论文标题
各向异性流动没有全球术语和双重orlicz christoffel-minkowski类型问题
Anisotropic flows without global terms and dual Orlicz Christoffel-Minkowski type problem
论文作者
论文摘要
在本文中，我们研究了一类各向异性非均匀曲率流的长期存在和渐近行为，而无需全球强迫术语。通过这种各向异性流的固定解决方案，我们获得了一类双重Orlicz ChristOffel-Minkowski类型问题的存在结果，这等同于求解PDE $ G（X，U_K，DU_K）F（d^2U_K+_KI）关于$ \ Mathbb s^n $和$ i $的标准度量，是订单$ n $的单位矩阵。该结果涵盖了$ l^p $双重Minkowski问题的许多已知解决方案，$ l^p $双重Christoffel-Minkowski问题以及一些双重ORLICZ MINKOWSKI问题等。与此同时，某些修改后的QuermassIntegrals的变异公式和一些相应的区域测量问题（ORLICZ CHRISTOFFERS）和INCERAFFIFISE（ORLICZ CHRISTOFFEL）andece，以及INEFEASSKI CHISTORFEL-MINKOWSSKI COLSECTES，是） QuermassIntegrals也会得出。作为推论，这给出了有关在Guan-Ren-Wang中提出的一般规定曲率问题的部分答案（CPAM，2015年）。
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).
