Duality and Perturbation Methods in Critical Point Theory
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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 earlier this century, and these are still being refined to overcome difficulties when applied to the theory of partial differential equations. Here, Professor Ghoussoub describes a new 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 a...
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 earlier this century, and these are still being refined to overcome difficulties when applied to the theory of partial differential equations. Here, Professor Ghoussoub describes a new 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 new 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:
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.
Professor Ghoussoub describes a new, general, approach to critical point theory. Building upon min-max methods, he also presents a whole new array of duality and perturbation methods. The book is reasonably self-contained: consequently, it should be accessible to all mathematicians, economists and engineers working in nonlinear analysis or optimization.
Describes a new approach to critical point theory and presents a whole new array of duality and perturbation methods.
Table of contents:
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.
Professor Ghoussoub describes a new, general, approach to critical point theory. Building upon min-max methods, he also presents a whole new array of duality and perturbation methods. The book is reasonably self-contained: consequently, it should be accessible to all mathematicians, economists and engineers working in nonlinear analysis or optimization.
Describes a new approach to critical point theory and presents a whole new array of duality and perturbation methods.
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