论文标题
关于Steklov特征值问题的一些最新发展
Some recent developments on the Steklov eigenvalue problem
论文作者
论文摘要
Steklov特征值问题是在125年前首次引入的，在过去的几十年中引起了人们的兴趣。这篇文章是对紧凑型riemannian歧管与流形的几何形状联系起来的一些最近的发展。 Topics include isoperimetric-type upper and lower bounds on Steklov eigenvalues (first in the case of surfaces and then in higher dimensions), stability and instability of eigenvalues under deformations of the Riemannian metric, optimisation of eigenvalues and connections to free boundary minimal surfaces in balls, inverse problems and isospectrality, discretisation, and the geometry特征函数。我们从背景材料和激励示例开始，对于读者的读者开始。在整个巡回演出中，我们经常将Steklov Spectrum的行为与Laplace Spectrum的行为进行比较和对比。我们在这个快速扩展的领域中包括许多开放问题。
The Steklov eigenvalue problem, first introduced over 125 years ago, has seen a surge of interest in the past few decades. This article is a tour of some of the recent developments linking the Steklov eigenvalues and eigenfunctions of compact Riemannian manifolds to the geometry of the manifolds. Topics include isoperimetric-type upper and lower bounds on Steklov eigenvalues (first in the case of surfaces and then in higher dimensions), stability and instability of eigenvalues under deformations of the Riemannian metric, optimisation of eigenvalues and connections to free boundary minimal surfaces in balls, inverse problems and isospectrality, discretisation, and the geometry of eigenfunctions. We begin with background material and motivating examples for readers that are new to the subject. Throughout the tour, we frequently compare and contrast the behavior of the Steklov spectrum with that of the Laplace spectrum. We include many open problems in this rapidly expanding area.
