Abstract:We study global convex solutions of the Monge-Ampère equation \[ \det D^2 u = \mu \quad \text{in } \mathbb{R}^n, \] where $\mu \not\equiv 0$ is a nonnegative locally finite periodic Borel measure on $\mathbb{R}^n$. We prove a Liouville-type theorem showing that every such solution admits a unique decomposition, up to an additive constant, as the sum of a quadratic polynomial and a periodic function. This extends earlier results of Caffarelli-Li and Li-Lu, which required $\mu$ to have a density with regular or bounded logarithm, to the full generality of periodic measures, allowing degeneracy and singularities. A key ingredient is a new dichotomous Harnack-type inequality for linearized Monge-Ampère equations with nonnegative periodic measures, which compensates for the failure of doubling and engulfing properties in the degenerate setting.
In the extremal example where $\mu$ is the periodic Dirac measure supported on the integer lattice, we show that the solutions, up to addition of a linear function, are in one-to-one correspondence with Dirichlet-Voronoi tilings of $\mathbb{R}^n$.
| Comments: | Edited the introduction. Added a section on the extremal example in which $μ$ is the periodic Dirac measure supported on the integer lattice, and showed that the solutions, up to addition of a linear function, are in one-to-one correspondence with Dirichlet-Voronoi tilings of $\mathbb{R}^n$ |
| Subjects: | Analysis of PDEs (math.AP) |
| Cite as: | arXiv:2511.15021 [math.AP] |
| (or arXiv:2511.15021v2 [math.AP] for this version) | |
| https://doi.org/10.48550/arXiv.2511.15021 arXiv-issued DOI via DataCite |
From: Tianling Jin [view email]
[v1]
Wed, 19 Nov 2025 01:38:59 UTC (25 KB)
[v2]
Fri, 22 May 2026 03:24:39 UTC (84 KB)
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