Global Monge-Ampere equation with asymptotically periodic data

TL;DR

This paper proves that convex solutions to the Monge-Ampère equation with asymptotically periodic data resemble a parabola plus a periodic function at infinity in dimensions three and higher.

Contribution

It establishes the asymptotic behavior of solutions to the Monge-Ampère equation with asymptotically periodic data, extending understanding of their structure at infinity.

Findings

Solutions differ from a parabola by a periodic function at infinity

The result holds for dimensions n ≥ 3

The data function f is asymptotically close to a periodic function

Abstract

Let $u$ be a convex solution to $\det(D^2u)=f$ in $\mathbb R^n$ where $f\in C^{1,\alpha}(\mathbb R^n)$ is asymptotically close to a periodic function $f_p$. We prove that the difference between $u$ and a parabola is asymptotically close to a periodic function at infinity, for dimension $n\ge 3$.