%0 Journal Article
%T Results on infinite dimensional topology and applications to the structure of the critical set of nonlinear Sturm-Liouville operators
%A Dan Burghelea
%A Nicolau C. Saldanha
%A Carlos Tomei
%J Mathematics
%D 2001
%I arXiv
%X We consider the nonlinear Sturm-Liouville differential operator $F(u) = -u'' + f(u)$ for $u \in H^2_D([0, \pi])$, a Sobolev space of functions satisfying Dirichlet boundary conditions. For a generic nonlinearity $f: \RR \to \RR$ we show that there is a diffeomorphism in the domain of $F$ converting the critical set $C$ of $F$ into a union of isolated parallel hyperplanes. For the proof, we show that the homotopy groups of connected components of $C$ are trivial and prove results which permit to replace homotopy equivalences of systems of infinite dimensional Hilbert manifolds by diffeomorphisms.
%U http://arxiv.org/abs/math/0107197v2