**Author:**
Guido Gentile, Vieri Mastropietro and Michela Procesi

**Title:** *
Periodic solutions for completely resonant nonlinear wave equations
*

**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

Universitą di Roma Tre

Largo San Leonardo Murialdo 1, 00146 Roma, Italy

e-mail: gentile@mat.uniroma3.it

Vieri Mastropietro

Dipartimento di Matematica

Universitą 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