2023/05/24 by Bin Chen, Weidong Wang, Chen, Bin +5
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2305.14830
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∈(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: fϕ(hK)|∇Ψ|p=τG for p∈(1,n) and some constant τ>0, where f is a positive function defined on the unit sphere Sn-1, ϕ is a continuous positive function defined in (0,+∞), and G is the Gauss curvature. In this paper, we confirm the existence of smooth solutions to p-capacitary Orlicz-Minkowski problem with p∈(1,n) for the first time by a class of inverse Gauss curvature flows, which converges smoothly to the solution of Equation (\ref0.1). Furthermore, we prove the uniqueness result for Equation (\ref0.1) in a special case.