Quantitative Studies of Solutions of Partial Differential Equations

拉普拉斯特征函数是由偏微分方程组中的拉普拉斯算子模拟的振动的基本模式。PDE被用来模拟物理世界中的各种现象。例如，在音乐中，本征函数是吉他弦的振动或鼓膜的振动。在数学中，本征函数是常见三角函数的高维类比。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。这些节点组是金属板振动最小的地方。在量子力学中，节点集是最不可能找到量子粒子的地方。理解给定能量(即本征值)的本征函数的轮廓取决于我们回答以下问题的能力：(I)本征函数的水平集相对于频率的大小有多大？以及(Ii)关于给定集合上的本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，以及针对K-12学生和普通公众的推广活动。该项目的研究目标集中在拉普拉斯算子特征函数水平集的定量研究和偏微分方程组求解的相关课题上。主要研究者(PI)的目的是研究光滑曲面上各种Laplace算子的节点集和奇异集的测度的上界。PI还将研究周期性椭圆齐化中特征函数的节点集的界限。这一研究将有助于理解非均质介质的定量性质。作为项目的一部分，另一个研究方向是研究具有正则势和奇异势的椭圆型偏微分方程解的定量唯一延拓。其结果将是更好地理解各种类型的偏微分方程强的唯一连续性质。这将导致在反问题和控制理论中的一些应用。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。

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.