Variational Methods Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems
Hilbert's talk at the second International Congress of 1900 in Paris marked the beginning of a new era in the calculus of variations. A development began which, within a few decades, brought tremendous success, highlighted by the 1929 theorem of Ljusternik and Schnirelman on the existence of th...
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Format: | eBook |
Language: | English |
Published: |
Berlin, Heidelberg
Springer Berlin Heidelberg
2000, 2000
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Edition: | 3rd ed. 2000 |
Series: | Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
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Subjects: | |
Online Access: | |
Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- I. The Direct Methods in the Calculus of Variations
- 1. Lower Semi-Continuity
- 2. Constraints
- 3. Compensated Compactness
- 4. The Concentration-Compactness Principle
- 5. Ekeland’s Variational Principle
- 6. Duality
- 7. Minimization Problems Depending on Parameters
- II. Minimax Methods
- 1. The Finite Dimensional Case
- 2. The Palais-Smale Condition
- 3. A General Deformation Lemma
- 4. The Minimax Principle
- 5. Index Theory
- 6. The Mountain Pass Lemma and its Variants
- 7. Perturbation Theory
- 8. Linking
- 9. Parameter Dependence
- 10. Critical Points of Mountain Pass Type
- 11. Non-Differentiable Functionals
- 12. Ljusternik-Schnirelman Theory on Convex Sets
- III. Limit Cases of the Palais-Smale Condition
- 1. Pohožaev’s Non-Existence Result
- 2. The Brezis-Nirenberg Result
- 3. The Effect of Topology
- 4. The Yamabe Problem
- 5. The Dirichlet Problem for the Equation of Constant Mean Curvature
- 6. Harmonic Maps of Riemannian Surfaces
- Appendix A
- Sobolev Spaces
- Hölder Spaces
- Imbedding Theorems
- Density Theorem
- Trace and Extension Theorems
- Poincaré Inequality
- Appendix B
- Schauder Estimates
- Weak Solutions
- A Regularity Result
- Maximum Principle
- Weak Maximum Principle
- Application
- Appendix C
- Fréchet Differentiability
- Natural Growth Conditions
- References