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

本项目旨在研究椭圆型偏微分方程（PDE）解及其梯度的局部性质。这样的解决方案的例子包括温度分布在一个身体，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，它描述了解的局部复杂性。该项目的超越目标是了解频率函数如何控制解决方案及其梯度的各种局部特征。椭圆方程解的定量性质在其他数学领域也有许多应用，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。主要研究者（PI）通过一系列关于频率函数方法的讲座和小型课程，积极传播作为该项目一部分获得的新思想和成果。作为本项目的一项教育倡议，PI将根据这些系列讲座准备一篇临时文章。频率函数是由Almgren引入的，Garofalo和Lin将其应用于椭圆型方程解的定量唯一延拓。最近的进展，在理解的行为的频率函数导致了证明Nadirashvili的猜想和部分解决的丘的猜想。这些结果和几何组合的方法，他们是基于开辟了新的可能性，在研究的解析，几何和拓扑性质的解决方案，二阶椭圆型偏微分方程。该项目的目标之一是研究有界频率的椭圆方程的解及其对拉普拉斯本征函数的应用，其中包括本征函数的限制估计和局部化性质。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，并研究这类函数的典型行为。对于一些确定性的问题，椭圆偏微分方程的解决方案的行为，目前是遥不可及的，PI研究相应的随机解的典型行为，这些随机解是通过采用具有独立高斯系数的斯捷克洛夫本征函数的随机组合来定义的。该奖项反映了NSF的法定使命，并且通过使用基金会的智力价值和更广泛的影响评审进行评估，被认为是值得支持的的搜索.

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.