The Minkowski problem with respect to the k-torsional rigidity associated with a k-Hessian equation

Abstract

P. Salani [Adv. Math., 229 (2012)] introduced the $k$-torsional rigidity associated with a $k$-Hessian equation and obtained the Brunn-Minkowski inequalities $w.r.t.$ the torsional rigidity in ${\mathbb{R}}^3$. We first construct, in the present paper, a Hadamard variational formula for the $k$-torsional rigidity with $1\le k\le n-1$, then we can deduce a $k$-torsional measure from the Hadamard variational formula. Based on the $k$-torsional measure, we propose the Minkowski problem for the $k$-torsional rigidity which can be equivalently converted to a nonlinear partial differential equation in smooth case: \begin{align}\label{eq00} f(x)=τ|Du(ν_Ω^{-1}(x))|^{k+1}σ_{n-k}(h_{ij}+hδ_{ij}), \end{align} where $\tau >0$ is a constant, $f$ is a positive smooth function defined on the unit sphere and ${\sigma }_{n-k}$ is the $(n-k)$-th elementary symmetric function of the principal curvature radii. Furthermore, we achieve a smooth solution to the Minkowski problem for the $k$-torsional rigidity using an appropriate Gauss curvature-type flow. A key ingredient of the argument for the existence of solutions to Gauss curvature-type flows is the uniform lower bound estimation given by the proof by contradiction in the $C^0$ estimation. The core contribution of the proof by contradiction is that it eliminates the additional conditions that the function $f$ must satisfy when studying such problems, such as symmetry by C. Chen, Y. Huang \& Y, Zhao [Math. Ann., 373(2019)], and by C. Haberl, E. Lutwak, D. Yang \& G. Y. Zhang [Adv. Math., 224(2010)] or having positive upper and lower bounds by Y. Liu, J. Lu [Trans. Amer. Math. Soc., 373(2020)].