Curvature bound for $L_p$ Minkowski problem
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866912032190627840 |
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| author | Choi, Kyeongsu Kim, Minhyun Lee, Taehun |
| author_facet | Choi, Kyeongsu Kim, Minhyun Lee, Taehun |
| contents | We establish curvature estimates for anisotropic Gauss curvature flows. By using this, we show that given a measure $μ$ with a positive smooth density $f$, any solution to the $L_p$ Minkowski problem in $\mathbb{R}^{n+1}$ with $p \le -n+2$ is a hypersurface of class $C^{1,1}$. This is a sharp result because for each $p\in [-n+2,1)$ there exists a convex hypersurface of class $C^{1,\frac{1}{n+p-1}}$ which is a solution to the $L_p$ Minkowski problem for a positive smooth density $f$. In particular, the $C^{1,1}$ regularity is optimal in the case $p=-n+2$ which includes the logarithmic Minkowski problem in $\mathbb{R}^3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_11617 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Curvature bound for $L_p$ Minkowski problem Choi, Kyeongsu Kim, Minhyun Lee, Taehun Differential Geometry Analysis of PDEs 53E99 (Primary) 35B65, 35C06, 35K96, 53A05 (Secondary) We establish curvature estimates for anisotropic Gauss curvature flows. By using this, we show that given a measure $μ$ with a positive smooth density $f$, any solution to the $L_p$ Minkowski problem in $\mathbb{R}^{n+1}$ with $p \le -n+2$ is a hypersurface of class $C^{1,1}$. This is a sharp result because for each $p\in [-n+2,1)$ there exists a convex hypersurface of class $C^{1,\frac{1}{n+p-1}}$ which is a solution to the $L_p$ Minkowski problem for a positive smooth density $f$. In particular, the $C^{1,1}$ regularity is optimal in the case $p=-n+2$ which includes the logarithmic Minkowski problem in $\mathbb{R}^3$. |
| title | Curvature bound for $L_p$ Minkowski problem |
| topic | Differential Geometry Analysis of PDEs 53E99 (Primary) 35B65, 35C06, 35K96, 53A05 (Secondary) |
| url | https://arxiv.org/abs/2304.11617 |