Geometric Measure Theory and Free Boundary Regularity Problems

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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.