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Mean curvature measure of free boundary

自由边界的平均曲率测量

基本信息

批准号：
EP/S03157X/1
负责人：
Aram Karakhanyan
金额：
$ 61.86万
依托单位：
University of Edinburgh
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2020
资助国家：
英国
起止时间：
2020 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FS03157X%2F1
关键词：
Mean curvature measure free boundary

项目摘要

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.

自由边界问题处理区域中的偏微分方程，其边界的一部分先验未知。为了确定区域，在边界的未知部分（称为自由边界）上施加一些附加条件。然后，人们试图确定自由边界和微分方程的解。相变和最佳形状的研究导致考虑各种泛函测量物理系统的总能量。变分技术使我们能够得出结论，弱解的问题存在。然后，我们可以着手建立解的正则性，然后，有希望地，研究自由边界本身的光滑性。物理系统往往具有最小的能量，因此我们寻求的域预计是最优的。这意味着区域的小扰动增加了能量，因此在非常小的尺度下，解和自由边界具有良好的结构。事实上，人们期望自由边界是一个几乎最小的表面相对于扰动的内部区域。尽管它的物理设置很简单，但数学公式却非常复杂。一个重要的模型是Alt-Caffarelli-Friedman（ACF）功能研究这三位作者在1984年。它是主要的自由边界问题之一，并为理论提供了关键的见解。此外，ACF功能，除其他外，模型的两个完美的流体或射流的平衡。最近PI观察到ACF问题与最小曲面理论密切相关。人们可以认为最小表面是将金属丝轮廓浸入肥皂溶液中后获得的肥皂膜。肥皂膜在跨越线边界的所有薄膜中具有最小的面积。事实上，极小曲面的小块以肥皂膜的形式出现，它们的平均曲率为零。人们可以自然地期望，有一个强大的并行与ACF问题和最小曲面。至少在三维空间中，ACF问题的所有粘性解都定义了一个由自由边界分量决定的具有多个端点的极小曲面。本项目的目的是研究由非线性偏微分方程驱动的自由边界问题，其处理方法与极小曲面理论有很大的不同，可求长的簇和极小簇。特别是，我们有兴趣在分类的整个粘度解决方案，这些问题（伯恩斯坦型定理）和估计的大小可能不规则点的Hausdorff和Minkowski的尺寸。