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

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: \tag{$\star$} f\phi(h_{K}) |\nabla\Psi|^{p}=\tau G 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.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.