Geometric Measure Theory and Free Boundary Regularity Problems

几何测度论与自由边界正则问题

• 批准号: 0244834
• 负责人: Tatiana Toro
• 金额: $ 9.3万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0244834&HistoricalAwards=false
• 关键词: Geometric; Measure; Theory; Free; Boundary

PI: Tatiana Toro, University of WashingtonDMS-0244834*****************************************************************************This proposal addresses two main questions. The first one concerns the free boundary regularity problem below the continuous threshold. In the search of the right formulation for the two-phase free boundary regularity problem below the continuous threshold, the PI and C. Kenig made an important discovery. The common theme to most of the results in the literature concerning the regularity of the free boundary is that near a flat point the free boundary is regular. The PI and her co-author found a global criterion which guarantees the regularity of the free boundary but which does not involve flatness. Motivated by this, they are in the process of developing a newset of techniques to prove regularity of the free boundary in several different setups. The second question addressed in this proposal concerns the existence of smooth solutions for the Schroedinger flow. In the last couple of years several authors have focused their attention on the Schroedinger flow, which is the geometric equivalent of a dispersive PDE. The approach of the PI and co-authors establishes a bridge between the theory of dispersive equations and the traditional techniques in geometric analysis.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 authors have studied the central problem of characterizing the regularity of the free boundary. For the last 8 years the investigator and C. Kenig have undertaken a joint program whose main goal has been to fully understand the boundary regularity problem below the continuous threshold (in the example above this corresponds to the case when the speed of the water is not a continuous function). The success of this program has enhanced the idea that weak notions of regularity are suitable to study problems that so far had only been considered in terms of classical notions of regularity. The approach proposed to study free boundary regularity problems should have an everlasting impact. It offers an alternative to the standard techniques used in geometric analysis to prove that a set is ``smooth'' which require that it be flat, in some appropriate sense, which is well adapted to the given problem. The proposed program concerning the Schroedinger flow is a significant step forward in the development of the area of geometric dispersive systems. The success of this project will benefit both geometric analysis and the field of dispersive equations. Furthermore by virtue of being related to the Heisenberg model for a ferromagnetic spin system it might yield some insight into this physical problem.

PI：Tatiana Toro，华盛顿大学DMS-0244834******************************************** ************************************** 该提案解决了两个主要问题。第一个涉及连续阈值以下的自由边界正则性问题。在寻找低于连续阈值的两相自由边界正则问题的正确公式时，PI 和 C. Kenig 取得了重要发现。关于自由边界规则性的文献中大多数结果的共同主题是，在平坦点附近，自由边界是规则的。 PI 和她的合著者找到了一个保证自由边界规则性但不涉及平坦度的全局标准。受此启发，他们正在开发一套新的技术，以证明几种不同设置中自由边界的规律性。该提案中解决的第二个问题涉及薛定谔流的平滑解的存在性。在过去几年中，一些作者将注意力集中在薛定谔流上，它是色散偏微分方程的几何等价物。 PI 和合著者的方法在色散方程理论和几何分析中的传统技术之间架起了一座桥梁。自由边界问题在物理和工程中自然出现。自由边界可以表现为流体和空气、或水和冰之间的界面。在过滤问题中，研究水如何从多孔介质（例如泥土）制成的水坝中过滤出来，自由边界将潮湿部分与干燥部分分开。许多作者研究了表征自由边界规律性的核心问题。在过去的 8 年里，研究者和 C. Kenig 开展了一项联合计划，其主要目标是充分理解连续阈值以下的边界规律性问题（在上面的示例中，这对应于水速不等于水流速度的情况）连续函数）。该计划的成功增强了这样一种观念，即弱正则性概念适合研究迄今为止仅根据经典正则性概念来考虑的问题。所提出的研究自由边界正则性问题的方法应该会产生持久的影响。它提供了几何分析中使用的标准技术的替代方法，以证明集合是“平滑的”，这要求它在某种适当的意义上是平坦的，这很好地适应了给定的问题。所提出的有关薛定谔流的计划是几何色散系统领域发展的重要一步。该项目的成功将有利于几何分析和色散方程领域。此外，由于与铁磁自旋系统的海森堡模型相关，它可能会对这个物理问题产生一些见解。