The $p$-th dual Minkowski problem for the $k$-torsional rigidity corresponding to a $k$-Hessian equation

Abstract

The study of the dual curvature measures [Y. Huang, E. Lutwak, D. Yang \& G. Y. Zhang, Acta. Math. 216 (2016): 325-388], which connects the cone-volume measure and Aleksandrov's integral curvature, and has created a precedent for the theoretical research of the dual Brunn-Minkowski theory. Motivated by the foregoing groundbreaking works, the present paper introduces the $p$-th dual $k$-torsional rigidity associated with a $k$-Hessian equation and establishes its Hadamard variational formula with $1\le k\le n-1$, which induces the $p$-th dual $k$-torsional measure. Further, based on the $p$-th dual $k$-torsional measure, this article, for the first time, proposes the $p$-th dual Minkowski problem of the $k$-torsional rigidity which can be equivalently converted to a nonlinear partial differential equation in smooth case: \begin{align}\label{eq01} f(x)=τ(|\nabla h|^2+h^2)^{\frac{p-n}{2}}h_Ω(x)|Du(ν^{-1}_Ω(x))|^{k+1}σ_{n-k}(h_{ij}(x)+h_Ω(x)δ_{ij}), \end{align} where $\tau >0$ is a constant, $f$ is a positive smooth function defined on $S^{n-1}$ and ${\sigma }_{n-k}$ is the $(n-k)$-th elementary symmetric function of the principal curvature radii. We confirm the existence of smooth non-even solution to the $p$-th dual Minkowski problem of the $k$-torsional rigidity for $p<n-2$ by the method of a curvature flow which converges smoothly to the solution of equation (\ref{eq01}). Specially, a novel approach for the uniform lower bound estimation in the $C^0$ estimation for the solution to the curvature flow is presented with the help of invariant functional $\Phi ({\Omega }_t)$.