Geometry of Measures

大多数人都同意欧几里德空间是平坦的，欧氏空间中的勒贝格测度是加倍的(即半径为2r的球的体积与半径为r的球的体积在上方有界)。Toro将研究这两个概念之间的关系：平坦度和加倍。她将展示这两个概念的深度交织程度。这个项目的主要目标之一是证明在一个度量的加倍性质和它的支集的平坦性之间存在等价性。这里讨论的一些问题自然地适合于具有弱边界条件的自由边界正则性问题的框架。为了解决这一建议中所描述的问题，需要从偏微分方程、调和分析和几何测度论等工具入手。这里讨论的想法属于一个更大的项目，该项目旨在建立正则性的弱概念用于许多目的足以回答分析和几何测度论中的基本问题。自由边界问题是物理学和工程学中自然而然产生的问题。自由边界可能表现为流体与空气或水与冰之间的界面。在过滤问题中，研究水是如何从由多孔介质(比如土壤)组成的大坝中过滤出来的，自由边界将湿部分和干部分分开。许多人关心自由边界的规律性问题。在这项提案中，调查员讨论了障碍问题自由边界的正则性问题。Alt和Caffarelli等人已经研究过这个问题。他们的假设限制了可以考虑的障碍的种类。在这个项目中，允许设置更大类别的障碍，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.