Monge-Amp\`ere equation with bounded periodic data

Abstract

We consider the Monge-Amp\`ere equation $\det(D^2u)=f$ in $\mathbb{R}^n$, where $f$ is a positive bounded periodic function. We prove that $u$ must be the sum of a quadratic polynomial and a periodic function. For $f\equiv 1$, this is the classic result by J\"orgens, Calabi and Pogorelov. For $f\in C^\alpha$, this was proved by Caffarelli and the first named author.