Geometry of Measures

AbstractToroThis奖是根据2009年美国复苏和再投资法案（公法111-5）资助的。这项建议的主题是如何从调和测度的正则性中恢复域的几何形状的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）放大镜下进行检查时，类比变得更加明显。这一建议的核心涉及三个问题。第一个目的是了解域在高维欧氏空间的调和措施方面，因为它已经做了2维取得了巨大成功。基本的论点是，在更高的维度GMT扮演的角色，复杂的分析在2维。第二个问题是用H\“older连续度量而不是光滑度量表示的变分问题极小解的存在性和正则性。这个问题包括对相应自由边界结构的理解。 一个副产品，这是一个问题的规律性准极小的功能研究的Alt和Caffarelli。第三个问题可以追溯到一个长期的兴趣PI关于存在良好的参数化子集的欧几里得空间。一个显着的特点是，这最后一个项目，这是纯粹的几何，动机是试图回答一个问题的潜在理论。调和分析和GMT之间的交叉授粉已经明显有利于这两个领域。变分理论一直是主要的理论工具，用于研究变分问题，往往涉及能量最小化。 能量最小化方法用于理解分子的平衡构型。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。建议的研究提供了新的出路GMT，数学领域，极大地促进了变分法和几何分析的发展。这一补助金的变革方面是数学的这一基本领域的振兴。在过去的几年里，在美国进入GMT的学生人数大大减少，而在欧洲却有所增加。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。该领域，这一直是支柱之一，其中一些领域的几何分析已经建成，提供了理论框架，研究各种各样的变分问题来自不同的场地的科学。

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.