Geometry of Measures

测量几何

• 批准号: 1361823
• 负责人: Tatiana Toro
• 金额: $ 24万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1361823&HistoricalAwards=false
• 关键词: Geometry; Measures

The title "Geometry of Measures" refers to the study of the regularity and the structure of measures (think of length and area and volume as examples of this concept). Problems both in smooth and fractal geometry, as well as questions arising in partial differential equations, fit under this umbrella. Along these lines, one of the projects the principal investigator studies concerns an energy minimization problem, where noise is taken into account. This provides a more realistic model for natural phenomena. The behavior one expects to verify is that, up to first order approximation, energy minimizers in the noisy setting behave exactly the same way as those in the noise-free environment. This project illustrates the idea that mathematical objects that can be well approximated by more regular ones inherit some of their regularity properties. Several applications of this principle are presented. Four main question are addressed in the project. The first concerns the regularity of almost minimizers associated with free-boundary variational problems involving Holder continuous Riemannian metrics, and the aim of studying th question is to understand the structure of the corresponding free boundary. New results concerning the free-boundary regularity for the minimizing problems with free boundary that were studied earlier by Alt-Caffarelli and Alt-Caffarelli-Friedman are expected. In this case the "good approximating objects" are solutions to the Laplacian in the corresponding Riemannian metric. The second question deals with the regularity of measures that are well approximated by flat measures (i.e., constant multiples of Lebesgue measure on planes) in the Wasserstein distance. Two distinct types of problems are considered, one geometric in nature, another that ties in very closely to the theory of weights in harmonic analysis. The third question concerns the existence of good parameterizations for two different types of subsets of Euclidean space. The fourth question ties in with an important branch of the principal investigator's research, namely, the regularity of elliptic measures associated with divergence-form elliptic operators on nonsmooth domains. The cross-pollination between harmonic analysis and geometric measure theory is one of the pillars of the proposed research.

标题“度量的几何形状”是指对措施的规律性和结构的研究（将长度，面积和数量作为此概念的例子）。在平滑和分形的几何形状中，以及偏微分方程中出现的问题，都适合此保护伞。沿着这些线路，主要研究者研究的一个项目涉及一个能量最小化问题，其中考虑了噪声。这为自然现象提供了更现实的模型。人们期望验证的行为是，直到一阶近似，嘈杂设置中的能量最小化的表现与无噪声环境中的能量完全相同。 该项目说明了一个想法，即数学对象可以通过更常规的对象继承其一些规律性属性。提出了该原则的几种应用。项目中提出了四个主要问题。首先是涉及几乎与涉及持有人连续riemannian指标的自由边缘变分问题相关的几乎最小化者的规律性，研究问题的目的是了解相应的自由边界的结构。关于最小化自由边界问题的自由边界规律性的新结果预计是由Alt-Caffarelli和Alt-Caffarelli-Friedman较早研究的。在这种情况下，“良好的近似对象”是相应的Riemannian公制中Laplacian的解决方案。第二个问题涉及在瓦斯坦斯坦距离中通过扁平措施（即，勒贝斯（Lebesgue）量度的恒定倍数）很好地近似的度量的规律性。考虑了两种不同类型的问题，一种本质上是一种几何，另一种是与谐波分析中的权重理论紧密联系的。第三个问题涉及两种不同类型的欧几里得空间子集的良好参数化。第四个问题与主要研究者研究的一个重要分支有关，即，与非滑动域上的椭圆形椭圆运算符相关的椭圆措施的规律性。谐波分析与几何测量理论之间的交叉授粉是拟议研究的支柱之一。