00-6 Bambusi, D.

Lyapunov Center Theorem for some Nonlinear PDE's: a simple proof (195K, ps) Jan 4, 00

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Abstract. We give a simple proof of existence of small oscillations in some nonlinear partial differential equations. The proof is based on the Lyapunov--Schmidt decomposition and the contraction mapping principle; the linear frequencies $\omega_j$ are assumed to satisfy a Diophantine type nonresonance condition (of the kind of the first Melnikov condition) slightly stronger than the usual one. If $\omega_j\sim j^d$ with $d>1$, such Diophantine condition will be proved to have full measure in a sense specified below; if $d=1$, we will prove that the condition is satisfied in a set of zero measure. Applications to nonlinear beam equations and to nonlinear wave equations with Dirichlet boundary condition are given. The result also applies to more general systems and boundary conditions (e.g. periodic).

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