Periodic solutions for completely resonant nonlinear wave equations with Dirichlet boundary conditions

We consider the nonlinear string equation with Dirichlet boundary conditions utt - uxx = φ(u), with φ(u) = Φu3 + O(u5) odd and analytic, Φ ¬= 0, and we construct small amplitude periodic solutions with frequency ω for a large Lebesgue measure set of ω 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 u11 - uxx + Mu = φ(u), M ¬= 0, is that not only the P equation but also the Q equation is infinite-dimensional.