A Liouville theorem for convex functions with periodic Monge-Ampère measure
Tianling Jin, YanYan Li, Hung V. Tran, Xushan Tu
TL;DR
This paper addresses Liouville-type characterizations for convex solutions of the Monge-Ampère equation $\det D^2 u = \mu$ in $\mathbb{R}^n$ with $\mu$ a nonzero periodic Borel measure. The authors show that any convex solution can be written as $u = v + P$, where $P(x) = \tfrac12 x^{\top} A x + b\cdot x + c$ is quadratic, $A \in \mathcal{S}_+^{n\times n}$ with $\det A = \mu(\mathbb{T}^n)$, and $v$ is $\mathbb{Z}^n$-periodic; for a fixed $P$ there is a unique periodic solution up to constants. They further prove that any global convex solution is itself periodic up to an additive constant, thereby answering a question of Li-Lu in full generality. The key innovations are a new dichotomous Harnack-type inequality for linearized MA with periodic measures, the φ-homogenization framework, and a semiconcavity analysis that yields precise quadratic blow-up limits and a complete decomposition. Collectively, these results extend the Liouville-type theorems of Caffarelli-Li and Li-Lu to general periodic measures and provide tools for homogenization of degenerate Monge-Ampère equations with periodic data.
Abstract
Let $μ\not\equiv 0$ be a nonnegative locally finite periodic Borel measure on $\mathbb{R}^n$. We show that any convex solution to the Monge-Ampère equation \[ \det D^2 u = μ\quad \text{in } \mathbb{R}^n \] 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 $μ=f(x)\,\mathrm{d} x$: when $\log f \in C^α$, 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-Ampère equations with nonnegative periodic measures.