Probabilistic Approach to Singular Free Boundary Problems and Applications

Princeton University, Princeton NJ

Investigators

Abstract

A quantitative understanding of many phenomena in science and engineering, such as the growth of crystals and human tissue, the phase segregation of mixtures, the collective behavior of neurons in the brain and of financial institutions, and the manufacturing of alloys, requires mathematical models of dynamically evolving surfaces. The first models of this kind, so-called free boundary problems, were proposed as early as 1831 and became ubiquitous across science and engineering in the second half of the 20th century. Despite the considerable attention devoted to free boundary problems, the understanding of the solutions they generate is still very limited. In this project, the investigator will use a novel approach on free boundary problems, grounded in modern probability theory to advance the understanding of this important class of mathematical models. A particular focus of the project will be on the desirable irregularities present in these models capturing, for example, the rapid growth of a crystal or neural synchronization, as well as the non-smooth nature of the surface of a tumor or between the components of an alloy. Integral parts of the project are supervision of graduate and undergraduate research, and graduate and undergraduate training in probability theory and stochastic analysis. Free boundary problems provide a universal mathematical framework for many phenomena in the natural and social sciences as well as engineering. While the mathematical formulation of free boundary problems dates back to the 19th century and their analysis has received much attention in the 20th century, the understanding of their solutions is still limited. This is largely due to the singularities oftentimes exhibited by the solutions, either in time, capturing for example the rapid growth of a crystal or neural synchronization, or in space, encapsulating for example the non-smooth nature of the surface of a tumor or between the components of an alloy. This project builds on the analysis for the one phase supercooled Stefan problem in one space dimension, a prototypical example of a singular free boundary problem. The aim is to expand the scope of the novel probabilistic solution concept they introduced to include a large set of singular free boundary problems, such as the one phase supercooled Stefan problem for the nonlinear heat equation and its two-phase analogue in one space dimension, the supercooled Stefan problem in multiple space dimensions, possibly with kinetic undercooling and/or surface tension, and the Hele-Shaw problem in multiple space dimensions. In addition, the investigator intends to identify these solutions with large scale limits of random growth models defined by means of interacting particle systems. 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.