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Cantor families of periodic solutions for completely resonant nonlinear wave equations

TitleCantor families of periodic solutions for completely resonant nonlinear wave equations
Publication TypeJournal Article
Year of Publication2006
AuthorsBerti, M, Bolle, P
JournalDuke Math. J. 134 (2006) 359-419
Abstract

We prove the existence of small amplitude, $2\\\\pi \\\\slash \\\\om$-periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions, for any frequency $ \\\\om $ belonging to a Cantor-like set of positive measure and for a new set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser Implicit Function Theorem. In spite of the complete resonance of the equation we show that we can still reduce the problem to a {\\\\it finite} dimensional bifurcation equation. Moreover, a new simple approach for the inversion of the linearized operators required by the Nash-Moser scheme is developed. It allows to deal also with nonlinearities which are not odd and with finite spatial regularity.

URLhttp://hdl.handle.net/1963/2161
DOI10.1215/S0012-7094-06-13424-5

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