The Frequency Function Method in Elliptic Partial Differential Equations and Harmonic Analysis

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This project is aimed at the study of local properties of solutions of elliptic partial differential equations (PDE) and their gradients. Examples of such solutions include the temperature distribution in a body, electromagnetic fields, and gravitational fields. One of the tools in the analysis of solutions to elliptic equations is the so-called frequency function, which describes the local complexity of the solution. The overreaching goal of the project is to understand how the frequency function controls various local characteristics of the solution and its gradient. The work on quantitative properties of solutions of elliptic equations has numerous applications in other areas of mathematics, including spectral geometry, geometric measure theory, control theory, and mathematical physics. The project provides research training opportunities for graduate students. The principal investigator (PI) is active in disseminating the new ideas and results obtained as part of this project through series of lectures and minicourses on the frequency function method. As an educational initiative within this project, the PI will prepare an expository article based on these series of lectures. The frequency function was introduced by Almgren and was applied to quantitative unique continuation for solutions of elliptic equations by Garofalo and Lin. Recent progress in the understanding of the behavior of the frequency function led to a proof of Nadirashvili's conjecture and a partial solution of Yau's conjecture. These results and the geometric combinatorial method on which they are based open up new possibilities in the study of analytic, geometric, and topological properties of solutions to second order elliptic PDE. One of the goals of the project concerns the study of solutions to elliptic equations with bounded frequency and their applications to Laplace eigenfunctions, which includes restriction estimates and localization properties of eigenfunctions. Another goal is to introduce a new framework to study random harmonic functions of bounded frequency and investigate the typical behavior of such functions. For a number of deterministic questions on the behavior of solutions of elliptic PDE that are currently out of reach, the PI studies the typical behavior of the corresponding random solutions defined by taking random combinations of the Steklov eigenfunctions with independent Gaussian coefficients.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.