Geometric Properties of Second Order Elliptic Partial Differential Equations

许多物理过程，例如热从点燃的蜡烛或散热器在整个房间中传播的方式，或者池塘中的两个波浪相互作用的方式，都是相当好理解的，我们有几个方程来描述这些过程。通常有三个主要问题被问到：（1）方程是否正确地描述了自然？(2)这个方程真的有解吗？(3)这个解能用手或计算机算出吗？这个项目的目的是调查一个不经常被问到的问题：（4）解决方案实际上是什么样子的？如果我们用四支蜡烛和一个散热器来加热一个房间，哪个地方最冷？这些简单的问题导致既有趣和美丽的数学以及令人惊讶的应用在实践中（Google搜索算法本质上基于这些类型的结构，“冷”网页的排名低于“热”网页）。该项目致力于研究（一致）椭圆二阶偏微分方程主要关注的是解决方案的几何属性以及这些属性如何与基础域的几何形状相互作用。将研究的三个明确的问题是（1）极值和临界点的位置，（2）水平集的几何和（3）椭圆算子的本征函数的几何。应用的主要类型将是（4）图的谱方法的分析和（5）数学物理中的局部化现象。主要工具将是从几何分析的基本事实，以几何方式重新解释解析估计，反之亦然，将椭圆方程解释为相关抛物方程的时间不动点（以及抛物方程的相关技术）和谱理论的各个方面。当在图形设置中将估算简化为玩具模型时，离散世界中的工具可能会很有用。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。

Many physical processes, for example the way heat spreads from a lit candle or a radiator throughout the room or the way two waves in a pond interact with each other, are rather well understood and we have several equations that describe the processes. There are usually three main questions that are being asked: (1) Is the equation correctly describing nature? (2) Does the equation actually have a solution? (3) Can the solution be computed, either by hand or on a computer? The purpose of this project is to investigate a question that is not asked quite as often: (4) what does the solution actually look like? If we are heating a room with, say, four candles and a radiator, which spot is going to be the coldest? These simple questions lead to both interesting and beautiful mathematics as well as surprising applications in practice (the Google Search Algorithm is essentially based on these types of structures, "cold" webpages are lower ranked than "hot" webpages).This project is dedicated to the study of (uniformly) elliptic second order partial differential equations, the main focus being on geometric properties of the solution and how those interact with the geometry of the underlying domain. Three explicit problems that will be studied are the (1) the location of extrema and critical points, (2) the geometry of level sets and (3) the geometry of eigenfunctions of an elliptic operator. The main types of applications will be (4) the analysis of spectral methods on graphs and (5) localization phenomena in mathematical physics. The main tools will be basic facts from geometry analysis to reinterpret analytic estimates geometrically and vice versa, the interpretation of elliptic equations as fixed points in time of an associated parabolic equation (as well as associated techniques from parabolic equations) and various aspects of spectral theory. Tools from the discrete world may be useful when reducing estimates to toy models in the graph setting.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.