 0438 Guido Gentile, Vieri Mastropietro, Michela Procesi
 Periodic solutions for completely resonant nonlinear wave equations
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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 zeromeasure set of frequencies
could be treated (the ones for which no small divisors appear).
The proof is based on combining the LyapunovSchmidt 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 infinitedimensional.
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