Paper · Oct 29, 2025
The -th dual Minkowski problem for the -torsional rigidity corresponding to a -Hessian equation
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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 -th dual -torsional rigidity associated with a -Hessian equation and establishes its Hadamard variational formula with , which induces the -th dual -torsional measure. Further, based on the -th dual -torsional measure, this article, for the first time, proposes the -th dual Minkowski problem of the -torsional rigidity which can be equivalently converted to a nonlinear partial differential equation in smooth case: \beginalign f(x)=τ(|\nabla h|^2+h^2)^{\fracp-n2}h_Ω(x)|Du(ν^-1_Ω(x))|^k+1σ_n-k(h_ij(x)+h_Ω(x)δ_ij), \endalign where is a constant, is a positive smooth function defined on and is the -th elementary symmetric function of the principal curvature radii. We confirm the existence of smooth non-even solution to the -th dual Minkowski problem of the -torsional rigidity for by the method of a curvature flow which converges smoothly to the solution of equation (). Specially, a novel approach for the uniform lower bound estimation in the estimation for the solution to the curvature flow is presented with the help of invariant functional .
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