Paper · May 24, 2023
An inverse Gauss curvature flow and its application to p-capacitary Orlicz-Minkowski problem
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Abstract
In [Calc. Var., 57:5 (2018)], Hong-Ye-Zhang proposed the -capacitary Orlicz-Minkowski problem and proved the existence of convex solutions to this problem by variational method for . However, the smoothness and uniqueness of solutions are still open. Notice that the -capacitary Orlicz-Minkowski problem can be converted equivalently to a Monge-Ampère type equation in smooth case: \beginalign fφ(h_K)|\nablaΨ|^p=τG \endalign for and some constant , where is a positive function defined on the unit sphere , is a continuous positive function defined in , and is the Gauss curvature. In this paper, we confirm the existence of smooth solutions to -capacitary Orlicz-Minkowski problem with for the first time by a class of inverse Gauss curvature flows, which converges smoothly to the solution of Equation (). Furthermore, we prove the uniqueness result for Equation () in a special case.
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