Exploiting Equivariance in the Reduced Bifurcation Equations

Abstract

Our aim here is to study and exploit equivariance properties of the reduced bifurcation equations for a class of elliptic boundary value problems. To illustrate the approach we consider specifically the problem

$$ G{\text{(}}u,\lambda ): = \Delta u + \lambda f{\text{(}}u{\text{)}} = {\text{0}} $$

((1.1))

with doubly periodic boundary conditions

$$ u{\text{(}}x + {\text{1}},y) = u{\text{(}}x,y{\text{ + 1) = }}u{\text{(}}x,y{\text{)for (}}x,y{\text{)}} \in R^2 $$

((1.2))

.

Partially supported by NSF Grant DMS-9104058.

Supported by Deutsch Forschungsgemeinschaft.