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.
|Titolo:||Cantor families of periodic solutions for completely resonant nonlinear wave equations|
|Autori:||M. BERTI; P. BOLLE|
|Rivista:||DUKE MATHEMATICAL JOURNAL|
|Data di pubblicazione:||2006|
|Digital Object Identifier (DOI):||10.1215/S0012-7094-06-13424-5|
|Appare nelle tipologie:||1.1 Journal article|