Rigidity for perimeter inequality under spherical symmetrisation

Some inequalities involving perimeter and torsional rigidity

We consider shape functionals of the form $F_q(Ω)=P(Ω)T^q(Ω)$ on the class of open sets of prescribed Lebesgue measure. Here $q>0$ is fixed, $P(Ω)$ denotes the perimeter of $Ω$ and $T(Ω)$ is the torsional rigidity of $Ω$. The minimization and maximization of $F_q(Ω)$ is considered on various classes of admissible domains $Ω$: in the class $\mathcal{A}_{all}$ of all domains, in the class $\mathcal{A}_{convex}$ of convex domains, and in the class $\mathcal{A}_{thin}$ of thin domains.

Rigidity of equality cases in Steiner's perimeter inequality

Characterization results for equality cases and for rigidity of equality cases in Steiner's perimeter inequality are presented. (By rigidity, we mean the situation when all equality cases are vertical translations of the Steiner's symmetral under consideration.) We achieve this through the introduction of a suitable measure-theoretic notion of connectedness and a fine analysis of barycenter functions for sets of finite perimeter having segments as orthogonal sections with respect to an hyperplane.

Essential connectedness and the rigidity problem for Gaussian symmetrization

We provide a geometric characterization of rigidity of equality cases in Ehrhard's symmetrization inequality for Gaussian perimeter. This condition is formulated in terms of a new measure-theoretic notion of connectedness for Borel sets, inspired by Federer's definition of indecomposable current.

关于涉及 $p$-容量与 $q$-扭转刚度的泛函

On functionals involving the $p$-capacity and the $q$-torsional rigidity

研究凸体在周长、测度及其组合约束下 $p$-容量与 $q$-扭转刚度幂次乘积的最优界，并在特定参数范围内证明球为唯一极值体。

带孔凸集上第一$p$-拉普拉斯特征值与$p$-扭转刚度的精确估计

Sharp estimates for the first $p$-Laplacian eigenvalue and for the $p$-torsional rigidity on convex sets with holes

该研究针对带孔凸集上满足外Robin、内Neumann边界条件的p-拉普拉斯问题，证明固定测度与外周长时圆环最大化第一特征值、最小化扭转刚度。

二维气泡的全局刚性

Global rigidity of two-dimensional bubbles

研究二维具有表面张力的静止空涡结构，证明小Weber数下单位圆的全局刚性，揭示了凸性约束变分问题的唯一解条件。

疏水性毛细泛函临界点的刚性

Rigidity of critical points of hydrophobic capillary functionals

本文研究半空间中毛细能量保体积临界点的刚性，在 $90^\circ$–$120^\circ$ 接触角下无需正则性假设，并在切向边界零测度条件下推广至整个疏水区间及各向异性情形。

切向多边形的波利亚-塞格定理

A Quantitative Pólya--Szegő Theorem for Tangential Polygons

该研究证明给定面积的切向N边形中，正N边形的扭转刚度最大，分解切向多边形为混合狄利克雷-纽曼直角三角形单元，推导了相关映射的严格凹性与亏空分解，还应用其证明等面积正多边形扭转刚度随边数增大并得到相关准则。

具有负Robin边界条件的拉普拉斯扭转刚度的精确估计

Sharp estimates for the Laplacian torsional rigidity with negative Robin boundary conditions

该研究通过证明在负Robin边界条件下，球体在特定约束下最大化扭转刚度，解决了Bandle和Wagner提出的开放问题。