Geometric Measure Theory and Free Boundary Regularity Problems

主要研究者：Tatiana Toro，University of Washington DMS-0244834**第一个是连续阈值下的自由边界正则性问题。在寻找连续阈值以下的两相自由边界正则性问题的正确表述时，PI和C。凯尼格有了一个重大发现。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，自由边界是正则的。PI和她的合著者发现了一个全局准则，该准则保证了自由边界的规则性，但不涉及平坦性。受此启发，他们正在开发一套新的技术来证明自由边界在几种不同设置中的规律性。第二个问题解决在这个建议涉及的存在性光滑的解决方案的薛定谔流。在过去的几年里，一些作者把他们的注意力集中在薛定谔流，这是一个色散PDE的几何等效。PI和合作者的方法在色散方程理论和传统几何分析技术之间建立了一座桥梁。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，研究水如何从由多孔介质（比如土）制成的堤坝中渗透出来，自由边界将湿的部分与干的部分分开。许多作者研究了自由边界正则性的中心问题。在过去的8年里，研究人员和C。凯尼格进行了一个联合项目，其主要目标是充分理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）。该计划的成功增强了这样一种想法，即弱正则性概念适用于研究迄今为止仅在正则性经典概念方面考虑的问题。提出的研究自由边界正则性问题的方法应该具有持久的影响。它提供了一个替代的标准技术中使用的几何分析，以证明一个集是"光滑“，这需要它是平坦的，在某种适当的意义上，这是很好地适应给定的问题。所提出的关于薛定谔流动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有关，它可能会对这个物理问题产生一些见解。

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.