Surface evolution equations : a level set approach /

Surface evolution equations : a level set approach /

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Bibliographic Details
Author / Creator:Giga, Yoshikazu.
Imprint:Basel, Switzerland ; Boston [Mass.] : Birkhäuser Verlag, c2006.
Description:1 online resource (xii, 264 p.) : ill.
Language:English
Series:Monographs in mathematics ; v. 99
Monographs in mathematics ; v. 99.
Subject:
Evolution equations.
Hamilton-Jacobi equations.
Curves, Algebraic.
Differential equations, Parabolic.
Curves, Algebraic.
Differential equations, Parabolic.
Evolution equations.
Hamilton-Jacobi equations.
Format: E-Resource Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/8880384
Hidden Bibliographic Details
ISBN:9783764373917
3764373911
9786610607952
6610607958
3764324309 (Cloth)
9783764324308 (Cloth)
Notes:Includes bibliographical references (p. [243]-260) and indexes.
Description based on print version record.
Summary:"This book presents a self-contained introduction to the analytic foundation of a level set approach for various surface evolution equations including curvature flow equations. These equations are important in many applications, such as material sciences, image processing and differential geometry. The goal is to introduce a generalized notion of solutions allowing singularities, and to solve the initial-value problem globally-in-time in a generalized sense. Various equivalent definitions of solutions are studied. Several new results on equivalence are also presented. Moreover, structures of level set equations are studied in detail. Further, a rather complete introduction to the theory of viscosity solutions is contained, which is a key tool for the level set approach."--Jacket.
Other form:Print version: Giga, Yoshikazu. Surface evolution equations. Basel, Switzerland ; Boston [Mass.] : Birkhäuser Verlag, c2006 3764324309 3764373911
Table of Contents:
  • Preface
  • Introduction
  • 1. Surface evolution equations
  • 1.1. Representation of a hypersurface
  • 1.2. Normal velocity
  • 1.3. Curvatures
  • 1.4. Expression of curvature tensors
  • 1.5. Examples of surface evolution equations
  • 1.5.1. General evolutions of isothermal interfaces
  • 1.5.2. Evolution by principal curvatures
  • 1.5.3. Other examples
  • 1.5.4. Boundary conditions
  • 1.6. Level set equations
  • 1.6.1. Examples
  • 1.6.2. General scaling invariance
  • 1.6.3. Ellipticity
  • 1.6.4. Geometric equations
  • 1.6.5. Singularities in level set equations
  • 1.7. Exact solutions
  • 1.7.1. Mean curvature flow equation
  • 1.7.2. Anisotropic version
  • 1.7.3. Anisotropic mean curvature of the Wulff shape
  • 1.7.4. Affine curvature flow equation
  • 1.8. Notes and comments
  • 2. Viscosity solutions
  • 2.1. Definitions and main expected properties
  • 2.1.1. Definition for arbitrary functions
  • 2.1.2. Expected properties of solutions
  • 2.1.3. Very singular equations
  • 2.2. Stability results
  • 2.2.1. Remarks on a class of test functions
  • 2.2.2. Convergence of maximum points
  • 2.2.3. Applications
  • 2.3. Boundary value problems
  • 2.4. Perron's method
  • 2.4.1. Closedness under supremum
  • 2.4.2. Maximal subsolution
  • 2.4.3. Adaptation for very singular equations
  • 2.4.4. Applicability
  • 2.5. Notes and comments
  • 3. Comparison principle
  • 3.1. Typical statements
  • 3.1.1. Bounded domains
  • 3.1.2. General domains
  • 3.1.3. Applicability
  • 3.2. Alternate definition of viscosity solutions
  • 3.2.1. Definition involving semijets
  • 3.2.2. Solutions on semiclosed time intervals
  • 3.3. General idea for the proof of comparison principles
  • 3.3.1. A typical problem
  • 3.3.2. Maximum principle for semicontinuous functions
  • 3.4. Proof of comparison principles for parabolic equations
  • 3.4.1. Proof for bounded domains
  • 3.4.2. Proof for unbounded domains
  • 3.5. Lipschitz preserving and convexity preserving properties
  • 3.6. Spatially inhomogeneous equations
  • 3.6.1. Inhomogeneity in first order perturbation
  • 3.6.2. Inhomogeneity in higher order terms
  • 3.7. Boundary value problems
  • 3.8. Notes and comments
  • 4. Classical level set method
  • 4.1. Brief sketch of a level set method
  • 4.2. Uniqueness of bounded evolutions
  • 4.2.1. Invariance under change of dependent variables
  • 4.2.2. Orientation-free surface evolution equations
  • 4.2.3. Uniqueness
  • 4.2.4. Unbounded evolutions
  • 4.3. Existence by Perron's method
  • 4.4. Existence by approximation
  • 4.5. Various properties of evolutions
  • 4.6. Convergence properties for level set equations
  • 4.7. Instant extinction
  • 4.8. Notes and comments
  • 5. Set-theoretic approach
  • 5.1. Set-theoretic solutions
  • 5.1.1. Definition and its characterization
  • 5.1.2. Characterization of solutions of level set equations
  • 5.1.3. Characterization by distance functions
  • 5.1.4. Comparison principle for sets
  • 5.1.5. Convergence of sets and functions
  • 5.2. Level Set solutions
  • 5.2.1. Nonuniqueness
  • 5.2.2. Definition of level set solutions
  • 5.2.3. Uniqueness of level set solutions
  • 5.3. Barrier solutions
  • 5.4. Consistency
  • 5.4.1. Nested family of subsolutions
  • 5.4.2. Applications
  • 5.4.3. Relation among various solutions
  • 5.5. Separation and comparison principle
  • 5.6. Notes and comments
  • Bibliography
  • Notation Index
  • Subject Index

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