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

by Alexandru Kristály , Vicenţiu D. Rădulescu , Csaba Varga

• ISBN13: 9780521117821
• ISBN10: 0521117828
• Format: Hardcover
• Copyright: 2010-10-11
• Publisher: Cambridge University Press

Summary

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 many figures to illustrate the abstract concepts. Its extensive bibliography, glossary and index also make this a valuable reference for researchers working in a variety of fields who are interested in partial differential equations and functional analysis.

Table of Contents

Foreword	p. x
Preface	p. xii
Variational principles in mathematical physics	p. 1
Variational principles	p. 3
Minimization techniques and Ekeland's variational principle	p. 3
Borwein-Preiss variational principle	p. 8
Minimax principles	p. 12
Ricceri's variational results	p. 19
H1 versus C1 local minimizers	p. 28
Szulkin-type functionals	p. 33
Pohozaev's fibering method	p. 38
Historical comments	p. 39
Variational inequalities	p. 42
Introduction	p. 42
Variational inequalities on unbounded strips	p. 43
Area-type variational inequalities	p. 55
Historical notes and comments	p. 78
Nonlinear eigenvalue problems	p. 81
Weighted Sobolev spaces	p. 82
Eigenvalue problems	p. 85
Superlinear case	p. 94
Sublinear case	p. 104
Comments and further perspectives	p. 115
Elliptic systems of gradient type	p. 117
Introduction	p. 117
Formulation of the problems	p. 117
Systems with superlinear potential	p. 119
Systems with sublinear potential	p. 127
Shift solutions for gradient systems	p. 134
Historical notes and comments	p. 144
Systems with arbitrary growth nonlinearities	p. 146
Introduction	p. 146
Elliptic systems with mountain pass geometry	p. 148
Elliptic systems with oscillatory terms	p. 153
Comments and perspectives	p. 160
Scalar field systems	p. 162
Introduction	p. 162
Multiple solutions of a double eigenvalue problem	p. 163
Scalar field systems with nonlinear oscillatory terms	p. 172
Applications	p. 178
Historical notes and comments	p. 182
Competition phenomena in Dirichlet problems	p. 183
Introduction	p. 184
Effects of the competition	p. 185
A general location property	p. 190
Nonlinearities with oscillation near the origin	p. 192
Nonlinearities with oscillation at infinity	p. 198
Perturbation from symmetry	p. 205
Historical notes and comments	p. 208
Problems to Part I	p. 210
Variational principles in geometry	p. 215
Sublinear problems on Riemannian manifolds	p. 217
Introduction	p. 217
Existence of two solutions	p. 218
Existence of many global minima	p. 224
Historical notes and comments	p. 227
Asymptotically critical problems on spheres	p. 228
Introduction	p. 228
Group-theoretical argument	p. 229
Arbitrarily small solutions	p. 235
Arbitrarily large solutions	p. 242
Historical notes, comments, and perspectives	p. 246
Equations with critical exponent	p. 248
Introduction	p. 248
Subcritical case	p. 250
Critical case	p. 252
Historical notes and comments	p. 259
Problems to Part II	p. 261
Variational principles in economics	p. 265
Mathematical preliminaries	p. 267
Metrics, geodesics, and flag curvature	p. 267
Busemann-type inequalities on Finsler manifolds	p. 271
Variational inequalities	p. 277
Minimization of cost-functions on manifolds	p. 278
Introduction	p. 278
A necessary condition	p. 280
Existence and uniqueness results	p. 282
Examples on the Finslerian-Poincaré disc	p. 285
Comments and further perspectives	p. 287
Best approximation problems on manifolds	p. 289
Introduction	p. 289
Existence of projections	p. 290
Geometric properties of projections	p. 291
Geodesic convexity and Chebyshev sets	p. 294
Optimal connection of two submanifolds	p. 297
Remarks and perspectives	p. 303
A variational approach to Nash equilibria	p. 304
Introduction	p. 304
Nash equilibria and variational inequalities	p. 305
Nash equilibria for set-valued maps	p. 308
Lack of convexity: a Riemannian approach	p. 313
Historical comments and perspectives	p. 319
Problems to Part III	p. 320
Elements of convex analysis	p. 322
Convex sets and convex functions	p. 322
Convex analysis in Banach spaces	p. 326
Function spaces	p. 328
Lebesgue spaces	p. 328
Sobolev spaces	p. 329
Compact embedding results	p. 330
Sobolev spaces on Riemann manifolds	p. 334
Category and genus	p. 337
Clarke and Degiovanni gradients	p. 339
Locally Lipschitz functionals	p. 339
Continuous or lower semi-continuous functionals	p. 341
Elements of set-valued analysis	p. 346
References	p. 349
Notation index	p. 361
Subject index	p. 363
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