Geometry of Measures

标题“几何的措施”是指研究的规则性和结构的措施（认为长度和面积和体积作为这个概念的例子）。光滑几何和分形几何中的问题，以及偏微分方程中出现的问题，都属于这个范畴。沿着这些路线，主要研究人员研究的项目之一涉及能量最小化问题，其中考虑了噪声。这为自然现象提供了一个更现实的模型。人们期望验证的行为是，直到一阶近似，在噪声环境中的能量最小化器的行为与无噪声环境中的能量最小化器的行为完全相同。 这个项目说明了这样一种思想，即数学对象可以很好地近似于更规则的对象，继承了它们的一些规则性属性。这一原则的几个应用。在该项目中解决了四个主要问题。第一个问题是关于包含保持器连续黎曼度量的自由边界变分问题的几乎极小解的正则性，研究这个问题的目的是为了了解相应的自由边界的结构。Alt-Caffarelli和Alt-Caffarelli-Friedman在自由边界极小化问题上的自由边界正则性的新结果是值得期待的。在这种情况下，“好的近似对象”是拉普拉斯算子在相应的黎曼度量中的解。第二个问题涉及由平坦测度很好地近似的测度的正则性（即，Lebesgue测度的常数倍数）在Wasserstein距离中。两种不同类型的问题被认为是，一个几何性质，另一个关系非常密切的调和分析理论的权重。第三个问题是关于欧几里得空间的两种不同类型子集的好参数化的存在性。第四个问题与主要研究者研究的一个重要分支有关，即与非光滑域上的发散型椭圆算子相关的椭圆测度的正则性。调和分析与几何测度理论的交叉是本文研究的重要内容之一。

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.