A flow method for the dual Orlicz–Minkowski problem

Abstract

In this paper the dual Orlicz-Minkowski problem, a generalization of the L p dual Minkowski problem, is studied. By studying a flow involving the Gauss curvature and support function, we obtain a new existence result of solutions to this problem for smooth measures.

Orlicz–Minkowski flows

“…(2) In [18, p. 42], using a logarithmic curvature flow, an existence result was obtained under the assumption (2) and that ϕ is non-increasing. See also [45], where a similar result was recently obtained.…”

“…For some choices of f and ϕ(s) = s 1− p , the flow (2.1) becomes homogeneous and was considered in [3,8,16,24,[34][35][36][37][38]44,53,55,57]. However, when it comes to non-homogeneous flows, the literature on geometric flows is not very rich and there are few works in this direction; e.g., [11,[16][17][18]42,45,50]. Since the Orlicz-Minkowski problem admits solutions in the non-homogeneous case, it is desirable to remove the homogeneity assumption in the flow.…”

Anisotropic Gauss curvature flows and their associated Dual Orlicz-Minkowski problems

“…Thus, theorem 1.2 recovers a parabolic proof in the smooth category for the existence of solutions to the even dual Minkowsi problem for which is given in [18]. Recently, Liu-Lu [19] also used the flow method to study the dual Orlicz-Minkowski problem and they obtained the existence result under the condition that which means if .…”

A Gauss curvature flow related to the Orlicz-Aleksandrov problem

“…We know that the Orlicz Aleksandrov problem is a generalization of the classical Recently, the argument of the smoothness of the even-solutions of Minkowski type problems via the geometric flow method has been made great progress(see [8,9,10,25,26] for details). In order to obtain the Theorem 1.1, our main idea is reflected in the following two folds: I).…”