Geometry of Measures

AbstractToro该奖项是根据 2009 年美国复苏和再投资法案（公法 111-5）资助的。该提案的主题是如何从调和测量的规律性中恢复域的几何形状的问题与自由边界规律性问题之间存在着密切的关系。值得注意的是，当在几何测度理论 (GMT) 放大镜下进行检查时，这些类比变得更加明显。该提案的核心解决了三个问题。第一个目标是根据调和度量来理解高维欧几里得空间中的域，因为它在二维中已经取得了巨大成功。基本论点是，在更高维度中，GMT 所扮演的角色与复杂分析在二维中所扮演的角色相同。第二个问题是用 H\" 较旧的连续度量而不是平滑度量表示的变分问题的最小化器的存在性和规律性。这个问题包括对相应自由边界的结构的理解。这个问题的一个副产品是关于 Alt 和 Caffarelli 研究的泛函的拟最小化器的规律性的问题。第三个问题可以追溯到 PI 对欧几里得空间子集的良好参数化的存在的长期兴趣。一个显着的特征最后一个项目纯粹是几何学的，其动机是试图回答势能理论中的一个问题。变分法理论一直是研究能量最小化问题的主要理论工具，其基本思想是分子系统的稳定状态应该对应于其势能的局部最小值。 GMT 是数学领域的一个分支，对变分法和几何分析的发展做出了巨大的贡献。在过去的几年里，美国进入 GMT 的学生数量大大减少，而欧洲的学生数量却有所增加。这项工作的一个重要特点是，虽然已经取得了一些成果，但我们特别希望研究生能够积极参与。该领域一直是几何分析某些领域的支柱之一，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.