Existence and nonexistence to exterior Dirichlet problem For Monge-Amp\`ere equation

TL;DR

This paper characterizes when solutions exist or do not exist for the exterior Dirichlet problem of the Monge-Ampère equation, based on their asymptotic behavior, extending previous work by Caffarelli and the first author.

Contribution

It completes the characterization of solution existence and nonexistence for the exterior Dirichlet problem of the Monge-Ampère equation in terms of asymptotic behavior.

Findings

Established criteria for existence of solutions.

Identified conditions leading to nonexistence.

Extended previous theoretical results.

Abstract

We consider the exterior Dirichlet problem for Monge-Amp\`ere equation with prescribed asymptotic behavior. Based on earlier work by Caffarelli and the first named author, we complete the characterization of the existence and nonexistence of solutions in terms of their asymptotic behaviors.