Nonminimizing and Min-max Solutions to Free Boundary Problems

Free boundary problems are models where one of the unknowns is a shape or interface rather than a function, with part of the model describing the rate at which the interface moves or the shape changes (much like how a differential equation describes how a function changes). This kind of model arises naturally in the study of fluids (the waves on the surface of a body of water), petroleum engineering (the evolution of a fully saturated region in a porous material), phase transitions (the shape of a melting block of ice), and combustion (the motion of a flame front in a forest fire). Our current mathematical tools work best for steady-state solutions to such problems, and moreover to ones which minimize an energy. The purpose of this project is to develop approaches to study moving interfaces and non-minimizing steady states. Better mathematical understanding may lead to smarter and safer approaches to the applied problems through rigorous approximation schemes, analysis of stability under perturbations, and rigid qualitative properties of solutions. At the same time, the project trains graduate and undergraduate students in a mathematical subject with important industrial applications. The specific topics covered by the project are compactness theorems for critical points to Bernoulli-type free boundaries, and applications of these to gravity water waves and other min-max constructions; general free boundary problems, arising from semilinear elliptic equations not admitting a strong maximum principle, with the goal of treating them all together with minimal assumptions on the structure of the nonlinearity and the minimality or positivity of solutions; and parabolic free boundaries of several types, including Bernoulli and free transmission, for which minimality is not even an available concept, and the task is to understand the structure of the free boundary to a level not previously possible because of this. The approach here represents a complete shift in perspective on nonconvex free boundaries and provides a framework to study the much richer global structure of nonminimizing solutions. Postdoctoral researchers, graduate students, and undergraduates are involved in the work of the project. 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.