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arXiv · 1304.2415

Monge-Ampere equation on exterior domains

Jiguang Bao·Haigang Li·Lei Zhang
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Abstract

We consider the Monge-Ampère equation $\det(D^2u)=f$ where $f$ is a positive function in $\mathbb R^n$ and $f=1+O(|x|^{-β})$ for some $β>2$ 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 $β>2$ is sharp for all the results in this article.

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Jiguang Bao, Haigang Li, Lei Zhang. 2013-04-08. Monge-Ampere equation on exterior domains. https://arxiv.org/abs/1304.2415

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