Geometric Measure Theory and Free Boundary Regularity Problems

PI：WashingtonDMS-0244834*****************************************************************************This大学的Tatiana Toro提案解决了两个主要问题。第一个问题是关于连续阈值以下的自由边界正则性问题。在寻找连续阈值以下两相自由边界正则性问题的正确公式的过程中，Pi和C.Kenig有了一个重要的发现。文献中关于自由边界正则性的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和她的合著者发现了一个全局准则，它保证了自由边界的正则性，但不涉及平坦性。受此启发，他们正在开发一套新的技术，以在几种不同的设置中证明自由边界的规律性。第二个问题涉及薛定谔流的光滑解的存在性。在过去的几年里，几位作者将注意力集中在薛定谔流上，这是色散偏微分方程组的几何等价。PI等人的方法在色散方程理论和几何分析的传统方法之间建立了一座桥梁。自由边界问题在物理和工程中自然而然地产生。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。许多作者研究了表征自由边界正则性的中心问题。在过去的8年里，调查员和C.Kenig进行了一项联合计划，其主要目标是充分了解连续阈值以下的边界正则性问题(在上面的例子中，这对应于水的速度不是连续函数的情况)。这个项目的成功增强了人们的想法，即弱正则性概念适合于研究迄今为止仅根据经典正则性概念来考虑的问题。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。它提供了一种替代几何分析中用来证明一个集合是“光滑”的标准技术，后者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。所提出的关于薛定谔流的程序是几何色散系统领域发展的重要一步。该项目的成功将使几何分析和色散方程领域受益。此外，由于它与铁磁自旋系统的海森堡模型有关，它可能会对这个物理问题有一些见解。

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.