The Lp Minkowski problem for q-torsional rigidity

TL;DR

This paper introduces a new $L_p$ $q$-torsional measure, establishes an $L_p$ variational formula for $q$-torsional rigidity, and proves existence of solutions to the $L_p$ Minkowski problem for convex bodies with various measures.

Contribution

It defines the $L_p$ $q$-torsional measure and proves existence results for the $L_p$ Minkowski problem related to $q$-torsional rigidity.

Findings

Introduced the $L_p$ $q$-torsional measure.

Established the $L_p$ variational formula for $q$-torsional rigidity.

Proved existence of solutions for the $L_p$ Minkowski problem for discrete and general measures.

Abstract

In this paper, we introduce the so-called $L_p$ $q$-torsional measure for $p\in\mathbb{R}$ and $q>1$ by establishing the $L_p$ variational formula for the $q$-torsional rigidity of convex bodies without smoothness conditions. Moreover, we achieve the existence of solutions to the $L_p$ Minkowski problem $w.r.t.$ the $q$-torsional rigidity for discrete measure and general measure when $0<p<1$ and $q>1$.