Geometry of Measures

“尺度几何”的标题指的是尺度的规律性和结构的研究（想想长度、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，都属于这一范畴。沿着这些思路，首席研究员研究的一个项目涉及能源最小化问题，其中考虑到噪音。这为自然现象提供了一个更现实的模型。我们期望验证的行为是，在一阶近似下，有噪声环境中的能量最小化器与无噪声环境中的能量最小化器的行为完全相同。这个项目说明了一个想法，即可以被更规则的对象很好地近似的数学对象继承了它们的一些规则属性。介绍了这一原理的几种应用。该项目解决了四个主要问题。第一部分是关于涉及Holder连续黎曼度量的自由边界变分问题的几乎极小值的正则性，研究该问题的目的是了解相应的自由边界的结构。对于早先由Alt-Caffarelli和Alt-Caffarelli- friedman研究的具有自由边界的最小化问题的自由边界正则性，期望得到新的结果。在这种情况下，“良好的近似对象”是相应黎曼度规中拉普拉斯量的解。第二个问题处理的是在Wasserstein距离内由平面测度（即平面上的勒贝格测度的常数倍）很好地近似的测度的规律性。考虑了两种不同类型的问题，一种本质上是几何问题，另一种与调和分析中的权重理论密切相关。第三个问题是关于两种不同类型的欧氏空间子集是否存在良好的参数化。第四个问题与首席研究员研究的一个重要分支有关，即与非光滑域上发散型椭圆算子相关的椭圆测度的正则性。谐波分析和几何测度理论之间的交叉授粉是本研究的支柱之一。

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.