Giuseppe Gaeta
Reduction and Equivariant Branching Lemma
without finite-dimensional reduction
(69K, Plain TeX)
ABSTRACT. In the bifurcation study of nonlinear
evolution PDEs with symmetry, one usually performs first a
reduction to a finite dimensional critical space, thus
obtaining the bifurcation equation (which inherits
symmetries properties from the original problem), and then
employs the symmetry -- tipically through the reduction
lemma and/or the equivariant branching lemma -- to study
this reduced problem. We argue that one could as well
proceed in the opposite way, i.e. apply bifurcation
analysis on a symmetry-reduced problem; this is done using
some general results of Palais for variational analysis of
$G$-invariant functionals. Such an approach presents some
delicate points, which we discuss in detail.