THE FACTS ON FILE PHYSICS HANDBOOK
P A R T I: VARIATIONAL PRINCIPLES IN MATHEMATICAL PHYSICS 1 1 VARIATIONAL PRINCIPLES 2 1 1 MINIMIZATION TECHNIQUES AND EKELAND VARIATIONAL PRINCIPLE 2 1 2 BORWEIN-PREISS VARIATIONAL PRINCIPLE 8 1 3 MINIMAX PRINCIPLES 12 1 3 1 MOUNTAIN PASS TYPE RESULTS 12 1 3 2 MINIMAX RESULTS VIA LJUSTERNIK-SCHNIRELMANN CATEGORY 15 1 4 RICCERI'S VARIATIONAL RESULTS 20 1 4 1 THREE CRITICAL POINT RESULTS 20 1 4 2 A GENERAL VARIATIONAL PRINCIPLE 27 1 5 H1 VERSUS C1 LOCAL MINIMIZERS 30 1 5 1 APPLICATION TO SUB AND SUPERSOLUTIONS 33 1 6 SZULKIN TYPE FUNCTIONALS 34 1 6 1 MINIMAX RESULTS OF SZULKIN TYPE 35 1 6 2 PRINCIPLE OF SYMMETRIC CRITICALITY 37 1 7 POHOZAEV'S ¯BERING METHOD 40 1 8 HISTORICAL COMMENTS 42 2 VARIATIONAL INEQUALITIES 44 2 1 INTRODUCTION 44 2 2 VARIATIONAL INEQUALITIES ON = ! £ RL 44 2 2 1 CASE L ¸ 2 45 2 2 2 CASE L = 1 50 2 3 AREA{TYPE VARIATIONAL INEQUALITIES 58 2 3 1 STATEMENT OF THE PROBLEM 60 2 3 2 AREA TYPE FUNCTIONALS 69 2 3 3 A RESULT OF CLARK TYPE 76 2 3 4 A SUPERLINEAR POTENTIAL 79 2 4 HISTORICAL NOTES AND COMMENTS 82 3 NONLINEAR EIGENVALUE PROBLEMS 83 3 1 WEIGHTED SOBOLEV SPACES 85 3 2 EIGENVALUE PROBLEMS 88 3 3 SUPERLINEAR CASE 96 3 4 SUBLINEAR CASE 108 3 5 COMMENTS AND FURTHER PERSPECTIVES 119 4 ELLIPTIC SYSTEMS OF GRADIENT TYPE 121 4 1 INTRODUCTION 121 4 2 FORMULATION OF THE PROBLEMS 121 4 3 SYSTEMS WITH SUPERLINEAR POTENTIAL 124 4 4 SYSTEMS WITH SUBLINEAR POTENTIAL 131 4 5 SHIFT SOLUTIONS FOR GRADIENT SYSTEMS 140 4 6 COMMENTS AND HISTORICAL NOTES 151 5 SYSTEMS WITH ARBITRARY GROWTH NONLINEARITIES 152 5 1 INTRODUCTION 152 5 2 ELLIPTIC SYSTEMS WITH OSCILLATORY TERMS 155 5 3 ELLIPTIC SYSTEMS WITH MOUNTAIN PASS GEOMETRY 163 5 4 COMMENTS AND PERSPECTIVES 170 6 SCALAR FIELD EQUATIONS WITH OSCILLATORY TERMS 171 6 1 INTRODUCTION 171 6 2 MULTIPLE SOLUTIONS OF A DOUBLE EIGENVALUE PROBLEM 172 6 3 SCALAR ¯ELD SYSTEMS WITH NONLINEAR OSCILLATORY TERMS 182 6 4 APPLICATIONS 188 6 5 COMMENTS AND HISTORICAL NOTES 192 7 COMPETITION PHENOMENA IN DIRICHLET PROBLEMS 193 7 1 INTRODUCTION 193 7 2 E®ECTS OF THE COMPETITION 194 7 3 A GENERAL LOCATION RESULT 200 7 4 NONLINEARITIES WITH OSCILLATION NEAR THE ORIGIN 203 7 5 NONLINEARITIES WITH OSCILLATION AT IN¯NITY 209 7 6 PERTURBATION FROM SYMMETRY 216 7 7 HISTORICAL NOTES, COMMENTS 221 8 PROBLEMS TO PART I 223 P A R T II: VARIATIONAL PRINCIPLES IN GEOMETRY 228 9 SUBLINEAR PROBLEMS ON RIEMANN MANIFOLDS 229 9 1 INTRODUCTION 229 9 2 EXISTENCE OF TWO SOLUTIONS 230 9 3 EXISTENCE OF MANY GLOBAL MINIMA 237 9 4 COMMENTS AND PERSPECTIVES 240 10 CRITICAL PROBLEMS ON SPHERES 241 10 1 INTRODUCTION 241 10 2 GROUP-THEORETICAL ARGUMENT 242 10 3 ARBITRARILY SMALL SOLUTIONS 248 10 4 ARBITRARILY LARGE SOLUTIONS 256 10 5 HISTORICAL NOTES, COMMENTS AND PERSPECTIVES 260 11 EQUATIONS WITH CRITICAL EXPONENT 263 11 1 INTRODUCTION 263 11 2 SUBCRITICAL CASE 268 11 3 CRITICAL CASE 270 11 4 COMMENTS AND HISTORICAL NOTES 277 12 PROBLEMS TO PART II 279 P A R T III: VARIATIONAL PRINCIPLES IN ECONOMICS 284 13 MATHEMATICAL PRELIMINARIES 284 13 1 METRICS, GEODESICS, °AG CURVATURE 284 13 2 BUSEMANN-TYPE INEQUALITIES ON FINSLER MANIFOLDS 289 13 3 VARIATIONAL INEQUALITIES 295 14 MINIMIZATION OF COST-FUNCTIONS ON MANIFOLDS 297 14 1 INTRODUCTION 297 14 2 A NECESSARY CONDITION 300 14 3 EXISTENCE AND UNIQUENESS RESULTS 301 14 4 EXAMPLES ON THE FINSLERIAN-POINCAR¶E DISC 305 14 5 COMMENTS AND FURTHER PERSPECTIVES 307 15 BEST APPROXIMATION PROBLEMS ON MANIFOLDS 310 15 1 INTRODUCTION 310 15 2 EXISTENCE OF PROJECTIONS 311 15 3 GEOMETRIC PROPERTIES OF PROJECTIONS 313 15 4 GEODESIC CONVEXITY AND CHEBYSHEV SETS 316 15 5 OPTIMAL CONNECTION OF TWO SUBMANIFOLDS 320 15 6 FINAL REMARKS AND PERSPECTIVES 327 16 A VARIATIONAL APPROACH OF THE NASH EQUILIBRIA 328 16 1 INTRODUCTION 328 16 2 NASH EQUILIBRIA AND VARIATIONAL INEQUALITIES 329 16 3 NASH EQUILIBRIA FOR SET-VALUED MAPS 332 16 4 LACK OF CONVEXITY: A RIEMANNIAN APPROACH 338 16 5 HISTORICAL COMMENTS AND PERSPECTIVES 344 17 PROBLEMS TO PART III