Geometric Properties of Second Order Elliptic Partial Differential Equations

许多物理过程，例如热量从点燃的蜡烛或散热器在整个房间中传播的方式，或者池塘中两个波相互作用的方式，都是相当清楚的，我们有几个描述这些过程的方程。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)方程实际上有解吗？(3)解是否可以手工计算或在计算机上计算？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？比方说，如果我们用四根蜡烛和一个暖气片来取暖，那么哪个地方会最冷？这些简单的问题导致了有趣和美丽的数学以及在实践中令人惊讶的应用(Google搜索算法本质上是基于这些类型的结构，“冷”网页比“热”网页排名较低)。这个项目致力于研究(一致)椭圆二阶偏微分方程组，主要集中在解的几何性质以及这些性质如何与潜在区域的几何相互作用。将研究的三个显式问题是(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。主要的应用类型将是(4)对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，将椭圆型方程解释为相关抛物型方程的时间不动点(以及来自抛物型方程的相关技术)，以及谱理论的各个方面。来自离散世界的工具在将评估减少到图表设置中的玩具模型时可能会很有用。这一奖项反映了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。

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.