Geometry of Measures

摘要：本奖项由2009年美国复苏与再投资法案（公法111-5）资助。本提案的主题是存在于如何从调和测度的正则性中恢复域的几何形状的问题和自由边界正则性问题之间的牢固关系。值得注意的是，在几何测量理论（GMT）的放大镜下，这种类比变得更加明显。这一建议的核心涉及三个问题。第一个目标是理解高维欧几里得空间中的域根据它们的调和测度就像在二维空间中所做的那样，并取得了巨大的成功。基本的论点是，在高维中，GMT扮演着复杂分析在二维中扮演的角色。第二个问题是关于变分问题中最小值的存在性和规律性的问题，这些变分问题是用旧的连续度量而不是光滑度量来表述的。这个问题包括对相应自由边界结构的理解。由此产生的一个问题是关于Alt和Caffarelli所研究的泛函的拟极小值的正则性问题。第三个问题回到PI的长期兴趣，即欧几里德空间子集的良好参数化的存在性。一个显著的特点是，这最后一个项目，这是纯粹的几何，是由试图回答一个问题在势理论的动机。调和分析和GMT之间的交叉授粉显然对两个地区都有利。变分理论一直是研究常与能量最小化有关的变分问题的主要理论工具。能量最小化方法被用来理解分子的平衡构型。基本思想是分子系统的稳定状态应该对应于它们的局部最小势能。提出的研究为GMT提供了新的出路，GMT是数学的一个领域，对变分演算和几何分析的发展做出了巨大贡献。这项拨款的变革性方面是对数学这一基础领域的鼓舞。在过去的几年里，美国进入GMT的学生数量大大减少，而在欧洲则有所增加。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。我们特别期待研究生和初级数学家的积极参与。该领域已经成为一些几何分析领域的支柱之一，为研究来自不同科学领域的各种变分问题提供了理论框架。

AbstractToroThis award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The 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 (GMT) magnifying glass. The core of this proposal addresses three questions. The first one aims to understand domains in higher dimensional Euclidean spaces in terms of their harmonic measure as it has been done in 2 dimensions with great success. The underlying thesis is that in higher dimensions GMT plays the role that complex analysis does in 2 dimensions. The second question is that of the existence and regularity of minimizers for variational problems stated in terms of H\"older continuous metrics rather than smooth metrics. This problem includes the understanding of the structure of the corresponding free boundary. A by-product of this, is a question concerning the regularity of quasi-minimizers of the functional studied by Alt and Caffarelli. The third question goes back to a long term interest of the PI concerning the existence of good parameterization for subsets of Euclidean space. A remarkable feature is that this last project, which is purely in geometry, was motivated by an attempt to answer a question in potential theory. The cross-pollenization between harmonic analysis and GMT has been clearly beneficial to both areas.The theory of calculus of variations has been the main theoretical tool used in the study of variational problems often concerning energy minimization. Energy minimization methods are used to understand the equilibrium configuration of molecules. The basic idea is that a stable state of a molecular system should correspond to a local minimum of their potential energy. The proposed research provides new outlets for GMT, a field of Mathematics that has contributed greatly to the development of the calculus of variations and geometric analysis. The transformative aspect of this grant is the invigoration of this fundamental area of Mathematics. In the last few years, the number of students going into GMT in the US, has greatly diminished while it has increased in Europe. 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. The field, which has been one of the pillars upon of which some areas of geometric analysis have been built, offers the theoretical framework to study a wide array of variational problems coming from different venues of science.