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

## TL;DR

This paper develops an infinite-dimensional KAM theorem and applies it to prove the existence of small amplitude quasi-periodic solutions for a resonant beam equation on a two-dimensional torus.

## Contribution

It introduces a new abstract KAM theorem for infinite dimensions and demonstrates its application to a specific resonant PDE, expanding the understanding of quasi-periodic solutions.

## Key findings

- Existence of Whitney smooth small amplitude quasi-periodic solutions.
- Application of the KAM theorem to a 2D resonant beam equation.
- Construction of solutions on finite-dimensional tori.

## Abstract

In this paper we consider the completely resonant beam equation on \T^2 with cubic nonlinearity on a subspace of L^2 (\T^2) which will be explained later. We establish an abstract infinite dimensional KAM theorem and apply it to the completely resonant beam equation. We prove the existence of a class of Whitney smooth small amplitude quasi-periodic solutions corresponding to finite dimensional tori.