  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.
