Free boundary Monge-Ampère equations

Abstract

In this paper, we consider a class of Monge-Ampère equations in a free boundary domain of ℝ2 where the value of the unknown function is prescribed on the free boundary. From a variational point of view, these equations describe an optimal transport problem from an a priori undetermined source domain to a prescribed target domain. We prove the existence and uniqueness of a variational solution to these Monge-Ampère equations under a singularity condition on the density function on the source domain. Furthermore, we provide regularity results under some conditions on the prescribed domain.