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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
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Abstract

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.

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Dan Burghelea, Nicolau C. Saldanha, Carlos Tomei. 2001-10-17. Results on infinite dimensional topology and applications to the structure of the critical set of nonlinear Sturm-Liouville operators. https://arxiv.org/abs/math/0107197

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