An Introduction to Variational Inequalities and Their Applications
Published on 23. April 1980
312 pages
978-0-08-087404-3 (ISBN)
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An Introduction to Variational Inequalities and Their Applications
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David Kinderlehrer
Introduction to Variational Inequalities and Their Applications
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07/1997
Academic Press
€99.04
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Content
- Front Cover
- An Introduction to Variational Inequalities and Their Applications
- Copyright Page
- Contents
- Preface
- Glossary of Notations
- Introduction
- Chapter I. Variational Inequalities in Rn
- 1. Fixed Points
- 2. The Characterization of the Projection onto a Convex Set
- 3. A First Theorem about Variational Inequalities
- 4. Variational Inequalities
- 5. Some Problem Which Lead to Variational Inequalities
- Comments and Bibliographical Notes
- Exercises
- Chapter II. Variational Inequalities in Hilbert Space
- 1. Bilinear Forms
- 2. Existence of a Solution
- 3. Truncation
- 4. Sobolev Spaces and Boundary Value Problems
- 5. The Weak Maximum Principle
- 6. The Obstacle Problem: First Properties
- 7. The Obstacle Problem in the One Dimensional Case
- Appendix A. Sobolev Spaces
- Appendix B. Solutions to Equations with Bounded Measurable Coefficients
- Appendix C. Local Estimates of Solutions
- Appendix D. Hölder Continuity of the Solutions
- Comments and Bibliographical Notes
- Exercises
- Chapter III. Variational Inequalities for Monotone Operators
- 1. An Abstract Existence Theorem
- 2. Noncoercive Operators
- 3. Semilinear Equations
- 4. Quasi-Linear Operators
- Comments and Bibliographical Notes
- Exercises
- Chapter IV. Problems of Regularity
- 1. Penalization
- 2. Dirichlet Integral
- 3. Coercive Vector Fields
- 4. Locally Coercive Vector Fields
- 5. Another Penalization
- 6. Limitation of Second Derivatives
- 7. Bounded Variation of Au
- 8. Lipschitz Obstacles
- 9. A Variational Inequality with Mixed Boundary Conditions
- Appendix A. Proof of Theorem 3.3
- Comments and Bibliographical Notes
- Exercises
- Chapter V. Free Boundary Problems and the Coincidence Set of the Solution
- I. Introduction
- 2. The Hodograph and Legendre Transformations
- 3. The Free Boundary in Two Dimensions
- 4. A Remark about Singularities
- 5. The Obstacle Problem for a Minimal Surface
- 6. The Topology of the Coincidence Set When the Obstacle Is Concave
- 7. A Remark about the Coincidence Set in Higher Dimensions
- Comments and Bibliographical Notes
- Exercises
- Chapter VI. Free Boundary Problems Governed by Elliptic Equations and Systems
- I. Introduction
- 2. Hodograph and Legendre Transforms: The Theory of a Single Equation
- 3. Elliptic Systems
- 4. A Reflection Problem
- 5. Elliptic Equations Sharing Cauchy Data
- 6. A Problem of Two Membranes
- Comments and Bibliographical Notes
- Exercises
- Chapter VII. Applications of Variational Inequalities
- 1. Introduction
- 2. A Problem in the Theory of Lubrication
- 3. The Filtration of a Liquid through a Porous Medium
- 4. The Resolution of the Filtration Problem by Variational Inequalities
- 5. The Filtration of a Liquid through a Porous Medium with Variable Cross Section
- 6. The Resolution of the Filtration Problem in Three Dimensions
- 7. Flow past a Given Profile: The Problem in the Physical Plane
- 8. Flow past a Given Profile: Resolution by Variational Inequalities
- 9. The Deflection of a Simply Supported Beam
- Comments and Bibliographical Notes
- Exercises
- Chapter VIII. A One Phase Stefan Problem
- 1. Introduction
- 2. Existence and Uniqueness of the Solution
- 3. Smoothness Properties of the Solution
- 4. The Legendre Transform
- Comments and Bibliographical Notes
- Bibliography
- Index
