Schechter Minimax Systems and Critical Point Theory

1. Auflage 2009
ISBN: 978-0-8176-4902-9
Verlag: Birkhäuser Boston
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 242 Seiten

This text starts at the foundations of the field, and is accessible with some background in functional analysis. As such, the book is ideal for classroom of self study. The new material covered also makes this book a must read for researchers in the theory of critical points.

Autoren/Hrsg.

Schechter, Martin

Inhaltsverzeichnis

1;Contents;6
2;Preface;10
3;Critical Points of Functionals;14
3.1;1.1 Introduction;14
3.2;1.2 Extrema;15
3.3;1.3 PalaisÒSmale sequences;15
3.4;1.4 Cerami sequences;16
3.5;1.5 Linking sets;16
3.6;1.6 Previous definitions of linking;17
3.7;1.7 Notes and remarks;18
4;Minimax Systems;19
4.1;2.1 Introduction;19
4.2;2.2 Definitions and theorems;20
4.3;2.3 Linking subsets;21
4.4;2.4 A variation;23
4.5;2.5 Weaker conditions;24
4.6;2.6 Some consequences;25
4.7;2.7 Notes and remarks;27
5;Examples of Minimax Systems;28
5.1;3.1 Introduction;28
5.2;3.2 A method using homeomorphisms;28
5.3;3.3 A method using metric spaces;30
5.4;3.4 A method using homotopy-stable families;30
5.5;3.5 Examples of linking sets;32
5.6;3.6 Various geometries;35
5.7;3.7 A sandwich theorem;37
5.8;3.8 Notes and remarks;40
6;Ordinary Differential Equations;41
6.1;4.1 Extensions of PicardÌs theorem;41
6.2;4.2 Estimating solutions;43
6.3;4.3 Extending solutions;44
6.4;4.4 The proofs;45
6.5;4.5 An important estimate;46
7;The Method Using Flows;48
7.1;5.1 Introduction;48
7.2;5.2 Theorem 2.4;48
7.3;5.3 Theorem 2.12;50
7.4;5.4 Theorem 2.14;53
7.5;5.5 Theorem 2.21;54
8;Finding Linking Sets;60
8.1;6.1 Introduction;60
8.2;6.2 The strong case;61
8.3;6.3 The remaining proofs;63
8.4;6.4 Notes and remarks;65
9;Sandwich Pairs;66
9.1;7.1 Introduction;66
9.2;7.2 Criteria;67
9.3;7.3 Notes and remarks;70
10;Semilinear Problems;71
10.1;8.1 Introduction;71
10.2;8.2 Bounded domains;71
10.3;8.3 Some useful quantities;77
10.4;8.4 Unbounded domains;79
10.5;8.5 Further applications;83
10.6;8.6 Special cases;88
10.7;8.7 The proofs;89
10.8;8.8 Notes and remarks;91
11;Superlinear Problems;92
11.1;9.1 Introduction;92
11.2;9.2 The main theorems;93
11.3;9.3 Preliminaries;95
11.4;9.4 Proofs;96
11.5;9.5 The parameter problem;99
11.6;9.6 The monotonicity trick;104
11.7;9.7 Notes and remarks;111
12;Weak Linking;113
12.1;10.1 Introduction;113
12.2;10.2 Another norm;114
12.3;10.3 Some examples;118
12.4;10.4 Some applications;120
12.5;10.5 Notes and remarks;132
13;Fucik Spectrum: Resonance;134
13.1;11.1 Introduction;134
13.2;11.2 The curves;136
13.3;11.3 Existence;140
13.4;11.4 Notes and remarks;144
14;Rotationally Invariant Solutions;145
14.1;12.1 Introduction;145
14.2;12.2 The spectrum of the linear operator;146
14.3;12.3 The nonlinear case;148
14.4;12.4 Notes and remarks;151
15;Semilinear Wave Equations;152
15.1;13.1 Introduction;152
15.2;13.2 Convexity and lower semi-continuity;152
15.3;13.3 Existence of saddle points;155
15.4;13.4 Criteria for convexity;158
15.5;13.5 Partial derivatives;159
15.6;13.6 The theorems;161
15.7;13.7 The proofs;161
15.8;13.8 Notes and remarks;164
16;Type (II) Regions;165
16.1;14.1 Introduction;165
16.2;14.2 The asymptotic equation;168
16.3;14.3 Local estimates;170
16.4;14.4 The solutions;174
16.5;14.5 Notes and remarks;177
17;Weak Sandwich Pairs;178
17.1;15.1 Introduction;178
17.2;15.2 Weak sandwich pairs;179
17.3;15.3 Applications;185
17.4;15.4 Notes and remarks;191
18;Multiple Solutions;192
18.1;16.1 Introduction;192
18.2;16.2 Two examples;192
18.3;16.3 Statement of the theorems;193
18.4;16.4 Some lemmas;195
18.5;16.5 Local linking;207
18.6;16.6 The proofs;209
18.7;16.7 Notes and remarks;210
19;Second-Order Periodic Systems;213
19.1;17.1 Introduction;213
19.2;17.2 Proofs of the theorems;215
19.3;17.3 Nonconstant solutions;221
19.4;17.4 Notes and remarks;226
20;Bibliography;228
21;Index;239