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Cantor families of periodic solutions for wave equations via a variational principle

TitleCantor families of periodic solutions for wave equations via a variational principle
Publication TypeJournal Article
Year of Publication2008
AuthorsBerti, M, Bolle, P
JournalAdvances in Mathematics
Volume217
Pagination1671-1727
ISSN00018708
Abstract

We prove existence of small amplitude periodic solutions of completely resonant wave equations with frequencies in a Cantor set of asymptotically full measure, via a variational principle. A Lyapunov-Schmidt decomposition reduces the problem to a finite dimensional bifurcation equation-variational in nature-defined on a Cantor set of non-resonant parameters. The Cantor gaps are due to "small divisors" phenomena. To solve the bifurcation equation we develop a suitable variational method. In particular, we do not require the typical "Arnold non-degeneracy condition" of the known theory on the nonlinear terms. As a consequence our existence results hold for new generic sets of nonlinearities. © 2007 Elsevier Inc. All rights reserved.

DOI10.1016/j.aim.2007.11.004

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