Monge-Ampere equation on exterior domains

TL;DR

This paper studies the Monge-Ampère equation on exterior domains, classifying solutions' asymptotics, solving exterior Dirichlet problems, and proving existence of solutions with prescribed behavior at infinity for dimensions three and higher.

Contribution

It provides a comprehensive analysis of solutions to the Monge-Ampère equation on exterior domains, including classification, existence, and boundary value problems, with sharp decay conditions.

Findings

Classified asymptotic behavior of solutions at infinity.

Solved exterior Dirichlet problems for the Monge-Ampère equation.

Proved existence of solutions with prescribed asymptotics in dimensions n ≥ 3.

Abstract

We consider the Monge-Amp\`ere equation $\det(D^2u)=f$ 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.