Laplace Eigenfunctions and Unique Continuation

Stanford University, Stanford CA

Investigators

Abstract

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.