The main goal of this project is to understand smoothness (or nonsmoothness) and other quantitative properties of level surfaces of solutions to nonlinear elliptic and parabolic partial differential equations. The PI's will consider semilinear equations that model flame fronts, free boundary problems that are the singular limits of semilinear equations, and other models mentioned in the next paragraph. The regularity for two-dimensional free boundaries in three-space was only recently established. The PI's propose to show that such regularity results extend to a broad class of equations in three space dimensions, including as many physically motivated examples as possible. In view of the strong analogy between the existence and regularity for free boundaries and the corresponding questions about minimal surfaces, it is suspected that regularity will break down in higher dimensions, where one expects to find singular energy-minimizing solutions, analogous to the celebrated examples of the Simons cone and of counterexamples to the Bernstein problem. Finally, the PI's will examine global behavior of level sets. For example, consider a Neumann eigenfunction corresponding to the smallest, nonzero eigenvalue in a convex planar domain. J. Rauch conjectures that all its level curves touch the boundary.
This project focuses on problems in nonlinear differential equations in which the boundary is unknown and has to be determined: a so-called free boundary. The classical Stefan problem of melting ice is an example. In the Stefan problem, the question of interest is the location, as a function of time, of the interface (``free boundary'') between water and ice. The particular problems to which the methods of the present proposal apply also include flame fronts, the interface between oil and water in a flow and the profile of the wake of a boat. The PI's treat both equilibrium and evolution problems. Recently PI Jerison established that certain equilibrium problems in three dimensions have well-behaved solutions, where previously only the two-dimensional case was understood. Because three is the dimension of physical space, this discovery opens the door to other physically meaningful mathematical models, such as models of compressible fluids and capillarity. Another kind of free boundary problem, one which is not at all physical in origin but to which mathematical free boundary theory applies, is the sort that arises in decision theory (PI Stroock). For example, one wants to know when continuing a medical trial is likely to cause more harm than good. Similarly, in a financial context, one wants to know when interest rates and stock prices indicate that it would be wise to buy or sell a stock option. Such questions arise when one is trying to price an American option, that is, an option that can be exercised at any time before it expires instead of at a fixed time.
