Quantitative Studies of Solutions of Partial Differential Equations

拉普拉斯本征函数是由偏微分方程中的拉普拉斯算子描述的振动的基本模态。偏微分方程用于模拟物理世界中的各种现象。在音乐中，特征函数是，例如，吉他弦的振动或鼓膜的振动。在数学中，特征函数是我们所熟悉的三角函数的高维类似物。在量子力学中，本征函数被称为能态。对节点集（即零水平集）的研究可以追溯到18世纪Chladni模式的发现。这些节点集是金属板振动最小的地方。在量子力学中，节点集是量子粒子最不可能被发现的地方。理解给定能量（即特征值）的特征函数的轮廓取决于我们回答以下问题的能力：(i)相对于频率，特征函数的水平集的大小有多大？（ii）关于某个给定集合上的特征函数的信息如何传播到附近的集合？该项目为本科生和研究生提供培训机会，并为K-12学生和公众提供外展活动。本课题的研究目标集中于拉普拉斯算子特征函数水平集的定量研究以及PDE解的相关课题。主要研究光滑表面上各种拉普拉斯算子的节点集和奇异集测度的上界。PI还将研究周期椭圆均匀化中特征函数节点集的界。这条研究路线将有助于理解异质介质的定量性质。项目的另一个研究方向是正则和奇异势椭圆偏微分方程的定量唯一延拓研究。结果将更好地理解各种类型PDE的强唯一延拓性质。这将导致许多应用在逆问题和控制理论。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。

Laplacian eigenfunctions are the fundamental modes of vibrations which are modeled by the Laplace operator in partial differential equations (PDE). PDE are used to model various phenomena in the physical world. In music, eigenfunctions are, for example, oscillations of a guitar's string or vibrations of a drum's membrane. In mathematics, eigenfunctions are the higher dimensional analogs of the familiar trigonometric functions. In quantum mechanics, eigenfunctions are known as the energy states. The study of nodal sets (that is, zero-level sets) dates back to the eighteenth century with the discovery of the Chladni patterns. These nodal sets are places where a metal plate vibrate the least. In quantum mechanics, nodal sets are where quantum particles are least likely to be found. Understanding the profile of eigenfunctions of a given energy (i.e., eigenvalue) hinges on our ability to answer the following questions: (i) How large are the sizes of level sets of eigenfunctions with respect to the frequency?; and (ii) How does the information about eigenfunctions on some given set propagate to nearby sets? This project provides training opportunities for undergraduate and graduate students, as well as outreach activities aimed at K-12 students and the general public.The research objectives of this project focus on quantitative studies of level sets of eigenfunctions for the Laplace operator and related topics for solutions of PDE. The principal investigator (PI) aims to study the upper bounds on the measure of nodal and singular sets for various Laplace operators on smooth surfaces. The PI will also study bounds on nodal sets of eigenfunctions in periodic elliptic homogenization. This line of investigation will contribute to the understanding of quantitative properties of heterogeneous media. Another research direction that will be pursued as part of the project is the study quantitative unique continuation for elliptic PDE with regular and singular potentials. The outcome will be a better understanding of the strong unique continuation property for various types of PDE. This will lead to a number of applications in inverse problems and control theory.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.