Mean curvature measure of free boundary

自由边界问题处理的是边界部分先验未知的区域内的偏微分方程。为了确定区域，在边界的未知部分（称为自由边界）上施加了一些附加条件。然后试图确定自由边界和微分方程的解。相变和最优形状的研究导致考虑各种功能，测量物理系统的总能量。变分技术使我们能够得出这个问题的弱解存在的结论。然后可以继续建立解的规则性，然后，希望，研究自由边界本身的平滑性。物理系统往往具有最小的能量，因此我们所寻求的领域被认为是最优的。这意味着区域的小扰动会增加能量，因此在非常小的尺度上解和自由边界具有很好的结构。事实上，人们期望自由边界是一个相对于域内部扰动的几乎最小的表面。尽管物理环境简单，但数学公式却非常复杂。一个重要的模型是这三位作者在1984年研究的Alt-Caffarelli-Friedman （ACF）泛函。它是主要的自由边界问题之一，为理论提供了关键的见解。此外，ACF函数，除其他外，模拟了两种完美流体或射流的平衡。最近，PI注意到ACF问题与最小曲面理论密切相关。人们可以把最小表面想象成将电线轮廓浸入肥皂溶液后得到的肥皂膜。肥皂膜在所有跨越电线边界的薄膜中具有最小的面积。事实上，最小表面的小块以肥皂膜的形式出现，它们的平均曲率为零。人们可以很自然地期望ACF问题和最小曲面之间存在很强的并行性。至少在三维空间中，ACF问题的每一个完整的粘度解都定义了一个由自由边界的分量决定的具有多个端点的最小表面。本课题的目的是研究由不同处理方式的非线性偏微分方程驱动的自由边界问题，它与最小曲面理论、可整流变分理论和最小变分理论以一种奇特的方式平行。特别是，我们感兴趣的是对这些问题的整个粘度解进行分类（Bernstein型定理），并根据Hausdorff和Minkowski的维度估计可能的不规则点的大小。

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.