Estimates on eigenvalues and eigenfunctions in convex settings

凸设置中特征值和特征函数的估计

• 批准号: 2042654
• 负责人: Thomas Beck
• 金额: $ 10.72万
• 依托单位: Fordham University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2024-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2042654&HistoricalAwards=false
• 关键词: Estimates; eigenvalues; eigenfunctions; convex; settings

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.

椭圆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，或在流体区域中运动的粒子的随机运动分布。除了在一些简单的模型案例中，这种行为还远未被理解。通过用偏微分方程重新表述这些问题，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定位在鼓的哪一部分。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物质的区域，例如海洋中水和空气之间的界面，或者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，理解它的形状和它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，促进了美国劳动力的发展。在这个项目中，PI将研究欧几里德空间和球面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特征函数的水平集。本文的出发点是水平集的一个定量性质，即水平集的凸性，目的是利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定域的定量性质。这将导致理解在没有显式公式可用的情况下特征函数局部化的定义域的区域。项目的另一部分涉及经典弗里德兰-海曼不等式的变体，该不等式涉及球体上的特征值。这将涉及从凸几何、等周不等式理论和Brenier的最优运输映射中开发工具。这将应用于证明相关自由边界问题的极小值的正则性。该奖项反映了美国国家科学基金会的法定使命，并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。

项目成果

Quantitative bounds on Impedance-to-Impedance operators with applications to fast direct solvers for PDEs

• DOI: 10.2140/paa.2022.4.225
• 发表时间: 2021-03
• 期刊: ArXiv
• 作者: Thomas Beck;Y. Canzani;J. Marzuola

Localisation of the first eigenfunction of a convex domain

凸域第一特征函数的局部化

• DOI: 10.1080/03605302.2020.1843050
• 发表时间: 2021
• 期刊: Communications in Partial Differential Equations
• 影响因子: 1.9
• 作者: Beck, Thomas

The Friedland–Hayman inequality and Caffarelli’s contraction theorem

弗里德兰海曼不等式和卡法雷利收缩定理

• DOI: 10.1063/5.0046058
• 发表时间: 2021
• 期刊: Journal of Mathematical Physics
• 影响因子: 1.3
• 作者: Beck, T.;Jerison, D.
• 通讯作者: Jerison, D.

Limiting eigenfunctions of Sturm–Liouville operators subject to a spectral flow

受谱流影响的 Sturm-Liouville 算子的极限本征函数

• DOI: 10.1007/s40316-020-00142-6
• 发表时间: 2020
• 期刊: Annales mathématiques du Québec
• 作者: Beck, Thomas;Bors, Isabel;Conte, Grace;Cox, Graham;Marzuola, Jeremy L.
• 通讯作者: Marzuola, Jeremy L.