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

Burghelea, Dan; Saldanha, Nicolau C.; Tomei, Carlos

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

Authors

Burghelea, Dan

Saldanha, Nicolau C.

Tomei, Carlos

Description

We consider the nonlinear Sturm-Liouville differential operator $F(u) = -u'' + f(u)$ for $u \in H^2_D([0, π])$, 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