Results on infinite dimensional topology and applications to the structure of the critical set of nonlinear Sturm-Liouville operators
Dan Burghelea, Nicolau C. Saldanha, Carlos Tomei
Abstract
We consider the nonlinear Sturm-Liouville differential operator F(u) = -u'' + f(u) for u ∈ H2D([0, π]), a Sobolev space of functions satisfying Dirichlet boundary conditions. For a generic nonlinearity f: 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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