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

Abstract

This paper explores the $p$-capacitary Orlicz–Minkowski problem. Note that the $p$-capacitary Orlicz–Minkowski problem can be converted equivalently to a Monge–Ampère type equation in the smooth case:

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.
We confirm for the first time the existence of smooth solutions to the $p$-capacitary Orlicz–Minkowski problem for $p\in (1,n)$ using a class of inverse Gauss curvature flows which converges smoothly to the solution of equation $(\star )$. Moreover, we prove uniqueness for equation $(\star )$ in a special case.