Free Boundary Regularity Problems in Harmonic Analysis

这一建议的主题是区域的几何如何从其调和测度的正则性中恢复的问题与自由边界正则性问题之间存在的强烈关系。值得注意的是，当在几何测量理论放大镜下观察时，这种类比变得更加明显。该提案的核心是在类似于下面所述的优化设计问题中详细描述自由边界的奇异集的问题。PI和N.Wickramasekera建议使用几何测度论中的工具，这些工具在几何分析中的类似情况下已经非常成功。他们指出，Weiss的新单调性公式提供的信息与能量最小化调和映射和面积最小化曲面的单调性公式所提供的信息相似。两相自由边界正则性问题是物理、化学和工程中自然产生的问题。在具有阈值电流的电化学加工的优化设计和最优控制问题中，阳极和电解液之间的表面起着自由边界的作用。在具有两种流体的对称平面射流模型中，外部射流被空气包围，因此流动区域有两个自由边界：外部流体边界和两种流体之间的边界。刻画自由边界正则性的中心问题已被许多作者研究过。在过去的三年里，PI和C.Kenig研究了两种流体在自由边界上速度之比的情况。这个问题以前没有被解决过。他们找到了一个全球标准来保证自由边界的规律性。这项提议的主题之一是使用这些新工具研究几个自由边界正则性问题。这项研究为几何测度论(GMT)提供了一个不同的出路，几何测度论是一个数学领域，对变分和几何分析的发展做出了巨大贡献。该项目的成功将为对将GMT技术应用于更具体问题感兴趣的新一代人提供路线图。拟议工作的一个重要特点是，虽然已经取得了一些成果，但仍有很大的扩大潜力。特别是，我们期待研究生和初级数学家的积极参与。

ABSTRACTThe theme of this proposal is the strong relationship that exists between the questions of how the geometry of a domain can be recovered from the regularity of its harmonic measure, and free boundary regularity problems. Remarkably the analogies become more apparent when examined under a geometric measure theory magnifying glass. The core of this proposal addresses the question of describing in detail the singular set of the free boundary in problems which are similar to the optimal design question described below. The PI and N. Wickramasekera propose to use tools from geometric measure theory, which have been very successful in similar situations in geometric analysis. They have remarked that Weiss' new monotonicity formulas yield similar information to the one provided by the monotonicity formulas for energy minimizing harmonic maps and area minimizing surfaces. The proposed program follows the schemes which have lead to the successful resolution of these 2 major problems.Two-phase free boundary regularity problems arise naturally in physics, chemistry and engineering. In optimal design and optimal control problems within the context of electrochemical machining with threshold current, the surface between the anode and the electrolytic solution plays the role of the free boundary. In the symmetric plane flow model of jets with two fluids the outer jet is surrounded by air, so the flow region has two free boundaries: the boundary of the outer fluid and the boundary between the two fluids. The central problem of characterizing the regularity of the free boundary has been studied by many authors. Over the last 3 years the PI and C. Kenig have studied the case in which one has information about the ratio of the speeds of the two fluids at the free boundary. This question had not been addressed before. They found a global criteria which guarantees the regularity of the free boundary. One the themes of this proposal is the study of several free boundary regularity problems using these new tools. The proposed research provides a different outlet for geometric measure theory (GMT), a field of Mathematics that has contributed greatly to the development of the calculus of variations and geometric analysis. The success of the project would provide a road map for a new generation interested in applying GMT techniques to more concrete problems. An important feature of the proposed work is that, while some results have already been obtained, there is great potential for expansion. In particular, we expect the active participation of graduate students and junior mathematicians.