Duality and Perturbation Methods in Critical Point Theory

by N. Ghoussoub

• ISBN13: 9780521071956
• ISBN10: 052107195X
• Edition: Revised
• Format: Paperback
• Copyright: 2008-08-14
• Publisher: Cambridge University Press

Summary

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.

Table of Contents

Lipschitz and smooth perturbed minimization principles

Linear and plurisubharmonic perturbed minimization principles

The classical min-max theorem

A strong form of the min-max principle

Relaxed boundary conditions in the presence of a dual set

The critical set in the mountain pass theorem

Group actions and multiplicity of critical points

The Palais-Smale condition around a dual set - examples

Morse indices of min-max critical points - the non-degenerate case

Morse indices of min-max critical points - the degenerate case

Morse-type information on Palais-Smale sequences

Appendix

References

Index

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