Continuous and Discrete Free Boundary Problems for Partial Differential Equations

Purdue University, West Lafayette IN

Investigators

Abstract

Free boundary problems are problems for partial differential equations (PDEs) which are defined in a domain with a boundary that is not known beforehand (i.e. "free"). A further quantitative condition must be then provided at the free boundary to exclude indeterminacy. Problems of this sort arise in a large number of areas of applied and industrial interest. The paradigm example is the classical Stefan problem describing the melting and solidification of the ice: the free boundary here is the moving interface between the regions occupied by the water and the ice. Other important examples occur in filtration through porous media, where free boundaries occur as fronts between saturated and unsaturated regions, and others come from combustion (propagation of the flame front), mathematical finance (optimal time for exercising an option), biology (regions occupied by different species) and so on. Because of the abundance of applications in various sciences and real world problems, free boundary problems are considered today as one of the most important directions in the mainstream of the analysis of partial differential equations and offer opportunities of collaboration among mathematicians, physicists, engineers, material scientists, finance practitioners and other industrial researchers, biologists, and other scientists. This project consists of two main parts. The first part concerns problems that naturally exhibit free boundaries of codimension two, which are often called thin free boundaries. Such problems appear in many applications, such as elasticity, math finance, boundary phase transitions. They are also related to problems for nonlocal integro-differential operators such as the fractional Laplacian through extension operators. The second part concerns the fascinating limiting configurations in various growth/aggregation models such as the Abelian sandpile growth model, which can be viewed as a discrete version of Poincare's balayage problem in potential theory and is closely related to the classical obstacle problem. While there has been a significant progress in such problems in recent years, there are still many important questions that remain to be answered and the current project aims to contribute in that directions. Particular questions to be studied include various thin free boundary problems: for parabolic PDEs with variable coefficients, for almost minimizers (elliptic case), properly stated two- and multi-phase problems, as well as the systematic study of the limits of discrete-valued least superharmonic majorants (so-called discrete-valued Perron method). This may lead to better understanding of the formation of polygonal and piecewise-quadratic limits in Abelian sandpile and other growth/aggregation models and related PDEs. 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.