Global solutions and exterior Dirichlet problem for Monge-Ampere equation in ${\mathbb{R}}^2$

Abstract

Monge-Ampère equation $\det (D^{2}u)=f$ in two dimensional spaces is different in nature from their counterparts in higher dimensional spaces. In this article we employ new ideas to establish two main results for the Monge-Ampère equation defined either globally in ${\mathbb{R}}^2$ or outside a convex set. First we prove the existence of a global solution that satisfies a prescribed asymptotic behavior at infinity, if $f$ is asymptotically close to a positive constant. Then we solve the exterior Dirichlet problem if data are given on the boundary of a convex set and at infinity.

17 pages