We prove the existence of small amplitude, (2π/ω)-periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for any frequency ω belonging to a Cantor-like set of asymptotically full 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 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 us to deal also with nonlinearities that are not odd and with finite spatial regularity.
Cantor families of periodic solutions for completely resonant nonlinear wave equations / Berti, M.; Bolle, P.. - In: DUKE MATHEMATICAL JOURNAL. - ISSN 0012-7094. - 134:2(2006), pp. 359-419. [10.1215/S0012-7094-06-13424-5]
Cantor families of periodic solutions for completely resonant nonlinear wave equations
Berti, M.;
2006-01-01
Abstract
We prove the existence of small amplitude, (2π/ω)-periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for any frequency ω belonging to a Cantor-like set of asymptotically full 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 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 us to deal also with nonlinearities that are not odd and with finite spatial regularity.File | Dimensione | Formato | |
---|---|---|---|
Berti-DUKE.pdf
non disponibili
Tipologia:
Versione Editoriale (PDF)
Licenza:
Non specificato
Dimensione
579.92 kB
Formato
Adobe PDF
|
579.92 kB | Adobe PDF | Visualizza/Apri Richiedi una copia |
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.