Morrey Inequalities, the Pressureless Euler System, and Semipermeable Obstacle Problems

University Of Pennsylvania, Philadelphia PA

Investigators

Abstract

This project addresses several questions regarding the solvability of certain partial differential equations (PDE). Solutions of these PDE are used to provide information about the models from which they are derived. One of the main challenges is to extract information about these solutions without knowing the functions explicitly. The Principal Investigator (PI) aims to refine and build upon techniques from PDE theory to overcome this challenge. The project focuses on an array of problems at the junction of several areas of mathematics. Moreover, it addresses fundamental problems involving function spaces, the theory of adhesion dynamics, and optimization. As part of this project, the PI mentors a postdoctoral scholar, disseminates results to a broad scientific audience, and continues his involvement in creating opportunities for members of underrepresented groups in mathematics. The methods of calculus of variations have been used to solve optimization problems in mathematics, physics, and engineering for hundreds of years. These methods continue to play an important role in science, and they also help bridge the gap between PDE theory and optimization. The topics studied by this project involve applications of this interplay to modern questions of interest in mathematical analysis. Morrey's inequality is one of the most important inequalities in the theory of Sobolev spaces. The PI has characterized Morrey extremals as solutions of a nonlinear PDE reminiscent of the equation which arises in the study of the classical electric dipole. In this project, he will develop ways to extend these considerations to Hardy-type inequalities in which the admissible functions are constrained to be supported in certain regions of Euclidean space. The pressureless Euler system is one of the basic model equations in cosmology. They were introduced a generation ago to understand low temperature settings in which galaxies form. The PI recently established an existence theorem for solutions in one spatial dimension. This will be further developed by establishing the uniqueness of solutions and by investigating the large time behavior of solutions. In addition, the project considers obstacle problems in which the competitor curve or shape can permeate the obstacle up to a given threshold. The PI aims to develop this optimization theory from scratch by considering the very basic problems which capture the essence of a semipermeable obstacle problem. In particular, he will study associated Hamilton-Jacobi equations and applications to minimal surfaces. 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.