Analysis and Geometry of Free Boundaries

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Partial Differential Equations (PDE) are the language of physics. In fact, the first partial differential equations naturally arose in physics when trying to describe the propagation of heat, the propagation of waves as well as electromagnetism. Many problems in engineering also rely on the theory of PDEs, as well as the ability to approximate their solutions. The scientific part of this project is concerned with the study of specific families of PDEs that model various physical phenomena. These include the melting of ice, combustion, chemical diffusions, liquid crystals, and image processing. One of the main scientific goals of the project is to develop new mathematical tools that can be used to better understand the physical phenomena being modeled. This will create new avenues to analyze them and enhance their comprehension. The investigator will also organize a week-long workshop focused on first-generation undergraduate students who are interested in mathematics. The students will participate in minicourses, attend research talks, and have informal conversations with mathematicians who work in different sectors. The workshop will contribute to the development of a mathematically well-versed and diverse workforce. This project is driven by questions arising in free boundary problems and geometric measure theory. In the applied sciences one often encounters free boundaries, which arise when the solution to a problem consists of a function (often satisfying a partial differential equation) and a set where this function has a specific behavior. The investigator will study a variety of problems that are motivated by the study of the regularity of the function and the geometry of the associated set. These are central questions that are ubiquitous in both theoretical and applied mathematics and can be directly used to model various physical phenomena. The project’s main goal is to contribute to a better understanding of problems involving nonlocal equations, almost minimizers with free boundaries, and minimizers for anisotropic energies. The first class of problems to be investigated involves PDE of fundamental importance for mathematical modeling. In particular, numerous applied phenomena give rise to nonlocal equations, such as nonlocal image processing and liquid crystals. The study of almost minimizers with free boundaries has an 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. The investigator will develop new tools which will address related questions at the interface of free boundary problems and geometric measure theory.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.