CAREER: Elliptic and Parabolic Partial Differential Equations

Montana State University, Bozeman MT

Investigators

Abstract

Partial differential equations (PDE) are mathematical tools that are used to model natural phenomena like electromagnetism, astronomy, and fluid dynamics, for example. This project is concerned with understanding how the solutions to such equations behave. The Laplace equation is the prototypical elliptic PDE, and it is used to model steady-state homogeneous systems. This equation is studied in the fields of PDE, complex analysis, harmonic analysis, geometry, and engineering; and therefore, the behavior of its solutions (known as harmonic functions) is very well-understood. However, many questions remain regarding the behavior of solutions to more complicated equations like those that model quantum behavior, systems with microscopic structure, and systems that are changing in time. The investigator’s knowledge of harmonic functions will be used to answer these questions, thereby advancing knowledge in the areas of elliptic and parabolic partial differential equations. Motivated by the goal of expanding participation in the mathematical sciences, as well as addressing common issues with retention in academia, this project integrates a workshop in PDE and harmonic analysis whose target audience includes junior mathematicians who are at difficult transitional stages in their careers. The Laplace equation is a PDE that models steady-state phenomena in a truly uniform environment. Since the world that we live in is not an isotropic vacuum, the mathematical equations that govern many natural phenomena 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. As such, there is a need to understand the properties of solutions to general elliptic PDEs. One component of this research project revolves around using known properties of harmonic functions to gain a better understanding of solutions to elliptic equations. Specifically, the investigator will explore how the presence of variable coefficients and lower-order terms affects the behavior of solutions to elliptic equations. This line of inquiry will be addressed through the perspectives of unique continuation and homogenization theory. Given that parabolic equations like the heat equation model the evolution of systems that are changing in time, it is also important to understand how the solutions to such PDE behave. Therefore, in another direction, the investigator will use elliptic theory to tackle problems related to parabolic PDE. More specifically, the investigator will construct a framework for using elliptic theory in high-dimensional settings to understand the properties of solutions to parabolic equations. 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.