Topological Methods for Variational Problems with Symmetries

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.