Asymptotic behavior at infinity of solutions of Monge-Ampère equations in half spaces

Abstract

We prove that any convex viscosity solution of $\det D^{2}u=1$ 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 $\mu |x{|}^2\le u\le {\mu }^{-1}|x{|}^2$ as $|x|\to \infty$ for some $0<\mu \le \frac{1}{2}$.

26 pages