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

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

2001-07-27

arxiv.org/abs/math/0107197v2

J. Differential Equations 188 (2003) 569-590

Mathematics

Functional Analysis

23 pages, 7 figures

Scientific paper

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.

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