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FM 2004-02 (mp_arc 04-38; arXiv math.DS/0402262)

Author: Guido Gentile, Vieri Mastropietro and Michela Procesi

Title: Periodic solutions for completely resonant nonlinear wave equations
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Abstract: We consider the nonlinear string equation, with Dirichlet boundary conditions and with an odd and analytic nonlinear term, and we construct small amplitude periodic solutions with frequency for a large Lebesgue measure set of frequencies close to 1. This extends previous results where only a zero-measure set of frequencies could be treated (the ones for which no small divisors appear). The proof is based on combining the Lyapunov-Schmidt decomposition, which leads to two separate sets of equations dealing with the resonant and non-resonant Fourier components, respectively the Q and the P equations, with resummation techniques of divergent powers series, allowing us to control the small divisors problem. The main difficulty with respect the nonlinear wave equations with a mass term is that not only the P equation but also the Q equation is infinite-dimensional.

Keywords: Nonlinear wave equation; String equation; Periodic solutions; Lindstedt series method; Tree formalism; Lyapunov-Schmidt decomposition; Counterterms; Renormalization Group; Diophantine conditions.

Guido Gentile
Dipartimento di Matematica
UniversitÓ di Roma Tre
Largo San Leonardo Murialdo 1, 00146 Roma, Italy
e-mail: gentile@mat.uniroma3.it

Vieri Mastropietro
Dipartimento di Matematica
UniversitÓ di Roma ``Tor Vergata"
Via della Ricerca Scientifica, Roma, Italy
e-mail: mastropi@mat.uniroma2.it

Michela Procesi
SISSA
Trieste, I-34014, Italy
e-mail: procesi@ma.sissa.it