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Topological Methods for Variational Problems with Symmetries
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Symmetry has a strong impact on the number and shape of§solutions to variational problems. This has been observed,§for instance, in the search for periodic solutions of§Hamiltonian systems or of the nonlinear wave equation; when§one is interested in elliptic equations on symmetric domains§or in the corresponding semiflows; and when one is looking§for "special" solutions of these problems.§This book is concerned with Lusternik-Schnirelmann theory§and Morse-Conley theory for group invariant functionals.§These topological methods are developed in detail with new§calculations of the equivariant Lusternik-Schnirelmann§category and versions of the Borsuk-Ulam theorem for very§general classes of symmetry groups. The Morse-Conley theory§is applied to bifurcation problems, in particular to the§bifurcation of steady states and hetero-clinic orbits of§O(3)-symmetric flows; and to the existence of periodic§solutions nearequilibria of symmetric Hamiltonian systems.§Some familiarity with the usualminimax theory and basic§algebraic topology is assumed.
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