The Mountain Pass Theorem: Variants, Generalizations and Some Applications (Encyclopedia of Mathematics and its Applications)

Cover......Page 1
Half-title......Page 3
Series-title......Page 5
Title......Page 7
Copyright......Page 8
Dedication......Page 9
Contents......Page 11
Who Should Read It?......Page 15
The Approach......Page 16
How Is the Book Organized?......Page 18
Acknowledgments......Page 20
An Algorithm for Finding Extrema by Fermat......Page 21
Appearance of Calculus......Page 22
Dirichlet Principle at the Roots of Modern Critical Point Theory......Page 23
Modern Critical Point Theory......Page 25
The Beginning of Postmodern Critical Point Theory......Page 26
I First Steps Toward the Mountains......Page 27
2.1 Definitions......Page 29
2.2 Examples......Page 31
2.3 Some Criteria for Checking (PS)......Page 33
2.II The Morse and Sard Theorems......Page 34
3 Obtaining “Almost Critical Points” – Variational Principle......Page 36
3.1 Ekeland’s Variational Principle......Page 37
3.I The Original Proof of Ekeland’s Principle......Page 41
3.II Smooth Extensions of Ekeland’s Principle......Page 42
3.III A Nice Generalization of Ekeland’s Principle......Page 43
3.IV Other Forms of Ekeland’s Principle......Page 44
3.V A Weighted Variational Principle......Page 46
4.1 Regularity and Topology of Level Sets......Page 47
4.2 A Technical Tool......Page 49
4.3 Deformation Lemma......Page 52
4.I Quantitative Deformation Lemma of Brezis and Nirenberg......Page 55
4.II The Deformation Lemma of Shafrir......Page 56
4.IV Extensions of Deformation Techniques to Nonsmooth Theories......Page 57
4.VI Uniqueness of the Steepest Descent Direction......Page 58
II Reaching the Mountain Pass Through Easy Climbs......Page 59
5 The Finite Dimensional MPT......Page 61
5.1 Finite Dimensional MPT......Page 63
5.2 Application......Page 66
5.I On Rolle’s Theorem and the MPT......Page 67
5.II The Deformation that Appears in the Original Proof of Courant......Page 68
5.IV Hadamard Global Inversion Theorem......Page 69
6.1 Some Preliminaries......Page 71
6.2 Topological MPT......Page 74
Comments and Additional Notes......Page 78
7.1 The Name of the Game......Page 79
7.2 The MPT......Page 80
7.3 A Superlinear Problem......Page 83
1. The Geometry of the MPT......Page 84
2. The Palais-Smale Condition......Page 85
7.4 A Problem of Ambrosetti-Prodi Type......Page 86
7.I A Third Proof of the MPT......Page 90
7.II The Situation of a Path…......Page 93
7.IV On Applications......Page 94
8.1 The Multidimensional MPT......Page 95
8.2 Application......Page 98
8.I Some Extensions of the Multidimensional MPT......Page 99
8.II Different Minimaxing Sets......Page 100
8.III The Dual MPT......Page 101
8.IV On a More General Minimax Principle......Page 102
III A Deeper Insight in Mountains Topology......Page 105
The Situation......Page 107
9.1.1 Some Tentatives......Page 108
9.1.3 The Infinite Dimensional Case......Page 110
9.2 Mountain Pass Principle......Page 113
Comments and Additional Notes......Page 114
9.II Higher Dimensional Links and the Limiting Case......Page 115
9.III Other MPT Versions with the Limiting Case......Page 116
10.1 (PS), Level Sets, and Coercivity......Page 117
10.2 On the Geometry of the MPT and (PS)......Page 122
10.3 Second-Order Information on (PS) Sequences in the MPT......Page 123
10.I The Concentration Compactness Principle......Page 125
10.II Hartree-Fock Equations for Coulomb Systems......Page 126
10.III Coercivity versus (PS) and Nonsmooth Critical Point Theory......Page 127
11.1 Ljusternik-Schnirelman Theory......Page 128
11.1.1 Index Theory......Page 129
The Krasnoselskii Genus......Page 130
11.2 Symmetric MPT......Page 132
Courant-Fischer Minimax Principle......Page 135
11.3 A Superlinear Problem with Odd Nonlinearity......Page 136
11.4 Inductive Symmetric MPTs......Page 138
11.5 The Fountain Theorem......Page 140
Comments and Additional Notes......Page 142
11.III Instability under Perturbation in the Ljusternik-Schnirelman Theorem......Page 143
11.V A Ljusternik-Schnirelmann Theorem without the Palais-Smale condition......Page 144
11.VI Further Developments of the Symmetric MPT......Page 145
11.VIII The Problem of Stability under Perturbation......Page 146
11.IX Reviews and Survey Papers......Page 147
12 The Structure of the Critical Set in the MPT......Page 148
12.1 Definitions and Terminology......Page 149
Mountain Pass Points......Page 150
12.2 On the Nature of K......Page 151
The Infinite Dimensional Case......Page 153
12.2.1 Fang’s Refinements......Page 154
12.I Morse Index at Critical Points in the MPT......Page 159
12.II Topological Degree at a Mountain Pass Point......Page 160
12.III A Counterexample......Page 161
12.VI A Strong Form of the MPT......Page 162
13 Weighted Palais-Smale Conditions......Page 163
13.1 “Weighted” Palais-Smale Conditions......Page 164
13.2 Changing the Metric......Page 170
13.I A Half MPT with a Loose End......Page 172
13.II Complementary Readings......Page 174
IV The Landscape Becoming Less Smooth......Page 175
14 The Semismooth MPT......Page 177
14.1 Preliminaries......Page 178
14.1.1 An Appropriate Deformation Lemma......Page 179
14.2 Semismooth MPT......Page 182
14.I The Original Proof of the Semismooth MPT......Page 184
14.III Some Applications of Semismooth MPT......Page 186
14.IV Another Semismooth Critical Point Theory......Page 187
15.1 Notions of Nonsmooth Analysis......Page 188
15.2 Nonsmooth MPT......Page 192
15.I Some Variants of Nonsmooth Critical Point Theory......Page 197
15.V A Critical Point Theory for Locally Lipschitz Functionals on Locally Convex Closed Subsets of Banach Spaces......Page 199
16.1.1 Critical Points of Continuous Functions in Metric Spaces......Page 200
16.1.2 Palais-Smale Condition for Continuous Functions......Page 202
16.2 Metric MPT......Page 203
16.II A Specific Subdifferential Calculus......Page 210
16.III More Definitions of Critical Points on Metric Spaces......Page 212
16.IV A Critical Point Theory for l.s.c. Functionals......Page 214
16.V Pseudo-Gradient Vector Field for Continuous Functionals on Metric Spaces......Page 215
16.VII Some Directions Where the Actual Metric Critical Point Theory May Be Extended......Page 216
16.X Some Additional References......Page 217
16.XII Extension of the Critical Point Theory for Continuous Functionals on Metric Spaces to Multivalued Mappings…......Page 218
V Speculating about the Mountain Pass Geometry......Page 219
17.1 Variational Inequalities as Extremal Conditions......Page 221
17.2.1 A First Form Using a Deformation Lemma......Page 222
17.2.2 A Second Form Using Ekeland’s Principle......Page 223
17.I A Minimization Result on Closed Convex Subsets......Page 226
17.II On Amann’s “Three Solution Problem” and the MPT on Convex Domains......Page 227
17.IV MPT Versions on Closed Convex Sets for Some Particular Functions......Page 228
17.VI An MPT for Continuous Convex Functionals......Page 229
18 MPT in Order Intervals......Page 230
18.1.1 Ordering, Cones and Positive Operators......Page 231
18.2 MPT in Order Intervals......Page 232
18.I Variational Results in OBS......Page 236
18.II Krein-Rutman Theorem......Page 238
18.III Some Complementary Information Using Morse Theory......Page 239
Linking in the Sense of Benci and Rabinowitz......Page 240
19.1 The Linking Principle......Page 241
19.2 Linking of Deformation Type......Page 245
19.3 Newer Extensions of the Notion of Linking......Page 247
19.II Saddle Point Theorem of Silva......Page 250
19.III Isotopic Linking......Page 251
19.VI Generalized Linking Theorem......Page 252
20 The Intrinsic MPT......Page 254
20.1 The Intrinsic MPT......Page 255
20.2 A Metric Extension......Page 256
20.I A Version of the Intrinsic MPT with No Regularity Assumptions at All......Page 260
21 Geometrically Constrained MPT......Page 262
21.1 A Bounded MPT without the (PS) Condition......Page 263
21.2 Mountain Impasse Theorem......Page 264
21.I Mountain Impasse Theorem for Strictly Differentiable Functionals......Page 268
VI Technical Climbs......Page 271
22.1 The Numerical Mountain Pass Algorithm......Page 273
22.1.1 Steepest Descent Direction......Page 274
But Practically, How Is It Implemented?......Page 275
22.1.2 The Numerical Algorithm......Page 277
22.2 A Partially Interactive Algorithm......Page 279
22.2.1 Description of the Algorithm......Page 280
Part I. Interactive Search for the Minimax......Page 281
Part II. Minimization of p(U) Using a Gradient Descent......Page 282
Description of the Algorithm......Page 283
Some Choices for the Direction D......Page 284
22.II Numerical PDE Solver......Page 285
22.V The Mountain Pass Algorithm in Various Applications......Page 286
22.VI On the Algorithm of Liotard and Penot......Page 288
23 Perturbation from Symmetry and the MPT......Page 289
23.1 “Standard” Method of Rabinowitz......Page 290
23.2 Homotopy-like MPT......Page 294
23.II A Z-Equivariant Ljusternik-Schnirelman Theory for Noneven Functionals......Page 300
23.III A Critical Point Theory for Nonsymmetric Perturbations of G-Invariant Functionals......Page 301
24.1 Preliminaries......Page 302
24.2 The Ljapunov-Schmidt Reduction......Page 304
24.3 The MPT in Bifurcation Problems......Page 306
Comments and Additional Notes......Page 309
25.I Directionally Constrained MPTs......Page 310
Duc Directionally Constrained MPT......Page 311
Constrained Critical Point Theory......Page 312
The Directionally Constrained MPT of Arcoya and Boccardo......Page 314
25.II Morse-Ekeland Index and the MPT......Page 315
Minimal Period Problem......Page 317
25.III Homological and Homotopical Linking......Page 318
25.V The MPT for Upper Semicontinuous Compact-Valued Mappings......Page 321
25.VI A Critical Point Theory for Multivalued Mappings with Closed Graph......Page 323
25.VII An MPT on Closed Convex Sets for Functionals Satisfying the Schauder Condition......Page 324
25.IX An MPT with a Retractable Property Instead of (PS)......Page 325
25.XII An MPT for Continuous Convex Functionals......Page 326
25.XVI The MPT in Quantum Chemistry......Page 327
25.XVIII Proof of GOD......Page 328
A.1.1 Definitions and Basic Properties......Page 329
A.1.2 Embedding Theorems......Page 331
A.1.4 Generalized and Classical Solutions......Page 332
A.2 Nemytskii Operators......Page 333
A.2.1 Continuity of Nemytskii Operators......Page 334
A.2.2 Potential Nemytskii Operators......Page 335
Bibliography......Page 337
Index......Page 379