Geometry of Sets and Measures

集合和测度的几何

• 批准号: 1500382
• 负责人: Matthew Badger
• 金额: $ 12万
• 依托单位: University of Connecticut
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500382&HistoricalAwards=false
• 关键词: Geometry; Sets; Measures

Geometric measure theory is a field of mathematics that developed starting in the 1920s and 1930s, growing out of a practical need to describe nonsmooth phenomena such as the formation of corners in soap bubble clusters. The term "measure" refers to an abstract generalization of length, area, or volume, which assigns a size value to every mathematical set. Traditional outlets for geometric measure theory, such as the calculus of variations and geometric analysis, have expanded in recent decades to include partial differential equations and harmonic analysis. The widespread utility and current use of geometric measure theory in different areas of analysis justifies its continued development. The proposed investigation on the geometry of sets and measures seeks to develop new techniques that will expand the toolbox that geometric measure theory provides for researchers in adjacent areas in analysis and geometry.This project focuses on two groups of questions about the geometry of sets and measures in Euclidean space. The first group of questions concerns rectifiable measures, one of the core objects of study in geometric measure theory. Specifically, these questions are aimed at increased understanding of rectifiable measures in the absence of a standing regularity assumption that has been assumed in the past. The main approach entails adapting quantitative techniques developed in the 1990s by Jones and David-Semmes to study the qualitative rectifiability of measures. The second group of questions are designed to examine the geometry of Reifenberg-type sets, which are sets that can be approximated at all locations and scales by one or more kinds of model sets. Instances where Reifenberg-type sets occur include geometric minimization problems and free boundary problems for elliptic partial differential equations. A general goal of this inquiry is to determine in what situations and to what extent good properties of solutions to problems in ideal models (smooth settings) persist under controlled perturbation (weak regularity).

几何测度论（英语：Geometric measure theory）是一个数学领域，在1920年代和1930年代开始发展，成长于描述非光滑现象的实际需要，例如肥皂泡簇中角的形成。 术语“测量”是指长度、面积或体积的抽象概括，它为每个数学集合分配一个大小值。 传统的几何测度理论，如变分法和几何分析，在最近几十年已经扩展到包括偏微分方程和调和分析。几何测度理论在不同分析领域的广泛应用和当前的使用证明了它的持续发展。 拟议的调查几何的集合和措施，旨在开发新的技术，将扩大工具箱，几何测度理论提供给研究人员在相邻地区的分析和geometrics.This project focused on two groups of questions about geometrics of sets and measures in Euclidean space. 第一组问题涉及可纠正措施，在几何测量理论的研究的核心对象之一。 具体而言，这些问题的目的是在缺乏过去一直假设的长期规律性假设的情况下，增加对可纠正措施的理解。 主要的方法需要适应琼斯和大卫-塞姆斯在20世纪90年代开发的定量技术来研究措施的定性可纠正性。 第二组问题旨在研究Reifenberg型集的几何形状，这些集可以在所有位置和尺度上由一种或多种模型集近似。 Reifenberg型集出现的问题包括椭圆型偏微分方程的几何极小化问题和自由边界问题。 本研究的一个总体目标是确定在什么情况下以及在什么程度上理想模型（光滑设置）中问题的解决方案的良好性质在受控扰动（弱正则性）下持续存在。