Related papers: The Minkowski problem for the $k$-torsional rigidi…

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…

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…

The Minkowski problem for torsional rigidity ($2$-torsional rigidity) was firstly studied by Colesanti and Fimiani \cite{CA} using variational method. Moreover, Hu \cite{HJ00} also studied this problem by the method of curvature flows and…

In this paper the Orlicz-Minkowski problem for torsional rigidity, a generalization of the classical Minkowski problem, is studied. Using the flow method, we obtain a new existence result of solutions to this problem for general measures.

We prove the existence and uniqueness up to translations of the solution to a Minkowski type problem for the torsional rigidity in the class of open bounded convex subsets of the $n$-dimensional Euclidean space. For the existence part we…

In this paper, we introduce the so-called $L_p$ $q$-torsional measure for $p\in\mathbb{R}$ and $q>1$ by establishing the $L_p$ variational formula for the $q$-torsional rigidity of convex bodies without smoothness conditions. Moreover, we…

The Minkowski problem of harmonic measures was first studied by Jerison [19]. Recently, Akman and Mukherjee [1] studied the Minkowski problem corresponding to $p$-harmonic measures on convex domains and generalized Jerison's results. In…

In this paper, the $L_q$-Minkowski problem of anisotropic $p$-torsional rigidity is considered. The existence of the solution of the $L_q$-Minkowski problem of anisotropic $p$-torsional rigidity with $0<q<1$ and $1<q\neq \frac{p}{p-1}+n$ is…

In this paper, we consider the Minkowski problem associated with the solution to the anisotropic $p$-Laplacian (or Finsler $p$-Laplacian) equation, namely, the Minkowski problem of anisotropic $p$-torsional rigidity. The sufficient and…

Using harmonic mean curvature flow, we establish a sharp Minkowski type lower bound for total mean curvature of convex surfaces with a given area in Cartan-Hadamard 3-manifolds. This inequality also improves the known estimates for total…

In the present paper, we first establish and verify a new sharp hyperbolic version of the Michael-Simon inequality for mean curvatures in hyperbolic space $\mathbb{H}^{n+1}$ based on the locally constrained inverse curvature flow introduced…

By using Minkowski addition of convex functions, we prove convexity and rearrangement properties of solutions to some Hessian equations in $\R^3$ and Brunn-Minkowski and isoperimetric inequalities for related functionals.

In this paper, we proved a sharp Minkowski type inequality in Cartan-Hadamard 3-spaces by harmonic mean curvature flow and improves the known estimates for total mean curvature in hyperbolic 3-space. In particular, we sharpened…

Brendle [6] successfully establishes the sharp Michael-Simon inequality for mean curvature on Riemannian manifolds with nonnegative sectional curvature ($\mathcal{K} \geq 0$), and the proof relies on the Alexandrov-Bakelman-Pucci method.…

In this paper, we study the $L_p$-Gaussian Minkowski problem, which arises in the $L_p$-Brunn-Minkowski theory in Gaussian probability space. We use Aleksandrov's variational method with Lagrange multipliers to prove the existence of the…

This paper deals with locally constrained inverse curvature flows in a broad class of Riemannian warped spaces. For a certain class of such flows we prove long time existence and smooth convergence to a radial coordinate slice. In the case…

In this paper, the $L_{p}$ chord Minkowski problem is concerned. Based on the results showed in \cite{HJ23}, we obtain a new existence result of solutions to this problem in terms of smooth measures by using a nonlocal Gauss curvature flow…

In this paper, we first prove the Hardy-Sobolev inequality for the Hessian integral by means of a descent gradient flow of certain Hessian functionals. As an application, we study the existence and regularity results of solutions to related…

In this paper, we study the $k$-Hessian curvature flow of noncompact spacelike hypersurfaces in Minkowski space. We first prove the existence of translating solutions with given asymptotic behavior. Then, we prove that for strictly convex…

In this paper, we derive the continuity of solutions to the $L_{p}$ torsional Minkowski problem for $p>1$. It is shown that the weak convergence of the $L_{p}$ torsional measure implies the convergence of the sequence of the corresponding…