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Variational and non-variational methods in nonlinear analysis and boundary value problems. (English) Zbl 1040.49001
Nonconvex Optimization and Its Applications 67. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1385-X/hbk). xii, 375 p. (2003).
The book presents some modern topics in nonlinear analysis such as multivalued elliptic problems with discontinuities, variational inequalities, hemivariational inequalities, evolution problems and applications of these theories to some classes of boundary value problems. The treatment relies on variational methods, monotonicity principles, topological arguments and optimization techniques. More precisely, the authors deal with the following topics:
– nonsmooth critical point theories due to Clarke and Degiovanni, Goeleven-Motreanu-Panagiotopoulos, Szulkin;
– abstract multiplicity theorem of Lusternik-Schnirelman type and a Schrödinger type equation with lack of compactness;
– extremal solutions for initial boundary value problems of parabolic type involving Clarke’s gradient;
– boundary value problems expressed by variational, hemivariational and variational-hemivariational inequalities;
– eigenvalue problems with symmetries;
– non-symmetric perturbations of symmetric eigenvalue problems;
– the location of critical points for nonsmooth functionals;
– first order evolution quasivariational inequalities and second order nonlinear evolution equations;
– stability results regarding variational inequalities;
– variational approach involving a functional not satisfying the Palais-Smale condition.

MSC:
49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
49Q20 Variational problems in a geometric measure-theoretic setting
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
49J40 Variational inequalities
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