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

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.