The first boundary value problem for Abreu's equation

Abstract

In this paper, we prove the existence and regularity of solutions to the first boundary value problem for Abreu's equation, which is a fourth-order nonlinear partial differential equation closely related to the Monge-Ampre equation. The first boundary value problem can be formulated as a variational problem for the energy functional. The existence and uniqueness of maximizers can be obtained by the concavity of the functional. The main ingredients of the paper are the a priori estimates and an approximation result, which enable us to prove that the maximizer is smooth in dimension 2.