Monge-Ampere equation on exterior domains

Abstract

We consider the Monge-Ampère equation $\det (D^{2}u)=f$ where $f$ is a positive function in ${\mathbb{R}}^n$ and $f=1+O(|x{|}^{-\beta })$ for some $\beta >2$ at infinity. If the equation is globally defined on ${\mathbb{R}}^n$ we classify the asymptotic behavior of solutions at infinity. If the equation is defined outside a convex bounded set we solve the corresponding exterior Dirichlet problem. Finally we prove for $n\ge 3$ the existence of global solutions with prescribed asymptotic behavior at infinity. The assumption $\beta >2$ is sharp for all the results in this article.

26 pages

References in corpus (1)

• A localization property at the boundary for the Monge-Ampere equation