Cover image for Methods in nonlinear analysis
Title:
Methods in nonlinear analysis
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Series:
Springer monographs in mathematics
Publication Information:
New York, NY : Springer-Verlag, 2005
ISBN:
9783540241331

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30000010138526 QA427 C42 2005 Open Access Book Book
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Summary

Summary

This book offers a systematic presentation of up-to-date material scattered throughout the literature from the methodology point of view. It reviews the basic theories and methods, with many interesting problems in partial and ordinary differential equations, differential geometry and mathematical physics as applications, and provides the necessary preparation for almost all important aspects in contemporary studies. All methods are illustrated by carefully chosen examples from mechanics, physics, engineering and geometry.


Table of Contents

1 Linearizationp. 1
1.1 Differential Calculus in Banach Spacesp. 1
1.1.1 Frechet Derivatives and Gateaux Derivativesp. 2
1.1.2 Nemytscki Operatorp. 7
1.1.3 High-Order Derivativesp. 9
1.2 Implicit Function Theorem and Continuity Methodp. 12
1.2.1 Inverse Function Theoremp. 12
1.2.2 Applicationsp. 17
1.2.3 Continuity Methodp. 23
1.3 Lyapunov-Schmidt Reduction and Bifurcationp. 30
1.3.1 Bifurcationp. 30
1.3.2 Lyapunov-Schmidt Reductionp. 33
1.3.3 A Perturbation Problemp. 43
1.3.4 Gluingp. 47
1.3.5 Transversalityp. 49
1.4 Hard Implicit Function Theoremp. 54
1.4.1 The Small Divisor Problemp. 55
1.4.2 Nash-Moser Iterationp. 62
2 Fixed-Point Theoremsp. 71
2.1 Order Methodp. 72
2.2 Convex Function and Its Subdifferentialsp. 80
2.2.1 Convex Functionsp. 80
2.2.2 Subdifferentialsp. 84
2.3 Convexity and Compactnessp. 87
2.4 Nonexpansive Mapsp. 104
2.5 Monotone Mappingsp. 109
2.6 Maximal Monotone Mappingp. 120
3 Degree Theory and Applicationsp. 127
3.1 The Notion of Topological Degreep. 128
3.2 Fundamental Properties and Calculations of Brouwer Degreesp. 137
3.3 Applications of Brouwer Degreep. 148
3.3.1 Brouwer Fixed-Point Theoremp. 148
3.3.2 The Borsuk-Ulam Theorem and Its Consequencesp. 148
3.3.3 Degrees for S[superscript 1] Equivariant Mappingsp. 151
3.3.4 Intersectionp. 153
3.4 Leray-Schauder Degreesp. 155
3.5 The Global Bifurcationp. 164
3.6 Applicationsp. 175
3.6.1 Degree Theory on Closed Convex Setsp. 175
3.6.2 Positive Solutions and the Scaling Methodp. 180
3.6.3 Krein-Rutman Theory for Positive Linear Operatorsp. 185
3.6.4 Multiple Solutionsp. 189
3.6.5 A Free Boundary Problemp. 192
3.6.6 Bridgingp. 193
3.7 Extensionsp. 195
3.7.1 Set-Valued Mappingsp. 195
3.7.2 Strict Set Contraction Mappings and Condensing Mappingsp. 198
3.7.3 Fredholm Mappingsp. 200
4 Minimization Methodsp. 205
4.1 Variational Principlesp. 206
4.1.1 Constraint Problemsp. 206
4.1.2 Euler-Lagrange Equationp. 209
4.1.3 Dual Variational Principlep. 212
4.2 Direct Methodp. 216
4.2.1 Fundamental Principlep. 216
4.2.2 Examplesp. 217
4.2.3 The Prescribing Gaussian Curvature Problem and the Schwarz Symmetric Rearrangementp. 223
4.3 Quasi-Convexityp. 231
4.3.1 Weak Continuity and Quasi-Convexityp. 232
4.3.2 Morrey Theoremp. 237
4.3.3 Nonlinear Elasticityp. 242
4.4 Relaxation and Young Measurep. 244
4.4.1 Relaxationsp. 245
4.4.2 Young Measurep. 251
4.5 Other Function Spacesp. 260
4.5.1 BV Spacep. 260
4.5.2 Hardy Space and BMO Spacep. 266
4.5.3 Compensation Compactnessp. 271
4.5.4 Applications to the Calculus of Variationsp. 274
4.6 Free Discontinuous Problemsp. 279
4.6.1 [Gamma]-convergencep. 279
4.6.2 A Phase Transition Problemp. 280
4.6.3 Segmentation and Mumford-Shah Problemp. 284
4.7 Concentration Compactnessp. 289
4.7.1 Concentration Functionp. 289
4.7.2 The Critical Sobolev Exponent and the Best Constantsp. 295
4.8 Minimax Methodsp. 301
4.8.1 Ekeland Variational Principlep. 301
4.8.2 Minimax Principlep. 303
4.8.3 Applicationsp. 306
5 Topological and Variational Methodsp. 315
5.1 Morse Theoryp. 317
5.1.1 Introductionp. 317
5.1.2 Deformation Theoremp. 319
5.1.3 Critical Groupsp. 327
5.1.4 Global Theoryp. 334
5.1.5 Applicationsp. 343
5.2 Minimax Principles (Revisited)p. 347
5.2.1 A Minimax Principlep. 347
5.2.2 Category and Ljusternik-Schnirelmann Multiplicity Theoremp. 349
5.2.3 Cap Productp. 354
5.2.4 Index Theoremp. 358
5.2.5 Applicationsp. 363
5.3 Periodic Orbits for Hamiltonian System and Weinstein Conjecturep. 371
5.3.1 Hamiltonian Operatorp. 373
5.3.2 Periodic Solutionsp. 374
5.3.3 Weinstein Conjecturep. 376
5.4 Prescribing Gaussian Curvature Problem on S[superscript 2]p. 380
5.4.1 The Conformal Group and the Best Constantp. 380
5.4.2 The Palais-Smale Sequencep. 387
5.4.3 Morse Theory for the Prescribing Gaussian Curvature Equation on S[superscript 2]p. 389
5.5 Conley Index Theoryp. 392
5.5.1 Isolated Invariant Setp. 393
5.5.2 Index Pair and Conley Indexp. 397
5.5.3 Morse Decomposition on Compact Invariant Sets and Its Extensionp. 408
Notesp. 419
Referencesp. 425