Geometric eigenvalue bounds for the Dirichlet-to-Neumann Operator

鼓以不同的频率振动。鼓的频率可以由一个称为拉普拉斯算子的椭圆算子的特征值来确定。椭圆算子的特征值的定义类似于欧几里德平面上线性映射的特征值的定义。现在想象另一种类型的鼓，其质量集中在边界上，即边界外的质量可以忽略不计。这种鼓的频率与另一个称为Dirichlet-to-Neumann （DtN）算子的椭圆算子的特征值有关。这些特征值被称为Steklov特征值，因为这个特征值问题是Steklov在1902年提出和研究的。流形的几何形状（如鼓的形状）对拉普拉斯特征值的影响已经得到了大量的研究。在1911年赫尔曼·魏尔（Hermann Weyl）关于拉普拉斯特征值的渐近行为的著名结果之后，出现了许多发展。拉普拉斯特征值问题的研究也扩展到图（顶点和边的集合）和概率空间的设置。它也对应用领域产生了重大影响。例如，最近研究图的拉普拉斯特征值问题的一个主要结果为计算机科学中的聚类算法提供了数学依据，并提供了有关其效率的信息。然而，近年来在基础空间几何与Steklov特征值之间的关系方面取得了许多进展。提出的研究项目旨在解决一些关于斯特克洛夫和拉普拉斯特征值与基础空间的几何不变量之间的联系的基本问题。底层空间可以是流形、图或概率空间。在流形设置中，研究将揭示未被拉普拉斯特征值捕获的几何/拓扑信息。在图和概率空间的设置中，该方法将基于一些最近开发的技术。该项目是跨学科的，其结果不仅在光谱几何和几何分析领域，而且在概率和计算机科学等其他领域都具有重要意义。DtN算子及其特征值在流体动力学、形状分析与图像处理、电阻抗层析成像（EIT）等领域的晃动问题研究中起着关键作用。因此，拟议项目的结果将是这些应用领域的根本利益。提出的研究项目还将解决DtN特征函数的节点域研究中的一些基本开放问题。DtN特征函数描述了质量集中在边界上的鼓的边界振动。在数学术语中，特征函数的零水平集称为节点集，它的补集称为节点域。提出的研究项目将研究节点域连接组件数量的界限。节点域和节点集的研究是数学和数学物理中一个引人入胜的研究领域。

A drum vibrates at distinct frequencies. The frequencies of a drum can be determined by eigenvalues of an elliptic operator called the Laplacian. The definition of the eigenvalues of an elliptic operator is similar to the definition of the eigenvalues of a linear map in the Euclidean plane. Now imagine another type of drum whose mass is concentrated on the boundary, i.e. the mass outside the boundary is negligible. The frequencies of such a drum are related to the eigenvalues of another elliptic operator called the Dirichlet-to-Neumann (DtN) operator. These eigenvalues are known as Steklov eigenvalues since this eigenvalue problem was introduced and studied by Steklov in 1902. The influence of the geometry of a manifold (e.g. the shape of a drum) on the Laplace eigenvalues has been greatly studied. Many developments came after the celebrated result of Hermann Weyl in 1911 on the asymptotic behaviour of the Laplace eigenvalues. The study of the Laplace eigenvalue problem has also extended to the setting of graphs (which are a collection of vertices and edges) and probability spaces. It also has had a significant influence on applied areas. For example, one of the main recent results in the study of the Laplace eigenvalue problem on graphs gave a mathematical justification for clustering algorithms in computer science and provided information about their efficiency. However, many developments on the relation between the geometry of underlying space and Steklov eigenvalues have been achieved during the last few years. The proposed research project aims to address some of the fundamental questions on connections between the Steklov and Laplace eigenvalues and geometric invariants of the underlying space. The underlying space can be a manifold, graph, or probability space. In the manifold setting, the study will reveal the geometric/topological information that is not captured by the Laplace eigenvalues. In the setting of a graph and probability space, the approach will be based on some of the recently developed techniques. The proposed project is intradisciplinary, and the results will be of significant importance not only in the areas of spectral geometry and geometric analysis but also in other areas such as probability and computer science. The DtN operator and its eigenvalues play a key role in the study of the sloshing problem in fluid dynamics, shape analysis and image processing, and Electrical Impedance Tomography (EIT). Hence, the outcome of the proposed project will be of fundamental interest in these applied areas. The proposed research project will also address some of the fundamental open problems in the study of nodal domains of the DtN eigenfunctions. The DtN eigenfunctions describe the vibration of the boundary of a drum whose mass concentrated on the boundary. In mathematical terminology, the zero-level set of an eigenfunction is called the nodal set, and its complement is the nodal domain. The proposed research project will investigate bounds on the number of the connected components of a nodal domain. The study of the nodal domains and nodal sets is a fascinating area of research in mathematics and mathematical physics.