04-342 M. Berti, P. Bolle

Cantor families of periodic solutions for completely resonant nonlinear wave equations (264K, dvi ) Oct 29, 04

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Abstract. We prove existence of small amplititude $2 \pi/omega$ periodic solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for any frequency $\omega$ belinging 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 proble to a finite dimensional bifurcation equation. Moreover, a new simple approach for the inversion of the linearized operators required by the Nash-Moser approach is developed. It allows to deal also with nonlinearities which are not off and with finite spatial regularity.

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