The Mountain Pass Theorem: Variants, Generalizations and Some Applications
Cambridge University Press, 15.09.2003
This 2003 book presents min-max methods through a study of the different faces of the celebrated Mountain Pass Theorem (MPT) of Ambrosetti and Rabinowitz. The reader is led from the most accessible results to the forefront of the theory, and at each step in this walk between the hills, the author presents the extensions and variants of the MPT in a complete and unified way. Coverage includes standard topics, but it also covers other topics covered nowhere else in book form: the non-smooth MPT; the geometrically constrained MPT; numerical approaches to the MPT; and even more exotic variants. Each chapter has a section with supplementary comments and bibliographical notes, and there is a rich bibliography and a detailed index to aid the reader. The book is suitable for researchers and graduate students. Nevertheless, the style and the choice of the material make it accessible to all newcomers to the field.
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Definitions and Examples
Obtaining Almost Critical Points Variational Principle
Obtaining Almost Critical Points The Deformation Lemma
The Finite Dimensional MPT
The Topological MPT
The Classical MPT
The Multidimensional MPT
The Metric MPT
The MPT on Convex Domains
MPT in Order Intervals
The Linking Principle
The Intrinsic MPT
Geometrically Constrained MPT
Numerical MPT Implementations
Perturbation from Symmetry and the MPT
The Limiting Case in the MPT
PalaisSmale Condition versus Asymptotic Behavior
Symmetry and the MPT
The Structure of the Critical Set in the MPT
Weighted PalaisSmale Conditions
The Semismooth MPT
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Ambrosetti applications Banach space boundary value problems bounded Brezis Chapter class C1 closed convex closed subset Consider continuous function contradiction convergent subsequence convex set Corollary Corvellec critical point theory critical set critical value definition Degiovanni and Marzocchi Denote differential equations Dirichlet problem dist eigenvalue Ekeland's principle Ekeland's variational principle elliptic problems example exists finite dimensional MPT Hence Hilbert space Hofer implies inequality inf max inf sup linear Lipschitz continuous Lipschitz functionals Ljusternik-Schnirelman locallv locally Lipschitz functionals mapping Math maximum metric space minimal minimax minimax theorems minimum Moreover Morse Morse index mountain pass point mountain pass theorem multiplicity neighborhood nonempty nonsmooth notion obtained Palais-Smale condition path perturbation proof Proposition proved pseudo-gradient vector field quantitative deformation lemma Rabinowitz saddle point satisfies PS)C Schechter sequence Sobolev Sobolev spaces Suppose symmetric MPT topological variational methods Willem
Seite 325 - A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann.
Seite 327 - V. Benci, A geometrical index for the group Sl and some applications to the study of periodic solutions of ordinary differential equations, Comm. Pure Appl. Math. 34 (1981), 393-432.