Duality and Perturbation Methods in Critical Point Theory

N. Ghoussoub (University of British Columbia, Vancouver)

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English

Cambridge University Press
14 August 2008

Calculus & mathematical analysis; Analytic topology

Series: Cambridge Tracts in Mathematics

The calculus of variations has been an active area of mathematics for over 300 years. Its main use is to find stable critical points of functions for the solution of problems. To find unstable values, new approaches (Morse theory and min-max methods) were developed, and these are still being refined to overcome difficulties when applied to the theory of partial differential equations. Here, Professor Ghoussoub describes a point of view that may help when dealing with such problems. Building upon min-max methods, he systematically develops a general theory that can be applied in a variety of situations. In so doing he also presents a whole array of duality and perturbation methods. The prerequisites for following this book are relatively few; an appendix sketching certain methods in analysis makes the book reasonably self-contained. Consequently, it should be accessible to all mathematicians, pure or applied, economists and engineers working in nonlinear analysis or optimization.

By: N. Ghoussoub (University of British Columbia Vancouver)
Imprint: Cambridge University Press
Country of Publication: United Kingdom
Volume: 107
Dimensions: Height: 228mm, Width: 151mm, Spine: 18mm
Weight: 440g
ISBN: 9780521071956
ISBN 10: 052107195X
Series: Cambridge Tracts in Mathematics
Pages: 280
Publication Date: 14 August 2008
Audience: Professional and scholarly , Undergraduate
Format: Paperback
Publisher's Status: Active

1. Lipschitz and smooth perturbed minimization principles; 2. Linear and plurisubharmonic perturbed minimization principles; 3. The classical min-max theorem; 4. A strong form of the min-max principle; 5. Relaxed boundary conditions in the presence of a dual set; 6. The critical set in the mountain pass theorem; 7. Group actions and multiplicity of critical points; 8. The Palais–Smale condition around a dual set - examples; 9. Morse indices of min-max critical points - the non-degenerate case; 10. Morse indices of min-max critical points - the degenerate case; 11. Morse-type information on Palais–Smale sequences; Appendix; References; Index.

Reviews for Duality and Perturbation Methods in Critical Point Theory

The excellent material presentation of this small book follows the tradition of the series. Jean Mawhin, SIAM Review ...I recommend Ghoussoub's book to anyone working on variational problems, in particular on LS-theory. The reader will find a systematic presentation and detailed exposition of useful techniques. T. Bartsch, Bulletin of the American Mathematical Society ...Each chapter ends with some historical notes and comments, which offer a very useful guide through the literature on the subjects mentioned. Maria Letizia Bertotti, Mathematical Reviews