Interfaces, Degenerate Partial Differential Equations, and Convexity

Northwestern University, Evanston IL

Investigators

Abstract

Partial differential equations (PDE) are essential mathematical objects for modeling physical processes. This project aims to understand the properties of some fundamental PDE models for the diffusion of gas, the shape of liquid droplets, and electric transmission in composite materials. Many such processes exhibit an interface, where the equation becomes degenerate or singular. In the case of a model of gas diffusion through a porous medium, the interface is the set which separates the region where there is gas from the region where there is no gas. The boundary of a liquid droplet is another example of an interface. The project will investigate qualitative and quantitative properties of these interfaces, including convexity and smoothness. For a composite material, consisting of two materials with different conductivity properties, the interface is where these materials meet. The Principal Investigator (PI) will study the behavior of the electric field when one of the materials has a very thin part. This project has possible implications for material failure, an important question in Engineering. Students and postdoctoral scholars will be trained on the techniques and theory of these PDE models. This project centers on four topics, connected by the themes of interfaces, degeneracies, and convexity/concavity. The porous medium equation is a nonlinear degenerate parabolic equation used to model the diffusion of gas. The PI will investigate questions of concavity and convexity of solutions and finding global optimal regularity estimates. Secondly, the PI will study linear PDE whose coefficients are discontinuous along two almost touching interfaces, a model for transmission problems and composite materials. In this setting, the PI will investigate new approaches to obtaining optimal gradient estimates in the thin region between the interfaces. A third project is to study linear equations which are parabolic on the interior on a fixed domain but are degenerate at the boundary. These equations arise as linearizations of the porous medium equation and the Gauss curvature flow. The PI will investigate optimal conditions for existence and uniqueness of smooth solutions. Finally, questions of concavity of solutions to the elliptic torsion problem with Dirichlet boundary conditions and the dynamic version of this equation, known as the quasi-static droplet model, will be studied. In addition, the PI will carry out summer research projects with undergraduates, exploring explicit solutions to these 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.