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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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