The first boundary value problem for Abreu's equation

TL;DR

This paper establishes existence and regularity results for the first boundary value problem of Abreu's equation, a complex nonlinear PDE related to the Monge-Ampere equation, using variational methods and a priori estimates.

Contribution

It provides the first proof of existence and regularity for solutions to this boundary value problem for Abreu's equation, including smoothness in two dimensions.

Findings

Existence and uniqueness of solutions via variational methods.

Solutions are smooth in dimension 2.

Development of a priori estimates for the nonlinear PDE.

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-Ampere 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.