Geometry of Measures

标题“测量的几何”是指研究测量的规律性和结构(以长度、面积和体积为例)。光滑几何学和分形学中的问题，以及偏微分方程式中出现的问题，都属于这一范畴。沿着这些思路，首席研究员研究的项目之一涉及能量最小化问题，其中考虑了噪音。这为自然现象提供了一个更逼真的模型。人们希望验证的行为是，在一阶近似下，能量最小化器在噪声环境中的行为与在无噪声环境中的行为完全相同。这个项目说明了这样一个想法，即可以被更规则的数学对象很好地近似的数学对象继承了它们的一些正则性属性。给出了这一原理的几个应用。该项目解决了四个主要问题。第一个问题是关于涉及Holder连续黎曼度量的自由边界变分问题的几乎极小元的正则性，研究这个问题的目的是了解相应的自由边界的结构。Alt-Caffarelli和Alt-Caffarelli-Friedman早先研究的自由边界极小化问题的自由边界正则性有望得到新的结果。在这种情况下，“好的近似对象”是相应的黎曼度量中的拉普拉斯解。第二个问题是关于由平面上的平坦测度(即平面上的勒贝格测度的常倍数)在Wasserstein距离上很好地逼近的测度的正则性。我们考虑了两种不同类型的问题，一种是几何性质的问题，另一种是与调和分析中的权重理论密切相关的问题。第三个问题是关于欧几里德空间的两种不同类型子集的良好参数的存在性。第四个问题与主要研究者研究的一个重要分支有关，即非光滑区域上散度型椭圆算子的椭圆测度的正则性。调和分析与几何测度论的交叉授粉是本研究的主要内容之一。

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.