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.

Cantor families of periodic solutions for wave equations via a variational principle / Berti, M.; Bolle, P.. - In: ADVANCES IN MATHEMATICS. - ISSN 0001-8708. - 217:4(2008), pp. 1671-1727. [10.1016/j.aim.2007.11.004]

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

Berti, M.;
2008-01-01

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.
2008
217
4
1671
1727
https://www.sciencedirect.com/science/article/pii/S0001870807003118?via=ihub
Berti, M.; Bolle, P.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/12497
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