Monge-Amp\`ere equation with bounded periodic data

TL;DR

This paper proves that solutions to the Monge-Ampère equation with bounded periodic data are composed of a quadratic polynomial plus a periodic function, extending classical results to broader classes of functions.

Contribution

The work generalizes known results by showing solutions are quadratic plus periodic for bounded periodic data, including less regular functions.

Findings

Solutions are quadratic plus periodic functions.

Extends classical results to bounded periodic data.

Includes cases where $f$ is less regular than smooth.

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.