Mathematical Sciences: Harmonic Measure Analytic Capacity and Rectifiable Sets

数学科学：调和测度分析能力和可整流集

• 批准号: 9100671
• 负责人: John Garnett
• 金额: $ 1.7万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-07-01 至 1992-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9100671&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Harmonic; Measure; Analytic

In this project the principal investigators will study the geometric properties of harmonic measure; in particular, they will focus on the relations between harmonic measure on domains in the plane and one-dimensional Hausdorff measure. The principal investigators will also look into problems involving harmonic measure and analytic capacity, the relationship between rectifiable sets and harmonic measure, and the estimation of eigenvalues of the Laplace operator on subdomains of the sphere in higher dimensions. The principal investigators are interested in studying the relationship between an analytical quantity known as the harmonic measure of a set and the geometry of this set. Harmonic measure is an important and useful concept that arises in probability theory and complex function theory. In this project the principal investigators will seek to characterize properties of the harmonic measure of a set in terms of geometric properties of the set such as its Hausdorff dimension.

在这个项目中，主要研究者将研究谐波测度的几何性质；特别是，他们将重点研究平面上域的谐波测度与一维豪斯多夫测度之间的关系。主要研究人员还将研究谐波测度与解析能力、可整流集与谐波测度之间的关系以及高维球面子域上拉普拉斯算子特征值的估计等问题。主要研究者感兴趣的是研究一个被称为集合的调和测度的解析量和这个集合的几何之间的关系。调和测度是概率论和复函数理论中出现的一个重要而有用的概念。在这个项目中，主要研究人员将根据集合的几何特性（如Hausdorff维）来描述集合的调和测度的特性。