Laplace Eigenfunctions and Unique Continuation

对拉普拉斯算子的特征函数及其零集的兴趣源于对振动膜的研究。目前，拉普拉斯特征函数的研究是偏微分方程理论、微分几何理论和谱理论交叉的一个快速发展的领域。与代数几何、遍历理论和数论等其他数学领域的众多联系使这一领域吸引了各种背景的研究人员。首席研究员（PI）计划继续研究一些长期存在的问题，包括紧流形上拉普拉斯特征函数的行为和椭圆型偏微分方程（PDEs）的解，使用已经产生了许多有趣结果的局部技术和方法。其中一种被称为定量唯一延拓。PI对椭圆偏微分方程解的定量性质的研究在其他数学领域有许多应用，包括节点几何、几何测量理论和数学物理。该项目为当前和未来的研究生制定了一些研究问题。PI通过一系列讲座和小型课程积极传播作为该项目的一部分所取得的成果。该项目的目标之一是支持向具有不同背景的初级研究人员介绍拉普拉斯特征函数理论的活动。倡议致力于鼓励妇女、残疾人和代表性不足的少数群体充分参与科学，促进学术界的多样性。调和函数的倍指数和拉普拉斯算子的本征函数的行为的最新进展导致了Nadirashvili猜想的证明和Yau猜想的部分解。PI将继续与A. Logunov合作解决与丘的猜想有关的问题。特别是，他们计划研究光滑黎曼度规流形和光滑度规曲面上的Dirichlet-Laplace特征函数的节点集。在许多问题中，拉普拉斯特征函数表现为具有相应幂的多项式。例如，Donnelly和Fefferman证明了特征函数的消失阶以特征值的平方根的倍数为界。PI和A. Logunov证明了特征函数的对数的BMO范数是由相同的量有界的。PI将继续研究这个类比；其中一个开放问题是得到特征函数的无维Bernstein不等式，推广Donnelly和Fefferman的结果。研究特征函数的局部方法与二阶偏微分方程解的性质有关。PI计划继续这项研究，首先解决解决梯度的小数量传播问题。本课题的另一个目标是研究有界势薛定谔方程解的消失阶数。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。

The interest in eigenfunctions of the Laplace operator and their zero sets stems from studies of vibrating membranes. Today, the study of Laplace eigenfunctions is a fast developing field which lies on the intersection of the theory of partial differential equations, differential geometry, and spectral theory. Numerous connections to other areas of mathematics, including algebraic geometry, ergodic theory and number theory make this field attractive to researches with various backgrounds. The principal investigator (PI) plans to continue working on a number of longstanding problems on behavior of Laplace eigenfunctions on compact manifolds and solutions to elliptic partial differential equations (PDEs), using local techniques and methods that already led to a number of interesting results. One of those is called quantitative unique continuation. The PI's research on quantitative properties of solutions of elliptic PDEs has numerous applications in other areas of mathematics, including nodal geometry, geometric measure theory, and mathematical physics. A number of research problems for current and prospective graduate students are formulated in the project. The PI is active in disseminating the results obtained as part of this project through series of lectures and mini-courses. One of the goals of the project is to support activities that introduce junior researchers with various backgrounds to the theory of Laplace eigenfunctions. The PI is committed to encouraging full participation of women, persons with disabilities, and underrepresented minorities in science, promoting diversity in academia. Recent progress in the understanding of the behavior of the doubling index of harmonic functions and eigenfunctions of the Laplace operator led to a proof of Nadirashvili's conjecture and a partial solution of Yau's conjecture. The PI will continue to collaborate with A. Logunov on problems related to Yau's conjecture. In particular, they plan to study the nodal sets of the Dirichlet-Laplace eigenfunctions on domains on manifolds with smooth Riemannian metric and on surfaces with smooth metric. In many questions, Laplace eigenfunctions behave as polynomials of a corresponding power. For example, Donnelly and Fefferman proved that the vanishing order of an eigenfunction is bounded by a multiple of the square root of the eigenvalue. The PI and A. Logunov showed that the BMO norm of the logarithm of an eigenfunction is bounded by the same quantity. The PI will continue to study this analogy; one of the open problems is to obtain dimension-free Bernstein's inequalities for eigenfunctions, generalizing results of Donnelly and Fefferman. Local methods developed to study eigenfunctions are connected to more general problems on the properties of solutions of second order PDE. The PI plans to continue this research, first addressing questions of quantitative propagation of smallness for the gradients of solutions. Another goal of the project is to study the order of vanishing of solutions to Schrodinger's equation with bounded potential.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.