Flow by Gauss Curvature to the Orlicz Minkowski Problem for q-torsional rigidity

TL;DR

This paper introduces the Orlicz Minkowski problem for q-torsional rigidity, extending previous work on the classical case, and proves the existence of smooth solutions using Gauss curvature flow methods.

Contribution

It formulates the Minkowski problem for q-torsional rigidity in the Orlicz setting and establishes the existence of smooth solutions for q>1 using curvature flow techniques.

Findings

Established the existence of smooth solutions for the Orlicz Minkowski problem for q-torsional rigidity.

Extended classical Minkowski problem results to the q-torsional rigidity context.

Applied Gauss curvature flow to solve the new geometric problem.

Abstract

The celebrated Minkowski problem for the torsional rigidity ($2$-torsional rigidity) was firstly studied by Colesanti and Fimiani \cite{CA} using variational method. Moreover, Hu, Liu and Ma \cite{HJ} also studied the Minkowski problem {\it w.r.t.} $2$-torsional rigidity by method of curvature flows and obtain the existence of smooth even solutions. Up to now, as far as we know, the study of the Minkowski problem for the $q$-torsional rigidity is still blank. In the present paper, we propose and investigate the Orlicz Minkowski problem for the $q$-torsional rigidity corresponding to the $q$-Laplace equation inspired by the foregoing works, and then confirm the existence of smooth non-even solutions to the Orlicz Minkowski problem for the $q$-torsional rigidity with $q>1$ by the method of a Gauss curvature flow.