04-38 Guido Gentile, Vieri Mastropietro, Michela Procesi

Periodic solutions for completely resonant nonlinear wave equations (764K, postscript) Feb 16, 04

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Abstract. We consider the nonlinear string equation with Dirichlet boundary conditions $u_{xx}-u_{tt}=\varphi(u)$, with $\varphi(u)= \Phi u^{3} + O(u^{5})$ odd and analytic, $\Phi\neq0$, 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 nonresonant 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 $u_{xx}-u_{tt}+ M u = \varphi(u)$, $M\neq0$, is that not only the P equation but also the Q equation is infinite-dimensional.

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