Heat content, torsional rigidity and generalized Hardy inequalities for complete Riemannian manifolds

The Semi-linear Torsional Rigidity on a Complete Riemannian Two-Manifold

This note is concerned with some essential properties (optimal isoperimetry, first variation, and monotonicity formula) of the so-called $[0,1)\niγ$-torsional rigidity $\mathcal{T}_{γ,\mathsf{g}}$ on a complete Riemannian two-manifold $(\mathbb M^2,\mathsf{g})$. Even in the special case of $\mathbb R^2$, major results are new.

Torsional Rigidity on Compact Riemannian Manifolds with lower Ricci Curvature Bounds

In this article we prove a reverse Hölder inequality for the fundamental eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with lower Ricci curvature bounds. We also prove an isoperimetric inequality for the torsional ridigity of such domains.

Torsional rigidity of submanifolds with controlled geometry

We prove explicit upper and lower bounds for the torsional rigidity of extrinsic domains of submanifolds P^m with controlled radial mean curvature in ambient Riemannian manifolds N^n with a pole p and with sectional curvatures bounded from above and from below, respectively. These bounds are given in terms of the torsional rigidities of corresponding Schwarz symmetrizations of the domains in warped product model spaces. Our main results are obtained using methods from previously established isoperimetric inequalities, as found in e.g. [MP4] and [MP5]. As in [MP4] we also characterize the geome…

关于 $q$-扭转刚度的对偶 Minkowski 问题

The dual Minkowski problem for $q$-torsional rigidity

本文针对 $q>1$ 提出了关于 $q$-扭转刚度的第 $p$ 次对偶 Minkowski 问题，并通过 Gauss 曲率流方法证明了 $p<n$ 时光滑偶解及 $p<0$ 时光滑非偶解的存在性。

k-扭性问题

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

本文基于k-扭性构造Hadamard变分公式，推导k-扭性测度，并提出k-扭性问题，通过曲率流方法证明其光滑非偶解的存在性。

Kohler-Jobin 与 Ehrhard 相遇：通过重排方法得到高斯扭转刚度固定时高斯主频率的精确下界

Kohler-Jobin meets Ehrhard: the sharp lower bound for the Gaussian principal frequency while the Gaussian torsional rigidity is fixed, via rearrangements

本文将Kohler-Jobin重排技术适配到高斯空间，证明了凸区域高斯扭转刚度固定时半空间高斯主频率最小的结论，揭示了高斯与勒贝格情形的意外类比。

Maximizing torsional rigidity on Riemannian manifolds

Let $\,(M,g)\,$ be a $n$-dimensional Riemannian manifold and $\,Ω\,$ be any compact connected domain in $\,M$. We study the problem of finding the {\em maxima} of the functional $\, {\mathcal E} (Ω)\,$ (known as {\em torsional rigidity} associated to $Ω$) among all domains of prescribed volume $v$. Our results show that for a given Riemannian manifold which is strictly isoperimetric at one of its points the maximum of such functional is realized by the geodesic ball centered at this point. More generally, we prove estimates for the functional $\, {\mathcal E} (Ω)\,$ by comparison with symmetri…

第p次对偶Minkowski问题与k-扭应力对应于k-Hessian方程

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

本文提出第p次对偶Minkowski问题，将其转化为非线性偏微分方程，并证明了光滑解的存在性。

一个泛函不等式与Riccati对：研究Hardy型不等式的一种替代方法

A generic functional inequality and Riccati pairs: an alternative approach to Hardy-type inequalities

本文在黎曼流形上提出加性与乘性泛函不等式，引入Riccati对扩展Bessel对，通过求解Riccati型ODE简洁证明多种Hardy型不等式，并在Cartan-Hadamard流形上证明精确不确定性原理。

Weighted Sobolev inequalities and Ricci flat manifolds

We prove a weighted Sobolev inequality and a Hardy inequality on manifolds with nonnegative Ricci curvature satisfying an inverse doubling volume condition. It enables us to obtain rigidity results for Ricci flat manifolds, generalizing earlier work of Bando, Kasue and Nakajima.

Talenti's comparison theorem for Poisson equation and applications on Riemannian manifold with nonnegative Ricci curvature

In this article, we prove Talenti's comparison theorem for Poisson equation on complete noncompact Riemannian manifold with nonnegative Ricci curvature. Furthermore, we obtain the Faber-Krahn inequality for the first eigenvalue of Dirichlet Laplacian, $L^1$- and $L^\infty$-moment spectrum, especially Saint-Venant theorem for torsional rigidity and a reverse Hölder inequality for eigenfunctions of Dirichlet Laplacian.

Finsler流形上意外的分析现象

Unexpected Analytic Phenomena on Finsler Manifolds

本文研究平坦Finsler Cartan-Hadamard流形上的分析性质，发现Nash嵌入定理失效、Sobolev空间非线性、Hardy和不确定性不等式完全失效，而CKN不等式存在参数依赖的尖锐阈值，并归因于S-曲率下界。

On the heat content functional and its critical domains

We study and classify smooth bounded domains in an analytic Riemannian manifold which are critical for the heat content at all times t>0. We do that by first computing the first variation of the heat content, and then showing that a domain is critical if and only if it has the so-called constant flow property, so that we can use a previous classification result established by the author. The outcome is that a domain is critical for the heat content at all times if and only if it admits an isoparametric foliation, that is, a foliation whose leaves are all parallel to the boundary and have const…