An inverse Gauss curvature flow and its application to p-capacitary Orlicz-Minkowski problem

Abstract

In [Calc. Var., 57:5 (2018)], Hong-Ye-Zhang proposed the $p$-capacitary Orlicz-Minkowski problem and proved the existence of convex solutions to this problem by variational method for $p\in (1,n)$. However, the smoothness and uniqueness of solutions are still open. Notice that the $p$-capacitary Orlicz-Minkowski problem can be converted equivalently to a Monge-Ampère type equation in smooth case: \begin{align}\label{0.1} fϕ(h_K)|\nablaΨ|^p=τG \end{align} for $p\in (1,n)$ and some constant $\tau >0$, where $f$ is a positive function defined on the unit sphere ${\mathcal{S}}^{n-1}$, $\varphi$ is a continuous positive function defined in $(0,+\infty )$, and $G$ is the Gauss curvature. In this paper, we confirm the existence of smooth solutions to $p$-capacitary Orlicz-Minkowski problem with $p\in (1,n)$ for the first time by a class of inverse Gauss curvature flows, which converges smoothly to the solution of Equation (\ref{0.1}). Furthermore, we prove the uniqueness result for Equation (\ref{0.1}) in a special case.