The main objective of this project is to study problems that naturally exhibit free boundaries of codimension two that live in an apriori given manifold. Free boundary problems of this type appear naturally in many applications, ranging from the theory of elasticity (the Signorini problem, or the thin obstacle problem), mathematical finance (American options), combustion, boundary heat control, and more generally, in problems with boundary phase transitions. An important source of such problems that has emerged recently is the study of free boundary problems governed by nonlocal integro-differential operators, such as the fractional Laplacian. There has been a significant progress in recent years in some problems with lower-dimensional free boundaries. However, there are still a number of fundamental questions that have yet to be addressed and the current project aims to do just that. Particular questions include monotonicity formulas for nonlocal operators, appropriate generalizations of the boundary Harnack principle, and the partial hodograph-Legendre transform, both in stationary and time-dependent situations.
Free boundary problems are problems for partial differential equations that are defined in domains whose boundaries are not known beforehand (i.e., are "free"). A further quantitative condition must be then provided at the free boundary to prevent indeterminacy. Problems of this sort arise in a large number of areas of applied and industrial interest. The paradigmatic example is the classical Stefan problem, which is to model the melting and solidification of 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 (e.g., an oil field), 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 forth. Because of the abundance of applications in various sciences and real world problems, free boundary problems are considered today to be one of the most important directions in the mainstream of the analysis of partial differential equations and offer opportunities for collaboration between mathematicians, physicists, engineers, materials scientists, financial practitioners and other industrial researchers, and biologists.
