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

The Minkowski problem for torsional rigidity (2-torsional rigidity) was firstly studied by Colesanti and Fimiani CA using variational method. Moreover, Hu, Liu and Ma HJ00 also studied this problem by method of curvature flow and obtain the existence of smooth solution. In addition, the Minkowski problem for 2-torsional rigidity was also extended to Lₚ version and Orlicz version. Recently, Hu and Zhang HJ2 introduced the concept of Orlicz mixed q-torsional rigidity and obtained Orlicz q-torsional measure through the variational method for q>1. Specially, they established the functional Orlicz Brunn-Minkowski inequality and the functional Orlicz Minkowski inequality. Motivated by the remarkable work by Hu and Zhang in HJ2, we can propose the Orlicz Minkowksi problem for q-torsional rigidity, and then confirm the existence of smooth even solutions to the Orlicz Minkowski problem for q-torsional rigidity with q>1 by method of a Gauss curvature flow.