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The Mountain Pass Theorem: Variants, Generalizations and Some Applicationsby Youssef Jabri
Synopses & Reviews
Joussef Jabri presents min-max methods through a comprehensive study of the different faces of the celebrated Mountain Pass Theorem (MPT) of Ambrosetti and Rabinowitz. Jabri clarifies the extensions and variants of the MPT in a complete and unified way and covers standard topics: the classical and dual MPT; second-order information from PS sequences; symmetry and topological index theory; perturbations from symmetry; convexity and more. He also covers the non-smooth MPT; the geometrically constrained MPT; numerical approaches to the MPT; and even more exotic variants. A bibliography and detailed index are also included.
This 2003 book presents min-max methods through a study of the different faces of the Mountain Pass Theorem of Ambrosetti and Rabinowitz.
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.
Includes bibliographical references (p. 323-365) and index.
Table of Contents
1. Retrospective; Part I. First Steps Toward the Mountains: 2. Palais-Smale condition. Definitions and examples; 3. Variational principle; 4. Deformation lemma; Part II. Reaching the Mountain Pass Through Easy Climbs: 5. The finite dimensional MPT; 6. The topological MPT; 7. The classical MPT; 8. The multidimensional MPT; Part III. A Deeper Insight in Mountain Topology: 9. The limiting case in the MPT; 10. Palais-Smale condition versus asymptotic behavior; 11. Symmetry and the MPT; 12. The structure of the critical set in the MPT; 13. Weighted Palais-Smale conditions; Part IV. The Landscape Becoming Less Smooth: 14. The semismooth MPT; 15. The nonsmooth MPT; 16. The metric MPT; Part V. Speculating about the Mountain Pass Geometry: 17. The MPT on convex domains; 18. A MPT in order intervals; 19. The linking principle; 20. The intrinsic MPT; 21. Geometrically contrained MPT; Part VI. Technical Climbs: 22. Numerical MPT implementations; 23. Perturbation from symmetry and the MPT; 24. Applying the MPT in bifurcation problems; 25. More climbs; A. Background material.
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