LEAPS-MPS: Elliptic theory for the Schrodinger operator

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The Laplace equation is the prototypical second-order elliptic partial differential equation (PDE). Consequently, solutions to the Laplace equation, known as harmonic functions, are a fundamental component of PDE theory. But these functions are also important to many other areas of science and engineering, like complex analysis, harmonic analysis, geometry, physics, and engineering. As such, harmonic functions have been extensively studied and are well understood. While the Laplace equation models steady-state phenomena in a uniform environment, the world that we live in is not an isotropic vacuum. The mathematical equations that govern many natural phenomena like electromagnetism, astronomy, and fluid dynamics are often more complicated than Laplace’s equation. For example, the Schrodinger equation describes the behavior of quantum-mechanical waves, while its generalizations describe even more complex settings. Therefore, there is a need to understand the properties of solutions to such general elliptic PDEs. This project combines mathematical pursuits in harmonic analysis with the goal of promoting the inclusion and retention of a diverse mathematical community. The latter objective will be achieved through an orientation program for incoming graduate students along with extra-curricular mentorship programs.In this project, the PI will explore how and to what extent the presence of lower-order terms and variable coefficients affects the behavior of solutions to elliptic equations. With the Schrodinger equation serving as the standard example, these effects will be examined through the three distinct perspectives of unique continuation, homogenization, and solvability. Harmonic functions have the following unique continuation properties: locally, they cannot vanish to infinite order; and if defined globally, Liouville’s Theorem asserts that they cannot be bounded everywhere. Motivated by Landis’ conjecture, one facet of this program seeks to precisely quantify these kinds of local and global behaviors for solutions to generalized Schrodinger equations. By going further and considering elliptic equations with periodic coefficients, this program also explores the interplay between homogenization theory and unique continuation. Carleman estimates and complex analysis techniques will be combined with compactness arguments to accomplish this feat. Work on the solvability of the Dirichlet and Neumann boundary value problems for the Laplace equation led to a huge development in the theory of PDEs and harmonic analysis. The PI’s previous work will be used to explore the questions of solvability for general systems of elliptic PDEs with lower order terms, and further knowledge will be gained while bringing together ideas from distinct areas of mathematics.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.