Laplace Eigenfunctions and Unique Continuation

对拉普拉斯算符及其零集的本征函数的兴趣源于对振动膜的研究。今天，拉普拉斯特征函数的研究是一个快速发展的领域，它是偏微分方程组理论、微分几何和谱理论的交叉点。与其他数学领域的大量联系，包括代数几何、遍历理论和数论，使这一领域对具有不同背景的研究人员具有吸引力。首席研究员(PI)计划继续研究紧致流形上拉普拉斯特征函数的性态和椭圆型偏微分方程(PDE)的解的一些长期存在的问题，使用当地的技巧和方法已经导致了一些有趣的结果。其中之一被称为定量唯一延拓。PI对椭圆型偏微分方程解的定量性质的研究在其他数学领域有着广泛的应用，包括节点几何、几何测度论和数学物理。该项目为当前和未来的研究生制定了一些研究问题。国际和平研究所通过一系列讲座和小型课程，积极传播作为该项目一部分取得的成果。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。该协会致力于鼓励妇女、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。在理解调和函数和拉普拉斯算子的本征函数的倍增指数行为方面的最新进展导致了Nadirashvili猜想的证明和Yau猜想的部分解。PI将继续与A·洛古诺夫合作，解决与邱氏猜想有关的问题。特别是，他们计划研究具有光滑黎曼度量的区域上和具有光滑度量的曲面上的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相应幂的多项式。例如，Donnelly和Fefferman证明了特征函数的消失阶是由特征根的平方根的倍数有界的。PI和A.Logunov证明了特征函数的对数的BMO范数有界于相同的量。PI将继续研究这种类比；悬而未决的问题之一是获得特征函数的无量纲Bernstein不等式，推广了Donnelly和Fefferman的结果。用来研究特征函数的局部方法与更一般的关于二阶偏微分方程解的性质的问题相联系。PI计划继续这项研究，首先解决解决方案梯度的小的定量传播问题。该项目的另一个目标是研究具有有限潜力的薛定谔方程的解的消失顺序。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。

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.