Giga Y. Surface Evolution Equations: A Level Set Approach
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Monographs in Mathematics
Vol. 99
Managing Editors:
H. Amann
Universität Zürich, Switzerland
J.-P. Bourguignon
IHES, Bures-sur-Yvette, France
K. Grove
University of Maryland, College Park, USA
P.-L. Lions
Université de Paris-Dauphine, France
Associate Editors:
H. Araki, Kyoto University
F. Brezzi, Università di Pavia
K.C. Chang, Peking University
N. Hitchin, University of Warwick
H. Hofer, Courant Institute, New York
H. Knörrer, ETH Zürich
K. Masuda, University of Tokyo
D. Zagier, Max-Planck-Institut Bonn
Yoshikazu Giga
Surface
Evolution
Equations
Birkhäuser Verlag
Basel Boston Berlin
A Level Set Approach
Author:
Yoshikazu Giga
Graduate School of Mathematical Sciences
University of Tokyo
Komaba 3-8-1. Meguro-ku
Tokyo 153-8914
Japan
e-mail: labgiga@ms.u-tokyo.ac.jp
2000 Mathematics Subject Classifi cation: Primary 53C44; Secondary 49L25, 35K10, 35K55, 74N20
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USA
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ISBN 3-7643-2430-9 Birkhäuser Verlag, Basel – Boston – Berlin
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To my parents, Kazuyo and Kenjiro
Contents
Preface xi
Introduction 1
1 Surface evolution equations 15
1.1 Representationofahypersurface ................... 15
1.2 Normalvelocity............................. 18
1.3 Curvatures ............................... 21
1.4 Expression of curvature tensors . . . . . . . . . . . . . . . . . . . . 26
1.5 Examplesofsurfaceevolutionequations ............... 33
1.5.1 General evolutions of isothermal interfaces . . . . . . . . . . 33
1.5.2 Evolutionbyprincipalcurvatures............... 34
1.5.3 Otherexamples......................... 35
1.5.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . 35
1.6 Levelsetequations........................... 36
1.6.1 Examples ............................ 36
1.6.2 Generalscalinginvariance................... 40
1.6.3 Ellipticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.6.4 Geometricequations...................... 46
1.6.5 Singularitiesinlevelsetequations .............. 49
1.7 Exactsolutions............................. 52
1.7.1 Meancurvatureflowequation................. 52
1.7.2 Anisotropicversion....................... 54
1.7.3 Anisotropic mean curvature of the Wulff shape . . . . . . . 58
1.7.4 Affinecurvatureflowequation ................ 62
1.8 Notesandcomments.......................... 63
2 Viscosity solutions 69
2.1 Definitions and main expected properties . . . . . . . . . . . . . . 69
2.1.1 Definitionforarbitraryfunctions............... 70
2.1.2 Expectedpropertiesofsolutions ............... 73
2.1.3 Verysingularequations .................... 77
viii Contents
2.2 Stability results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.2.1 Remarks on a class of test functions . . . . . . . . . . . . . 83
2.2.2 Convergence of maximum points . . . . . . . . . . . . . . . 85
2.2.3 Applications .......................... 87
2.3 Boundary value problems . . . . . . . . . . . . . . . . . . . . . . . 92
2.4 Perron’smethod ............................ 98
2.4.1 Closedness under supremum . . . . . . . . . . . . . . . . . . 100
2.4.2 Maximalsubsolution...................... 101
2.4.3 Adaptation for very singular equations . . . . . . . . . . . . 103
2.4.4 Applicability . . . . . . . . . . . . . . . . . . . . . . . . . . 105
2.5 Notesandcomments.......................... 105
3 Comparison principle 109
3.1 Typicalstatements........................... 109
3.1.1 Bounded domains . . . . . . . . . . . . . . . . . . . . . . . 110
3.1.2 Generaldomains ........................ 112
3.1.3 Applicability . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.2 Alternatedefinitionofviscositysolutions............... 113
3.2.1 Definitioninvolvingsemijets.................. 113
3.2.2 Solutions on semiclosed time intervals . . . . . . . . . . . . 119
3.3 General idea for the proof of comparison principles . . . . . . . . . 123
3.3.1 Atypicalproblem ....................... 123
3.3.2 Maximum principle for semicontinuous functions . . . . . . 126
3.4 Proof of comparison principles for parabolic equations . . . . . . . 128
3.4.1 Proof for bounded domains . . . . . . . . . . . . . . . . . . 129
3.4.2 Proof for unbounded domains . . . . . . . . . . . . . . . . . 134
3.5 Lipschitz preserving and convexity preserving properties . . . . . . 139
3.6 Spatially inhomogeneous equations . . . . . . . . . . . . . . . . . . 148
3.6.1 Inhomogeneity in first order perturbation . . . . . . . . . . 148
3.6.2 Inhomogeneityinhigherorderterms............. 150
3.7 Boundary value problems . . . . . . . . . . . . . . . . . . . . . . . 155
3.8 Notesandcomments.......................... 158
4 Classical level set method 163
4.1 Briefsketchofalevelsetmethod................... 163
4.2 Uniqueness of bounded evolutions . . . . . . . . . . . . . . . . . . . 166
4.2.1 Invariance under change of dependent variables . . . . . . . 166
4.2.2 Orientation-free surface evolution equations . . . . . . . . . 171
4.2.3 Uniqueness ........................... 172
4.2.4 Unbounded evolutions . . . . . . . . . . . . . . . . . . . . . 174
4.3 ExistencebyPerron’smethod..................... 175
4.4 Existencebyapproximation...................... 180
4.5 Variouspropertiesofevolutions.................... 182
4.6 Convergencepropertiesforlevelsetequations............ 192
Contents ix
4.7 Instantextinction............................ 198
4.8 Notesandcomments.......................... 201
5 Set-theoretic approach 207
5.1 Set-theoreticsolutions ......................... 207
5.1.1 Definition and its characterization . . . . . . . . . . . . . . 208
5.1.2 Characterization of solutions of level set equations . . . . . 211
5.1.3 Characterization by distance functions . . . . . . . . . . . . 213
5.1.4 Comparisonprincipleforsets ................. 216
5.1.5 Convergence of sets and functions . . . . . . . . . . . . . . 219
5.2 Levelsetsolutions ........................... 219
5.2.1 Nonuniqueness ......................... 219
5.2.2 Definitionoflevelsetsolutions ................ 221
5.2.3 Uniquenessoflevelsetsolutions ............... 223
5.3 Barriersolutions ............................ 226
5.4 Consistency............................... 231
5.4.1 Nestedfamilyofsubsolutions................. 231
5.4.2 Applications .......................... 233
5.4.3 Relationamongvarioussolutions............... 234
5.5 Separation and comparison principle . . . . . . . . . . . . . . . . . 236
5.6 Notesandcomments.......................... 238
Bibliography 243
Notation Index
Subject Index
261
263