Geometry of Measures

该奖项由2009年《美国复苏和再投资法案》(公法111-5)资助。这项提议的主题是如何从调和测度的正则性中恢复区域的几何形状问题与自由边界正则性问题之间存在的密切关系。值得注意的是，当在几何测量理论(GMT)的放大镜下观察时，这种类比变得更加明显。该提案的核心涉及三个问题。第一个目的是根据它们的调和度量来理解高维欧几里德空间中的域，就像在2维空间中所做的那样，并且取得了巨大的成功。其基本论点是，在更高的维度上，GMT扮演着复杂分析在2个维度上所起的作用。第二个问题是关于变分问题的极小元的存在性和正则性，它是用H“较老的连续度量而不是光滑度量来表示的。这个问题包括对相应自由边界结构的理解。由此产生的一个副产品是关于Alt和Caffarelli研究的泛函的拟极小子的正则性的问题。第三个问题回到PI的长期利益，即欧几里德空间的子集是否存在良好的参数化。一个显著的特点是，这个纯几何的最后一个项目的动机是试图回答位势理论中的一个问题。调和分析和GMT之间的交叉极化显然对这两个领域都有好处。变分理论一直是研究经常涉及能量最小化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势能的局部最小值。这项拟议的研究为GMT提供了新的出路，GMT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。这笔赠款的变革性方面是对这一数学基础领域的振兴。在过去的几年里，美国进入格林威治标准时间考试的学生人数大幅减少，而欧洲的学生人数却有所增加。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。特别是，我们期待研究生和初级数学家的积极参与。这一领域一直是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.