Estimates on eigenvalues and eigenfunctions in convex settings

椭圆微分方程被用来描述各种各样的物理现象。该项目的主要目标是研究这些方程中的一个特定类别，这些方程可用于模拟鼓或非均匀材料的振动行为，热导体的温度分布，或在流体区域中移动的颗粒的随机运动分布。除了在一些简单的模型案例中，这种行为仍然远未被理解。通过将这些问题重新表述为偏微分方程，本项目将研究这种行为。一个特别感兴趣的问题是鼓的形状如何影响其振动通常局限于鼓的哪个部分。另一个目的是研究自由边界问题。自由边界是分隔两种不同材料的区域，例如海洋中水和空气之间的界面或绝缘材料和空气之间的界面。自由边界可以通过求解微分方程来建模，并且了解其形状和其看起来光滑的尺度可以应用于最佳形状设计，电磁学和流体流动。该项目通过对本科生的指导，为美国劳动力的发展做出贡献。在该项目中，PI将研究欧几里德空间和球面中凸域上的特征值和特征函数。该项目的一个主要目标是进一步理解本征函数的水平集。起点是一个定量属性，即水平集的凸性，目的是使用和发展椭圆和微分几何理论的技术，以建立涉及其形状和位置的定量属性。这将导致理解在没有显式公式可用的情况下本征函数局部化的域的区域。该项目的另一部分涉及经典的Friedland-Hayman不等式的变体，涉及球面上的特征值。这将涉及开发工具，从凸几何，等周不等式理论，和Brenier的最佳运输映射。这将用于证明相关自由边界问题最小化者的规律性。该奖项反映了NSF的法定使命，并且通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Elliptic differential equations are used to describe a wide variety of physical phenomena. A main goal of this project is to study a particular class of these equations that can be used to model the behavior of vibrations of a drum or an inhomogeneous material, the temperature distribution of a thermal conductor, or the distribution of the random motion of particles moving in a region of fluid. Other than in a few simple model cases, this behavior is still far from understood. By reformulating these problems in terms of partial differential equations, this project will study this behavior. One particular question of interest is how does the shape of a drum influence which part of the drum its vibrations are typically localized to. Another aim is to study free boundary problems. A free boundary is the region separating two different materials, such as the interface between water and the air in the ocean or between an insulating material and the air. The free boundary can be modeled via solving differential equations, and understanding its shape and the scale at which it appears smooth has applications to optimal shape design, electromagnetism, and fluid flow. This project contributes to the development of the US workforce through mentoring of undergraduate students.In this project, the PI will study eigenvalues and eigenfunctions on convex domains in Euclidean space and the sphere. A main goal of the project is to further the understanding of the level sets of the eigenfunction. The starting point is a quantitative property, namely the convexity of the level sets, and the aim is to use and develop techniques from elliptic and differential geometry theory to establish quantitative properties involving their shape and location with the domain. This would lead to understanding the region of the domain where the eigenfunctions localize in cases where no explicit formulae are available. Another part of the project involves variants of the classical Friedland-Hayman inequality concerning eigenvalues on the sphere. This will involve developing tools from convex geometry, isoperimetric inequality theory, and Brenier's optimal transportation mappings. This will be applied to prove regularity properties of minimizers of associated free boundary problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.