Variational Methods : Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,9783540664796

Variational Methods : Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems

by ; ; ; ; ; ; ;
ISBN13: 9783540664796
ISBN10: 3540664793
Format: Hardcover
Pub. Date: 1999-12-01
Publisher(s): Springer Verlag
List Price: $208.65

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Summary

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 three distinct prime closed geodesics on any compact surface of genus zero, and the 1930/31 solution of Plateau's problem by Douglas and Radó. The book gives a concise introduction to variational methods and presents an overview of areas of current research in the field. The third edition gives a survey on new developments in the field. References have been updated and a small number of mistakes have been rectified.

Table of Contents

The Direct Methods in the Calculus of Variations
1(73)
Lower Semi-Continuity
2(11)
Degenerate Elliptic Equations
4(2)
Minimal Partitioning Hypersurfaces
6(1)
Minimal Hypersurfaces in Riemannian Manifolds
7(1)
A General Lower Semi-Continuity Result
8(5)
Constraints
13(12)
Semi-Linear Elliptic Boundary Value Problems
14(2)
Perron's Method in a Variational Guise
16(3)
The Classical Plateau Problem
19(6)
Compensated Compactness
25(11)
Applications in Elasticity
29(3)
Convergence Results for Nonlinear Elliptic Equations
32(3)
Hardy space methods
35(1)
The Concentration-Compactness Principle
36(15)
Existence of Extremal Functions for Sobolev Embeddings
42(9)
Ekeland's Variational Principle
51(6)
Existence of Minimizers for Quasi-Convex Functionals
54(3)
Duality
57(12)
Hamiltonian Systems
60(5)
Periodic Solutions of Nonlinear Wave-Equations
65(4)
Minimization Problems Depending on Parameters
69(5)
Harmonic maps with singularities
71(3)
Minimax Methods
74(95)
The Finite Dimensional Case
74(3)
The Palais-Smale Condition
77(4)
A General Deformation Lemma
81(6)
Pseudo-Gradient Flows on Banach Spaces
81(4)
Pseudo-Gradient Flows on Manifolds
85(2)
The Minimax Principle
87(7)
Closed Geodesics on Spheres
89(5)
Index Theory
94(14)
Krasnoselskii Genus
94(2)
Minimax Principles for Even Functionals
96(2)
Applications to Semilinear Elliptic Problems
98(1)
General Index Theories
99(1)
Ljusternik-Schnirelman Category
100(1)
A Geometrical S1-Index
101(2)
Multiple Periodic Orbits of Hamiltonian Systems
103(5)
The Mountain Pass Lemma and its Variants
108(10)
Applications to Semilinear Elliptic Boundary Value Problems
110(2)
The Symmetric Mountain Pass Lemma
112(4)
Application to Semilinear Equations with Symmetry
116(2)
Perturbation Theory
118(7)
Applications to Semilinear Elliptic Equations
120(5)
Linking
125(12)
Applications to Semilinear Elliptic Equations
128(2)
Applications to Hamiltonian Systems
130(7)
Parameter Dependence
137(6)
Critical Points of Mountain Pass Type
143(7)
Multiple Solutions of Coercive Elliptic Problems
147(3)
Non-Differentiable Functionals
150(12)
Ljusternik-Schnirelman Theory on Convex Sets
162(7)
Applications to Semilinear Elliptic Boundary Value Problems
166(3)
Limit Cases of the Palais-Smale Condition
169(68)
Pohozaev's Non-Existence Result
170(3)
The Brezis-Nirenberg Result
173(10)
Constrained Minimization
174(1)
The Unconstrained Case: Local Compactness
175(5)
Multiple Solutions
180(3)
The Effect of Topology
183(10)
A Global Compactness Result
184(6)
Positive Solutions on Annular-Shaped Regions
190(3)
The Yamabe Problem
193(10)
The Dirichlet Problem for the Equation of Constant Mean Curvature
203(11)
Small Solutions
204(2)
The Volume Functional
206(2)
Wente's Uniqueness Result
208(1)
Local Compactness
209(3)
Large Solutions
212(2)
Harmonic Maps of Riemannian Surfaces
214(23)
The Euler-Lagrange Equations for Harmonic Maps
215(2)
Bochner identity
217(1)
The Homotopy Problem and its Functional Analytic Setting
217(3)
Existence and Non-Existence Results
220(1)
The Evolution of Harmonic Maps
221(16)
Appendix A 237(5)
Sobolev Spaces
237(1)
Holder Spaces
238(1)
Imbedding Theorems
238(1)
Density Theorem
239(1)
Trace and Extension Theorems
239(1)
Poincare Inequality
240(2)
Appendix B 242(6)
Schauder Estimates
242(1)
Lp-Theory
242(1)
Weak Solutions
243(1)
A Regularity Result
243(2)
Maximum Principle
245(1)
Weak Maximum Principle
246(1)
Application
247(1)
Appendix C 248(3)
Frechet Differentiability
248(2)
Natural Growth Conditions
250(1)
References 251(22)
Index 273

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