Liouville theorem and sharp solvability for solutions of the parabolic Monge-Ampère equation with periodic data
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866912982269689856 |
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| author | Yan, Kui Bao, Jiguang |
| author_facet | Yan, Kui Bao, Jiguang |
| contents | We prove a Liouville Theorem for ancient solutions of the parabolic Monge-Ampère equation with smooth periodic data, generalizing Caffarelli-Li's result \cite{cl04} in 2004 to the parabolic background. To achieve this, we obtain a necessary and sufficient condition for the existence of the smooth periodic solution of the equation $\left(1-u_t\right)\det \left(D_x^2u+I\right)=f$ in $\mathbb{R}^{n+1}$, where $f$ is smooth and periodic in both spatial and temporal variables. This parabolic existence theorem parallels the elliptic counterpart established by Li \cite{l90} in 1990. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24479 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Liouville theorem and sharp solvability for solutions of the parabolic Monge-Ampère equation with periodic data Yan, Kui Bao, Jiguang Analysis of PDEs 35B40, 35A01, 35K55 We prove a Liouville Theorem for ancient solutions of the parabolic Monge-Ampère equation with smooth periodic data, generalizing Caffarelli-Li's result \cite{cl04} in 2004 to the parabolic background. To achieve this, we obtain a necessary and sufficient condition for the existence of the smooth periodic solution of the equation $\left(1-u_t\right)\det \left(D_x^2u+I\right)=f$ in $\mathbb{R}^{n+1}$, where $f$ is smooth and periodic in both spatial and temporal variables. This parabolic existence theorem parallels the elliptic counterpart established by Li \cite{l90} in 1990. |
| title | Liouville theorem and sharp solvability for solutions of the parabolic Monge-Ampère equation with periodic data |
| topic | Analysis of PDEs 35B40, 35A01, 35K55 |
| url | https://arxiv.org/abs/2603.24479 |