A Remark on Monge-Amp\`ere 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-Amp\`{e}re equation $det(D^2u)=1$ on $\mathbb{R}^n\backslash K$ for $n\geq 3$. We prove that the error term $O(|x|^{2-n})$ can be refined to $d (\sqrt{x'Ax})^{2-n}+O(|x|^{1-n})$ with $d=Res[u]$ the residue of $u$.