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：塔蒂亚娜·托罗（Tatiana Toro），华盛顿大学-0244834 *******************************************************************************************************************************这两个主要问题。第一个涉及在连续阈值以下的自由边界规则性问题。在寻找在连续阈值以下的两阶段无边界规则性问题的正确配方时，PI和C. Kenig提出了重要的发现。关于自由边界规律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和她的合着者找到了一个全球标准，该标准保证了自由边界的规律性，但不涉及平坦。由此激励，他们正在开发一项技术的新闻网，以证明几种不同的设置中的自由边界的规律性。该提案中解决的第二个问题涉及施罗辛格流的平滑解决方案。在过去的几年中，几位作者将注意力集中在Schroedinger流上，这是分散PDE的几何等效物。 PI和合着者的方法在几何分析中的分散方程理论与传统技术之间建立了一个桥梁。在物理和工程学中自然出现了无边界问题。自由边界可能显示为流体和空气之间的界面，或水和冰。在过滤问题中，研究水如何从多孔培养基（例如地球）制成的大坝中过滤，自由边界将湿部分与干部分分开。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年中，研究者和C. Kenig进行了一个联合计划，其主要目标是完全了解低于连续阈值的边界规则性问题（在上面的示例中，这与水的速度不是连续功能相对应）。该计划的成功增强了这样一个观念，即弱规则性概念适合研究到目前为止仅在经典的规律性概念方面考虑的问题。提议研究自由边界规则性问题的方法应具有永恒的影响。它提供了用于几何分析中使用的标准技术的替代方法，以证明一组是``平滑''，它要求它在某种意义上是平坦的，这很适合给定的问题。关于施罗丁格流的提议的计划是几何分散系统领域发展的重要一步。该项目的成功将有益于几何分析和分散方程的领域。此外，由于与Heisenberg模型有关铁磁旋转系统的关系，它可能会对这个物理问题产生一些见解。