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-Amp\`{e}re type equation in smooth case: \begin{align}\label{0.1} f\phi(h_K)|\nabla\Psi|^p=\tau G \end{align} for $p\in(1,n)$ and some constant $\tau>0$, where $f$ is a positive function defined on the unit sphere $\mathcal{S}^{n-1}$, $\phi$ 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.