A Liouville theorem for ancient solutions of the parabolic Monge-Ampère equation with periodic data
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915891529121792 |
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| author | Yan, Kui Bao, Jiguang |
| author_facet | Yan, Kui Bao, Jiguang |
| contents | This article is concerned with the parabolic Monge-Ampère equation $-u_t\det D_x^2u=f$, where $f=f_1(x)f_2(t)$ and $f_1,f_2$ are positive periodic functions. We prove that any classical parabolically convex ancient solution $u$ must be of the form $-τt+p(x)+v(x,t)$, where $τ$ is a positive constant, $p(x)$ is a convex quadratic polynomial, and $v$ inherits both the spatial and temporal periodicity from $f$. This work extends previous contributions by Caffarelli-Li \cite{cl04} on periodic frameworks for the elliptic Monge-Ampère equations, and generalizes Zhang-Bao \cite{zb18}'s Liouville theorem for $f_2\equiv1$ in parabolic case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24452 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Liouville theorem for ancient solutions of the parabolic Monge-Ampère equation with periodic data Yan, Kui Bao, Jiguang Analysis of PDEs 35B40, 35K96, 35K55 This article is concerned with the parabolic Monge-Ampère equation $-u_t\det D_x^2u=f$, where $f=f_1(x)f_2(t)$ and $f_1,f_2$ are positive periodic functions. We prove that any classical parabolically convex ancient solution $u$ must be of the form $-τt+p(x)+v(x,t)$, where $τ$ is a positive constant, $p(x)$ is a convex quadratic polynomial, and $v$ inherits both the spatial and temporal periodicity from $f$. This work extends previous contributions by Caffarelli-Li \cite{cl04} on periodic frameworks for the elliptic Monge-Ampère equations, and generalizes Zhang-Bao \cite{zb18}'s Liouville theorem for $f_2\equiv1$ in parabolic case. |
| title | A Liouville theorem for ancient solutions of the parabolic Monge-Ampère equation with periodic data |
| topic | Analysis of PDEs 35B40, 35K96, 35K55 |
| url | https://arxiv.org/abs/2603.24452 |