Geometry of Measures

大多数人都同意欧几里得空间是平坦的，欧几里得空间中的勒贝格测度是加倍的（即半径为2r的球的体积以半径为r的球的体积为界）。托罗将研究这两个概念之间的关系：平面和加倍。她将展示这两个概念深深交织在一起的程度。这个项目的主要目标之一是证明在一个测度的加倍性质和它的支撑的平坦度之间存在等价性。这里讨论的一些问题很自然地符合具有弱边界条件的自由边界正则性问题的框架。为了解决本建议中描述的问题，需要从偏微分方程，调和分析和几何测量理论中获得工具。这里讨论的想法属于一个更大的项目，旨在建立弱规则概念在许多目的下足以回答分析和几何测量理论中的基本问题。自由边界问题在物理学和工程学中自然出现。自由边界可以表现为流体与空气或水与冰之间的界面。在研究水如何从多孔介质（如土）筑成的大坝中过滤的过滤问题中，自由边界将湿部分与干部分分开。许多人关心自由边界的规律性问题。在这个建议中，研究者解决了障碍问题的自由边界的规则性问题。这个问题已经被Alt和Caffarelli等人研究过。他们的假设限制了可以考虑的障碍种类。在这个项目中，允许更大类别的障碍，希望潜在的结果将涵盖更广泛的物理问题。

Most people agree that Euclidean space is flat and that the Lebesgue measure in Euclidean space is doubling (i.e the volume of the ball of radius 2r is bounded above by the volume of the ball of radius r). Toro will investigate the relationship between these two notions: flatness and doubling. She will show the extent to which these two concepts are deeply intertwined. One of the main goals of this project is to show that there exists an equivalence between the doubling properties of a measure and the flatness of its support. Some of the questions discussed here fit naturally into the framework of free boundary regularity problems with weak boundary conditions. In order to address the questions described in this proposal tools from partial differential equations, harmonic analysis and geometric measure theory are needed. The ideas discussed here belong to a larger project that intends to establish that weak notions of regularity are for many purposes sufficient to answer basic questions in analysis and geometric measure theory. Free boundary problems arise naturally in physics and engineering. The free boundary may appear as the interface between a fluid and the air, or water and ice. In the filtration problem, which studies how water filtrates from a dam made of a porous medium (say earth), the free boundary separates the wet part from the dry part. Many people are concerned with the question of the regularity of the free boundary. In this proposal the investigator addresses the question of the regularity of the free boundary for the obstacle problem. This problem has been studied by Alt and Caffarelli among others. Their hypothesis restrict the sort of obstacles that can be considered. In this project a larger class of obstacles is allowed with the hope that the potential results will cover a broader spectrum of physical problems.