Estimates on eigenvalues and eigenfunctions in convex settings

椭圆型微分方程被用来描述各种各样的物理现象。这个项目的一个主要目标是研究一类特殊的方程，这些方程可以用来模拟滚筒或非均匀材料的振动行为，热导体的温度分布，或在流体区域中运动的颗粒的随机运动的分布。除了在几个简单的模型案例中，这种行为还远未被理解。通过将这些问题重新表述为偏微分方程，本项目将研究这一行为。一个特别令人感兴趣的问题是，滚筒的形状如何影响其振动通常局限在滚筒的哪个部分。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模拟，了解它的形状和它看起来光滑的尺度对于优化形状设计、电磁和流体流动都有应用。这个项目通过指导本科生来促进美国劳动力的发展。在这个项目中，PI将学习欧几里德空间和球面上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了解本征函数的水平集。起点是定量性质，即水平集的凸性，目的是使用和发展椭圆和微分几何理论中的技术来建立涉及它们的形状和位置与域的定量性质。这将导致理解在没有显式公式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。这将涉及到从凸几何、等周不等论和Brenier的最优运输映射中开发工具。这将被用来证明相关自由边界问题极小化的正则性。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势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.