The first boundary value problem for Abreu's equation

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.