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Selected new aspects of the calculus of variations in the large. (English) Zbl 1064.35054
The authors discuss several topics in the calculus of variations. General principles are illustrated by examples. Semilinear elliptic PDEs, Hamiltonian systems and symplectic topology, second order systems, closed geodesics problems and Hartree-Fock equations are discussed. This very nice paper should be read by all those who are interested in the calculus of variations. However, the reader should not forget that the authors had to be selective. Some important developments are omitted, like for instance the fundamental work of Schoen on semilinear elliptic equations of critical Sobolev growth.

MSC:
35J60 Nonlinear elliptic equations
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
49J27 Existence theories for problems in abstract spaces
35J20 Variational methods for second-order elliptic equations
58E30 Variational principles in infinite-dimensional spaces
47J30 Variational methods involving nonlinear operators
57R17 Symplectic and contact topology in high or arbitrary dimension
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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