A Remark on Monge-Ampère equation over exterior domains

Abstract

We improve the result of Caffarelli-Li [CL03] on the asymptotic behavior at infinity of the exterior solution $u$ to Monge-Ampère equation $det(D^{2}u)=1$ on ${\mathbb{R}}^n\backslash K$ for $n\ge 3$. We prove that the error term $O(|x{|}^{2-n})$ can be refined to $d(\sqrt{x^{\prime }Ax}{)}^{2-n}+O(|x{|}^{1-n})$ with $d=Res[u]$ the residue of $u$.

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