Flow by Gauss curvature to the Minkowski problem of p-harmonic measure

TL;DR

This paper proves the existence of smooth solutions to the Minkowski problem for p-harmonic measures on convex domains using Gauss curvature flow, extending previous work on harmonic and p-harmonic measures.

Contribution

It introduces a novel approach employing Gauss curvature flow to solve the Minkowski problem for p-harmonic measures, providing new existence results.

Findings

Established existence of smooth solutions for the Minkowski problem in this setting

Extended previous results from harmonic to p-harmonic measures

Applied Gauss curvature flow as a key method

Abstract

The Minkowski problem of harmonic measures was first studied by Jerison [19]. Recently, Akman and Mukherjee [1] studied the Minkowski problem corresponding to $p$-harmonic measures on convex domains and generalized Jerison's results. In this paper, we prove the existence of the smooth solution to the Minkowski problem for the $p$-harmonic measure by method of the Gauss curvature flow.