Nonsmooth Variational Problems and Their Inequalities (eBook)

Comparison Principles and Applications

Siegfried Carl, Vy Khoi Le, Dumitru Motreanu (Autoren)

eBook Download: PDF

2007 | 2007
X, 398 Seiten
Springer US (Verlag)
978-0-387-46252-3 (ISBN)

Infos

This monograph focuses primarily on nonsmooth variational problems that arise from boundary value problems with nonsmooth data and/or nonsmooth constraints, such as multivalued elliptic problems, variational inequalities, hemivariational inequalities, and their corresponding evolution problems. It provides a systematic and unified exposition of comparison principles based on a suitably extended sub-supersolution method.

Inhalt

Preface 5
Contents 7
1 Introduction 11
2 Mathematical Preliminaries 21
2.1 Basic Functional Analysis 21
2.2 Sobolev Spaces 38
2.3 Operators of Monotone Type 49
2.4 First-Order Evolution Equations 59
2.5 Nonsmooth Analysis 73
3 Variational Equations 91
3.1 Semilinear Elliptic Equations 91
3.2 Quasilinear Elliptic Equations 103
3.3 Quasilinear Parabolic Equations 115
3.4 Sign-Changing Solutions via Fucik Spectrum 133
3.5 Quasilinear Elliptic Problems of Periodic Type 144
3.6 Notes and Comments 151
4 Multivalued Variational Equations 153
4.1 Motivation and Introductory Examples 153
4.2 Inclusions with Global Growth on Clarke’s Gradient 165
4.3 Inclusions with Local Growth on Clarke’s Gradient 177
4.4 Application: Difference of Multifunctions 190
4.5 Parabolic Inclusions with Local Growth 200
4.6 An Alternative Concept of Sub-Supersolutions 218
4.7 Notes and Comments 219
5 Variational Inequalities 221
5.1 Variational Inequalities on Closed Convex Sets 223
5.2 Variational Inequalities with Convex Functionals 244
5.3 Evolutionary Variational Inequalities 256
5.4 Sub-Supersolutions and Monotone Penalty Approximations 267
5.5 Systems of Variational Inequalities 277
5.6 Notes and Comments 287
6 Hemivariational Inequalities 289
6.1 Notion of Sub-Supersolution 291
6.2 Quasilinear Elliptic Hemivariational Inequalities 295
6.3 Evolutionary Hemivariational Inequalities 309
6.4 Notes and Comments 326
7 Variational–Hemivariational Inequalities 329
7.1 Elliptic Variational–Hemivariational Inequalities 329
7.2 Evolution Variational–Hemivariational Inequalities 346
7.3 Nonsmooth Critical Point Theory 365
7.4 A Constraint Hemivariational Inequality 372
7.5 Eigenvalue Problem for a Variational– Hemivariational Inequality 378
7.6 Notes and Comments 385
List of Symbols 388
References 390
Index 401

Produkt-Details

• Erscheint lt. Verlag: 7.6.2007
• Reihe/Serie: Springer Monographs in Mathematics
• Reihe/Serie: Springer Monographs in Mathematics
• Zusatzinfo: XII, 395 p. 30 illus.
• Verlagsort: New York
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• Schlagworte: Boundary value problem • differential equation • Functional Analysis • Mathematica • partial differential equation • Partial differential equations
• ISBN-10: 0-387-46252-X / 038746252X
• ISBN-13: 978-0-387-46252-3 / 9780387462523