Results on infinite-dimensional topology and applications to the structure of the critical set of nonlinear Sturm-Liouville operators |
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Authors: | Dan Burghelea Carlos Tomei |
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Affiliation: | a Department of Mathematics, Ohio State University, 231 West 18th Ave, Columbus, OH 43210-1174, USA b Departamento de Matemática, PUC-Rio R. Marquês de S. Vicente 225, Rio de Janeiro, RJ 22453-900, Brazil |
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Abstract: | We consider the nonlinear Sturm-Liouville differential operator F(u)=−u″+f(u) for u∈HD2([0,π]), a Sobolev space of functions satisfying Dirichlet boundary conditions. For a generic nonlinearity 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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Keywords: | primary 34L30 58B05 secondary 34B15 46T05 |
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