Mean curvature measure of free boundary

点击翻译按钮获取中文摘要

Free boundary problems deal with partial differential equations in a domain, a part of whose boundary is a priori unknown. In order to determine the domain some additional conditions are imposed on the unknown part of the boundary which is called a free boundary. One then seeks to determine both the free boundary and the solution of the differential equations. The study of phase transitions and optimal shapes leads to the consideration of various functionals which measure the total energy of the physical system. The variational techniques enables us to conclude that a weak solution to the problem exists. One can then proceed to establish the regularity of the solution and then, hopefully, study the smoothness of the free boundary itself. Physical systems tend to have minimal energy and hence the domain we seek is expected to be optimal. This means that small perturbations of the domain increase the energy and hence the solution and the free boundary at very small scales have nice structure. In fact, one expects that the free boundary is an almost minimal surface with respect to the perturbation from the interior of the domain. Despite its simple physical setting the mathematical formulation is very complicated. An important model is the Alt-Caffarelli-Friedman (ACF) functional studied by these three authors in 1984. It is one of the chief free boundary problems and provides key insights into the theory. Moreover, the ACF functional, among other things, models the equilibrium of two perfect fluids or jet flows. Recently the PI observed that the ACF problem is very closely related to the minimal surface theory. One can think of minimal surfaces as soap films obtained after dipping a wire contour into a soap solution. The soap film has the smallest area among all thin films that span the wire boundary. In fact, small pieces of a minimal surface occur as soap films and they have zero mean curvature. One can naturally expect that there is a strong parallelism with the ACF problem and the minimal surfaces. At least in the three dimensions it is true that every entire viscosity solutions of the ACF problem defines a minimal surface with multiple ends, determined by the components of the free boundary.The aim of this project is to study free boundary problems driven by nonlinear partial differential equations with considerably different treatment, which is parallel in a curious way with the theory of minimal surfaces, rectifiable varifolds and minimal varieties. In particular, we are interested in classifying the entire viscosity solutions of these problems (Bernstein type theorems) and estimating the size of possible irregular points in terms of Hausdorff's and Minkowski's dimensions.