A Liouville theorem for convex functions with periodic Monge-Amp\`ere measure

TL;DR

This paper proves a Liouville-type theorem for convex solutions of the Monge-Ampère equation with periodic measures, showing they decompose into a quadratic polynomial plus a periodic function, extending previous results.

Contribution

It generalizes earlier results by establishing a decomposition for solutions with arbitrary nonnegative periodic measures, answering a question by Li and Lu.

Findings

Convex solutions decompose into quadratic plus periodic functions.

Introduces a new Harnack-type inequality for linearized Monge-Ampère equations.

Extends prior work to full generality for periodic measures.

Abstract

Let $\mu \not\equiv 0$ be a nonnegative locally finite periodic Borel measure on $\mathbb{R}^n$. We show that any convex solution to the Monge-Amp\`ere equation \[ \det D^2 u = \mu \quad \text{in } \mathbb{R}^n \] admits a unique decomposition (up to addition of constants) as the sum of a quadratic polynomial and a periodic function. This result extends, in full generality, the earlier works for the case $\mu=f(x)\,\mathrm{d} x$: when $\log f \in C^\alpha$, it was established by Caffarelli and Li; and when $\log f$ is merely bounded, it was proved by Li and Lu. Our result thus answers a question raised by Li and Lu. A key ingredient in the proof is a new dichotomous Harnack-type inequality for linearized Monge-Amp\`ere equations with nonnegative periodic measures.