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Surface Evolution Equations : A Level Set Approach - Yoshikazu Giga

Surface Evolution Equations

A Level Set Approach

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This book is intended to be a self-contained introduction to analytic foundations of a level set method for various surface evolution equations including curvature ?ow equations. These equations are important in various ?elds including material sciences, image processing and di?erential geometry. The goal of this book is to introduce a generalized notion of solutions allowing singularities and solve the initial-value problem globally-in-time in a generalized sense. Various equivalent de?nitions of solutions are studied. Several new results on equivalence are also presented. Wepresentherearathercompleteintroductiontothetheoryofviscosityso- tionswhichis a keytoolforthe levelsetmethod. Alsoa self-containedexplanation isgivenforgeneralsurfaceevolutionequationsofthe secondorder.Althoughmost ofthe resultsin this book aremoreor lessknown,they arescatteredinseveralr- erences, sometimes without proof. This book presents these results in a synthetic way with full proofs. However, the references are not exhaustive at all. The book is suitable for applied researchers who would like to know the detail of the theory as well as its ?avour.No familiarity with di?erential geometry and the theory of viscosity solutions is required. The prerequisites are calculus, linear algebra and some familiarity with semicontinuous functions. This book is also suitable for upper level under graduate students who are interested in the ?eld.

In view of its detailed and thorough presentation this book will be a valuable source for everyone interested in the level set approach to surface evolution equations.

Zentralblatt MATH

Prefacep. xi
Introductionp. 1
Surface evolution equationsp. 15
Representation of a hypersurfacep. 15
Normal velocityp. 18
Curvaturesp. 21
Expression of curvature tensorsp. 26
Examples of surface evolution equationsp. 33
General evolutions of isothermal interfacesp. 33
Evolution by principal curvaturesp. 34
Other examplesp. 35
Boundary conditionsp. 35
Level set equationsp. 36
Examplesp. 36
General scaling invariancep. 40
Ellipticityp. 42
Geometric equationsp. 46
Singularities in level set equationsp. 49
Exact solutionsp. 52
Mean curvature flow equationp. 52
Anisotropic versionp. 54
Anisotropic mean curvature of the Wulff shapep. 58
Affine curvature flow equationp. 62
Notes and commentsp. 63
Viscosity solutionsp. 69
Definitions and main expected propertiesp. 69
Definition for arbitrary functionsp. 70
Expected properties of solutionsp. 73
Very singular equationsp. 77
Stability resultsp. 82
Remarks on a class of test functionsp. 83
Convergence of maximum pointsp. 85
Applicationsp. 87
Boundary value problemsp. 92
Perron's methodp. 98
Closedness under supremump. 100
Maximal subsolutionp. 101
Adaptation for very singular equationsp. 103
Applicabilityp. 105
Notes and commentsp. 105
Comparison principlep. 109
Typical statementsp. 109
Bounded domainsp. 110
General domainsp. 112
Applicabilityp. 112
Alternate definition of viscosity solutionsp. 113
Definition involving semijetsp. 113
Solutions on semiclosed time intervalsp. 119
General idea for the proof of comparison principlesp. 123
A typical problemp. 123
Maximum principle for semicontinuous functionsp. 126
Proof of comparison principles for parabolic equationsp. 128
Proof for bounded domainsp. 129
Proof for unbounded domainsp. 134
Lipschitz preserving and convexity preserving propertiesp. 139
Spatially inhomogeneous equationsp. 148
Inhomogeneity in first order perturbationp. 148
Inhomogeneity in higher order termsp. 150
Boundary value problemsp. 155
Notes and commentsp. 158
Classical level set methodp. 163
Brief sketch of a level set methodp. 163
Uniqueness of bounded evolutionsp. 166
Invariance under change of dependent variablesp. 166
Orientation-free surface evolution equationsp. 171
Uniquenessp. 172
Unbounded evolutionsp. 174
Existence by Perron's methodp. 175
Existence by approximationp. 180
Various properties of evolutionsp. 182
Convergence properties for level set equationsp. 192
Instant extinctionp. 198
Notes and commentsp. 201
Set-theoretic approachp. 207
Set-theoretic solutionsp. 207
Definition and its characterizationp. 208
Characterization of solutions of level set equationsp. 211
Characterization by distance functionsp. 213
Comparison principle for setsp. 216
Convergence of sets and functionsp. 219
Level Set solutionsp. 219
Nonuniquenessp. 219
Definition of level set solutionsp. 221
Uniqueness of level set solutionsp. 223
Barrier solutionsp. 226
Consistencyp. 231
Nested family of subsolutionsp. 231
Applicationsp. 233
Relation among various solutionsp. 234
Separation and comparison principlep. 236
Notes and commentsp. 238
Bibliographyp. 243
Notation Indexp. 261
Subject Indexp. 263
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9783764324308
ISBN-10: 3764324309
Series: Monographs in Mathematics
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 264
Published: 30th August 2006
Country of Publication: CH
Dimensions (cm): 24.4 x 17.0  x 1.91
Weight (kg): 0.66