A Liouville theorem for convex functions with periodic Monge-Ampère measure

2025/11/19 by Jin, Tianling, Li, YanYan, Tran, Hung V. +1
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2511.15021

Abstract

Let μ\not≡ 0 be a nonnegative locally finite periodic Borel measure on ℝⁿ. We show that any convex solution to the Monge-Ampère equation det D² u = μ in ℝⁿ 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) d x: when log f ∈ 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.

Citations

• Monge-Ampère equation with bounded periodic data
• Some remarks on singular solutions of nonlinear elliptic equations. III: viscosity solutions, including parabolic operators

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