Free Boundary Regularity Problems in Harmonic Analysis

调和分析中的自由边界正则性问题

• 批准号: 0600915
• 负责人: Tatiana Toro
• 金额: $ 14.07万
• 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: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0600915&HistoricalAwards=false
• 关键词: Free; Boundary; Regularity; Problems; Harmonic

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.

这个建议的主题是强烈的关系，存在的问题之间的关系如何域的几何形状可以恢复其调和措施的规律性，和自由边界的正则性问题。值得注意的是，当在几何测度论的放大镜下考察时，这种类比变得更加明显。这个建议的核心解决了详细描述问题的奇异集的自由边界的问题，这是类似于下面描述的最优设计问题。PI和N。Wickramasekera建议使用几何测量理论的工具，这些工具在几何分析中的类似情况下非常成功。他们指出，韦斯的新单调性公式产生类似的信息提供的单调性公式的能量最小化调和映射和面积最小化表面。所提出的程序遵循的方案，导致成功地解决这两个主要问题。两相自由边界正则性问题自然出现在物理，化学和工程。在具有阈值电流的电化学加工的最优设计和最优控制问题中，阳极和电解液之间的表面起自由边界的作用。在双流体射流的平面对称流动模型中，由于外部射流被空气包围，所以流动区域有两个自由边界：外部流体边界和两流体边界。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。凯尼格研究了这样一种情况，即人们有关于两种流体在自由边界处速度之比的信息。这个问题以前没有得到处理。他们找到了一个保证自由边界规则性的全局准则。这个建议的主题之一是研究几个自由边界的正则性问题，使用这些新的工具。拟议的研究提供了一个不同的出口几何测量理论（GMT），数学领域，极大地促进了变分法和几何分析的发展。该项目的成功将为有兴趣将GMT技术应用于更具体问题的新一代提供路线图。拟议工作的一个重要特点是，虽然已经取得了一些成果，但还有很大的扩展潜力。特别是，我们希望研究生和初级数学家的积极参与。