Regularity Problems in Elliptic Equations

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In the physical sciences, as well as economics, finance, and computer science, the fundamental laws are written in terms of partial differential equations (PDEs). For the quantity under study, these PDEs reveal the relation between the rates of change in different directions, and dictate the properties and evolution of the quantity. At the heart of the study of PDEs is a particular characteristic of solutions known as regularity. Theoretically, regularity is necessary to justify the models, and is the stepping stone for other properties of the solution (existence, uniqueness, and long term behavior). In practice, applications of the models involve numerical solutions of the PDEs and to obtain reasonable results requires stability of the solutions, which is often a consequence of regularity. Broadly speaking, objects that might seem rough often turn out to be regular if they satisfy an elliptic equation or constraint. The investigator will investigate three specific phenomena under this general philosophy: free interface problems, optimal transport maps, and free boundary problems. The results of this project will have numerous applications in other fields such as computer vision, data mining, machine learning, material sciences and biology.The first project will investigate interfaces arising in the study of liquid drops exposed to electric fields. The goal is to show that these interfaces are smooth in low spatial dimensions. This is related to an important conjecture in the physics literature. The second project concerns the optimal transport map between convex domains in Euclidean spaces. The plan is to show that without any extra assumptions on the domains, these maps can be very regular "up to the boundary". The third project is about free boundary problems, in particular problems involving multiple interacting free boundaries. In all three projects, new techniques are needed to incorporate geometric information. These techniques will be useful in addressing a wide range of problems that have been so far out of reach using current techniques.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.