A Liouville theorem for convex functions with periodic Monge-Ampère measure
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arXiv
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| Format: | Preprint |
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2025
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| author | Jin, Tianling Li, YanYan Tran, Hung V. Tu, Xushan |
| author_facet | Jin, Tianling Li, YanYan Tran, Hung V. Tu, Xushan |
| contents | We study global convex solutions of the Monge-Ampère equation \[ \det D^2 u = μ\quad \text{in } \mathbb{R}^n, \] where $μ\not\equiv 0$ 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 $μ$ 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 $μ$ 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 $\mathbb{R}^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_15021 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Liouville theorem for convex functions with periodic Monge-Ampère measure Jin, Tianling Li, YanYan Tran, Hung V. Tu, Xushan Analysis of PDEs We study global convex solutions of the Monge-Ampère equation \[ \det D^2 u = μ\quad \text{in } \mathbb{R}^n, \] where $μ\not\equiv 0$ 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 $μ$ 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 $μ$ 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 $\mathbb{R}^n$. |
| title | A Liouville theorem for convex functions with periodic Monge-Ampère measure |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2511.15021 |