Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems

by Alexandru Kristály, Vicentiu D. Radulescu, Csaba Varga

Overview

This comprehensive introduction to the calculus of variations and its main principles also presents their real-life applications in various contexts: mathematical physics, differential geometry, and optimization in economics. Based on the authors' original work, it provides an overview of the field, with examples and exercises suitable for graduate students entering research. The method of presentation will appeal to readers with diverse backgrounds in functional analysis, differential geometry and partial differential equations. Each chapter includes detailed heuristic arguments, providing thorough motivation for the material developed later in the text. Since much of the material has a strong geometric flavor, the authors have supplemented the text with figures to illustrate the abstract concepts. Its extensive reference list and index also make this a valuable resource for researchers working in a variety of fields who are interested in partial differential equations and functional analysis.

Product Details

ISBN-13:	9780521117821
Publisher:	Cambridge University Press
Publication date:	08/19/2010
Series:	Encyclopedia of Mathematics and its Applications , #136
Pages:	386
Product dimensions:	6.10(w) x 9.30(h) x 1.10(d)

About the Author

Alexandru Kristály is Associate Professor in the Department of Economics at the University of Babeş-Bolyai in Cluj-Napoca, Romania.

Vicenţiu D. Rădulescu is Professor in the Department of Mathematics at the University of Craiova, Romania.

Csaba Gyorgy Varga is Professor in the Department of Mathematics at the University of Babeş-Bolyai in Cluj-Napoca, Romania.

Table of Contents

Foreword x

Preface xii

Part I Variational principles in mathematical physics 1

1 Variational principles 3

1.1 Minimization techniques and Ekeland's variational principle 3

1.2 Borwein-Preiss variational principle 8

1.3 Minimax principles 12

1.4 Ricceri's variational results 19

1.5 H¹ versus C¹ local minimizers 28

1.6 Szulkin-type functionals 33

1.7 Pohozaev's fibering method 38

1.8 Historical comments 39

2 Variational inequalities 42

2.1 Introduction 42

2.2 Variational inequalities on unbounded strips 43

2.3 Area-type variational inequalities 55

2.4 Historical notes and comments 78

3 Nonlinear eigenvalue problems 81

3.1 Weighted Sobolev spaces 82

3.2 Eigenvalue problems 85

3.3 Superlinear case 94

3.4 Sublinear case 104

3.5 Comments and further perspectives 115

4 Elliptic systems of gradient type 117

4.1 Introduction 117

4.2 Formulation of the problems 117

4.3 Systems with superlinear potential 119

4.4 Systems with sublinear potential 127

4.5 Shift solutions for gradient systems 134

4.6 Historical notes and comments 144

5 Systems with arbitrary growth nonlinearities 146

5.1 Introduction 146

5.2 Elliptic systems with mountain pass geometry 148

5.3 Elliptic systems with oscillatory terms 153

5.4 Comments and perspectives 160

6 Scalar field systems 162

6.1 Introduction 162

6.2 Multiple solutions of a double eigenvalue problem 163

6.3 Scalar field systems with nonlinear oscillatory terms 172

6.4 Applications 178

6.5 Historical notes and comments 182

7 Competition phenomena in Dirichlet problems 183

7.1 Introduction 184

7.2 Effects of the competition 185

7.3 A general location property 190

7.4 Nonlinearities with oscillation near the origin 192

7.5 Nonlinearities with oscillation at infinity 198

7.6 Perturbation from symmetry 205

7.7 Historical notes and comments 208

8 Problems to Part I 210

Part II Variational principles in geometry 215

9 Sublinear problems on Riemannian manifolds 217

9.1 Introduction 217

9.2 Existence of two solutions 218

9.3 Existence of many global minima 224

9.4 Historical notes and comments 227

10 Asymptotically critical problems on spheres 228

10.1 Introduction 228

10.2 Group-theoretical argument 229

10.3 Arbitrarily small solutions 235

10.4 Arbitrarily large solutions 242

10.5 Historical notes, comments, and perspectives 246

11 Equations with critical exponent 248

11.1 Introduction 248

11.2 Subcritical case 250

11.3 Critical case 252

11.4 Historical notes and comments 259

12 Problems to Part II 261

Part III Variational principles in economics 265

13 Mathematical preliminaries 267

13.1 Metrics, geodesics, and flag curvature 267

13.2 Busemann-type inequalities on Finsler manifolds 271

13.3 Variational inequalities 277

14 Minimization of cost-functions on manifolds 278

14.1 Introduction 278

14.2 A necessary condition 280

14.3 Existence and uniqueness results 282

14.4 Examples on the Finslerian-Poincaré disc 285

14.5 Comments and further perspectives 287

15 Best approximation problems on manifolds 289

15.1 Introduction 289

15.2 Existence of projections 290

15.3 Geometric properties of projections 291

15.4 Geodesic convexity and Chebyshev sets 294

15.5 Optimal connection of two submanifolds 297

15.6 Remarks and perspectives 303

16 A variational approach to Nash equilibria 304

16.1 Introduction 304

16.2 Nash equilibria and variational inequalities 305

16.3 Nash equilibria for set-valued maps 308

16.4 Lack of convexity: a Riemannian approach 313

16.5 Historical comments and perspectives 319

17 Problems to Part III 320

Appendix A Elements of convex analysis 322

A.1 Convex sets and convex functions 322

A.2 Convex analysis in Banach spaces 326

Appendix B Function spaces 328

B.1 Lebesgue spaces 328

B.2 Sobolev spaces 329

B.3 Compact embedding results 330

B.4 Sobolev spaces on Riemann manifolds 334

Appendix C Category and genus 337

Appendix D Clarke and Degiovanni gradients 339

D.1 Locally Lipschitz functionals 339

D.2 Continuous or lower semi-continuous functionals 341

Appendix E Elements of set-valued analysis 346

References 349

Notation index 361

Subject index 363

Editorial Reviews

"This book is a pleasure to read. It will be an excellent source, allowing the reader to build a proper intuition and to understand the basic facts of the theory. I deeply believe that the present book is one of the most important works in its field that have appeared until now. To conclude, this book provides the reader with an in-depth introduction to a rich and rapidly developing research area that has already produced remarkable results. It is beautifully written and the reviewer strongly recommends this book to anyone seeking a stylish, balanced, up-to-date survey of this central area of nonlinear partial differential equations."
Constantin Costara, Ovidius University of Constanta for Analele Stiintifice ale Universitatii Ovidius Constanta

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