An infinite dimensional KAM theorem with application to two dimensional completely resonant beam equation

Abstract

In this paper, we consider the two dimensional completely resonant beam equation with cubic nonlinearity on T2. We prove the existence of the quasi-periodic solutions, which lie in a special subspace of L2(T2). After some transformations, we write the Hamiltonian of the equation as an angle-dependent block-diagonal normal form plus a small perturbation with some regularity. By establishing an abstract KAM (Kolmogorov-Arnold-Moser) theorem, we prove the existence of a class of invariant tori, which implies the existence of a class of small-amplitude quasi-periodic solutions. In each step of the KAM iteration, the measure estimate could be fulfilled by making use of the regularity of the nonlinearity.