Periodic solutions of completely resonant nonlinear wave equations.

We consider the nonlinear string equation with Dirichlet boundary conditions utt-uxx=phi v(u), with phi v(u)=Phi u3+O(u5) odd and analytic, Phi>0, and we construct small amplitude periodic solutions with frequency omega for a large Lebesgue measure set of omega 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 to the nonlinear wave equations utt-uxx+Mu=phi v(u), M different from 0, is that not only the P equation but also the Q equation is infinite-dimensional