Bambusi, D.
Lyapunov Center Theorem for some Nonlinear PDE's: a simple proof
(195K, ps)
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).