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

摘要

In this paper, we consider the two dimensional completely resonant beam equation with cubic nonlinearity on T-2. We prove the existence of the quasi-periodic solutions, which lie in a special subspace of L-2(T-2). 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.