The Frequency Function Method in Elliptic Partial Differential Equations and Harmonic Analysis

椭圆偏微分方程与调和分析中的频率函数法

• 批准号: 2247185
• 负责人: Eugenia Malinnikova
• 金额: $ 50.99万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247185&HistoricalAwards=false
• 关键词: Frequency; Function; Method; Elliptic; Partial

This project is aimed at the study of local properties of solutions of elliptic partial differential equations (PDE) and their gradients. Examples of such solutions include the temperature distribution in a body, electromagnetic fields, and gravitational fields. One of the tools in the analysis of solutions to elliptic equations is the so-called frequency function, which describes the local complexity of the solution. The overreaching goal of the project is to understand how the frequency function controls various local characteristics of the solution and its gradient. The work on quantitative properties of solutions of elliptic equations has numerous applications in other areas of mathematics, including spectral geometry, geometric measure theory, control theory, and mathematical physics. The project provides research training opportunities for graduate students. The principal investigator (PI) is active in disseminating the new ideas and results obtained as part of this project through series of lectures and minicourses on the frequency function method. As an educational initiative within this project, the PI will prepare an expository article based on these series of lectures. The frequency function was introduced by Almgren and was applied to quantitative unique continuation for solutions of elliptic equations by Garofalo and Lin. Recent progress in the understanding of the behavior of the frequency function led to a proof of Nadirashvili's conjecture and a partial solution of Yau's conjecture. These results and the geometric combinatorial method on which they are based open up new possibilities in the study of analytic, geometric, and topological properties of solutions to second order elliptic PDE. One of the goals of the project concerns the study of solutions to elliptic equations with bounded frequency and their applications to Laplace eigenfunctions, which includes restriction estimates and localization properties of eigenfunctions. Another goal is to introduce a new framework to study random harmonic functions of bounded frequency and investigate the typical behavior of such functions. For a number of deterministic questions on the behavior of solutions of elliptic PDE that are currently out of reach, the PI studies the typical behavior of the corresponding random solutions defined by taking random combinations of the Steklov eigenfunctions with independent Gaussian coefficients.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.

该项目旨在研究椭圆偏微分方程（PDE）及其梯度解的局部性质。此类解决方案的示例包括体内的温度分布、电磁场和重力场。分析椭圆方程解的工具之一是所谓的频率函数，它描述解的局部复杂性。该项目的总体目标是了解频率函数如何控制解及其梯度的各种局部特征。关于椭圆方程解的定量性质的研究在数学的其他领域有许多应用，包括谱几何、几何测度论、控制论和数学物理。该项目为研究生提供研究培训机会。首席研究员（PI）通过一系列关于频率函数方法的讲座和迷你课程，积极传播作为该项目的一部分获得的新想法和结果。作为该项目中的一项教育举措，PI 将根据这一系列讲座准备一篇说明性文章。频率函数由 Almgren 引入，并由 Garofalo 和 Lin 应用于椭圆方程解的定量唯一连续。对频率函数行为的理解的最新进展导致了 Nadirashvili 猜想的证明和 Yau 猜想的部分解决。这些结果以及它们所基于的几何组合方法为二阶椭圆偏微分方程解的解析、几何和拓扑性质的研究开辟了新的可能性。该项目的目标之一涉及研究有界频率椭圆方程的解及其在拉普拉斯本征函数中的应用，其中包括本征函数的限制估计和局部化特性。另一个目标是引入一个新的框架来研究有界频率的随机调和函数并研究此类函数的典型行为。对于目前无法解决的有关椭圆偏微分方程解的行为的许多确定性问题，PI 研究了通过采用 Steklov 特征函数与独立高斯系数的随机组合定义的相应随机解的典型行为。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。