RUI: Topics in Free Boundary Problems

Western Washington University, Bellingham WA

Investigators

Abstract

Partial Differential Equations (PDE) describe many physical phenomena, including heat or wave propagation, and electromagnetism. The scientific part of this project focuses on families of PDE that model stochastic control, image processing, chemical diffusion, and combustion. The investigator develops new tools which allow her to better understand questions at the interface of mathematics and other sciences, leading to a deeper understanding of the problems being modeled. Furthermore, the investigator organizes a week-long mathematics workshop focused on first-generation freshmen and sophomore students, addressing a large group that is severely underserved. The students participate in minicourses, attend research talks, and have informal conversations with mathematicians who work in different sectors. The workshop is designed to maximize the chance of success of these students, promoting the progress of science and contributing to the development of a mathematically well-versed and diverse workforce. Finally, the investigator organizes a yearly event to help advanced undergraduates and masters’ students prepare their applications for graduate school in mathematics. This project focuses on questions arising in free boundary problems and geometric measure theory. Free boundaries often appear in the applied sciences, in situations where the solution to a problem consists of a pair: a function (often satisfying a PDE), and a set related to this function. The main questions investigated by this project are related to the regularity of the function and the geometry of the associated set. The investigator answers these questions for problems modeled by nonlocal equations, almost minimizers with free boundaries, and minimizers for anisotropic energies. The first class of problems involves PDE which have fundamental importance for mathematical modeling. In particular, numerous applied phenomena give rise to nonlocal equations, such as nonlocal image processing and liquid crystals. In this part of the project, the investigator develops a technique to obtain results for parabolic, nonlocal equations, from their elliptic counterparts. Secondly, the study of almost minimizers with free boundaries has outstanding potential to treat a new group of physically motivated problems, as the almost minimizing property can be understood as a minimizing problem with noise. Finally, minimizers for anisotropic energies lead to non-uniformly elliptic PDE, generating new, challenging questions in geometric PDE. 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.