The Mountain Pass Theorem
Variants, Generalizations and Some Applications
Résumé
Variational methods are very powerful techniques in
nonlinear analysis and are extensively used. They are
extensively used in many disciplines of pure and applied
mathematics (including ordinary and partial differential
equations, mathematical physics, gauge theory, geometrical
analysis). This book presents min-max methods through a
comprehensive study of the different faces of the
celebrated Mountain Pass Theorem (MPT) of Ambrosetti and
Rabinowitz. The reader is gently 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: the classical and dual
MPT; second-order information from PS sequences; symmetry
and topological index theory; perturbations from symmetry;
convexity and more. 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.
Contents
- Retrospective
- Part I. First Steps Toward the Mountains:
- Palais-Smale condition. Definitions and examples
- Variational principle
- Deformation lemma
- Part II. Reaching the Mountain Pass Through Easy
Climbs: 5. The finite dimensional MPT
- The topological MPT
- The classical MPT
- The multidimensional MPT
- Part III. A Deeper Insight in Mountain Topology:
- The limiting case in the MPT
- Palais-Smale condition versus asymptotic behavior
- Symmetry and the MPT
- The structure of the critical set in the MPT
- Weighted Palais-Smale conditions
- Part IV. The Landscape Becoming Less Smooth:
- The semismooth MPT
- The nonsmooth MPT
- The metric MPT
- Part V. Speculating about the Mountain Pass Geometry:
17. The MPT on convex domains
- A MPT in order intervals
- The linking principle
- The intrinsic MPT
- Geometrically contrained MPT
- Part VI. Technical Climbs:
- Numerical MPT implementations
- Perturbation from symmetry and the MPT
- Applying the MPT in bifurcation problems
- More climbs
- A. Background material.
Caractéristiques techniques
| PAPIER | |
| Éditeur(s) | Cambridge University Press |
| Auteur(s) | Youssef Jabri |
| Parution | 07/11/2003 |
| Nb. de pages | 382 |
| Format | 16 x 24 |
| Couverture | Relié |
| Poids | 650g |
| Intérieur | Noir et Blanc |
| EAN13 | 9780521827218 |
| ISBN13 | 978-0-521-82721-8 |
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