Studies on Quantitative Properties of Solutions of Partial Differential Equations

Vibrations are modeled by the Laplace operator on partial differential equations (PDE). Laplacian eigenfunctions are the fundamental modes of vibrations. In music, eigenfunctions are oscillations of a guitar string or vibrations of a drum's membrane. In mathematics, eigenfunctions are the higher dimensional analogs of the familiar trigonometric functions for the circle. Just as in music, one expects different shapes or concentrations to become more apparent as the frequency becomes larger. Thus, in mathematics, it is essential to understand how these changes of shapes or concentrations depend on the high frequency and how the amplitude of vibrations propagates from one region to other regions. These lead to the quantitative studies of solutions of PDE. 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 the project focus on the sizes of the level sets such as zero-level sets for different PDE models, quantitative propagation of smallness, and their applications to other subjects. The Principal Investigator (PI) studies the sizes of the level sets of eigenfunctions for second order elliptic equations, higher order elliptic equations, and periodic elliptic homogenization. The studies of eigenfunctions advance the progress of PDE in general. The PI also studies the quantitative Cauchy uniqueness and propagation of smallness for elliptic equations with singular weights on measurable sets of positive measure. These problems are essential in the study of Laplacian eigenfunctions and PDE. The applications of quantitative unique continuation to null controllability for heat equations and energy density in spectral theory are also investigated. The progress on the quantitative studies of eigenfunctions and other PDE models will also benefit other related subjects in mathematics, such as control theory and spectral 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.