A type of anisotropic flows and dual Orlicz Christoffel-Minkowski type equations

Abstract

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 $G(x,u_K,D{u}_K)F\left(\right[D^2{u}_K+u_{K}I\left]\right)=1$ on $S^n$ for a convex body K, where D is the covariant derivative with respect to the standard metric on $S^n$ 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 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 also provides a sufficient condition for the existence to the general prescribed curvature equation $F_{⁎}\left(\kappa \right)=G(\nu ,X)$ raised in [20].