Title:
Methods in nonlinear analysis
Personal Author:
Series:
Springer monographs in mathematics
Publication Information:
New York, NY : Springer-Verlag, 2005
ISBN:
9783540241331
Available:*
Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
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Searching... | 30000010138526 | QA427 C42 2005 | Open Access Book | Book | Searching... |
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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 Linearization | p. 1 |
1.1 Differential Calculus in Banach Spaces | p. 1 |
1.1.1 Frechet Derivatives and Gateaux Derivatives | p. 2 |
1.1.2 Nemytscki Operator | p. 7 |
1.1.3 High-Order Derivatives | p. 9 |
1.2 Implicit Function Theorem and Continuity Method | p. 12 |
1.2.1 Inverse Function Theorem | p. 12 |
1.2.2 Applications | p. 17 |
1.2.3 Continuity Method | p. 23 |
1.3 Lyapunov-Schmidt Reduction and Bifurcation | p. 30 |
1.3.1 Bifurcation | p. 30 |
1.3.2 Lyapunov-Schmidt Reduction | p. 33 |
1.3.3 A Perturbation Problem | p. 43 |
1.3.4 Gluing | p. 47 |
1.3.5 Transversality | p. 49 |
1.4 Hard Implicit Function Theorem | p. 54 |
1.4.1 The Small Divisor Problem | p. 55 |
1.4.2 Nash-Moser Iteration | p. 62 |
2 Fixed-Point Theorems | p. 71 |
2.1 Order Method | p. 72 |
2.2 Convex Function and Its Subdifferentials | p. 80 |
2.2.1 Convex Functions | p. 80 |
2.2.2 Subdifferentials | p. 84 |
2.3 Convexity and Compactness | p. 87 |
2.4 Nonexpansive Maps | p. 104 |
2.5 Monotone Mappings | p. 109 |
2.6 Maximal Monotone Mapping | p. 120 |
3 Degree Theory and Applications | p. 127 |
3.1 The Notion of Topological Degree | p. 128 |
3.2 Fundamental Properties and Calculations of Brouwer Degrees | p. 137 |
3.3 Applications of Brouwer Degree | p. 148 |
3.3.1 Brouwer Fixed-Point Theorem | p. 148 |
3.3.2 The Borsuk-Ulam Theorem and Its Consequences | p. 148 |
3.3.3 Degrees for S[superscript 1] Equivariant Mappings | p. 151 |
3.3.4 Intersection | p. 153 |
3.4 Leray-Schauder Degrees | p. 155 |
3.5 The Global Bifurcation | p. 164 |
3.6 Applications | p. 175 |
3.6.1 Degree Theory on Closed Convex Sets | p. 175 |
3.6.2 Positive Solutions and the Scaling Method | p. 180 |
3.6.3 Krein-Rutman Theory for Positive Linear Operators | p. 185 |
3.6.4 Multiple Solutions | p. 189 |
3.6.5 A Free Boundary Problem | p. 192 |
3.6.6 Bridging | p. 193 |
3.7 Extensions | p. 195 |
3.7.1 Set-Valued Mappings | p. 195 |
3.7.2 Strict Set Contraction Mappings and Condensing Mappings | p. 198 |
3.7.3 Fredholm Mappings | p. 200 |
4 Minimization Methods | p. 205 |
4.1 Variational Principles | p. 206 |
4.1.1 Constraint Problems | p. 206 |
4.1.2 Euler-Lagrange Equation | p. 209 |
4.1.3 Dual Variational Principle | p. 212 |
4.2 Direct Method | p. 216 |
4.2.1 Fundamental Principle | p. 216 |
4.2.2 Examples | p. 217 |
4.2.3 The Prescribing Gaussian Curvature Problem and the Schwarz Symmetric Rearrangement | p. 223 |
4.3 Quasi-Convexity | p. 231 |
4.3.1 Weak Continuity and Quasi-Convexity | p. 232 |
4.3.2 Morrey Theorem | p. 237 |
4.3.3 Nonlinear Elasticity | p. 242 |
4.4 Relaxation and Young Measure | p. 244 |
4.4.1 Relaxations | p. 245 |
4.4.2 Young Measure | p. 251 |
4.5 Other Function Spaces | p. 260 |
4.5.1 BV Space | p. 260 |
4.5.2 Hardy Space and BMO Space | p. 266 |
4.5.3 Compensation Compactness | p. 271 |
4.5.4 Applications to the Calculus of Variations | p. 274 |
4.6 Free Discontinuous Problems | p. 279 |
4.6.1 [Gamma]-convergence | p. 279 |
4.6.2 A Phase Transition Problem | p. 280 |
4.6.3 Segmentation and Mumford-Shah Problem | p. 284 |
4.7 Concentration Compactness | p. 289 |
4.7.1 Concentration Function | p. 289 |
4.7.2 The Critical Sobolev Exponent and the Best Constants | p. 295 |
4.8 Minimax Methods | p. 301 |
4.8.1 Ekeland Variational Principle | p. 301 |
4.8.2 Minimax Principle | p. 303 |
4.8.3 Applications | p. 306 |
5 Topological and Variational Methods | p. 315 |
5.1 Morse Theory | p. 317 |
5.1.1 Introduction | p. 317 |
5.1.2 Deformation Theorem | p. 319 |
5.1.3 Critical Groups | p. 327 |
5.1.4 Global Theory | p. 334 |
5.1.5 Applications | p. 343 |
5.2 Minimax Principles (Revisited) | p. 347 |
5.2.1 A Minimax Principle | p. 347 |
5.2.2 Category and Ljusternik-Schnirelmann Multiplicity Theorem | p. 349 |
5.2.3 Cap Product | p. 354 |
5.2.4 Index Theorem | p. 358 |
5.2.5 Applications | p. 363 |
5.3 Periodic Orbits for Hamiltonian System and Weinstein Conjecture | p. 371 |
5.3.1 Hamiltonian Operator | p. 373 |
5.3.2 Periodic Solutions | p. 374 |
5.3.3 Weinstein Conjecture | p. 376 |
5.4 Prescribing Gaussian Curvature Problem on S[superscript 2] | p. 380 |
5.4.1 The Conformal Group and the Best Constant | p. 380 |
5.4.2 The Palais-Smale Sequence | p. 387 |
5.4.3 Morse Theory for the Prescribing Gaussian Curvature Equation on S[superscript 2] | p. 389 |
5.5 Conley Index Theory | p. 392 |
5.5.1 Isolated Invariant Set | p. 393 |
5.5.2 Index Pair and Conley Index | p. 397 |
5.5.3 Morse Decomposition on Compact Invariant Sets and Its Extension | p. 408 |
Notes | p. 419 |
References | p. 425 |