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Tables of Contents for Variational Methods
Chapter/Section Title
Page #
Page Count
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