The Geometry of Measures and Regularity of Associated Operators

措施的几何性和关联算子的规律性

• 批准号: 1500881
• 负责人: Benjamin Jaye
• 金额: $ 11.91万
• 依托单位: Kent State University
• 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: 2015-09-01 至 2018-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500881&HistoricalAwards=false
• 关键词: Geometry; Measures; Regularity; Associated; Operators

This project concerns a study of the some of the mathematics behind the following basic physical question: To what extent can the geometry of a body be determined from information about a force field associated to the body (for instance, its gravitational field)? Such inverse problems in potential theory have a rich history, but the mathematical tools needed to properly answer this question, especially in the case when the operator relating the force field to the mass distribution of the body is sensitive to long-range interactions, are currently underdeveloped. In this project, the principal investigator will develop tools to further understand this problem, concentrating especially on what can be said if one knows only that the field has bounded magnitude.More specifically, the project primarily concerns the relationship between the geometry of a measure and the regularity of an associated differential or singular integral operator. This is is a question that has attracted mathematicians ever since the Cauchy and Riesz transforms were introduced as tools to study the behavior of analytic and harmonic functions, respectively. An integrated approach to such problems is proposed that goes through the study of reflectionless measures. This approach has recently yielded several new results and could potentially address a number of open problems, especially those concerning the smoothness of the support of a measure that has a bounded Riesz transform. Here new tools in quantitative geometry and higher order partial differential equations need to be developed in order to make progress. Furthermore, the principal investigator seeks to build upon recent innovations in the theory of quasilinear differential equations to consider analogous problems for a wide range of nonlinear differential operators, where no integral representation is available.

该项目涉及以下基本物理问题背后的一些数学研究：在多大程度上可以根据与物体相关的力场（例如，其引力场）信息来确定物体的几何形状？ 势论中的此类反问题有着丰富的历史，但是正确回答这个问题所需的数学工具，特别是在将力场与物体的质量分布相关联的算子对长程相互作用敏感的情况下，目前正在研究中。不发达。 在这个项目中，首席研究员将开发工具来进一步理解这个问题，特别关注如果只知道场有界大小的话可以说些什么。更具体地说，该项目主要关注测量的几何形状与测量的几何形状之间的关系。相关微分或奇异积分算子的正则性。 自从柯西变换和里斯变换分别作为研究解析函数和调和函数行为的工具被引入以来，这个问题就一直吸引着数学家。 提出了解决此类问题的综合方法，该方法通过无反射措施的研究。 这种方法最近产生了一些新结果，并可能解决许多开放性问题，特别是那些涉及有界 Riesz 变换的测度支持平滑性的问题。 为了取得进展，需要开发定量几何和高阶偏微分方程方面的新工具。此外，主要研究者试图以拟线性微分方程理论的最新创新为基础，考虑各种非线性微分算子的类似问题，其中没有可用的积分表示。