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

TL;DR

This paper investigates the asymptotic behavior of convex solutions to the Monge-Ampère equation in half spaces, showing they approach quadratic polynomials at infinity with a specific rate under certain boundary and growth conditions.

Contribution

It establishes the asymptotic quadratic polynomial behavior of solutions in half spaces with precise decay rates, extending understanding of Monge-Ampère solutions at infinity.

Findings

Solutions tend to quadratic polynomials at infinity.

The rate of convergence is at least rac{x_n}{|x|^n}.

Results apply to solutions with quadratic boundary conditions and growth bounds.

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 $0<\mu\leq \frac{1}{2}$.