The $L_q$ Minkowski problem for $\mathbf{p}$-harmonic measure

TL;DR

This paper introduces a new measure derived from a variational formula related to the $ extbf{p}$-harmonic measure and proves the existence of solutions to the $L_q$ Minkowski problem for this measure when $0<q<1$ and $ extbf{p} eq n+1$.

Contribution

It establishes the existence of solutions to the $L_q$ Minkowski problem for a novel measure linked to $ extbf{p}$-harmonic measure, expanding the scope of Minkowski problems.

Findings

Existence of solutions for $0<q<1$ and $ extbf{p} eq n+1$.

Development of a variational formula for the $ extbf{p}$-harmonic measure.

Introduction of a new measure related to $ extbf{p}$-harmonic measure.

Abstract

In this paper, we consider an extremal problem associated with the solution to a boundary value problem. Our main focus is on establishing a variational formula for a functional related to the $\mathbf{p}$-harmonic measure, from which a new measure is derived. This further motivates us to study the Minkowski problem for this new measure. As a main result, we prove the existence of solutions to the $L_q$ Minkowski problem associated with the $\mathbf{p}$-harmonic measure for $0<q<1$ and $1<\mathbf{p}\ne n+1$.