
â«
RN
f(t, x) dx dt for T > 0 .
Moreover, by the Duhamel formula,
âv(t)â1 †âg(t) â v0â1 +
â« t
0
âg(tâ Ï) â f(Ï)â1 dÏ â€ âv0â1 +
â« â
0
âf(Ï)â1 dÏ
for t â„ 0 and thus v â Lâ(0,â; L1(RN )).We now prove (6.4). Let T > 0 and t â (T,â). By the Duhamel formula,
v(t) = g(t â T ) â v(T ) +
â« t
T
g(t â Ï) â f(Ï) dÏ ,
so that, by the Young inequality,
âv(t) â g(t â T ) â v(T )â1 â€â« t
T
âg(t â Ï)â1 âf(Ï)â1 dÏ â€â« â
T
âf(Ï)â1 dÏ .
Then
âv(t) â Kâ g(t)â1 †âv(t) â g(t â T ) â v(T )â1 + â(g(t â T ) â g(t)) â v(T )â1
+ âg(t) â v(T ) â K(T ) g(t)â1 + |K(T )â Kâ| âg(t)â1
†2
â« â
T
âf(Ï)â1 dÏ + supÏâ„0
âv(Ï)â1 â(g(t â T ) â g(t))â1
+ âg(t) â v(T ) â K(T ) g(t)â1 .
Owing to Proposition A.1 and the properties of g, we may let t â â in the aboveinequality and deduce that
lim suptââ
âv(t) â Kâ g(t)â1 †2
â« â
T
âf(Ï)â1 dÏ .
We next use the time integrability of f to pass to the limit as T â â and completethe proof of (6.4).
We next assume (6.5) and prove (6.7). Let T > 0 and t â (T,â). By theDuhamel formula,
âv(t) = âg(t â T ) â v(T ) +
â« t
T
âg(t â Ï) â f(Ï) dÏ .
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1. THE DIFFUSION-DOMINATED CASE: Ï = 1 153
It follows from the Young inequality that
t(N/2)(1â1/p)+1/2 ââv(t) ââg(t â T ) â v(T )âp
†C t(N/2)(1â1/p)+1/2
â« (T+t)/2
T
(t â Ï)â(N/2)(1â1/p)â1/2 âf(Ï)â1 dÏ
+ C t(N/2)(1â1/p)+1/2
â« t
(T+t)/2
(t â Ï)â1/2 âf(Ï)âp dÏ
†C
(t
t â T
)(N/2)(1â1/p)+1/2 â« â
T
âf(Ï)â1 dÏ
+ C supÏâ„T
Ï (N/2)(1â1/p)+1 âf(Ï)âp
â« t
(T+t)/2
(t â Ï)â1/2 Ïâ1/2 dÏ
†C
(t
t â T
)(N/2)(1â1/p)+1/2 â« â
T
âf(Ï)â1 dÏ
+ C supÏâ„T
Ï (N/2)(1â1/p)+1 âf(Ï)âp
.
It also follows from Proposition A.1 and the properties of g that
limtââ
t(N/2)(1â1/p)+1/2 ââg(t â T ) â v(T ) â K(T ) âg(t â T )âp = 0 ,
and
limtââ
t(N/2)(1â1/p)+1/2 ââg(t â T ) ââg(t)âp = 0
for every p â [1,â]. Since
ââv(t) â Kâ âg(t)âp †ââv(t) ââg(t â T ) â v(T )âp
+ ââg(t â T ) â v(T ) â K(T ) âg(t â T )âp
+ |K(T ) â Kâ| ââg(t â T )âp
+ |Kâ| ââg(t â T ) ââg(t)âp ,
the previous relations imply that
lim suptââ
t(N/2)(1â1/p)+1/2 ââv(t) â Kâ âg(t)âp
†C
(â« â
T
âf(Ï)â1 dÏ + supÏâ„T
Ï (N/2)(1â1/p)+1 âf(Ï)âp
).
The above inequality being valid for any T > 0, we may let T â â and concludethat (6.7) holds true. The assertion (6.6) then follows from (6.4) and (6.7) by theGagliardoâNirenberg inequality.
Proof of Theorem 6.1. Consider p â [1,â] and t > 0. Since Ï = 1, u is asubsolution to the linear heat equation and therefore satisfies
âu(t)âp †âetâu0âp †C tâ(N/2)(1â1/p) âu0â1
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154 6. CONVERGENCE TO SELF-SIMILARITY
as a consequence of the comparison principle. Combining this estimate with (2.6)(with s = t/2) and (4.12) gives
â|âu(t)|qâp =
(q
q â 1
)q â„â„â„u(t)âŁâŁâŁâu(qâ1)/q(t)
âŁâŁâŁqâ„â„â„
p
â€(
q
q â 1
)q â„â„â„âu(qâ1)/q(t)â„â„â„
q
ââu(t)âp
†C âu0â1
â„â„â„â„u(
t
2
)â„â„â„â„qâ1
â
tâq/2 tâN(pâ1)/2p
†C âu0âq1 tâ(q+N(qâ1))/2 tâN(pâ1)/2p ,
whence
t(N/2)(1â1/p)+1 â|âu(t)|qâp †C âu0âq1 t(N+1)(qââq)/2 ââ
tââ0
because q > qâ. Theorem 6.1 then readily follows from Lemma 6.2 with f(t, x) =â|âu(t, x)|q.
Remark 6.3. The convergence (6.2) in Theorem 6.1 is also true when Ï = â1and either q â„ 2 or q â (qâ, 2) and u0 is suitably small. Indeed, in both cases, I1(â)is positive and finite by Proposition 5.3 and Proposition 5.5, respectively. The proofagain relies on Lemma 6.2 but establishing that f = |âu|q fulfils the condition (6.5)for all p â [1,â] is more delicate [9].
2. The reaction-dominated case: Ï = â1
The aim of this section is to show that, when Ï = â1 and Iâ(â) > 0, thelarge time behaviour of the solution u to (1.1), (1.2) is dominated by the nonlinearHamiltonâJacobi term in the sense that u behaves as a self-similar solution to (1.9)for large times (recall that Iâ(â) is the limit of âu(t)ââ as t â â, see (5.5)).Such a result however holds for initial data decaying to zero as |x| â â and wedefine the space C0(R
N ) of such functions by
C0(RN ) :=
z â BC(RN ) : lim
Rââsup
|x|â„R|z(x)| = 0
.
Theorem 6.4. Assume that Ï = â1 and q â (1, 2). Let u0 â C0(RN ) â©
W 1,â(RN ) be a non-negative function and denote by u the corresponding classicalsolution to (1.1), (1.2). Assume further that
Iâ(â) := limtââ
âu(t)ââ > 0 . (6.8)
Then
limtââ
âu(t) â hâ(t)ââ = 0 , (6.9)
where hâ is given by
hâ(t, x) := Hâ
( x
t1/q
)and Hâ(x) :=
(Iâ(â) â Îłq |x|q/(qâ1)
)
+
for (t, x) â Qâ and Îłq := (q â 1) qâq/(qâ1).
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2. THE REACTION-DOMINATED CASE: Ï = â1 155
A similar result is valid for viscosity solutions to (1.9), see Section 2 below.Thus Theorem 6.4 clearly asserts the sole domination of the HamiltonâJacobi termfor large times. In fact, hâ is the unique viscosity solution in BUC(Qâ) to (1.9)with the bounded and upper semicontinuous initial condition hâ(0, x) = 0 if x 6= 0and hâ(0, 0) = Iâ(â).
We next point out that there is no loss of generality in assuming that q < 2 inTheorem 6.4. Indeed, according to Proposition 5.3, we always have Iâ(â) = 0 ifq â„ 2.
Proof of Theorem 6.4. We change the variables and the unknown functionso that the convergence (6.9) reduces to a convergence to a steady state. Moreprecisely, we introduce the self-similar (or scaling) variables
Ï =1
qlog (1 + t) , y =
x
(1 + t)1/q,
and the new unknown function v defined by
u(t, x) = v
(log (1 + t)
q,
x
(1 + t)1/q
), (t, x) â [0,â) Ă R
N .
Then v(Ï, y) = u (eqÏ â 1, yeÏ) and it follows from (1.1), (1.2) that v solves
âÏv â y · âv â q |âv|q â q eâ(2âq)Ï âv = 0 , (Ï, y) â Qâ , (6.10)
v(0) = u0 , y â RN . (6.11)
We also infer from (2.1), (2.3), (2.5) and (2.9) that there is a positive constant K1
depending only on N , q and u0 such that
âv(Ï)ââ + ââv(Ï)ââ †K1 , Ï â„ 0 , (6.12)
âv(Ï, y) â„ âK1 , (Ï, y) â [1,â) Ă RN . (6.13)
At this point, since q < 2, it is clear that the diffusion in (6.10) becomesweaker and weaker as time increases to infinity and the behaviour of v for largetimes is expected to look like that of the solution to the HamiltonâJacobi equationâÏz â y · âz â q |âz|q = 0. To investigate the large time behaviour of first-order HamiltonâJacobi equations, a useful technique has been developed in [43, 45]which relies on the relaxed half-limits method introduced in [7]. More precisely, for(Ï, y) â Qâ we define the relaxed half-limits vâ and vâ by
vâ(y) := lim inf(Ï,z,s)â(Ï,y,â)
v(Ï + s, z) and vâ(y) := lim sup(Ï,z,s)â(Ï,y,â)
v(Ï + s, z) (6.14)
and first note that the right-hand sides of the above definitions indeed do not dependon Ï > 0. In addition,
0 †vâ(x) †vâ(x) †Iâ(â) for x â RN (6.15)
by (5.5) and (6.14), while (6.12) and the Rademacher theorem clearly ensure thatvâ and vâ both belong to W 1,â(RN ). Finally, by [6, Theoreme 4.1] applied toequation (6.10), vâ and vâ are viscosity subsolution and supersolution, respectively,to the HamiltonâJacobi equation
H(y,âz) := ây · âz â q |âz|q = 0 in RN . (6.16)
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156 6. CONVERGENCE TO SELF-SIMILARITY
The next step is to show that vâ and vâ actually coincide. However, the equation(6.16) has several solutions as y 7ââ
(c â Îłq |y|q/(qâ1)
)+
solves (6.16) for any c > 0.
The information obtained so far on vâ and vâ are thus not sufficient. The next tworesults provide the lacking information.
Lemma 6.5. Given Δ â (0, 1), there is RΔ > 1/Δ such that
v(Ï, y) †Δ for Ï â„ 0 and y â RN \ B(0, RΔ) . (6.17)
Proof. We first construct a supersolution to (6.10) in (0,â) Ă RN \ B(0, R)
for R large enough. To this end, consider R â„(4 + q 2qâ1 âu0âqâ1
â
)1/qand define
ÎŁR(y) = âu0ââ R2 |y|â2 for y â RN . Let L be the parabolic operator defined by
Lw(Ï, y) := âÏw(Ï, y) â y · âw(Ï, y) â q |âw(Ï, y)|q â q eâ(2âq)Ï âw(Ï, y)
for (Ï, y) â Qâ (so that Lv = 0). Then, if y â RN \ B(0, R), we have
LÎŁR(y) = 2 âu0ââR2
|y|2 â q 2q âu0âqâ
R2q
|y|3qâ 2 (4 â N) âu0ââ
R2
|y|4 eâ(2âq)Ï
= 2 âu0ââR2
|y|2
(1 +
N â 4
|y|2 eâ(2âq)Ï â q(2 âu0ââ)qâ1
|y|q(
R2
|y|2)qâ1
)
â„ 2 âu0ââR2
|y|2(
1 â 4
|y|2 eâ(2âq)Ï â q(2 âu0ââ)qâ1
Rq
)
â„ 2 âu0ââR2
|y|2(1 â
(4 + q (2 âu0ââ)qâ1
)Râq
)
â„ 0 .
Consequently, for R â„(4 + q 2qâ1 âu0âqâ1
â
)1/q, ÎŁR is a supersolution to (6.10) in
(0,â) Ă RN \ B(0, R).
Consider next Δ â (0, 1). Since u0 â C0(RN ), there is rΔ â„
(4 + q 2qâ1 âu0âqâ1
â
)1/q,
rΔ â„ 1/Δ such that u0(y) †Δ/2 if |y| â„ rΔ. Then, if y â RN \ B(0, rΔ), we have
u0(y)âΔ/2 †0 †ΣrΔ(y) while v(Ï, y)âΔ/2 †âu0ââ = ÎŁrΔ(y) if Ï â„ 0 and |y| = rΔ.Since ÎŁrΔ is a supersolution to (6.10) as previously established and v â Δ/2 is a so-lution to (6.10), the comparison principle entails that v(Ï, y) â Δ/2 †ΣrΔ(y) forÏ â„ 0 and y â R
N \ B(0, rΔ). Since ÎŁrΔ(y) â 0 as |y| â â, there is RΔ â„ rΔ suchthat ÎŁrΔ(y) †Δ/2 for |y| â„ RΔ which completes the proof of (6.17).
It actually follows from Lemma 6.5 that v(Ï) belongs to C0(RN ) for every Ï â„ 0
in a way which is âuniformâ with respect to Ï â„ 0.
Lemma 6.6. For y â RN , we have
Hâ(y) †vâ(y) †vâ(y) . (6.18)
Proof. For Ï â„ 0, y â RN , Ο0 â R and Ο â R
N , we set H(Ï, y, Ο0, Ο) :=Ο0 â y · Ο â q |Ο|q + q K1 eâ(2âq)Ï , the constant K1 being defined in (6.12) and(6.13). Since âv(Ï, y) â„ âK1 for (Ï, y) â (1,â) Ă R
N by (6.13), we deduce
âtopics-in-mathematical-modelingâ â 2008/12/5 â 8:30 â page 157 â #169
2. THE REACTION-DOMINATED CASE: Ï = â1 157
from (6.10) that H(Ï, y, âÏv(Ï, y),âv(Ï, y)) = q eâ(2âq)Ï (âv(Ï, y) + K1) â„ 0 for(Ï, y) â (1,â) Ă R
N . Consequently,
v is a supersolution to H(Ï, y, âÏz,âz) = 0 in (1,â) Ă RN . (6.19)
Next fix Ï0 > 1 and denote by w the (viscosity) solution to
âÏw â y · âw â q |âw|q = 0 , (Ï, y) â (Ï0,â) Ă RN ,
w(Ï0) = v(Ï0) , y â RN .
On the one hand, the function w defined by
w(Ï, y) := w(Ï, y) â q K1
â« Ï
Ï0
eâ(2âq)s ds , (Ï, y) â (Ï0,â) Ă RN ,
is the (viscosity) solution to H(Ï, y, âÏ w,âw) = 0 in (Ï0,â) Ă RN with initial
condition w(Ï0) = v(Ï0). Recalling (6.19), we infer from the comparison principlethat
w(Ï, y) †v(Ï, y) for (Ï, y) â (Ï0,â) Ă RN . (6.20)
On the other hand, it follows from Corollary A.4 that
limÏââ
supyâRN
âŁâŁâŁâŁw(Ï) â(âv(Ï0)ââ â Îłq |y|q/(qâ1)
)
+
âŁâŁâŁâŁ = 0 .
We may then pass to the limit as Ï â â in (6.20) to conclude that(âv(Ï0)ââ â Îłq |y|q/(qâ1)
)
+â q K1
â« â
Ï0
eâ(2âq)s ds †vâ(y) †vâ(y)
for y â RN . Letting Ï0 â â in the above inequality gives (6.18).
We are now in a position to complete the proof of Theorem 6.4. Fix Δ â (0, 1).On the one hand, it readily follows from Lemma 6.5 that vâ(y) †Δ for |y| â„ RΔ â„1/Δ. On the other hand, Hâ(0) = Iâ(â) and the continuity of Hâ ensures thatthere is rΔ â (0, Δ) such that Hâ(y) â„ Iâ(â)âΔ for y â B(0, rΔ). Recalling (6.15),we conclude that
vâ(y) â Δ †0 †Hâ(y) if |y| = RΔ ,
vâ(y) â Δ †Iâ(â) â Δ †Hâ(y) if |y| = rΔ .(6.21)
Now, introducing Ï(y) = âÎłq |y|q/(qâ1)/2 and recalling that H is defined in (6.16),we have
H(y,âÏ(y)) =qÎłq
2(q â 1)|y|q/(qâ1)
(1 â 1
2qâ1
)> 0 if rΔ < |y| < RΔ . (6.22)
Summarizing, putting ΩΔ :=y â R
N : rΔ < |y| < RΔ
, we have shown that Hâ
and vâ â Δ are supersolution and subsolution to (6.16) in ΩΔ, respectively, withvâ â Δ †Hâ on âΩΔ by (6.21). Owing to (6.22) and the concavity of H withrespect to its second variable, we may apply [36, Theorem 1] to conclude thatvâ â Δ †Hâ in ΩΔ. Letting Δ â 0, we end up with vâ †Hâ in R
N . Recalling(6.18), we have thus established that vâ = vâ = Hâ in R
N . In addition, owing to[4, Lemma V.1.9] or [6, Lemme 4.1], the equality vâ = vâ and (6.14) provide the
âtopics-in-mathematical-modelingâ â 2008/12/5 â 8:30 â page 158 â #170
158 6. CONVERGENCE TO SELF-SIMILARITY
convergence of (v(Ï))Ï>0 towards Hâ uniformly on every compact subset of RN as
Ï â â. Combining this last property with Lemma 6.5 implies that
limÏââ
âv(Ï) â Hâââ = 0 . (6.23)
Theorem 6.4 then readily follows after writing the convergence (6.23) in the originalvariables (t, x) and noticing that âhâ(1 + t) â hâ(t)ââ ââ 0 as t â â.
Bibliographical notes
Theorem 6.1 was first proved for p = 1 in [23] when Ï = 1 and [39] whenÏ = â1. The extension to the other Lp-norms and the gradient was done in [9].Theorem 6.4 is proved in [9, Theorem 2.6] with a different method which uses ratherthe Hopf-Lax-Oleinik representation formula than the relaxed half-limits technique.
When Ï = 1 and q â (1, qâ), the large time behaviour of non-negative solutionsto (1.1), (1.2) is also self-similar (at least for initial data vanishing sufficientlyrapidly for large values of x) but neither the diffusion nor the absorption dominatefor large times. It is actually a self-similar solution W to (1.1) which characterisesthe large time behaviour of non-negative solutions u to (1.1), (1.2) with an initialcondition satisfying (1.4) and the decay condition
ess lim|x|ââ
|x|(2âq)/(qâ1) u0(x) = 0 .
More precisely, it has been shown in [9] that
limtââ
t(N+1)(qââq)/(2(qâ1)) âu(t) â W (t)â1 + t(2âq)/(2(qâ1)) âu(t) â W (t)ââ
= 0
and W satisfies W (t, x) = tâ(2âq)/(2(qâ1)) W(1, xtâ1/2
)for (t, x) â Qâ. In fact,
W is a very singular solution to (1.1) in the sense of [24], that is, W is a classicalsolution to (1.1) in Qâ, W (t) â L1(RN ) for each t > 0,
limtâ0
â«
B(0,r)
W (t, x) dx = â and limtâ0
â«
|x|â„r
W (t, x) dx = 0
for every r > 0. The existence and uniqueness of such a solution have been studiedin [10, 12, 28, 44].
Finally, convergence to self-similarity also takes place for the critical exponentq = qâ when Ï = 1, the scaling profile being the heat kernel g as for q > qâ (seeTheorem 6.1) but with an additional logarithmic scaling factor. More precisely, itis shown in [30] that, if q = qâ and u0 satisfies (1.4) together with u0 â L2(RN ; (1+|x|2m) dx) for some m > N/2, the corresponding solution u to (1.1), (1.2) satisfies
limtââ
t(N(pâ1))/2p (log t)N+1
â„â„â„â„u(t) â Mâ
(log t)N+1g(t)
â„â„â„â„p
= 0
with Mâ := (N + 1)N+1 ââGââ(N+2)qâ (recall that g and G are defined in (1.7)).
âtopics-in-mathematical-modelingâ â 2008/12/5 â 8:30 â page 159 â #171
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âtopics-in-mathematical-modelingâ â 2008/12/5 â 8:30 â page 163 â #175
APPENDIX A
Self-similar large time behaviour
1. Convergence to self-similarity for the heat equation
We recall the following well-known result.
Proposition A.1. Consider u0 â L1(RN ) and put
M0 :=
â«
RN
u0(x) dx .
Then, for p â [1,â], we have
limtââ
tN(pâ1)/(2p)â„â„etâu0 â M0 g(t)
â„â„p
= 0 , (A.1)
limtââ
tN(pâ1)/(2p)+(1/2)â„â„âetâu0 â M0 âg(t)
â„â„p
= 0 , (A.2)
where the functions g and G are given by (1.7).
Proof. For (t, x) â Qâ, we put
c(t, x) :=(etâu0
)(x) =
1
tN/2
â«
RN
G
(x â y
t1/2
)u0(y) dy
and z(t, x) := c(t, x) â M0 g(t, x). An alternative formula for z reads
z(t, x) =1
tN/2
â«
RN
[G
(x â y
t1/2
)â G
( x
t1/2
)]u0(y) dy .
On the one hand,
âz(t)â1 â€â«
RN
|u0(y)|â«
RN
âŁâŁâŁG(x) â G(x â ytâ1/2
)âŁâŁâŁ dxdy .
Noting that â«
RN
âŁâŁâŁG(x) â G(x â ytâ1/2
)âŁâŁâŁ dx †2 âGâ1 < â
and
limtââ
â«
RN
âŁâŁâŁG(x) â G(x â ytâ1/2
)âŁâŁâŁ dx = 0
for each y â RN , we infer from the Lebesgue dominated convergence theorem that
limtââ
âz(t)â1 = 0 . (A.3)
On the other hand, since c(t) = e(t/2)âc(t/2) and g(t) = e(t/2)âg(t/2), we have
|z(t, x)| â€âŁâŁâŁâŁ
1
(2Ït)N/2
â«
RN
z
(t
2, y
)exp
â|x â y|2
2t
dy
âŁâŁâŁâŁ â€1
(2Ït)N/2
â„â„â„â„z(
t
2
)â„â„â„â„1
,
163
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164 A. SELF-SIMILAR LARGE TIME BEHAVIOUR
hence
tN/2 âz(t)ââ †1
(2Ï)N/2
â„â„â„â„z(
t
2
)â„â„â„â„1
. (A.4)
Combining (A.3) and (A.4) yields (A.1) for p = â. The result for p â (1,â) thenfollows by interpolation with (A.3).
Similarly,
âz(t, x) =2(N+1)/2
t(N+1)/2
â«
RN
âG
(21/2(x â y)
t1/2
)z
(t
2, y
)dy
and we infer from the Young inequality that
ââz(t)âp †21/2
t1/2ââGâ1
â„â„â„â„z(
t
2
)â„â„â„â„p
for p â [1,â]. We now use (A.1) to estimate the right-hand side of the previousinequality and obtain (A.2).
2. Convergence to self-similarity for HamiltonâJacobi equations
This section is devoted to the large time behaviour of non-negative (viscosity)solutions to the HamiltonâJacobi equation (1.9). We begin with the absorptioncase Ï = 1 and establish the following result for compactly supported initial data.
Theorem A.2. Assume that q > 1 and Ï = 1. Consider a non-negativefunction u0 â C(RN ) with compact support and denote by h the correspondingviscosity solution to the HamiltonâJacobi equation (1.9). Then
limtââ
â„â„â„t1/(qâ1) h(t) â hâ
â„â„â„â
= 0 (A.5)
with
hâ(x) = Îłq d(x, RN \ P0
)q/(qâ1), x â R
N ,
Îłq := (q â 1) qâq/(qâ1) and P0 :=x â R
N : u0(x) > 0.
Proof. We first recall that h is given by the HopfâLaxâOleinik formula (1.10)
h(t, x) := infyâRN
u0(y) + Îłq |x â y|q/(qâ1) tâ1/(qâ1)
for (t, x) â Qâ . (A.6)
We next introduce the positivity set P(t) of h at time t > 0 and claim that
P(t) :=x â R
N : h(t, x) > 0
= P0 . (A.7)
Indeed, if t > 0 and x 6â P0, we have u0(x) = 0 and the choice y = x in theright-hand side of (A.6) ensures that 0 †h(t, x) †u0(x) = 0, whence x 6â P(t).Next, if t > 0 and x â P0, we assume for contradiction that h(t, x) = 0. Sinceu0(y) + Îłq |x â y|q/(qâ1) tâ1/(qâ1) ââ â as |y| â â, the infimum in (A.6) isattained and there is y â R
N such that
0 = h(t, x) = u0(y) + Îłq |x â y|q/(qâ1) tâ1/(qâ1) .
This readily implies that y = x and u0(y) = u0(x) = 0, and a contradiction.Therefore, x â P(t) and the proof of (A.7) is complete.
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2. CONVERGENCE TO SELF-SIMILARITY FOR HAMILTONâJACOBI EQUATIONS 165
We next introduce â(t, x) := t1/(qâ1) h(t, x) so that
â(t, x) := infyâRN
t1/(qâ1) u0(y) + Îłq |x â y|q/(qâ1)
(A.8)
for (t, x) â [0,â) Ă RN . We first note that
0 †â(t1, x) †â(t2, x) †γq d(x, RN \ P0
)q/(qâ1)= hâ(x) (A.9)
for x â RN and 0 < t1 < t2, the last inequality being deduced from (A.8) by taking
the infimum only on RN \ P0 where u0 vanishes. We next consider x â P0 and
Δ â(0, d
(x, RN \ P0
)/2). Setting
mΔ := infu0(y) : d
(y, RN \ P0
)℠Δ
> 0 , tΔ :=
(Îłq
mΔ
)qâ1
d(x, RN \ P0
)q
and choosing t â„ tΔ, we have the following alternative for y â RN : either d
(y, RN \ P0
)â„
Δ and
t1/(qâ1) u0(y) + Îłq |x â y|q/(qâ1) â„ mΔ t1/(qâ1)Δ â„ Îłq d
(x, RN \ P0
)q/(qâ1)
or d(y, RN \ P0
)< Δ and
t1/(qâ1) u0(y) + Îłq |x â y|q/(qâ1) â„ Îłq
(d(x, RN \ P0
)â d
(y, RN \ P0
))q/(qâ1)
â„ Îłq
(d(x, RN \ P0
)â Δ)q/(qâ1)
.
Consequently,
â(t, x) â„ Îłq
(d(x, RN \ P0
)â Δ)q/(qâ1)
for t â [tΔ,â) Ă RN . (A.10)
The convergence (A.5) then readily follows from (A.9) and (A.10) after passing tothe limit Δ â 0.
We next turn to the case Ï = â1 and prove the following result.
Theorem A.3. Assume that q > 1 and Ï = â1. Consider a non-negativefunction u0 â C0(R
N ) and denote by h the corresponding viscosity solution to theHamiltonâJacobi equation (1.9). Then
limtââ
supxâRN
âŁâŁâŁh(t, x) â Hâ
( x
t1/q
)âŁâŁâŁ = 0 (A.11)
with
Hâ(x) =(âu0ââ â Îłq |x|q/(qâ1)
)
+, x â R
N .
Proof. As before, h is given by the HopfâLaxâOleinik formula (1.10)
h(t, x) := supyâRN
u0(y) â Îłq |x â y|q/(qâ1) tâ1/(qâ1)
for (t, x) â Qâ . (A.12)
We first show that
âh(t)ââ = âu0ââ , t â„ 0 . (A.13)
Indeed, recall that h(t, x) â„ u0(x) for (t, x) â [0,â) Ă RN which ensures that
âh(t)ââ â„ âu0ââ while the reverse inequality readily follows from the representa-tion formula (A.12).
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166 A. SELF-SIMILAR LARGE TIME BEHAVIOUR
Next, it is rather easy to deduce from (A.12) that h(t, x) ââ âu0ââ as t â âfor each x â R
N . The goal is now to study more precisely this convergence. To thisend we introduce the self-similar (or scaling) variables
s =1
qlog (1 + t) , Ο =
x
(1 + t)1/q,
and the new unknown function â defined by
h(t, x) = â
(log (1 + t)
q,
x
(1 + t)1/q
), (t, x) â [0,â) Ă R
N .
Then â(s, Ο) = h (eqs â 1, Οes) and it follows from (A.12) that
â(s, Ο) = supyâRN
u0(y) â Îłq
âŁâŁÎŸ â y eâsâŁâŁq/(qâ1) (
1 â eâqs)â1/(qâ1)
for (s, Ο) â [0,â) Ă RN . Since â(s, Ο) â„ u0(Οe
s) â„ 0, we have in fact
â(s, Ο) = supyâRN
(u0(y) â Îłq
âŁâŁÎŸ â y eâsâŁâŁq/(qâ1) (
1 â eâqs)â1/(qâ1)
)
+
for (s, Ο) â [0,â) Ă RN . Let us point out here that, as a consequence of (1.9), â is
actually the viscosity solution to
âsâ = Ο · ââ + q |ââ|q , (s, Ο) â Qâ , (A.14)
â(0) = u0 , Ο â RN . (A.15)
Consider now Δ â (0, 1). As u0 â C0(RN ), there is RΔ > 0 such that
u0(y) †Δ for |y| ℠RΔ . (A.16)
On the one hand, if (s, Ο) â [log RΔ,â)Ă RN and y â R
N , we have either |y| ℠RΔ
and
âŁâŁâŁâŁâŁ
(u0(y) â Îłq
|Ο â y eâs|q/(qâ1)
(1 â eâqs)1/(qâ1)
)
+
â(u0(y) â Îłq |Ο|q/(qâ1)
)
+
âŁâŁâŁâŁâŁ
â€(
u0(y) â Îłq|Ο â y eâs|q/(qâ1)
(1 â eâqs)1/(qâ1)
)
+
+(u0(y) â Îłq |Ο|q/(qâ1)
)
+
†2Δ
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2. CONVERGENCE TO SELF-SIMILARITY FOR HAMILTONâJACOBI EQUATIONS 167
by (A.16) or y â B(0, RΔ) andâŁâŁâŁâŁâŁ
(u0(y) â Îłq
|Ο â y eâs|q/(qâ1)
(1 â eâqs)1/(qâ1)
)
+
â(u0(y) â Îłq |Ο|q/(qâ1)
)
+
âŁâŁâŁâŁâŁ
†γq
âŁâŁÎŸ â y eâsâŁâŁq/(qâ1)
(1 â eâqs
)â1/(qâ1) â 1
+ Îłq
âŁâŁâŁâŁâŁÎŸ â y eâs
âŁâŁq/(qâ1) â |Ο|q/(qâ1)âŁâŁâŁ
†γq
(|Ο| + RΔ eâs
)q/(qâ1)(
1 â eâqs)â1/(qâ1) â 1
+q Îłq
q â 1
(|Ο| + |y| eâs
)1/(qâ1) |y| eâs
†γq (1 + |Ο|)1/(qâ1)
q
q â 1+ 1 + |Ο|
(1 â eâqs
)â1/(qâ1) â 1 + RΔ eâs
as s â„ log RΔ. Combining the above two estimates givesâŁâŁâŁâŁâŁâ(s, Ο) â sup
yâRN
(u0(y) â Îłq |Ο|q/(qâ1)
)
+
âŁâŁâŁâŁâŁ
†C (|Ο| + 1)q/(qâ1)
(1 â eâqs
)â1/(qâ1) â 1 + RΔ eâs
+ 2Δ ,
whence
|â(s, Ο) â Hâ(Ο)| (A.17)
†C (|Ο| + 1)q/(qâ1)
(1 â eâqs
)â1/(qâ1) â 1 + RΔ eâs
+ 2Δ
for (s, Ο) â [log RΔ,â) Ă RN . On the other hand, if s â„ log(RΔ), |Ο| â„ Î :=
1 + (âu0ââ/Îłq)(qâ1)/q and y â R
N , we have either |Ο â y eâs| â„ Î â 1 and
u0(y) â Îłq
âŁâŁÎŸ â y eâsâŁâŁq/(qâ1) (
1 â eâqs)â1/(qâ1)
â€(1 â eâqs
)â1/(qâ1)âu0ââ
(1 â eâqs
)1/(qâ1) â Îłq
âŁâŁÎŸ â y eâsâŁâŁq/(qâ1)
â€(1 â eâqs
)â1/(qâ1)âu0ââ â Îłq (Î â 1)q/(qâ1)
†0 ,
or |Ο â y eâs| < Î â 1 and
|y| â„ |Ο es| â |y â Ο es| â„ Î es â (Î â 1) es = es â„ RΔ ,
so that
u0(y) â Îłq
âŁâŁÎŸ â y eâsâŁâŁq/(qâ1) (
1 â eâqs)â1/(qâ1) †Δ
by (A.16). Therefore,
â(s, Ο) †Δ for (s, Ο) â [log RΔ,â) Ă RN \ B(0, Î) . (A.18)
It then follows easily from (A.17) and (A.18) that ââ(s) â Hâââ ââ 0 ass â â. Returning to the original variables (t, x), we conclude that
limtââ
supxâRN
âŁâŁâŁâŁh(t, x) â Hâ
(x
(1 + t)1/q
)âŁâŁâŁâŁ = 0 .
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168 A. SELF-SIMILAR LARGE TIME BEHAVIOUR
Since
limtââ
supxâRN
âŁâŁâŁâŁHâ
(x
(1 + t)1/q
)â Hâ
( x
t1/q
)âŁâŁâŁâŁ = 0 ,
the convergence (A.11) follows.
We have actually shown the following result.
Corollary A.4. Let q > 1 and consider a non-negative function u0 â C0(RN ).
If â denotes the corresponding viscosity solution to (A.14), (A.15), then
limsââ
ââ(s) â Hâââ = 0 ,
the function Hâ being defined in Theorem A.3.
Remark A.5. Related results may be found in [5, Section II] for the HamiltonâJacobi equation âtw + H(âw) = 0 in Qâ, the function H being non-negative andcontinuous with suitable properties.
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Part 4
An area-preserving crystalline
curvature flow equation
Shigetoshi Yazaki
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2000 Mathematics Subject Classification. 34A34, 34K25, 39A12, 52A10, 53A04,74N05, 34K12, 34K28, 65L12, 65M06, 65M12, 65D99, 41A25, 82D25
Key words and phrases. evolving interfaces, polygonal curves, curvature,curve-shortening, crystalline curvature, anisotropy, interfacial energy, the Frank
diagram, the Wulff shape, admissibility, area-preserving, single ice crystal,internal melting, Tyndall figure, negative crystal, vapor figure, crystal growth
Abstract. In this note, we will show how to derive crystalline curvature flowequation and its area-preserving version. This flow is regarded as a gradientflow of total interfacial energy with the singular anisotropy, then the usualdifferential calculation is not allowed in the classical sense. Because of this, a
difference calculation is used instead of differential calculation and polygonalcurves are used instead of smooth curves. This consideration will be focusedin the former and main part of this note, which includes the following threetopics:
âą plane curves, geometric quantities on them, and evolution of them;âą anisotropy, the Frank diagram, and the Wulff shape, etc.; andâą crystalline curvature flow equation and its area-preserving version.
In the latter part, we will see the following two applications:âą a numerical scheme of the area-preserving crystalline curvature flow; andâą a modeling perspective on producing negative ice crystals.
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Contents
Preface 173
Chapter 1. An area-preserving crystalline curvature flow equation:introduction to mathematical aspects, numerical computations,and a modeling perspective 175
1. Introduction 1752. Plane curve 1753. Moving plane curve 1784. Curvature flow equations 1795. Anisotropy 1816. The Frank diagram and the Wulff shape 1837. Crystalline energy 1878. Crystalline curvature flow equations 1899. An area-preserving motion by crystalline curvature 19510. Scenario of the proof of Theorem 1.50 19611. Numerical scheme 19812. Towards modeling the formation of negative ice crystals or vapor
figures produced by freezing of internal melt figures 202
Bibliography 207
Index 211
Appendix A. 2131. Strange examples 2132. A non-concave curve 2133. Anisotropic inequalityâproof of Lemma 1.53â 213
171
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Preface
The so-called crystalline curvature flow equation appeared at the end of 1980âsin the field of mathematical sciences. This equation is regarded as a weak form ofan equation of interfacial motion with a singular interfacial energy. The equationis described as a system of ODEs and governs motion of polygonal curves in twodimensional case. If the number of vertices of curves is large, the motion canapproximate the one described by PDE with a smooth interfacial energy. On theother hand, if the number is small, the motion is sometimes strikingly different fromthe one in the smooth case. This phenomena suggest that research of this field willcontinue to a wide range of applications.
The lecture note will target beginners of the research field. Therefore, only twodimensional closed curves will be focused and I will put emphasis on a fundamentalpart of mathematical introduction to crystalline curvature flow equations. I hope allthe readers will follow several calculations and challenge to solve given exercises. Inaddition, we will touch on topics of a numerical scheme and a modeling of negativecrystal, and they may give directions of applications.
The note is based on the lectures which were held at Czech Technical Universityin Prague, September 4 and 6, 2006, and December 4, 6, 11 and 13, 2007, while theauthor was visiting Czech Republic within the period from August to September,2006, and from March, 2007 to March, 2008; this visit was sponsored by Faculty ofNuclear Sciences and Physical Engineering, Czech Technical University in Prague,within the Jindrich Necas Center for Mathematical Modeling (Project of the CzechMinistry of Education, Youth and Sports LC 06052). The author would like tothank Professor Michal Benes for giving me this opportunity not only to have theseries of lectures but also to visit Czech Republic as a young researcher.
Shigetoshi YazakiMiyazaki, June 2008
173
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CHAPTER 1
An area-preserving crystalline curvature flow
equation: introduction to mathematical aspects,
numerical computations, and a modeling
perspective
1. Introduction
Motion by curvature is generally referred as a motion of curves in the plane orsurfaces in space which change its shape in time and depend on its bend, especiallyon its curvature. They are also called curvature flows, since one tracks flows of thefamily of curves and surfaces parametrized in time. The curvature flow equationis a general term which describes such flows, and has been investigated by manyscientists and mathematicians since the 1950âs. At the end of 1980âs, J. E. Tay-lor, and S. Angenent and M. E. Gurtin focused on motion of polygonal curves bycrystalline curvature in the plane, and since then crystalline curvature flow equa-tion has been studied under various kinds of evolution law by several authors. Werefer the reader to the pioneer works Taylor [Tay91a, Tay93] and Angenent andGurtin [AG89], and the surveys by Taylor, Cahn and Handwerker [TCH92] and thebooks by Gurtin [Gur93] for a geometric and physical background. Also one canfind essentially the same method of crystalline as a numerical scheme for curvatureflow equation in Roberts [Rob93]. We refer the reader Almgren and Taylor [AT95]for detailed history. Besides this crystalline strategy, other strategies by subdiffer-ential and level-set method have been extensively studied. See Giga [Gig00] andreferences therein.
This note will focus mainly on motion of smooth and piecewise linear curvesin the plane, and touch on a numerical scheme of crystalline curvature flow equa-tion which is a kind of so-called direct approach. On motion of surfaces, indirectapproach (ex. level set methods), other curvature flows, and physical background,the reader is referred to the books by Gurtin [Gur93] and Sethian [Set96], andthe surveys by [TCH92] and Giga [Gig95, Gig98, Gig00], and references therein,respectively. In the last section, a modeling perspective will be reported.
2. Plane curve
In this section we will explain plane curve and several geometric quantitieson it.
A plane curve is defined as the following map x:
Î = x(Ï) â R2; Ï â Q,
175
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176 1. AN AREA-PRESERVING CRYSTALLINE CURVATURE FLOW EQUATION
where Q â R is an interval1, and the map x : Q 7â Î â R2 is continuous and
surjective2. The plane curve x is said to be closed if Q = [a, b] and x(a) = x(b).If x is not closed, then it is called open. The plane curve x is said to be simple ifthe map3 x is injective4. A simple and closed plane curve is called Jordan curve.If x is a Jordan curve, then Î is also called Jordan curve.
Remark 1.1 (Jordan curve theorem). If Î is a Jordan curve, then R2\Î has
two components5, where Î is the boundary of each.
In what follows, curves mean plane curves, and we will omit the term âplaneâ.A curve x is said to be Ck-curve (k = 1, 2, . . .) if the map x is Ck-class function.We put
g(Ï) = |âÏx(Ï)|.Here and hereafter, we denote âΟF = âF/âΟ. A curve x is said to be immersed ifx is C1-curve and g(Ï) 6= 0 holds for all Ï.
For closed curves, we take ÎŁ = R/Z with parameter Ï (modulo 1), and use theinterval ÎŁ = [0, 1]. Then a closed plane curve x can be described as
Î = x(Ï) â R2; Ï â ÎŁ.
Remark 1.2. If x is a closed immersed curve, then
s(Ï) =
â« Ï
0
g(Ï) dÏ
is increasing C1-function for Ï â ÎŁ, since g is continuous function and âÏs(Ï) =g(Ï) > 0 holds. Therefore, by virtue of an implicit function theorem, Ï is a func-tion of s, and the closed immersed curve x can be parametrized by the arc-lengthparameter s:
Î = x(s) â R2; s â [0, L], L =
â«
Î
ds =
â«
ÎŁ
g(Ï) dÏ,
where L is the total length of Î.
A curve x is said to be embedded if x is immersed and simple. Hence, if Î hasself-intersection or self-tangency, then x is not embedded curve. Note that all theabove terminologies are defined for the map x, not for the set Î. This distinctionis important. See section 1.
Let x be an immersed curve. The unit tangent vector is defined as t =âÏx/|âÏx| = âsx, and the unit normal vector n is defined such as det(n, t) = 1,or t = nâ„, where (a, b)â„ = (âb, a). Note that the derivative with respect tothe arc-length âs along the curve is uniquely defined, while the arc-length s isunique only up to a constant. The positive direction of x is the direction whichÏ is increasing. If the curve is a Jordan curve, we take the positive direction as
1Non-empty and connected subset in R.2Onto map, i.e., for any x â Î there exists a Ï â Q for which x(Ï) = x.3If x is closed, we take Q = [a, b).4One-to-one map, i.e., for any Ï1, Ï2 â Q if Ï1 6= Ï2, then x(Ï1) 6= x(Ï2) holds.5So-called âinsideâ D and âoutsideâ E. Both D and E are connected, respectively, but its
direct sum D â E is not connected.
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2. PLANE CURVE 177
anticlockwise, i.e., the direction which one can always see enclosed region on the leftside when traveling along the curve in the direction of Ï increasing. See Figure 1.
Let x be an immersed C2-curve. The curvature K of x is defined as
âst = âKn and âsn = Kt, (1.1)
and it is often called the curvature in the direction of ân. The signed conventionof the curvature K is that K = 1 if and only if x is the unit circle. See Figure 1.
K > 0
K < 0
n
t
positive direction
Figure 1. The normal vector n, the tangent vector t, and thecurvature K on a closed embedded curve.
Remark 1.3. Equations (1.1) are called FrenetâSerret formulae for planecurves. From n · n = 1, we obtain âs(n · n) = 2(âsn) · n = 0. Here and hereafter,a · b is the Euclidean inner product between a and b. Hence there exists a functionc(s) such as âsn = ct. Similarly, from t · t = 1 and t · n = 0, we obtain âs(t · t) =2(âst) · t = 0 and âs(t · n) = (âst) · n + t · (âsn) = (âst) · n + c = 0. Thereforeâst = âcn holds. We defined c as the curvature K.
Let Î be a Jordan curve and let Ω be a region enclosed by Î6. The Jordan curveÎ is said to be convex or oval if for any Ï1, Ï2 â ÎŁ (Ï1 6= Ï2) the line segment7
S = [x(Ï1), x(Ï2)] is not contained outside of Î, i.e., S â Ω holds. In particular, ifS â© Î = x(Ï1), x(Ï2) holds for any Ï1, Ï2 â ÎŁ, then Î is called strictly convex.The curve is called concave or nonconvex, if it is not convex.
Lemma 1.4. A closed embedded C2-curve is convex if and only if K(Ï) â„ 0holds for Ï â ÎŁ. Equivalently, the curve is concave if and only if K takes negativevalue at some point.
Lemma 1.5. A closed embedded C2-curve is strictly convex, if K(Ï) > 0 holdsfor Ï â ÎŁ.
Exercise 1.6. Prove the above two lemmas.
Remark 1.7. Convex curves admit including a line segment (K = 0 on someportion of ÎŁ), while strictly convex curve does not admit it. This does not meanthat the converse of Lemma 1.5 is true. See section 2.
6Ω is called inside of Î, and outside of Î is R2\Ω.7In detail, S = (1 â λ)x(Ï1) + λx(Ï2); λ â [0, 1].
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178 1. AN AREA-PRESERVING CRYSTALLINE CURVATURE FLOW EQUATION
3. Moving plane curve
We consider a moving curve with the time t in some interval, for instancet â [0, T ) with some T > 0. Let a smooth-closed-immersed moving curve be
Î(t) = x(Ï, t) â R2; Ï â ÎŁ.
By V we denote the growth speed at each point of Î(t) in the direction of n:
V = âtx · n.
It is well known that information of V determines the shape of Î(t), while thetangential component v does not effect on the shape [EG87]:
âtx = V n + vt.
Here and hereafter âtF means âtF(Ï, t), and âsF is given in terms of Ï:
âsF = gâ1âÏF, g =â
(âÏx)2 + (âÏy)2 = |âÏx|, x(Ï, t) = (x(Ï, t), y(Ï, t)).
That is, the arc-length element is ds = g dÏ.In what follows, we will see that fundamental properties of evolution of geo-
metric quantities.Since ât and âÏ can commute: âtâÏ = âÏât, using the FrenetâSerret formulae
we obtain:
Lemma 1.8. âtg = (KV + âsv)g.
This lemma and t = gâ1âÏx yield:
Lemma 1.9. âtt = (âsV â Kv)n and âtn = â(âsV â Kv)t.
Let Ξ = Ξ(Ï, t) be the normal angle such as n = (cos Ξ, sin Ξ). Then we have:
Lemma 1.10. âsΞ = K and âtΞ = ââsV + Kv.
By using the above lemmas, we have the evolution of K:
Lemma 1.11. âtK = â(â2sV + K2V ) + vâsK.
Since s depends on t, ât and âs can not commute:
Lemma 1.12. âtâsF = âsâtF â (KV + âsv)âsF.
We denote by L(t) the total length of Î(t): L =â«Î ds =
â«ÎŁ g dÏ. The evolution
of L(t) is given by:
Lemma 1.13. âtL =â«Î
KV ds.
We denote by Ω(t) the enclosed region of Î(t) and by A(t) the area of Ω(t).Using the relation
A =1
2
â«
Ω
div(xy
)dxdy =
1
2
â«
Î
x · n ds =1
2
â«
ÎŁ
(x · n)g dÏ,
we have the evolution of A(t):
Lemma 1.14. âtA =â«Î
V ds.
Exercise 1.15. Show above lemmas.
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4. CURVATURE FLOW EQUATIONS 179
4. Curvature flow equations
To catch mathematical characteristics of curvature flows, we shall formulatethe typical problem. We mainly focus on smooth, embedded and closed curve, andsometimes mention the case where curves are immersed.
Classical curvature flow. The classical and the typical equation which mo-tivates investigation of curvature flow equations is
V = âK. (1.2)
This is called the classical curvature flow equation. Figure 2 indicates evolutionof curve by (1.2). Convex part (K > 0) of nonconvex curve Î(t) moves towards thedirection of ân (inward), and concave part (K < 0) moves towards the directionof n (outward). Any embedded curve becomes convex in finite time [Gra87], andany convex curve shrinks to a single point in finite time and its asymptotic shapeis a circle [GH86]. In the case where the solution curve is immersed, for example,
Figure 2. Evolution of a simple closed curve by (1.2) (from left to right).
the curve has a single small loop as in Figure 3, we can observe that the singleloop shrinks and the curve may have cusp in finite time. A detailed analysis of thesingularity near the extinction time was done by Angenent and Velazquez [AV95].
Figure 3. Evolution of a immersed closed curve by (1.2) (fromleft to right). The small loop disappears in finite time.
Example of graph. In the case where Î(t) is described by a graph y = u(x, t),
we have the partial differential equation âtu =â2
xu1+(âxu)2 equivalent to (1.2) with
V = (0, âtu)T · n, n = (ââxu,1)Tâ1+(âxu)2
and K =ââ2
xu
(1+(âxu)2)3/2 , where (a, b)T =
(ab
).
Figure 4 indicates its numerical example.Gradient flow. As Figure 2 and Figure 4 suggested, the circumference L(t)
of the solution curve Î(t) is decreasing in time. This is the reason why (1.2) isoften called the curve-shortening equation. Actually, (1.2) is characterized bythe gradient flow of L as follows. By Lemma 1.13, the rate of change of the totallength L = L(Î(t)) of Î(t) is
âtL(Î(t)) =
â«
Î
KV ds.
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180 1. AN AREA-PRESERVING CRYSTALLINE CURVATURE FLOW EQUATION
Figure 4. Evolution of a graph y = u(x, t) by the curvature flow
equation âtu =â2
xu1+(âxu)2 (0 < x < 1) (from left to right). The
boundary condition is u(0) = u(1) = 0, and the initial curve isgiven by u(x, 0) = 0.1 sin(Ïx) â 0.3 sin(2Ïx) + 0.2 sin(5Ïx).
Let the inner product on Î be
ăF, GăÎ =
â«
Î
FG ds.
Then we have âtL(Î(t)) = ăK, V ăÎ. If the left-hand-side is regarded as a linearfunctional of V in a suitable space with the metric ă·, ·ăÎ, then the gradient flow isgiven by V = âK, and âtL(Î(t)) = âăK, KăΠ†0. Hence, (1.2) is the equationwhich requires motion of curve in the most decreasing direction of L in the senseof the above metric.
Exercise 1.16. Show âtA(t) = â2Ï.
Area-preserving curvature flow. Besides the curve-shortening flow, therehas also been interest in area-preserving flows. The typical flow is the gradient flowof L along curves which enclose a fixed area as follows:
V = K â K, (1.3)
where K is the average of K: K =â«Î K ds
/ â«Î ds = 2Ï/L if Î is simple and closed,
K = 2ηÏ/L if Î is immersed with the rotation number being η = 2, 3, . . . We have(1.3) as the gradient flow of L in the metric ă·, ·ăÎ under the constraint for V :ă1, V ăÎ = 0.
Figure 5. Evolution of a convex curve by (1.3) (from left to right).
Exercise 1.17. Show that âtL(t) †0 and âtA(t) = 0.
Gage [Gag86] proved that any convex curve converges to a circle as time tendsto infinity (see Figure 5), and conjectured that a nonconvex curve may intersect;this conjecture was proved rigorously by Mayer and Simonett [MS00].
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5. ANISOTROPY 181
Figure 6. Evolution of a immersed closed curve by (1.3) (fromleft to right). The small loop disappears in finite time (at leastnumerically).
Open problems. The following two problems are still open: One is the prob-lem whether the small loop may shrink or not. Figure 6 is a numerical exam-ple, and it suggests that extinction occurs in finite time. The other one is theproblem whether any embedded curve becomes eventually convex or not, even ifself-intersection occurred. In the curve-shortening case, these problems have beenalready analyzed, as it was mentioned above.
5. Anisotropy
In the field of material sciences and crystallography, we need to explain theanisotropyâphenomenon of interface motion which depends on the normal direc-tion, i.e., model equations have to contain effects of anisotropy of materials orcrystals.
Note. There are two kinds of anisotropy: one is kinetic anisotropy which ap-pears in the problem of growth form, and the other is equilibrium anisotropy whichappears in the problem of equilibrium form.
Interfacial energy. For example, if the crystal growth of snow flakes is re-garded as motion of closed plane curve, then some points on the interface Î areeasy to growth in the case where their normal directions are in the six special direc-tions. To explain these phenomena, it is convenient to define an interfacial energyon the curve Î which has line density Îł(n) > 0. Integration of Îł over Î is the totalinterfacial energy:
EÎł =
â«
Î
Îł(n) ds.
In the case where γ ⥠1, Eγ is nothing but the total length L, and its gradient flowwas (1.2). For a general γ, what is the gradient flow of Eγ?
Extension of Îł. The function Îł(n) can be extended to the function x â R2
by putting
Îł(x) =
|x| Îł
(x|x|
), x 6= 0,
0, x = 0.
This extension is called the extension of positively homogeneous of degree 1 (in short,PHD1) , since
γ(λx) = λγ(x)
holds for λ â„ 0 and x â R2. We will use the same notation Îł for the extended
function. Hereafter we assume that Îł is positive for x 6= 0 and positively homoge-neous of degree 1 function. Following Kobayashi and Giga [KG01], we call such Îła PPHD1 function.
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182 1. AN AREA-PRESERVING CRYSTALLINE CURVATURE FLOW EQUATION
Remark 1.18. The isotropic case γ(n) ⥠1 holds if and only if γ(x) = |x|.
We assume that Îł â C2(R2\0). Then the following properties hold:
âą âÎł(λx) = âÎł(x) (âÎł is homogeneous of degree 0),âą Îł(x) = x · âÎł(x) (In particular, Îł(n) = n · âÎł(n) holds),âą Hess Îł(λx) = 1
λ Hess Îł(x) (Hess Îł is homogeneous of degree â1),
where Hess Îł =
(Îł11 Îł12
Îł21 Îł22
)is the Hessian of Îł (Îłij = â2Îł/âxiâxj for i, j â 1, 2
and x = (x1, x2)).
Exercise 1.19. Show these properties. (The answer is in [KG01, Appendix 1].)
Weighted curvature flow. The gradient flow of the total interfacial energy
EÎł(Î(t)) =
â«
Î
Îł(n) ds =
â«
ÎŁ
Îł(n)g dÏ
is given byV = âÎÎł(n), ÎÎł(n) = (Hess Îł(n)nâ„) · nâ„K. (1.4)
This ÎÎł(n) is called weighted curvature or anisotropic curvature , and (1.4) iscalled the weighted curvature flow equation or anisotropic curvature flowequation .
Exercise 1.20. Show âtEÎł(Î(t)) =â«Î
ÎÎł(n)V ds.
Exercise 1.21. Try to calculate the gradient flow of FÎł =â«Î
Îł(x) ds.
Exercise 1.22. Try to calculate the gradient flow of GÎł =â«Î
ÎÎł(n)α ds,α = 1, 2.
Weighted curvature. Meaning of the weighted curvature ÎÎł(n) will be cleareras follows: Let Ξ be the exterior normal angle such as n = n(Ξ) = (cos Ξ, sin Ξ) andt = t(Ξ) = (â sin Ξ, cos Ξ). Put Îł(Ξ) = Îł(n(Ξ)). Then we obtain
ÎÎł(n(Ξ)) = (Îł(Ξ) + â2Ξ Îł(Ξ))K.
Exercise 1.23. Prove this equation.
Example of Îł. Let us construct an example of characteristic Îł. For a naturalnumber M , we extend the line segment y = x (x â [0, 2Ï/M ]) to 2Ï/M -periodicfunction p(x) such as
p(x) =2
M
(arctan
(tan
(M
2x â Ï
2
))+
Ï
2
),
and by η(x) we denote a concave and symmetric function with respect to the centralline x = Ï/M defined on [0, 2Ï/M ]. For example,
η(x) =sin x + sin(2Ï/M â x)
sin(2Ï/M).
The following function Îłp is a 2Ï/M -periodic function with the number of peakbeing M on [0, 2Ï/M ] (Figure 8 (left)).
Îłp(Ξ) = η(p(Ξ)), Ξ â R.
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6. THE FRANK DIAGRAM AND THE WULFF SHAPE 183
Note that Îłp is not differentiable at 2Ïk/M (k â Z). Then by using Ï”(x) =âx2 + ”â2, we construct a smooth function γ” (Figure 8 (left)):
γ”(Ξ) = Ï”(Îłp(Ξ) â 1) + 1, Ξ â R.
As ” tends to infinity, Ï”(x) converges to |x|, and then γ” converges to Îłp formally.In general, if the curve Î(t) is strictly convex, then the solution K of (1.4)
satisfies the partial differential equation âtK = K2(â2Ξ (âV )+ (âV )). See the book
of Gurtin [Gur93] or Exercise 1.24 below. Hence if Îł + â2Ξ Îł > 0 holds, then the
PDE is quasi-linear strictly parabolic type and its Cauchy problem is solvable. Wenote that γ” satisfies γ” + â2
Ξγ” > 0.
Exercise 1.24. Check the following story: Using the height function h = x ·n,the curve Î = x is described as x = hn + (âΞh)t, if Î is strictly convex. Henceâ2
Ξh = Kâ1 â h holds, and then we have ât(Kâ1) = â2
ΞV + V by V = âth.
Figure 7 indicates evolution of solution curves of (1.4) by using the interfacialenergy density γ = γ” in the case where M = 6 and ” = 15. The asymptotic shapeseems to be a round hexagon which is strikingly different from the one in Figure 2.
Figure 7. Evolution of a simple closed curve by (1.4) (from leftto right). The initial curve is the same as in Figure 2.
6. The Frank diagram and the Wulff shape
We will consider γp and γ” by graphical constructions.The Frank diagram. Figure 8 (left) indicates the graph of γ = γp(Ξ) and
γ = γp(Ξ), respectively; and Figure 8 (middle) indicates the polar coordinate ofeach γ:
CÎł =Îł(n(Ξ))n(Ξ); Ξ â S1
.
Incidentally, to observe the characteristic of Îł, the following Frank diagram(Figure 8 (right)) is more useful than a graph of Îł and CÎł :
FÎł =
n(Ξ)
Îł(n(Ξ)); Ξ â S1
=x â R
2; Îł(x) = 1
.
From this definition FÎł = C1/Îł holds.
Exercise 1.25. Show the following three proposition are equivalent, if Îł is aPPHD1 function. (The answer is in [KG01, Appendix 2].)
(i) We denote the set enclosed by FÎł by FÎł =x â R
2; γ(x) †1. Then
FÎł is convex.(ii) The function Îł satisfies Îł((1 â λ)x + λy) †(1 â λ)Îł(x) + λγ(y) for
x, y â R2 and λ â [0, 1]. (That is, Îł is convex.)
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184 1. AN AREA-PRESERVING CRYSTALLINE CURVATURE FLOW EQUATION
Figure 8. In each figure, the solid line is Îłp(Ξ), and the dottedline is γ”(Ξ) (” = 15), respectively. M = 6 in any case. (Left):graphs of Îłp(Ξ) and γ”(Ξ). The horizontal axis is 0 †Ξ †2Ï, andthe vertical axis is magnified for easy to observe. At each pointΞ = 2Ïk/M (k â Z), the function Îłp(Ξ) (resp. γ”(Ξ)) achieves theminimum value 1 (resp. 1+1/”). (Middle): CÎłp and Cγ” . (Right):the Frank diagram FÎłp and Fγ” . FÎłp is a regular M -polygon.
(iii) The function Îł satisfies Îł(x + y) †γ(x) + Îł(y) for x, y â R2. (That is,
Îł is subadditive.)
The interfacial energy Îł can be classified by the Frank diagram FÎł . Since thesign of the curvature of the curve FÎł agrees with the sign of Îł + â2
Ξ γ, then asmentioned above, the initial value problem of (1.4) can be caught in the frameworkof parabolic partial differential equations. The function γ = γ” is such an example.However, in the case where γ = γp, its Frank diagram Fγp is a regular M -polygon,
and on the each edge Îłp + â2Ξ Îłp = 0, and Îłp may not be differentiable at each
vertex. To treat such interfacial energy, crystalline curvature flow in the title willenter the stage. It will be mentioned later. Before it, let us define another shapeof characterizing Îł.
The Wulff shape. The following shape is called the Wulff shape:
WÎł =â
ΞâS1
x â R
2; x · n(Ξ) †γ(n(Ξ))
.
Figure 9 indicates the Wulff shape of γp and γ”, respectively, in the case whereM = 4 and M = 6.
The Wulff shape is a convex set by means of constructions. In particular, ifÎł is smooth and FÎł is strictly convex, then WÎł is also a strictly convex set witha smooth boundary. In this case, the distance from the origin (which is inside ofWÎł) to LΞ equals to Îł(n(Ξ)) = Îł(Ξ), where LΞ is the tangent line which is passingthrough the point on âWÎł with its outward normal vector being n(Ξ). From thisfact, we note that the curvature of âWÎł is given by (Îł + â2
Ξ Îł)â1.
Exercise 1.26. Show that the curvature of âWÎł is (Îł + â2Ξ Îł)â1 if WÎł is a
strictly convex set with a smooth boundary.
In the physical context, WÎł describes the equilibrium of crystal, i.e., in theplane case, WÎł is the answer of the following âȘWulff problemâ« on the equilibriumshape of crystals (by Gibbs (1878), Curie (1885), and Wulff (1901)):
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6. THE FRANK DIAGRAM AND THE WULFF SHAPE 185
Figure 9. Comparison of γp(Ξ) and γ”(Ξ) (” = 15) by the Wulffshape. (Left, middle left): The Wulff shapes Wγp and Wγ” withM = 4. (Middle right, right): The Wulff shapes Wγp and Wγ”
with M = 6. Two WÎłp âs are regular M -polygons, and two Wγ” âsare round regular M -polygons, respectively.
What is the shape which has the least total interfacial energyEÎł of the curve for the fixed enclosed area?
This suggests that the asymptotic shape of a solution curve of the gradient flowof EÎł (1.4) is âWγ” in the case Îł = γ” (Figure 7). However, in the mathematicalcontext, there are not so many known results.
Open problems. Convexified phenomena or formation of convexity is one ofthe interesting problems. In the isotropic case Îł ⥠1, Grayson [Gra87] provedthat convexity is formed (see Figure 2). However in the anisotropic case, even ifÎł is smooth, the formation of convexity has not been completed, except in thesymmetric case Îł(Ξ + Ï) = Îł(Ξ) by Chou and Zhu [CZ99].
Incidentally, the Frank diagram FÎł has been considered after the Wulff shapeWÎł has appeared (Frank (1963) and Meijering (1963)). See the book on crystalgrowth in detail. We also refer the reader to Kobayashi and Giga [KG01] for a lotof examples of CÎł , FÎł and WÎł , and for discussion of anisotropy and the effect ofcurvature.
Numerical examples. Now, going back the story, we observe the behavior ofa solution curve y = u(x, t) by (1.4) with γ = γ” (” = 1000) from a view point ofan approximation of γ = γp.
Since the total interfacial energy is given by
EÎł =
â« 1
0
W (âxu) dx, W (Ο) = Îł(Ξ(Ο))â
1 + Ο2, Ξ(Ο) = â arctan1
Ο,
we obtain the partial differential equation
âtu =â
1 + (âxu)2 âx(âΟW (âxu)) =Îł + â2
Ξ γ
1 + (âxu)2â2
xu,
which is equivalent to (1.4).Figure 10 indicates a numerical example of the case where M = 4 and ” = 1000.
As time goes by, one can observe that line segment parts satisfying âxu = 0 expands,and the other parts (which are put between two parallel line segments satisfyingâxu = 0) do not move, and will disappear in finite time. Actually, since γ” + â2
Ξ γ”
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186 1. AN AREA-PRESERVING CRYSTALLINE CURVATURE FLOW EQUATION
takes a huge value only at Ξ = Ïk/2 (k â Z) and almost zero at other points, themovement of points satisfying âxu ⌠0 can be observed and other points do notseem to move.
Figure 10. Evolution of a graph y = u(x, t) by the curvature flow
equation âtu =â
1 + (âxu)2 âx(âΟW (âxu)) (0 < x < 1) (from left
to right). Here W (Ο) = γ”(Ξ(Ο))â
1 + Ο2 (” = 1000, M = 4),Ξ(Ο) = Ï/2+arctan Ο. The boundary conditions are u(0) = u(1) =0, and the initial curve is the same as the one in Figure 4.
Figure 11 indicates a numerical example of the case where M = 6 and ” = 1000.In this case, WÎł is almost a regular hexagon (it almost agrees with WÎłp in Figure9 (middle right), and in the set of normal angles of Wγ” one can observe themovement of points on the graph satisfying that their normal angles almost agreewith Ξ = ±Ïk/3 (k = 1, 2), and other points do not seem to move. Since the endpoints are fixed, Figure 11 (far right) is the final shape.
Figure 11. Evolution of a graph y = u(x, t) by the curvature flow
equation âtu =â
1 + (âxu)2 âx(âΟW (âxu)) (0 < x < 1) (from left
to right). Here W (Ο) = f”(Ξ(Ο))â
1 + Ο2 (” = 1000, M = 6),Ξ(Ο) = Ï/2+arctan Ο. The boundary conditions are u(0) = u(1) =0, and the initial curve is the same as the one in Figure 4.
When ” tends to infinity, how does the equation
âtu =â
1 + (âxu)2âx(âΟW (âxu))
change? In the case where M = 4, Îłp(Ξ) = | cos Ξ | + | sin Ξ | holds, and then wehave W (Ο) = |Ο| + 1. From this we obtain formally â2
Ο W (Ο) = 2Ύ(Ο), i.e., the
partial differential equation becomes âtu = 2â
1 + (âxu)2 ÎŽ(âxu)â2xu, where ÎŽ is
Dirac delta function. As Figure 10 suggested, the right-hand-side is considerableas a nonlocal quantity [Gig97].
By the observation of Figure 10 and Figure 11, it seems that there is no problemif curves are restricted to the following special class of polygonal curves: the set ofnormal vectors of the curves agrees with the one of the Wulff shape.
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7. CRYSTALLINE ENERGY 187
7. Crystalline energy
If the Frank diagram FÎł is a convex polygon, Îł is called crystalline en-ergy. When FÎł is a J-sided convex polygon, there exists a set of angles Ïi |Ï1 <Ï2 < · · · < ÏJ < Ï1 + 2Ï such that the position vectors of vertices are labeledn(Ïi)/Îł(n(Ïi)) in an anticlockwise order (ÏJ+1 = Ï1, Ï0 = ÏJ) and Ïi+1 â Ïi â(0, Ï) holds for i = 1, 2, . . . , J :
FÎł =
Jâ
i=1
[Îœi
Îł(Îœi),
Îœi+1
Îł(Îœi+1)
].
Here and hereafter, we denote Îœi = n(Ïi) (âi). See Figure 12 (left). In this case,the Wulff shape is also a J-sided convex polygon with the outward normal vectorof the i-th edge being Îœi:
WÎł =
Jâ
i=1
x â R
2; x · Îœi †γ(Îœi)
.
See Figure 12 (right). We define a set of normal vectors of WÎł by NÎł = Îœ1, Îœ2, . . . , ÎœJ.
Μ1
Μ2Μ3
Μ4
Μ5 Μ6
Figure 12. Example of the Frank diagram (left) and the Wulffshape (right) in the case where Îł is a crystalline energy and J = 6.This Frank diagram is the same regular hexagon FÎłp as in Figure8 (right), and this Wulff shape is the same regular hexagon WÎłp
as in Figure 9 (middle right).
Remark 1.27. A point on the (i â 1)-th edge of J-sided polygon FÎł can bedescribed as
Μ
Îł(Îœ)= (1 â ”)
Îœiâ1
Îł(Îœiâ1)+ ”
Îœi
Îł(Îœi)
for some ” â (0, 1), where Îœ = n(Ï) with some Ï â (Ïiâ1, Ïi); or one can describeÎœ without using Ï:
Îœ =(1 â λ)Îœiâ1 + λΜi
|(1 â λ)Îœiâ1 + λΜi|, λ =
”γ(Îœiâ1)
(1 â ”)Îł(Îœi) + ”γ(Îœiâ1).
Exercise 1.28. Show this expression of Μ.
Let Li be the straight line passing through Îł(Îœi)Îœi in the direction Îœâ„i for all
i, and let yi be the intersection point of Liâ1 and Li:
yi = Îł(Îœi)Îœi â lâÎł Îœâ„i , lâÎł =
Îł(Îœiâ1) â (Îœiâ1 · Îœi)Îł(Îœi)
âÎœiâ1 · Îœâ„i
.
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188 1. AN AREA-PRESERVING CRYSTALLINE CURVATURE FLOW EQUATION
It is easy to check that the following claim.
Proposition 1.29. yi · Îœ = Îł(Îœ) holds for each i.
Exercise 1.30. Prove Proposition 1.29.
Proposition 1.29 asserts that the straight line passing through Îł(Îœ)Îœ in thedirection Μ℠passes through yi.
We have
lâÎł =1 â aâ
bâ, aâ = (Îœiâ1 · Îœi)
Îł(Îœi)
Îł(Îœiâ1), bâ = âÎœiâ1 · Îœâ„
i
Îł(Îœiâ1).
In the same way, we have
yi+1 = Îł(Îœi)Îœi + l+Îł Îœâ„i , l+Îł =
Îł(Îœi+1) â (Îœi+1 · Îœi)Îł(Îœi)
Îœi+1 · Îœâ„i
=1 â a+
b+,
where
a+ = (Îœi+1 · Îœi)Îł(Îœi)
Îł(Îœi+1), b+ =
Îœi+1 · Îœâ„i
Îł(Îœi+1).
Since the angle between Îœiâ1 and Îœi satisfies Ïiâ1 â Ïi â (0, Ï), bâ > 0 holds, andalso b+ > 0 holds by the similar reason.
Proposition 1.31. (yi+1 â yi) · Îœâ„i = lâÎł + l+Îł > 0 holds.
Since FÎł is a J-sided convex polygon, there exists ” â (0, 1) such that for each i
η = (1 â ”)Îœiâ1
Îł(Îœiâ1)+ ”
Îœi+1
Îł(Îœi+1)
holds, and that
η · Μi <1
Îł(Îœi), η · Îœâ„
i = 0
hold. From the inequality and equation, and from the facts bâ > 0 and b+ > 0, wehave
” =bâ
bâ + b+, aâ â ”(aâ â a+) =
aâb+ + a+bâ
bâ + b+< 1.
Hence it holds that
lâÎł + l+Îł =1
bâb+(bâ + b+ â (aâb+ + a+bâ)) > 0.
Therefore WÎł is J-sided convex polygon with the i-th vertex being yi and the thelength of the i-th edge being lâÎł + l+Îł > 0 for each i.
Exercise 1.32. Show that if FÎł is the regular J-sided polygon, WÎł is also theregular J-sided polygon.
Remark 1.33. There exist two crystalline energies Îł1 and Îł2 such that WÎł1
agrees with WÎł2except the position of the center. See Figure 13.
Remark 1.34. As we saw the above, if the Frank diagram is a J-sided convexpolygon, then the Wulff shape is also the J-sided convex polygon. But the converseis not true. See Figure 14.
Remark 1.35. Remark 1.33 and Remark 1.34 suggest that WÎł does not haveall the information of Îł. Actually, WÎł = WÎł holds, where Îł is convexification of Îł.See [KG01].
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8. CRYSTALLINE CURVATURE FLOW EQUATIONS 189
FÎł WÎł
Îł = Îł1:
Îł = Îł2:
Figure 13. Example of the same shape of the Wulff shape withdifferent Îł: Îł1 6= Îł2.
8. Crystalline curvature flow equations
We will define crystalline curvature flow equations.Polygonal curves. Let P be a simple closed N -sided polygonal curve in the
plane R2, and label the position vector of vertices pi (i = 1, 2, . . . , N) in an anti-
clockwise order:
P =
Nâ
i=1
Si,
where Si = [pi, pi+1] is the i-th edge (pN+1 = p1, p0 = pN ). The length of Si
is di = |pi+1 â pi|, and then the i-th unit tangent vector is ti = (pi+1 â pi)/di
and the i-th unit outward normal vector is ni = âtâ„i . We define a set of normalvectors of P by N = n1, n2, . . . , nN. Let Ξi be the exterior normal angle ofSi. Then ni = n(Ξi) and ti = t(Ξi) hold. We define the i-th hight functionhi = pi · ni = pi+1 · ni. See Figure 15. By using N -tuple h = (h1, h2, . . . , hN ), di
is described as di = Di(h), where
Di(h) =Ïiâ1,iâ
1 â (niâ1 · ni)2(hiâ1 â (niâ1 · ni)hi)
+Ïi,i+1â
1 â (ni · ni+1)2(hi+1 â (ni · ni+1)hi),
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190 1. AN AREA-PRESERVING CRYSTALLINE CURVATURE FLOW EQUATION
FÎł WÎł
Îł = Îł1:
Îł = Îł2:
Figure 14. The Wulff shapes are both regular hexagons, whilethe corresponding Frank diagrams are not polygons with differentÎł: Îł1 6= Îł2.
where Ïi,j = sgn(det(ni, nj)) for i = 1, 2, . . . , N (hN+1 = h1, h0 = hN). Sinceni · nj = cos(Ξi â Ξj), we have another expression:
Di(h) = â(cotÏi + cotÏi+1)hi + hiâ1 cosecÏi + hi+1 cosecÏi+1, (1.5)
where Ïi = Ξi â Ξiâ1 for i = 1, 2, . . . , N . Note that 0 < |Ïi| < Ï holds for all i.Furthermore, the i-th vertex pi (i = 1, 2, . . . , N) is described as follows:
pi = hini +hiâ1 â (niâ1 · ni)hi
niâ1 · titi. (1.6)
Exercise 1.36. From the relation nTiâ1pi = hiâ1 and nT
i pi = hi, we obtain
pi =
(nT
iâ1
nTi
)â1(hiâ1
hi
).
From this, show (1.6). This idea can be found in [BKY08].
Exercise 1.37. From (1.6) and the relation ti · ni+1 = sinÏi+1, show (1.5).
Exercise 1.38. From (1.6), check that di = |pi+1âpi| and di = (pi+1âpi) ·ti
hold.
Exercise 1.39. Fix any a â R2. By ki = a · ni we denote distance from the
origin to a in the direction ni, and by k = (k1, k2, . . . , kN ) we denote their N -tuple(kN+1 = k1, k0 = kN ). Show that Di(k) = 0 holds for i = 1, 2, . . . , N . From this
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8. CRYSTALLINE CURVATURE FLOW EQUATIONS 191
pi+1
Si
pi
ΞiΞj
nj
tj
hj
dj
Figure 15. Example of polygonal curves (N = 56).
fact, Di(h + k) = Di(h) + Di(k) = di holds for all i. This means that the shape ofcurve does not change for a shift of the origin.
Admissible curves. Following [HGGD05], we call curve P essentially ad-missible if and only if the consecutive outward unit normal vectors ni, ni+1 â N(nN+1 = n1, n0 = nN ) satisfy
ni+λ =(1 â λ)ni + λni+1
|(1 â λ)ni + λni+1|6â NÎł
for λ â (0, 1) and i = 1, 2, . . . , N . See Figure 16 (middle).
Exercise 1.40. If P is an essentially admissible curve, then N â NÎł holds.But the converse is not true. Make a counterexample.
Remark 1.41. In the case where P is a convex polygon, P is an essentiallyadmissible if and only if N â NÎł holds.
Exercise 1.42. Check this remark.
We call curve P admissible if and only if P is an essentially admissible curveand N = NÎł (especially, N â NÎł) holds. In other words, P is admissible if and onlyif N = NÎł holds and any adjacent two normal vectors in the set NÎł , for exampleÎœi and Îœi+1 are also adjacent in the set N , i.e., Îœi, Îœi+1 = nj , nj+1 â N holdsfor some i and j (ÎœJ+1 = Îœ1, nN+1 = n1). See Figure 16 (left).
Gradient flow. We consider a moving essentially admissible and an N -sidedcurve P(t) with the time t in some interval and with the N -tuple of hight func-tions h(t) = (h1(t), h2(t), . . . , hN(t)). Then we have di(t) = Di(h(t)). The total
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192 1. AN AREA-PRESERVING CRYSTALLINE CURVATURE FLOW EQUATION
n1
(a) Admissible polygonal curve(N = 10).
n1
n3
Μ2Μ3
Μ3
n9
Μ4
Μ5
n12
Μ6
(b) Essentially admissible polygo-nal curve (N = 13). Note thatNÎł â n3, n9, n12 = N holds.
n1
Μ2
n3
(Μ3)
Μ4Μ2
n6
(Μ3)
Μ4
n9Μ5
(Μ6)
Μ1
(c) This polygonal curve (N = 9) isnot admissible, nor essentially admis-sible, since NÎł 6â N (Îœ6 6â N ) holds,and in addition there exist λ, λâČ, λâČâČ â(0, 1) such that n3+λ = n6+λâČ =Îœ3, n9+λâČâČ = Îœ6 â NÎł holds.
Figure 16. Three typical examples of polygonal curves for theWulff shape in Figure 12 (right).
crystalline energy on P(t) is defined as EÎł(P(t)) =âN
i=1 Îł(ni)di(t), and the rateof change of it is given as
âtEÎł(P(t)) =
Nâ
i=1
Îł(ni)âtDi(h) =
Nâ
i=1
Îł(ni)Di(V ),
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8. CRYSTALLINE CURVATURE FLOW EQUATIONS 193
where V = (V1, V2, . . . , VN ) and Vi = âthi(t) is the velocity on Si in the direction ofni for i = 1, 2, . . . , N . Here we have used the relation âtdi(t) = Di(V ) from (1.5),which is equivalent to the time derivative of (1.6):
âtpi = Vini + viti, vi =Viâ1 â (niâ1 · ni)Vi
niâ1 · ti, i = 1, 2, . . . , N. (1.7)
We denote Îł = (Îł(n1), Îł(n2), . . . , Îł(nN )), and then we have
Nâ
i=1
Îł(ni)Di(V ) =
Nâ
i=1
ViDi(Îł) =
Nâ
i=1
Di(Îł)
diVidi.
Hence we have
âtEÎł(P(t)) =Nâ
i=1
ÎÎł(ni)Vidi, ÎÎł(ni) =Di(Îł)
di, i = 1, 2, . . . , N.
Here ÎÎł(ni) is called crystalline curvature on the i-th edge Si, which is thecrystalline version of weighted curvature derived from âtEÎł(Î(t)) =
â«Î ÎÎł(n)V ds.
For two N -tuples F = (F1, F2, . . . , FN ), G = (G1, G2, . . . , GN ) â RN , let us define
the inner product on P as follows:
ăF, GăP =
Nâ
i=1
FiGidi.
Therefore by analogue of gradient flow of EÎł(Î(t)), we have the gradient flow ofEÎł(P(t)) such as
Vi = âÎÎł(ni), i = 1, 2, . . . , N.
This is called crystalline curvature flow equation.The numerator of crystalline curvature ÎÎł(ni) is described as
Di(Îł) = ÏilÎł(ni), (1.8)
where Ïi = (Ïiâ1,i + Ïi,i+1)/2 takes +1 (resp. â1) if P is convex (resp. concave)around Si in the direction of âni, otherwise Ïi = 0; and lÎł(n) is the length of thej-th edge of WÎł if n = Îœj for some j, otherwise lÎł(n) = 0.
Remark 1.43. Equation (1.8) can be derived as follows. We define
lâÎł (ÏâČ, Ï) =Îł(n(ÏâČ)) â (n(Ï) · n(ÏâČ))Îł(n(Ï))
ân(ÏâČ) · n(Ï)â„=
Îł(n(ÏâČ)) â Îł(n(Ï)) cos(Ï â ÏâČ)
sin(Ï â ÏâČ),
and l+Îł (Ï, ÏâČ) = âlâÎł (ÏâČ, Ï). Then we have
lÎł(Îœj) = lâÎł (Ïjâ1, Ïj) + l+Îł (Ïj , Ïj+1), Di(Îł) = lâÎł (Ξiâ1, Ξi) + l+Îł (Ξi, Ξi+1).
By geometric consideration, we have
lâÎł (Ξ, Ïj) =
lâÎł (Ïjâ1, Ïj), Ξ â [Ïjâ1, Ïj),âl+Îł (Ïj , Ξ) = âl+Îł (Ïj , Ïj+1), Ξ â (Ïj , Ïj+1],
and
l+Îł (Ïj , Ξ) =
l+Îł (Ïj , Ïj+1), Ξ â (Ïj , Ïj+1],âlâÎł (Ξ, Ïj) = âlâÎł (Ïjâ1, Ïj), Ξ â [Ïjâ1, Ïj).
The two cases are possible. The first case is ni â NÎł , i.e., there exists j such thatni = Îœj and Ξi = Ïj; and we have four subcases as follows:
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194 1. AN AREA-PRESERVING CRYSTALLINE CURVATURE FLOW EQUATION
(i) Ξiâ1 â [Ïjâ1, Ïj) and Ξi+1 â (Ïj , Ïj+1],(ii) Ξiâ1 â (Ïj , Ïj+1] and Ξi+1 â [Ïjâ1, Ïj),(iii) Ξiâ1, Ξi+1 â (Ïj , Ïj+1], and(iv) Ξiâ1, Ξi+1 â [Ïjâ1, Ïj).
Hence, Di(Îł) is equal to
(i) lÎł(Îœj),(ii) âlÎł(Îœj),(iii) 0, and(iv) 0, respectively.
The second case is ni 6â NÎł, i.e., there exists j such that Ξi â (Ïjâ1, Ïj); and wehave four subcases similarly as above. In this case, we regard that there is a zero-length edge of WÎł between the (j â 1)-th and the j-th edges, and for all subcasesDi(Îł) = 0 holds. Thus we obtain (1.8).
Exercise 1.44. Follow this remark.
aΜ1
b
Μ5 Μ6n12
(a) a is the length of edgelÎł(n1) > 0 and b is thelength of degenerate edgelÎł(n12) = 0.
d1 n1
d12
n12
(b) d1 (resp. d12) is the length ofthe edge whose normal vector is n1
(resp. n12). Hence the crystallinecurvature is ÎÎł(n1) = lÎł(n1)/d1
(resp. ÎÎł(n12) = 0).
Figure 17. The Wulff shape (left) is the same hexagon as in Fig-ure 12 (right). Essentially admissible polygonal curve (right) is thesame curve as in Figure 16(b).
Remark 1.45. If P is an admissible and convex polygon, then ni = Îœi andÏi = 1 for all i = 1, 2, . . . , N = J ; and moreover, if P = âWÎł, then the crystallinecurvature is 1.
Note. To handle the gradient flow of EÎł (1.4) in the case where Îł is crys-talline, we have restricted smooth curve to admissible curve and have introducedthe crystalline curvature defined on each edge. This strategy was proposed by Tay-lor [Tay90, Tay91a, Tay91b, Tay93] and independently by Angenent and Gurtin[AG89]. Also one can find essentially the same method as a numerical scheme for
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9. AN AREA-PRESERVING MOTION BY CRYSTALLINE CURVATURE 195
curvature flow equation in Roberts [Rob93]. We refer the reader Almgren and Tay-lor [AT95] for detailed history. In the physical context, the region enclosed by Prepresents the crystal. See also Taylor, Cahn and Handwerker [TCH92] and Gurtin[Gur93] for physical background. Besides this crystalline strategy, other strategiesby subdifferential and level-set method have been extensively studied. See Giga[Gig97, Gig98, Gig00] and references therein.
9. An area-preserving motion by crystalline curvature
The enclosed area A(t) of P(t) is given by
A =1
2
Nâ
i=1
hidi,
and its rate of change is
âtA(t) =1
2
Nâ
i=1
Vidi +1
2
Nâ
i=1
hiDi(V ) =
Nâ
i=1
Vidi.
Then the gradient flow of EÎł along P which encloses a fixed area is
Vi = ÎÎł â ÎÎł(ni), i = 1, 2, . . . , N, (1.9)
where
ÎÎł =
âNi=1 ÎÎł(ni)diâN
k=1 dk
=
âNi=1 Di(Îł)
L=
âNi=1 ÏilÎł(ni)
L
is the average of the crystalline curvature. The gradient flow (1.9) is in the metrică·, ·ăP under the constraint for V : ă1, V ăP = 0, 1 = (1, 1, . . . , 1).
Exercise 1.46. Check the following two basic properties: âtEÎł(P(t)) †0 andâtA(t) = 0.
Problem 1. For a given N -sided essentially admissible closed curve P0, find afamily of N -sided essentially admissible curves
â0â€t<T P(t) satisfying
Vi(t) = ÎÎł â ÎÎł(ni), 0 †t < T, i = 1, 2, . . . , N,P(0) = P0.
Remark 1.47. Problem 1 is equivalent to âtdi(t) = Di(V ), or equivalent to(1.7); and since it is described as the N -system of ordinary differential equations,the maximal existence time is positive: T > 0.
Remark 1.48. Problem 1 is closely related to general area-preserving motionof polygonal curves. See [BKY07, BKY08].
Known results. What might happen to P(t) as t tends to T †â? For thisquestion, we have the following three results. The first result is the case wheremotion is isotropic and polygon is admissible.
Theorem 1.49. Let the interfacial energy be isotropic Îł ⥠1. Assume the initialpolygon P0 is an N -sided admissible convex polygon. Then a solution admissiblepolygon P(t) exists globally in time keeping the area enclosed by the polygon constantA, and P(t) converges to the shape of the boundary of the Wulff shape âWÎłâ
in the
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196 1. AN AREA-PRESERVING CRYSTALLINE CURVATURE FLOW EQUATION
Hausdorff metric as t tends to infinity, where Îłâ =â
2A/âN
k=1 l1(nk) is constant.
In particular, if P0 is centrally symmetric with respect to the origin, then we havean exponential rate of convergence.
This theorem is proved by Yazaki [Yaz02] by using the isoperimetric inequalityand the theory of dynamical systems. We note that âWÎłâ
is the circumscribedpolygon of a circle with radius Îłâ, and then this result is a semidiscrete version ofGage [Gag86].
The second result is the case where motion is anisotropic and polygon is ad-missible.
Theorem 1.50. Let the crystalline energy be Îł > 0. Assume the initial polygonP0 is an N -sided admissible convex polygon. Then a solution admissible polygonP(t) exists globally in time keeping the area enclosed by the polygon constant A,and P(t) converges to the shape of the boundary of the Wulff shape âWÎłâ
in the
Hausdorff metric as t tends to infinity, where Îłâ(ni) = Îł(ni)/wâ, wâ =â|WÎł |/A
for i = 1, 2, . . . , N and |WÎł | =âN
k=1 Îł(nk)lÎł(nk)/2 is enclosed area of WÎł .
This theorem is proved in Yazaki [Yaz04, Part I] by using the anisoperimet-ric inequality or Brunn and Minkowskiâs inequality and the theory of dynamicalsystems. For readerâs convenience, the proof will be shown in the next section.
The last result is the case where motion is anisotropic and polygon is essentiallyadmissible.
Theorem 1.51. Let the crystalline energy be Îł > 0. Assume the initial polygonP0 is an N -sided essentially admissible convex polygon. If the maximal existencetime of a solution essentially admissible polygon P(t) is finite T < â, then thereexists the i-th edge Si such that limtâT di(t) = 0 and lÎł(ni) = 0 hold. That is,the normal vector of vanishing edge does not belong to NÎł, and inf0<t<T dk(t) > 0holds for all nk â NÎł .
This theorem is proved in [Yaz07b].Open problems. For any essentially admissible convex polygon P0, is T a
finite value? This is still open. If the answer of this question is yes, then wehave the finite time sequence T1 < T2 < · · · < TM such that P(Ti) is essentiallyadmissible for i = 1, 2, . . . , M â 1 and P(TM ) is admissible. In the general casewhere Vi = F (ni, ÎÎł(ni)) for all i under certain conditions of F , the answer of the
above question is yes. See [Yaz07a]. However, F does not include ÎÎł .Open problems in the case where P0 is nonconvex. Even if P0 is ad-
missible, we have no theoretical results at this stage. See section 11 in detail.
10. Scenario of the proof of Theorem 1.50
Put the crystalline curvature such as
wi(t) = ÎÎł(ni) =lÎł(ni)
di, i = 1, 2, . . . , N.
Then we have
âtwi(t) = âlâ1i wi(t)
2Di(V ), Di(V ) = Di(1)ÎÎł â Di(w),
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10. SCENARIO OF THE PROOF OF THEOREM 1.50 197
where li = lÎł(ni), 1 = (1, 1, . . . , 1) and w = (w1, w2, . . . , wN ). Note that
Di(1) = tanÏi
2+ tan
Ïi+1
2
holds for i = 1, 2, . . . , N . Hence from the average ÎÎł =âN
k=1 lk/L and the length
L =âN
j=1 ljwj(t)â1, we can restate Problem 1 as the following Problem 2.
Problem 2. Find a function w(t) = (w1(t), w2(t), . . . , wN (t)) â (C[0, T ) â©C1(0, T ))N satisfying
âtwi(t) = lâ1i Di(w(t))wi(t)
2 â lâ1i Di(1)
âNk=1 lkâN
j=1 ljwj(t)â1wi(t)
2,
i = 1, 2, . . . , N, 0 < t < T
and
wi(0) = w0i , i = 1, 2, . . . , N,
wN+1(t) = w1(t), w0(t) = wN (t), 0 < t < T,
where w0i is the i-th initial crystalline curvature of P0.
Problem 1 and Problem 2 are equivalent except the indefiniteness of positionof a solution polygon. See [Yaz02, Remark 2.1]. Since Problem 2 is the initialvalue problem of ordinary differential equations, there exists a unique time localsolution. Moreover, by using a similar argument as in Taylor [Tay93, Proposition3.1], Ishii and Soner [IS99, Lemma 3.4], Yazaki [Yaz02, Lemma 3.1], we obtain thetime global solvability.
Lemma 1.52 (time global existence). A solution w of Problem 2 and a solutionpolygon of Problem 1 exist globally in time, i.e., a solution polygon does not developsingularities in a finite time.
In the following, we will show three lemmas, which play an important role in ascenario of the proof of Theorem 1.50.
Let the anisoperimetric ratio be
JÎł(P) =EÎł(P)2
4|WÎł |Afor an N -sided admissible convex polygon P associated with WÎł . Here A (resp.|WÎł |) is the enclosed area of P (resp. the area of WÎł). The first key lemma is theanisotropic version of the isoperimetric inequality.
Lemma 1.53 (anisoperimetric inequality). For a polygon P associated with WÎł,the anisoperimetric inequality
JÎł(P) â„ 1
holds. The equality JÎł(P) ⥠1 holds if and only if wi = ÎÎł(ni) ⥠const. for all i,i.e., Ω (the enclosed region of P) satisfies Ω = kWÎł for some constant k > 0.
We will prove this lemma in Appendix 3 using the mixed area and the Brunnand Minkowskiâs inequality.
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198 1. AN AREA-PRESERVING CRYSTALLINE CURVATURE FLOW EQUATION
From this lemma and lemma 1.52, for a solution polygon P(t),
JÎł(P(t)) =EÎł(P(t))2
4|WÎł |Aâ„ 1
holds for t â„ 0. Moreover, we have the following second key lemma.
Lemma 1.54. limtââ EÎł(P(t)) = 2â|WÎł |A and limtââ JÎł(P(t)) = 1 hold.
Proof of this lemma closely follows [Yaz02, Lemma 5.6].From this lemma, if a solution polygon P(t) is N -sided polygon at the time
infinity, then Ω(t) approaches to kWÎł (k > 0) as t tends to infinity. There-fore, to complete the proof of Theorem 1.50, it is required to show the estimateinf0<t<â min1â€iâ€N di(t) > 0. As a matter of fact, the following the third keylemma holds. This lemma is a strong assertion compared with the above estimate.
Lemma 1.55. Let wâ be the equilibrium point of the first evolution equation inProblem 2. Then wâ =
â|WÎł |/A holds. Moreover, the equilibrium point wâ is
asymptotically stable and
limtââ
wi(t) = wâ
holds for i = 1, 2, . . . , N .
One can prove this lemma by the general theory of dynamical systems or theLyapunov theorem (see [Yaz02, Lemma 5.8]).
Proof of Theorem 1.50. From Lemma 1.55, we have
limtââ
di(t) =liwâ
, i = 1, 2, . . . , N.
From this limit and the theory of the generalized eigenvalue space, there exists avector c(t) â R
2 such that hi(t) â c(t) · ni converges to Îłâ(ni) = Îł(ni)/wâ forall i as t tends to infinity. Hence for any Δ > 0 there exists tâČ > 0 such thatP(t) â W(1+Δ)Îłâ
\W(1âΔ)Îłâholds for t â„ tâČ. Then the assertion holds.
11. Numerical scheme
The aims are to construct a numerical scheme which enjoys two basic propertiesâtEÎł †0 and âtA = 0 (see Exercise 1.46), and to investigate what might happen tosolution P(t) of the evolution equation (1.9) as t tends to T †â. We discretize thesystem of ordinary equations âtdi(t) = Di(V ) or (1.7) with the initial essentiallyadmissible closed curve P0 = P0 at the time t0 = 0. Let m = 0, 1, 2, . . . be a step
number. By am, we denote the approximation of a(tm) at the time tm =âmâ1
i=0 Ïi
(the time step Ïm will be defined in the following procedure).The following procedure is an extension of [UY04].Procedure 1. Fix parameters ” â [0, 1] and λ, Δ â (0, 1). For a given essen-
tially admissible N -sided closed curve Pm =âN
i=1[pmi , pm
i+1], we define Pm+1 =âNi=1[p
m+1i , pm+1
i+1 ] as follows:
(i) the i-th length: dmi = |pm
i+1 â pmi | (âi);
(ii) the m-th variable time step: Ïm = Ï(dmmin)
2/â, whereÏ = Δ(1â”λ)minλ, 1â”λ, â = 2|ÏlÎł(n)|max(2/| sinÏ|min+| tan(Ï/2)|max);
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11. NUMERICAL SCHEME 199
(iii) the i-th length dm+1i :
(DÏd)mi = â(cotÏi + cotÏi+1)V
m+”i + V m+”
iâ1 cosecÏi + V m+”i+1 cosecÏi+1
(âi),
V m+”i =
PNj=1
Ïj lÎł(nj)PN
k=1dm+”
k
â ÏilÎł(ni)
dm+”i
, dm+”i = (1 â ”)dm
i + ”dm+1i ;
(iv) the i-th hight hm+1i : (DÏh)m
i = V m+”i (âi);
(v) the i-th vertex: pm+1i = hm+1
i ni +hm+1
iâ1â(niâ1·ni)h
m+1
i
niâ1·titi (âi);
(vi) the new time: tm+1 = tm + Ïm.
Here we have used the notation: amin = min1â€iâ€N ai, |a|min = min1â€iâ€N |ai|,|a|max = max1â€iâ€N |ai|, (DÏa)m
i = (am+1i âam
i )/Ïm, and the periodicity FN+1 = F1,F0 = FN .
Two basic properties. One is that the total energy EmÎł is decreasing in steps:
(DÏEÎł)m †0 for any ” â [0, 1]. The other is that the enclosed area Am of Pm ispreserved: (DÏA)m = 0 if ” = 1/2.
Iteration. In (3), if ” â (0, 1], we solve the following iteration starting fromz0
i = dmi :
zk+1i â z0
i
Ïm= â(cotÏi + cotÏi+1)V
m+”i + V m+”
iâ1 cosecÏi + V m+”i+1 cosecÏi+1,
V m+”i =
âNj=1 Ïj lÎł(nj)âN
k=1 dm+”k
â ÏilÎł(ni)
dm+”i
, dm+”i = (1 â ”)z0
i + ”zki , k = 0, 1, . . .
Convergence limkââ zki = dm+1
i and positivity dm+1i â„ (1âλ)dm
i > 0 hold for all i([UY04]).
The maximal existence time. Since the above positivity dmi > 0 holds, we
can keep iterating Procedure 1 in finitely many steps (even if Pm self-intersects at astep m, we can continue). Then the maximal existence time is tâ = lim
mââ
âmk=0 Ïk.
Here we have two questions: one is whether tâ is finite or not, and the other is whatmight happen to Pm as m tends to infinity. It is known that tâ = â holds if WÎł
is an N -sided regular polygon and P0 is an admissible convex polygon ([UY04]).Extension. At the maximal existence time tâ, it is possible that at least one
edge, for instance the i-th edge, may disappear. If Ïi = 0 or lÎł(ni) = 0, then Pâ
is still essentially admissible. Hence we can continue Procedure 1 starting from theinitial curve Pâ. In practice, if some edges are small enough, we eliminate themartificially as the following procedure.
Procedure 2. Put a positive parameter ÎŽ âȘ 1. For every step m, do thefollowings:
(i) define D = mindmi ; Ïi 6= 0, lÎł(ni) > 0 (this is well-defined);
(ii) find the value k such that dmk = mindm
i ; Ïi = 0 (if it exists);find the value j such that dm
j = mindmi ; Ïi 6= 0, lÎł(ni) = 0 (if it exists);
(iii) if k or j exists, check the followings:(a) if dm
k /D < Ύ and if dmk †dm
j or the value j does not exist, theneliminate the k-th edge (see Figure 18 (left));
(b) if dmj /D < ÎŽ and if dm
j < dmk or the value k does not exist, then
eliminate the j-th edge (see Figure 18 (right));(c) otherwise, exit from Procedure 2;
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200 1. AN AREA-PRESERVING CRYSTALLINE CURVATURE FLOW EQUATION
(iv) in (3), if (a) occurred, then do renumbering within the new number N :=N â 2;else if (b) occurred, then do renumbering within the new number N :=N â 1.
Numerical computation will be continued repeating Procedure 1 and 2.
k
(a) In the type of (3) (a), thenumber of particles decreasesfrom N to N â 2.
j
(b) In the type of (3) (b), thenumber of particles decreasesfrom N to N â 1.
Figure 18. Two types of elimination of some edges in Procedure 2 (3).
Numerical simulations. In the following 8 figures on each line, from left toright, they indicate Wγ , P0,Pm1 , . . . ,Pm6 (0 < m1 < · · · < m6).
The case where P0 is convex and admissible. Figure 19 and Figure 20indicate that asymptotic behavior of solution polygon which does not break theresults of Theorem 1.49 and Theorem 1.50. See also Gage [Gag86] for the smoothcase. On the convergence between Î(tm) and P(tm), see Ushijima and Yazaki[UY04].
Figure 19. A numerical example of Theorem 1.49: Convergenceof a hexagon to the regular hexagon WÎł .
Figure 20. A numerical example of Theorem 1.50: Even whenthe initial polygon is quite different shape from WÎł , it convergesto WÎł .
The case where P0 is convex and essentially admissible. Known result isonly Theorem 1.51. Figure 21 suggests possibility of positive answer of the questionin open problem just after Theorem 1.51.
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11. NUMERICAL SCHEME 201
Figure 21. A numerical example of Theorem 1.51: An essentiallyadmissible convex polygon converges to WÎł in a finite time.
Figure 22. A numerical example of convexified phenomena.
The case where P0 is nonconvex and admissible. Figure 22 suggests thatconvexified phenomena holds.
An example of self-intersection. In smooth case, a self-intersection is con-jectured in Gage [Gag86], and is proved in Mayer and Simonett [MS00]. Figure 23suggests that self-intersection is possible to occur.
Figure 23. A numerical example of self-intersection. The motionis delicate. See the symbolic motion in Figure 24.
Open problems. At this stage, we have the following three open problems:
(i) Does P(t) become convex in finite time?(ii) Will the admissibility be preserved?(iii) Does P(t) self-intersect?
We have information related to these three questions. (1) In the case whereVi = âÎł(ni)|ÎÎł(ni)|αâ1ÎÎł(ni), there exist α â (0, 1), Îł and P0 such that non-convex solution curve P(t) shrinks homothetically, i.e., there exists a nonconvexself-similar solution polygonal curve. See Ishiwata, Ushijima, Yagisita and Yazaki[IUYY04]. (2) In the case where Vi = âa(ni)|ÎÎł(ni)|αâ1ÎÎł(ni), for any α â„ 1, a(·)and P0 if Îł is symmetric, then the solution curve keeps admissibility. See Giga andGiga [GGig00]. However, there exist α â (0, 1), a(·) and P0 such that the admissi-bility collapses in finite time, i.e., we have the examples that admissible nonconvexpolygonal curve becomes nonadmissible in finite time. See Hirota, Ishiwata andYazaki [HIY06, HIY07]. (3) Figure 23 suggests the possibility of self-intersection.
Remark 1.56. Thanks to the tangential velocity vi in (1.7), numerical compu-tation of crystalline curvature flow equations is quite stable. The vi corresponds to
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202 1. AN AREA-PRESERVING CRYSTALLINE CURVATURE FLOW EQUATION
p1
p2p3
p4 p5
p6p7
p8p9
p10 p11
p12Ï
Figure 24. Symbolic motion of a numerical example in Figure 23.The initial 12-sided nonconvex polygonal curve P0 in Figure 23 is(far left). Let Ï be the distance between the 6-th and the 12-thedges (left), which is small enough. Then Ï disappears immedi-ately, and moreover, becomes negative (middle left), and becomespositive again (middle right). After that, the vertical two edges,the thick line segments in (right), disappear. Finally, the horizon-tal two edges, the thick line segments in (far right), disappear, andbecomes rectangle such as Figure 23 (far right).
v = âsV/K (this is equivalent to âtΞ = 0 by Lemma 1.10.) in the continuous case(see [Yaz07c]), and it is utilized as redistribution of grid points on evolving curve.See [SY07, SY, BKPSTY07].
12. Towards modeling the formation of negative ice crystals or vaporfigures produced by freezing of internal melt figures
This section is based on the work with Tetsuya Ishiwata [IY07, IY08].When a block of ice is exposed to solar beams or other radiation, internal
melting of ice occurs. That is, internal melting starts from some interior pointsof ice without melting the exterior portions, and each water region forms a flowerof six petals, which is called âTyndall figureâ (see Figure 25). The figure is filled
Figure 25. Tyndall figures seen from the direction of 45 to c-axis[Nak56, No. 17].
with water except for a vapor bubble (black spot in Figure 25). This phenomenonwas first observed by Tyndall (1858). When Tyndall figure is refrozen, the vaporbubble remains in the ice as a hexagonal disk (see Figure 26). This hexagonaldisk is filled with water vapor saturated at that temperature and surrounded by
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12. FORMATION OF NEGATIVE ICE CRYSTALS OR VAPOR FIGURES 203
Figure 26. Natural negative crystals in an ice single crystal[Nak56, No. 1].
ice. McConnel found these disks in the ice of Davos lake [McC89]. Nakaya calledthis hexagonal disk âvapor figureâ and investigated its properties precisely [Nak56].Adams and Lewis (1934) called it ânegative crystal.â (Although Nakaya said âthisterm does not seem adequateâ with a certain reason, hereafter we use the termnegative crystal for avoiding confusion.)
Negative crystal is useful to determine the structure and orientation of iceor solids. Because, within a single ice crystal, all negative crystals are similarlyoriented, that is, corresponding edges of hexagon are parallel each other (see Fig-ure 27). Furukawa and Kohata made hexagonal prisms experimentally in a single
Figure 27. A cluster of minute negative crystals [Nak56, No. 53].
ice crystal, and investigated the habit change of negative crystals with respect tothe temperatures and the evaporation mechanisms of ice surfaces [FK93].
To the best of authorâs knowledge, after the Furukawa and Kohataâs experimen-tal research, there have been no published results on negative crystals, and thereare no dynamical model equations describing the process of formation of negativecrystals. In the present talk, we will focus on the process of formation of negativecrystals after Tyndall figures are refrozen, and try to propose a model equation ofinterfacial motion which tracks the deformation of negative crystals in time.
âtopics-in-mathematical-modelingâ â 2008/12/5 â 8:30 â page 204 â #216
204 1. AN AREA-PRESERVING CRYSTALLINE CURVATURE FLOW EQUATION
Formation of negative crystals. Figure 28 indicates aftereffect of freezingof Tyndall figures from the initial stage of refrozen process to the final stage of theformation of negative crystals. The aim of this talk is to propose a model equation,
Figure 28. From left to right, upper to lower: (a) Start of freez-ing, t = 0min. (b) Freezing proceeds, t = 3min. (c) Freezing pro-ceeds further, t = 11min. (d) The bubble is separated, t = 17min.(e) The separated liquid film migrates, t = 28min. (f) After freez-ing, cloudy layers and a vapor figure are left, t = 1hr 21min.[Nak56, No. 52aâ52f].
revealed the process in Figure 28 from (e) to (f). This process may be described asthe following:
Negative crystal changes the shape from oval to hexagon.
Thus, our model will be assumed that
(i) water vapor region is simply connected and bounded region in the planeR
2 (we denote it by Ω);(ii) Ω is surrounded by a single ice crystal (i.e., ice region is R
2\Ω);
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12. FORMATION OF NEGATIVE ICE CRYSTALS OR VAPOR FIGURES 205
(iii) moving boundary âΩ is interface of the water vapor region and the singleice crystal; and
(iv) c-axis (main axis) of the single ice crystal is perpendicular to the plane.
The evolution law of moving interface is similar to the growth of snow crystal,since deformation of negative crystal is regarded as crystal growth in the air. As amodel of snow crystal growth, we refer the Yokoyama-Kuroda model [YK90], whichis based on the diffusion process and the surface kinetic process by Burton-Cabrera-Frank (BCF) theory [BCF51]. Meanwhile, we assume the existence of interfacialenergy (density) on the boundary âΩ. The equilibrium shape of negative crystalis a regular hexagon, and if the region Ω is very close to a regular hexagon, thenthe evolution process may be described as a gradient flow of total interfacial energysubject to a fixed enclosed area. Therefore, we can divide the process of formationof negative crystals into two stages as follows:
(i) in the former stage by the diffusion and the surface kinetic, oval Ω changesto a hexagon; and
(ii) in the latter stage by a gradient flow of interfacial energy, the hexagonconverges to a regular hexagon.
A simple modeling of these stages is proposed as an area-preserving crystallinecurvature flow for ânegativeâ polygonal curves. See [IY07, IY08].
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[AT95] F. Almgren and J. E. Taylor, Flat flow is motion by crystalline curvature
for curves with crystalline energies, J. Diff. Geom. 42 (1995) 1â22.[AG89] S. Angenent and M. E. Gurtin, Multiphase thermomechanics with interfa-
cial structure, 2. Evolution of an isothermal interface, Arch. Rational Mech.Anal. 108 (1989) 323â391.
[AV95] S. Angenent and J. J. L Velazquez, Asymptotic shape of cusp singularities
in curve shortening, Duke Math. J. 77 (1995) 71â110.[BKPSTY07] M. Benes, M. Kimura, P. Paus, D. Sevcovic, T. Tsujikawa and
S. Yazaki, Application of a curvature adjusted method in image segmen-
tation, submitted to Bulletin of Inst. of Mathematics, Academia Sinica atTaipei (arXiv:0712.2334v1).
[BKY07] M. Benes, M. Kimura and S. Yazaki, Analytical and numerical aspects on
motion of polygonal curves with constant area speed, Proceedings of Slovak-Austrian Mathematical Congress (Podbanske, 2007), within MAGIA 2007,Dept. of Mathematics and Descriptive Geometry, Faculty of Civil Engineer-ing, Slovak University of Technology, ISBN 978-80-227-2796-9 (2007) 127â141.(arXiv:0802.2204v3 is a new version including a minor change.)
[BKY08] M. Benes, M. Kimura and S. Yazaki, Second order numerical scheme
for motion of polygonal curves with constant area speed, preprint(arXiv:0805.4469v1).
[BCF51] W. K. Burton, N. Cabrera and F. C. Frank, The growth of crystals and
the equilibrium structure of their surfaces, Philos. Trans. R. Soc. Landon,Ser. A 243 (1951) 299â358.
[CZ99] K.-S. Chou and X.-P. Zhu, A convexity theorem for a class of anisotropic
flows of plane curves, Indiana Univ. Math. J. 48 (1999) 139â154.[EG87] C. L. Epstein and M. Gage, The curve shortening flow, Wave motion: the-
ory, modelling, and computation (Berkeley, Calif., 1986), Math. Sci. Res. Inst.Publ., 7, Springer, New York (1987) 15â59.
[FK93] Y. Furukawa and S. Kohata, Temperature dependence of the growth form
of negative crystal in an ice single crystal and evaporation kinetics for its
surfaces, J. Crystal Growth 129 (1993) 571â581.[Gag86] M. Gage, On an area-preserving evolution equations for plane curves, Con-
temporary Math. 51 (1986) 51â62.[GH86] M. Gage and R. S. Hamilton, The heat equation shrinking convex plane
curves, J. Diff. Geom. 23 (1986) 69â96.[Gig95] Y. Giga, A level set method for surface evolution equations, Sugaku 47
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[Gig97] Y. Giga, Interesting diffusion equations, Sugaku 49 (1997) 193â196.(Japanese)
[Gig98] Y. Giga, Interface dynamics â effects of curvature, Hokkaido UniversityTechnical Report Series 56 (1998). (Japanese)
[Gig00] Y. Giga, Anisotropic curvature effects in interface dynamics, Sugaku 52(2000) 113â127; English transl., Sugaku Expositions 16 (2003) 135â152.
[GGig00] M.-H. Giga and Y. Giga, Crystalline and level set flow â convergence of
a crystalline algorithm for a general anisotropic curvature flow in the plane,Free boundary problems: theory and applications, I (Chiba, 1999), GAKUTOInternat. Ser. Math. Sci. Appl., Gakkotosho, Tokyo 13 (2000) 64â79.
[Gra87] M. A. Grayson, The heat equation shrinks emmbedded plane curves to
round points, J. Diff. Geom. 26 (1987) 285â314.[Gur93] M. E. Gurtin, Thermomechanics of evolving phase boundaries in the plane,
Oxford, Clarendon Press (1993).[HIY06] C. Hirota, T. Ishiwata and S. Yazaki, Note on the asymptotic behavior of
solutions to an anisotropic crystalline curvature flow, Recent Advances on El-liptic and Parabolic Issues (Editors: Michel Chipot and Hirokazu Ninomiya),Proceedings of the 2004 Swiss-Japan Seminar, World Scientific (2006) 129â143.
[HIY07] C. Hirota, T. Ishiwata and S. Yazaki, Numerical study and examples on
singularities of solutions to anisotropic crystalline curvature flows of noncon-
vex polygonal curves, Proceedings of MSJ-IRI 2005 âAsymptotic Analysisand Singularitiesâ (Sendai, 2005), Advanced Studies in Pure Mathematics(ASPM) 47 (2007) 543â564.
[HGGD05] H. Hontani, M.-H. Giga, Y. Giga and K. Deguchi, Expanding selfsimilar
solutions of a crystalline flow with applications to contour figure analysis,Discrete Applied Mathematics 147 (2005) 265â285.
[IS99] K. Ishii and H. M. Soner , Regularity and convergence of crystalline motion,SIAM J. Math. Anal. 30 (1999) 19â37.
[IUYY04] T. Ishiwata, T. K. Ushijima, H. Yagisita and S. Yazaki, Two examples of
nonconvex self-similar solution curves for a crystalline curvature flow, Proc.Japan Academy 80, Ser. A, No. 8 (2004) 151â154.
[IY07] T. Ishiwata and S. Yazaki, Towards modelling the formation of negative
ice crystals or vapor figures produced by freezing of internal melt figures.Symposium on Mathematical Models of Phenomena and Evolution Equations(Kyoto 2006). RIMS Kokyuroku 1542 (2007) 1â11.
[IY08] T. Ishiwata and S. Yazaki, Interface motion of a negative crystal and its
analysis. Nonlinear Evolution Equations and Mathematical Modeling (Kyoto2007). RIMS Kokyuroku 1588 (2008) 23â29.
[KG01] R. Kobayashi and Y. Giga, On anisotropy and curvature effects for growing
crystals, Japan J. Indust. Appl. Math. 18 (2001) 207â230.[MS00] U. F. Mayer and G. Simonett, Self-intersections for the surface diffusion and
the volume-preserving mean curvature flow, Differential Integral Equations 13(2000) 1189â1199.
[McC89] J. C. McConnel, The crystallization of lake ice, Nature 39 (1889) 367.[Nak56] U. Nayaka, Properties of single crystals of ice, revealed by internal melting,
SIPRE (Snow, Ice and Permafrost Reseach Establishment) Research Paper
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13 (1956).[Rob93] S. Roberts, A line element algorithm for curve flow problems in the plane,
CMA Research Report 58 (1989); J. Austral. Math. Soc. Ser. B 35 (1993)244â261.
[Sch93] R. Schneider, Convex bodies: the Brunn-Minkowski theory, CambridgeUniversity Press (1993).
[Set96] J. A. Sethian, Level set methods: evolving interfaces in geometry, fluid
mechanics, computer vision, and materials science, Cambridge Monographson Applied and Computational Mathematics 3, Cambridge University Press,Cambridge (1996).
[SY07] D. Sevcovic and S. Yazaki, On a motion of plane curves with a curvature
adjusted tangential velocity, submitted to Proceedings of Equadiff 2007 Con-ference (arXiv:0711.2568v1).
[SY] D. Sevcovic and S. Yazaki, Evolution of plane curves with curvature adjusted
tangential redistribution velocity, in preparation.[Tay90] J. E. Taylor, Crystals, in equilibrium and otherwise, videotape of 1989
AMS-MAA lecture, Selected Lectures in Math., Amer. Math. Soc. (1990).[Tay91a] J. E. Taylor, Constructions and conjectures in crystalline nondifferential
geometry, Proceedings of the Conference on Differential Geometry, Rio deJaneiro, Pitman Monographs Surveys Pure Appl. Math. 52 (1991) 321â336,Pitman London.
[Tay91b] J. E. Taylor, Motion by crystalline curvature, Computing Optimal Ge-ometries, Selected Lectures in Math., Amer. Math. Soc. (1991) 63â65 plusvideo.
[Tay93] J. E. Taylor, Motion of curves by crystalline curvature, including triple
junctions and boundary points, Diff. Geom.: partial diff. eqs. on manifolds(Los Angeles, CA, 1990), Proc. Sympos. Pure Math., 54 (1993), Part I, 417â438, AMS, Providencd, RI.
[TCH92] J. E. Taylor, J. Cahn and C. Handwerker, Geometric models of crystal
growth, Acta Metall., 40 (1992) 1443â1474.[Tot72] L. F. Toth, Lagerungen in der Ebene auf der Kugel und im Raum, Springer-
Verlag, Heidelberg (1972); Japanese transl., Misuzu Shobo (1983).[UY04] T. K. Ushijima and S. Yazaki, Convergence of a crystalline approxima-
tion for an area-preserving motion, Journal of Computational and AppliedMathematics 166 (2004) 427â452.
[Yaz02] S. Yazaki, On an area-preserving crystalline motion, Calc. Var. 14 (2002)85â105.
[Yaz04] S. Yazaki, On an anisotropic area-preserving crystalline motion and motion
of nonadmissible polygons by crystalline curvature, RIMS Kokyuroku 1356(2004) 44â58.
[Yaz07a] S. Yazaki, Motion of nonadmissible convex polygons by crystalline curva-
ture, Publications of Research Institute for Mathematical Sciences 43 (2007),155â170.
[Yaz07b] S. Yazaki, Asymptotic behavior of solutions to an area-preserving motion
by crystalline curvature, Kybernetika 43 (2007) 903â912.
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[Yaz07c] S. Yazaki, On the tangential velocity arising in a crystalline approximation
of evolving plane curves, Kybernetika 43 (2007) 913â918.[YK90] E. Yokoyama and T. Kuroda, Pattern formation in growth of snow crystals
occurring in the surface kinetic process and the diffusion process, PhysicalReview A 41(4) (1990) 2038â2049.
âtopics-in-mathematical-modelingâ â 2008/12/5 â 8:30 â page 211 â #223
Index
anisotropy, 49
crystalnegative, 71, 72
crystals
ice, 70curvature, 45
anisotropic, 50crystalline, 61
weighted, 50curve
Ck , 44admissible, 59
closed, 44concave, 45convex, 45
embedded, 44essentially admissible, 59immersed, 44Jordan, 44
noncovex, 45open, 44oval, 45plane, 44
polygonal, 57simple, 44strictly convex, 45
diagram
Frank, 51
energycrystalline, 55interfacial, 49
equationanisotropic curvature flow, 50classical curvature flow, 47
crystalline curvature flow, 57, 61curve shortening, 47weighted curvature flow, 50
extension
positively homogeneous of degree 1, 49
figureTyndall, 70
vapor, 70flow
area preserving, 48classical curvature, 47gradient, 47, 59weighted curvature, 50
formulaeFrenetâSerret, 45
inequalityanisoperimetric, 65
motionarea preserving, 63
PHD1, 49PPHD1, 49
shapeWulff, 52
vector
positive direction of, 45unit tangent, 44
211
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APPENDIX A
1. Strange examples
The following set Î describes the unit circle:
Î = x(Ï) = (cos f(Ï), sin f(Ï)); Ï â [0, 1],if we take the range of f is greater than or equals 2Ï. However, the following circlesare strange in the sense of contrary to intuition.
Non-closed circle: If f(Ï) = (2Ï + 1)Ï, then x is not closed, since x(0) 6=x(1).
Non-simple circle: If f(Ï) = Ï(2Ï â 1)(4(2Ï â 1)2 â 1)/3, then the mapf : [0, 1] â [0, 2Ï] is surjective but not injective. Hence x is not simple.
Non-immersed circle: If f(Ï) = Ï((2Ïâ1)3+1), then the map f : [0, 1] â[0, 2Ï] is bijective. Hence x is simple and C1-closed curve but not im-mersed, since g(Ï) = |âÏf(Ï)| = 0 holds at Ï = 1/2.
Non-embedded circle: If f(Ï) = 2nÏÏ (n = 2, 3, . . .), then x is the n-multiply covered unit circle. Since the map x : ÎŁ â Î is not injective, x
is immersed but not embedded.
2. A non-concave curve
We need some attention concerning the converse of Lemma 1.5. Strictly convexcurves admit finite number of zeros of K (such points are considered as degenerateline segment). Figure 1 indicates such an example. Thus we can say that
A closed embedded C2-curve is strictly convex, if and only ifK > 0 holds in ÎŁ except finite number of zero points.
3. Anisotropic inequalityâproof of Lemma 1.53â
The result of Lemma 1.53 follows from a classical convex geometry by using aconcept of mixed area and the Brunn and Minkowskiâs inequality.
Let P0 and P1 be N -sided admissible convex polygons associated with WÎł .
For i = 0, 1, by h(i)j and d
(i)j , we denote the j-th hight function and the length
of the j-th edge, respectively. Note that d(i)j = Dj(h
(i)) holds for i = 1, 2, . . . , N ,
h(i) = (h(i)1 , h
(i)2 , . . . , h
(i)N ). Let the enclosed region of Pi be Ωi =
âNj=1x â R
2; x ·nj †h
(i)j for i = 0, 1. Define the linear interpolant of Ω0 and Ω1 by Ω” =
(1 â ”)Ω0 + ”Ω1 = (1 â ”)x + ”y; x â Ω0, y â Ω1, and P” = (1 â ”)P0 + ”P1
for 0 †” †1. Then the j-th height function and the length of the j-th edge of P”
213
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214 A
1+Ï/2
2+Ï/2
3+Ï/2
4+Ï/2
5+Ï/2
6+Ï/2
3Ï/2
5Ï/2
0 0.19 0.29 0.43 0.68 0.78 0.90 10.5
Ï
Ξ
Ï/2
Ï=1 (Ï=0)
Ï=0.190811Ï=0.294048
Ï=0.431905
Ï=0.675084Ï=0.779306
Ï=0.902935
Ï=0.5
Figure 1. Graph of Ξ versus Ï (left), and the C2-class strictlyconvex curve Î = x(Ï); Ï â [0, 1] (right). Here Ξ(Ï) =η(Ï) = Ï(2 â cos(2ÏÏ))/2 if Ï â [0, 0.5), Ξ(Ï) = η(Ï â 0.5) + Ïif Ï â [0.5, 1], and x(Ï) =
â« Ï
0 (cos Ξ(Ï), sin Ξ(Ï)) dÏ. In theright figure, the set Ï; Ξ(Ï) â Î is indicated, where Î =i + Ï/2 (i = 1, 2, . . . , 6), 3Ï/2, 5Ï/2. Note that Ξ(0.5) = 3Ï/2and Ξ(1) = 5Ï/2. Since g(Ï) ⥠1, the curvature is given byK(Ï) = âÏΞ = Ï2| sin(2ÏÏ)|. Hence K(Ï) is positive except twopoints Ï = 0.5 and Ï = 1 (Ï = 0).
are given as h(”)j = (1 â ”)h
(0)j + ”h
(1)j and d
(”)j = (1 â ”)d
(0)j + ”d
(1)j , respectively.
Let A(Ω”) be the area of Ω”. Then we have
A(Ω”) =1
2
Nâ
j=1
d(s)j h
(s)j = (1â”)2A(Ω0)+”2A(Ω1)+
”(1 â ”)
2
Nâ
j=1
(d(1)j h
(0)j + d
(0)j h
(1)j
).
By summation by parts:
Nâ
j=1
d(1)j h
(0)j =
Nâ
j=1
Dj(h(1))h
(0)j =
Nâ
j=1
Dj(h(0))h
(1)j =
Nâ
j=1
d(0)j h
(1)j , (A.1)
we have
A(Ω”) =1
2
Nâ
j=1
d(”)j h
(”)j = (1 â ”)2A(Ω0) + ”2A(Ω1) + 2”(1 â ”)
1
2
Nâ
j=1
d(0)j h
(1)j .
The coefficient of 2”(1 â ”) is called the mixed area of Ω0 and Ω1, and denotedby
A(Ω0, Ω1) =1
2
Nâ
j=1
d(0)j h
(1)j .
Note that A(Ω0, Ω1) = A(Ω1, Ω0) holds by (A.1).H. Brunn and H. Minkowski proved the following inequality:
âA(Ω”) â„ (1 â ”)
âA(Ω0) + ”
âA(Ω1), 0 †” †1. (A.2)
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3. ANISOTROPIC INEQUALITYâPROOF OF LEMMA 1.53â 215
Equality holds if and only if Ω0 = kΩ1 (k > 0). The Brunn and Minkowskiâsinequality (A.2) is equivalent to the following inequality:
A(Ω0, Ω1) â„â
A(Ω0)A(Ω1). (A.3)
From this inequality, we obtain the anisoperimetric inequality as follows. LetΩ = Ω0 be the enclosed region of a polygon P and let the enclosed area be A(Ω) = A.For the crystalline energy γ > 0, let Ω1 be the Wulff shape Wγ . Then the area of
WÎł is A(WÎł) = |WÎł | =âN
k=1 Îł(nk)lÎł(nk)/2, and the mixed area is A(Ω,WÎł) =âNj=1 Îł(nj)dj/2 = EÎł(P)/2, which is a half of the total interfacial energy on P .
Hence by (A.3), EÎł(P)/2 â„â
A|WÎł |, namely,
JÎł(P) =EÎł(P)2
4|WÎł |Aâ„ 1. (A.4)
The equality JÎł(P) ⥠1 holds if and only if Ω = kWÎł for some constant k > 0, i.e.,ÎÎł(nj) ⥠const. for j = 1, 2, . . . , N .
In particular, if Îł ⥠1, then EÎł = L and |WÎł | =âN
j=1 lÎł(nj)/2, so we have
I =L2
2âN
j=1 l1(nj)Aâ„ 1, (A.5)
which is the isoperimetric inequality of polygons. Note that L. Fejes Toth writesin his book [Tot72] such that this inequality was done by S. Lhuilier. We referR. Schneider [Sch93] for general theory of convex bodies. See also [Yaz02] for theproof of (A.5) by using a solution of crystalline curvature flow equation.
The isoperimetric inequality (A.5) represents the variational problem: what isthe shape which has the least total length of a polygon for the fixed enclosed area?The answer (it corresponds to the case where I = 1) is the boundary of the Wulffshape âWÎłâ
, which is the circumscribed polygon of the circle with radius Îłâ =â2A/
âNi=1 l1(ni) (cf. Theorem 1.49). Similarly, the anisoperimetric inequality
(A.4) represents the variational problem: what is the shape which has the leasttotal interfacial energy of a polygon for the fixed enclosed area? The answer (itcorresponds to the case where JÎł = 1) is the boundary of the Wulff shape âWÎłâ
(cf. Theorem 1.50).
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(Eds.) M. Benes and E. Feireisl
Topics in mathematical modeling
Published byMATFYZPRESS
Publishing House of the Faculty of Mathematics and PhysicsCharles University, Prague
Sokolovska 83, CZ â 186 75 Praha 8as the 254. publication
The volume was typeset by the authors using LATEX
Printed byReproduction center UK MFF
Sokolovska 83, CZ â 186 75 Praha 8
First edition
Praha 2008
ISBN 978-80-7378-060-9
âtopics-in-mathematical-modelingâ â 2008/12/5 â 8:30 â page 218 â #230
âtopics-in-mathematical-modelingâ â 2008/12/5 â 8:30 â page 219 â #231
Jindrich Necas
Jindrich Necas was born in Prague on December 14th, 1929. He studied mathe-matics at the Faculty of Natural Sciences at the Charles University from 1948 to1952. After a brief stint as a member of the Faculty of Civil Engineering at theCzech Technical University, he joined the Czechoslovak Academy of Sciences wherehe served as the Head of the Department of Partial Differential Equations. He heldjoint appointments at the Czechoslovak Academy of Sciences and the Charles Uni-versity from 1967 and became a full time member of the Faculty of Mathematicsand Physics at the Charles University in 1977. He spent the rest of his life there, asignificant portion of it as the Head of the Department of Mathematical Analysisand the Department of Mathematical Modeling.
His initial interest in continuum mechanics led naturally to his abiding passionto various aspects of the applications of mathematics. He can be rightfully consid-ered as the father of modern methods in partial differential equations in the CzechRepublic, both through his contributions and through those of his numerous stu-dents. He has made significant contributions to both linear and non-linear theoriesof partial differential equations. That which immediately strikes a person conver-sant with his contributions is their breadth without the depth being compromisedin the least bit. He made seminal contributions to the study of Rellich identities andinequalities, proved an infinite dimensional version of Sardâs Theorem for analyticfunctionals, established important results of the type of Fredholm alternative, andmost importantly established a significant body of work concerning the regularityof partial differential equations that had a bearing on both elliptic and parabolicequations. At the same time, Necas also made important contributions to rigorousstudies in mechanics. Notice must be made of his work, with his collaborators, onthe linearized elastic and inelastic response of solids, the challenging field of contactmechanics, a variety of aspects of the Navier-Stokes theory that includes regularityissues as well as important results concerning transonic flows, and finally non-linearfluid theories that include fluids with shear-rate dependent viscosities, multi-polarfluids, and finally incompressible fluids with pressure dependent viscosities.
Necas was a prolific writer. He authored or co-authored eight books. Specialmention must be made of his book âLes methodes directes en theorie des equationselliptiquesâ which has already had tremendous impact on the progress of the subjectand will have a lasting influence in the field. He has written a hundred and fortyseven papers in archival journals as well as numerous papers in the proceedings ofconferences all of which have had a significant impact in various areas of applicationsof mathematics and mechanics.
Jindrich Necas passed away on December 5th, 2002. However, the legacy thatNecas has left behind will be cherished by generations of mathematicians in theCzech Republic in particular, and the world of mathematical analysts in general.
âtopics-in-mathematical-modelingâ â 2008/12/5 â 8:30 â page 220 â #232
Jindrich Necas Center for Mathematical Modeling
The Necas Center for Mathematical Modeling is a collaborative effort between theFaculty of Mathematics and Physics of the Charles University, the Institute ofMathematics of the Academy of Sciences of the Czech Republic and the Faculty ofNuclear Sciences and Physical Engineering of the Czech Technical University.
The goal of the Center is to provide a place for interaction between mathemati-cians, physicists, and engineers with a view towards achieving a better understand-ing of, and to develop a better mathematical representation of the world that welive in. The Center provides a forum for experts from different parts of the worldto interact and exchange ideas with Czech scientists.
The main focus of the Center is in the following areas, though not restrictedonly to them: non-linear theoretical, numerical and computer analysis of problemsin the physics of continua; thermodynamics of compressible and incompressiblefluids and solids; the mathematics of interacting continua; analysis of the equationsgoverning biochemical reactions; modeling of the non-linear response of materials.
The Jindrich Necas Center conducts workshops, house post-doctoral scholarsfor periods up to one year and senior scientists for durations up to one term. TheCenter is expected to become world renowned in its intended field of interest.
ISBN 978-80-7378-060-9