zbMATH — the first resource for mathematics

Duality and perturbation methods in critical point theory. (English) Zbl 0790.58002
Cambridge Tracts in Mathematics and Mathematical Physics. 107. Cambridge: Cambridge University Press. xviii, 258 p. (1993).
This rich book develops in a systematic way duality and perturbation methods in order to deal with problems in critical point theory. In particular, it deals with problems where neither ‘the usual compactness condition à la Palais-Smale nor the nondegeneracy conditions à la Fredholm’ are satisfied, by presenting various new variational principles. An aim is to make these methods accessible to non-linear analysts. The following list of the chapters may give an impression of the content (within the text, a wealth of examples is worked out; to name just a few: the Hartree-Fock equation for Coulomb systems, infinite- dimensional Hamilton-Jacobi equations, the forced double pendulum, solutions of elliptic equations involving the critical Sobolev exponent…):
Lipschitz and smooth perturbed minimization problems, 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 and the degenerate case, Morse-type information on Palais-Smale sequences \(+ 5\) Appendices on background material.
This impressive research monograph provides an excellent basis for an advanced course or a seminar on problems of nonlinear analysis. However, I doubt that besides mathematicians it is also ‘accessible to economists and engineers’ as claimed in the cover text. It is written in a quite concise style, for example Ekeland’s variational principle is proven on p. 2.

58-02 Research exposition (monographs, survey articles) pertaining to global analysis
49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces