Asymptotic behavior at infinity of solutions of Monge-Amp\`ere equations in half spaces

Abstract

We prove that any convex viscosity solution of $\det D^2u=1 $ outside a bounded domain of $\mathbb{R}^n_+$ tends to a quadratic polynomial at infinity with rate at least $\frac{x_n}{|x|^{n}}$ if $u$ is a quadratic polynomial on $\{x_n=0\}$ and satisfies $ \mu|x|^2\leq u\leq \mu^{-1}|x|^2$ as $|x|\rightarrow \infty$ for some $0<\mu\leq \frac{1}{2}$.