Anisotropic flows without global terms and dual Orlicz Christoffel-Minkowski type problem
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866910285417152512 |
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| author | Ding, Shanwei Li, Guanghan |
| author_facet | Ding, Shanwei Li, Guanghan |
| contents | In this paper, we study the long-time existence and asymptotic behavior for a class of anisotropic non-homogeneous curvature flows without global forcing terms. By the stationary solutions of such anisotropic flows, we obtain existence results for a class of dual Orlicz Christoffel-Minkowski type problems, which is equivalent to solve the PDE $G(x,u_K,Du_K)F(D^2u_K+u_KI)=1$ on $\mathbb S^n$ for a convex body $K$, where $D$ is the covariant derivative with respect to the standard metric on $\mathbb S^n$ and $I$ is the unit matrix of order $n$. This result covers many previous known solutions to $L^p$ dual Minkowski problem, $L^p$ dual Christoffel-Minkowski problem, and some dual Orlicz Minkowski problem etc.. Meanwhile, the variational formula of some modified quermassintegrals and the corresponding prescribed area measure problem (Orlicz Christoffel-Minkowski type problem) are considered, and inequalities involving modified quermassintegrals are also derived. As corollary, this gives a partial answer about the general prescribed curvature problem raised in Guan-Ren-Wang (CPAM, 2015). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_03114 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Anisotropic flows without global terms and dual Orlicz Christoffel-Minkowski type problem Ding, Shanwei Li, Guanghan Differential Geometry Analysis of PDEs In this paper, we study the long-time existence and asymptotic behavior for a class of anisotropic non-homogeneous curvature flows without global forcing terms. By the stationary solutions of such anisotropic flows, we obtain existence results for a class of dual Orlicz Christoffel-Minkowski type problems, which is equivalent to solve the PDE $G(x,u_K,Du_K)F(D^2u_K+u_KI)=1$ on $\mathbb S^n$ for a convex body $K$, where $D$ is the covariant derivative with respect to the standard metric on $\mathbb S^n$ and $I$ is the unit matrix of order $n$. This result covers many previous known solutions to $L^p$ dual Minkowski problem, $L^p$ dual Christoffel-Minkowski problem, and some dual Orlicz Minkowski problem etc.. Meanwhile, the variational formula of some modified quermassintegrals and the corresponding prescribed area measure problem (Orlicz Christoffel-Minkowski type problem) are considered, and inequalities involving modified quermassintegrals are also derived. As corollary, this gives a partial answer about the general prescribed curvature problem raised in Guan-Ren-Wang (CPAM, 2015). |
| title | Anisotropic flows without global terms and dual Orlicz Christoffel-Minkowski type problem |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2207.03114 |