Results on infinite dimensional topology and applications to the structure of the critical set of nonlinear Sturm-Liouville operators

Abstract

We consider the nonlinear Sturm-Liouville differential operator $F(u)=-u^{\prime \prime }+f(u)$ for $u\in H_D^2([0,\pi ])$, 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.

23 pages, 7 figures