An introduction to variational inequalities and their applications. 0124073506, 9780124073500

Table of contents :
An Introduction to Variational Inequalities and Their Applications......Page 4
Copyright Page......Page 5
Contents......Page 8
Preface......Page 12
Glossary of Notations......Page 14
Introduction......Page 16
1. Fixed Points......Page 22
2. The Characterization of the Projection onto a Convex Set......Page 23
3. A First Theorem about Variational Inequalities......Page 26
4. Variational Inequalities......Page 28
5. Some Problem Which Lead to Variational Inequalities......Page 30
Exercises......Page 33
1. Bilinear Forms......Page 38
2. Existence of a Solution......Page 39
3. Truncation......Page 42
4. Sobolev Spaces and Boundary Value Problems......Page 43
5. The Weak Maximum Principle......Page 50
6. The Obstacle Problem: First Properties......Page 55
7. The Obstacle Problem in the One Dimensional Case......Page 62
Appendix A. Sobolev Spaces......Page 64
Appendix B. Solutions to Equations with Bounded Measurable Coefficients......Page 77
Appendix C. Local Estimates of Solutions......Page 81
Appendix D. Hölder Continuity of the Solutions......Page 87
Comments and Bibliographical Notes......Page 91
Exercises......Page 92
1. An Abstract Existence Theorem......Page 98
2. Noncoercive Operators......Page 102
3. Semilinear Equations......Page 108
4. Quasi-Linear Operators......Page 109
Comments and Bibliographical Notes......Page 115
Exercises......Page 116
1. Penalization......Page 120
2. Dirichlet Integral......Page 121
3. Coercive Vector Fields......Page 128
4. Locally Coercive Vector Fields......Page 131
5. Another Penalization......Page 135
6. Limitation of Second Derivatives......Page 139
7. Bounded Variation of Au......Page 145
8. Lipschitz Obstacles......Page 149
9. A Variational Inequality with Mixed Boundary Conditions......Page 154
Appendix A. Proof of Theorem 3.3......Page 158
Comments and Bibliographical Notes......Page 161
Exercises......Page 162
I. Introduction......Page 164
2. The Hodograph and Legendre Transformations......Page 168
3. The Free Boundary in Two Dimensions......Page 170
4. A Remark about Singularities......Page 181
5. The Obstacle Problem for a Minimal Surface......Page 182
6. The Topology of the Coincidence Set When the Obstacle Is Concave......Page 188
7. A Remark about the Coincidence Set in Higher Dimensions......Page 193
Comments and Bibliographical Notes......Page 196
Exercises......Page 197
I. Introduction......Page 199
2. Hodograph and Legendre Transforms: The Theory of a Single Equation......Page 200
3. Elliptic Systems......Page 205
4. A Reflection Problem......Page 217
5. Elliptic Equations Sharing Cauchy Data......Page 219
6. A Problem of Two Membranes......Page 227
Exercises......Page 233
1. Introduction......Page 237
2. A Problem in the Theory of Lubrication......Page 238
3. The Filtration of a Liquid through a Porous Medium......Page 242
4. The Resolution of the Filtration Problem by Variational Inequalities......Page 250
5. The Filtration of a Liquid through a Porous Medium with Variable Cross Section......Page 257
6. The Resolution of the Filtration Problem in Three Dimensions......Page 264
7. Flow past a Given Profile: The Problem in the Physical Plane......Page 272
8. Flow past a Given Profile: Resolution by Variational Inequalities......Page 275
9. The Deflection of a Simply Supported Beam......Page 285
Comments and Bibliographical Notes......Page 288
Exercises......Page 289
1. Introduction......Page 293
2. Existence and Uniqueness of the Solution......Page 296
3. Smoothness Properties of the Solution......Page 304
4. The Legendre Transform......Page 312
Comments and Bibliographical Notes......Page 314
Bibliography......Page 315
Index......Page 324