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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
; | | Date: |
27 Jul 2001 | | Journal: | J. Differential Equations 188 (2003) 569-590 | | Subject: | Functional Analysis MSC-class: Primary 34L30, 58B05, Secondary 34B15, 46T05 | math.FA | | Abstract: | 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 $f: RR o RR$ 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. | | Source: | arXiv, math.FA/0107197 | | Services: | Forum | Review | PDF | Favorites | |
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