RII Track-4: Uniqueness and Quantitative Uniqueness of Solutions to Partial Differential Equations

Louisiana State University, Baton Rouge LA

Investigators

Abstract

Nontechnical Description Mathematics provides essential tools for the description and investigation of numerous real-life problems. Rate of change, which is the derivative in mathematics, is fundamental in these problems. Partial differential equations (PDEs) are differential equations that relate unknown multivariable functions and their partial derivatives. They describe a wide variety of seemingly distinct physical phenomena such as sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, or quantum mechanics. Because of this, PDEs play a prominent role in many disciplines including engineering, physics, economics, and biology. One of the central problems in the study of PDEs is about uniqueness and its quantitative properties. The fellowship builds a collaboration between Louisiana State University (LSU) and University of Chicago (UChi) by enabling the PI to make extended research visits to UChi. This project will lead to advancements in understanding uniqueness of PDEs, stimulate the PI's research capacity, strengthen the research program in PDEs and analysis at LSU, and benefit its undergraduate and graduate education. Technical Description The goal of this research is to investigate the quantitative and qualitative properties of uniqueness of solutions to PDEs. Providing quantitative and qualitative information for the solutions is essential in the study of PDEs, which lies in the core of mathematical analysis. Often the most effective way to obtain such information is to first explore the quantitative and qualitative properties of solutions of the equations and then to develop algorithms in accordance; this is a beneficial alternative to the challenge of instead solving PDEs computationally with sufficient accuracy. Quantitative uniqueness, a recent fast-developing area, describes the quantitative behavior of the strong unique continuation property. The proposed research is to study the quantitative uniqueness for elliptic PDEs with singular weights. The outcome will provide a deeper understanding of the strong unique continuation property for this category of PDEs. It is important and interesting to study how exactly the information on a small open set propagates to any other open sets in the domain, which quantifies the weak unique continuation property. Such exploration will lead to many applications in inverse problems and control theory. The PI will explore how the coefficient functions determine the uniqueness and how the shape of the domain influences the uniqueness of solutions for parabolic PDEs. The project will contribute the fundamental understanding of elliptic and parabolic partial differential 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.