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E-raamat: Topics in Critical Point Theory

(Florida Institute of Technology), (University of California, Irvine)
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This book introduces the reader to powerful methods of critical point theory and details successful contemporary approaches to many problems, some of which had proved resistant to attack by older methods. Topics covered include Morse theory, critical groups, the minimax principle, various notions of linking, jumping nonlinearities and the Fuík spectrum in an abstract setting, sandwich pairs and the cohomological index. Applications to semilinear elliptic boundary value problems, p-Laplacian problems and anisotropic systems are given. Written for graduate students and research scientists, the book includes numerous examples and presents more recent developments in the subject to bring the reader up to date with the latest research.

Arvustused

'The authors have presented extremely powerful methods in critical point theory. It can be presumed that researchers in these subjects had been awaiting such an excellent source and here they have it. It is undoubtedly an excellent reference for research scientists in mathematics, physics and engineering.' Dhruba Adhikari, MAA Reviews

Muu info

Provides an introduction to critical point theory and shows how it solves many difficult problems.
Preface ix
1 Morse theory
1(29)
1.1 Introduction
1(8)
1.2 Compactness conditions
9(1)
1.3 Deformation lemmas
9(9)
1.4 Critical groups
18(4)
1.5 Minimizers
22(1)
1.6 Nontrivial critical points
22(2)
1.7 Mountain pass points
24(1)
1.8 Three critical points theorem
24(1)
1.9 Generalized local linking
25(1)
1.10 p-Laplacian
25(5)
2 Linking
30(17)
2.1 Introduction
30(1)
2.2 Minimax principle
30(1)
2.3 Homotopical linking
31(2)
2.4 Homological linking
33(3)
2.5 Schechter and Tintarev's notion of linking
36(4)
2.6 Pairs of critical points with nontrivial critical groups
40(2)
2.7 Nonstandard geometries
42(5)
3 Applications to semilinear problems
47(20)
3.1 Introduction
47(1)
3.2 Local nature of critical groups
48(2)
3.3 Critical groups at zero
50(7)
3.4 Asymptotically linear problems
57(6)
3.5 Problems with concave nonlinearities
63(4)
4 Fucik spectrum
67(40)
4.1 Introduction
67(2)
4.2 Examples
69(3)
4.3 Preliminaries on operators
72(2)
4.4 Variational formulation
74(1)
4.5 Some estimates
75(2)
4.6 Convexity and concavity
77(1)
4.7 Minimal and maximal curves
78(15)
4.8 Null manifold
93(7)
4.9 Type II regions
100(1)
4.10 Simple eigenvalues
101(1)
4.11 Critical groups
101(6)
5 Jumping nonlinearities
107(18)
5.1 Introduction
107(2)
5.2 Compactness
109(1)
5.3 Critical groups at infinity
110(5)
5.4 Solvability
115(1)
5.5 Critical groups at zero
116(6)
5.6 Nonlinearities crossing the Fucik spectrum
122(3)
6 Sandwich pairs
125(18)
6.1 Introduction
125(1)
6.2 Flows
125(1)
6.3 Cohomological index
126(5)
6.4 Semilinear problems
131(2)
6.5 p-Laplacian problems
133(3)
6.6 Anisotropic systems
136(7)
Appendix Sobolev spaces
143(4)
A.1 Sobolev inequality
143(1)
A.2 Sobolev spaces
144(3)
Bibliography 147(9)
Index 156
Kanishka Perera is Professor in the Department of Mathematical Sciences at Florida Institute of Technology. Martin Schechter is Professor in the Department of Mathematics at the University of California, Irvine.
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