Spectral gap for stable process on convex planar double symmetric domains

平面凸域上拉普拉斯算子Neumann特征值的两个新通用不等式

Two new universal inequalities for Neumann eigenvalues of the Laplacian on a planar convex domain

本文提出两个新通用不等式，用于平面凸域中拉普拉斯算子的Neumann特征值，扩展了相关理论的应用范围。

具有小特征值的凸双曲平面域中的热点

Hot spots in convex hyperbolic planar domains with small eigenvalues

本文证明了在具有小Neumann或混合Dirichlet-Neumann特征值的双曲平面域中，Rauch热点猜想的一个变种，得出结论：在双曲平面上足够大的面积有界凸域中，第二Neumann拉普拉斯特征函数没有内部临界点。

On the Hardy constant of some non-convex planar domains

The Hardy constant of a simply connected domain $Ω\subset\mathbf{R}^2$ is the best constant for the inequality \[ \int_Ω|\nabla u|^2dx \geq c\int_Ω \frac{u^2}{{\rm dist}(x,\partialΩ)^2}\, dx \; , \;\;\quad u\in C^{\infty}_c(Ω). \] After the work of Ancona where the universal lower bound 1/16 was obtained, there has been a substantial interest on computing or estimating the Hardy constant of planar domains. In \cite{BT} we have determined the Hardy constant of an arbitrary quadrilateral in the plane. In this work we continue our investigation and we compute the Hardy constant for other non-conv…

关于平面凸域中Neumann问题的Pólya猜想

On the Pólya conjecture for the Neumann problem in planar convex domains

本文证明了平面凸域中Neumann问题谱计数函数满足Pólya猜想的一个弱化下界，系数为$(2\sqrt{3}j_0^2)^{-1}$，其中$j_0$为Bessel函数$J_0$的首个零点。

Convexity properties of the difference over the real axis between the Steklov zeta functions of a smooth planar domain with $2π$ perimeter and of the unit disk

We consider the zeta function $ζ_Ω$ for the Dirichlet-to-Neumann operator of a simply connected planar domain $Ω$ bounded by a smooth closed curve of perimeter $2π$. We prove that $ζ_Ω''(0)\ge ζ_{\mathbb{D}}''(0)$ with equality if and only if $Ω$ is a disk where $\mathbb{D}$ denotes the closed unit disk. We also provide an elementary proof that for a fixed real $s$ satisfying $s\le-1$ the estimate $ζ_Ω''(s)\ge ζ_{\mathbb{D}}''(s)$ holds with equality if and only if $Ω$ is a disk. We then bring examples of domains $Ω$ close to the unit disk where this estimate fails to be extended to the interv…