Uniform rectifiability, Singular Integrals and Harmonic Measure

均匀可整流性、奇异积分和谐波测量

• 批准号: 1101244
• 负责人: Steven Hofmann
• 金额: $ 29.47万
• 依托单位: University of Missouri-Columbia
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101244&HistoricalAwards=false
• 关键词: Uniform; rectifiability; Singular; Integrals; Harmonic

The investigator plans to investigate several questions linked by the common themes of local Tb Theorems, and the interplay between singular integral estimates, Poisson kernel estimates, square function estimates, and the regularity of boundaries. Among the main directions of the proposed research are: 1. To investigate the relationships among boundedness of layer potentials, properties of harmonic measure, and uniform rectifiability. 2. To investigate the structure of uniformly rectifiable sets. 3. To develop and apply "local" Tb theorems to study the regularity of free boundaries and the solvability of elliptic boundary value problems. 4. To develop techniques to study the solvability of boundary value problems for complex elliptic equations, or more generally, for strongly elliptic systems, with bounded measurable coefficients.The project lies within the field of harmonic analysis and its application to, and interaction with, geometric measure theory and the theory of elliptic partial differential equations. Roughly speaking, in harmonic analysis one investigates properties of functions and "operators" (i.e., mappings which transform one function into another) by decomposing them into smaller, constituent pieces, which are easier to understand, and then reassembling the pieces. The name itself arose by analogy to the decomposition of a musical sound into its various frequency components ("harmonics"). Geometric measure theory involves the study of the relationship between geometric properties of sets, and their "measures" (the latter are generalizations of the notions of length, area, and volume). Partial differential equations and systems of elliptic type describe a wide variety of phenoma in the real world, including electrostatics, and steady-state temperature distributions and elastic deformations. A particular focus of the present proposal is to explore further the relationship between geometry and properties of "harmonic measure." The latter area of investigation has already found application in recent work in acoustical engineering, in particular in the design of a room with desirable acoustic properties. Progress on the problems to be considered would in all likelihood open up further avenues of investigation. All such progress will be disseminated by the investigator via lectures at conferences, seminars and graduate courses, and via electronic preprints posted on his website and on the ArXiv web site. The investigator will involve Ph.D. students and a postdoc on problems related to the proposed work. Two former postdocs are already involved in the project.

研究人员计划调查与本地结核病定理的共同主题相关的几个问题，以及奇异积分估计值，泊松内核估计值，平方函数估计和界限的规则性之间的相互作用。拟议研究的主要方向之一是：1。研究层势的有限性，谐波测量的特性以及统一的可重新可相关性之间的关系。 2。研究均匀整流集的结构。 3。开发和应用“局部”结核病定理以研究自由边界的规律性和椭圆边界值问题的可溶性。 4。为了研究复杂椭圆方程的边界价值问题的解决性，或更普遍地，对于强椭圆形系统，具有有限的可测量系数。该项目在谐波分析领域及其在谐波分析领域内及其与几何和与几何学测量理论以及椭圆形偏微分方程的理论相互作用。 粗略地说，在谐波分析中，一个人通过将它们分解为较小的组成部分，调查了功能的属性和“操作员”（即，将一个函数转换为另一个函数的映射），这些部分易于理解，然后重新组装零件。 该名称本身与音乐声音分解为各种频率组件的分解（“谐波”）。 几何措施理论涉及对集合的几何特性之间的关系及其“度量”（后者是长度，面积和体积的概念的概括）。 椭圆类型的部分微分方程和系统描述了现实世界中各种各样的现象，包括静电和稳态温度分布和弹性变形。 本提案的一个特定重点是进一步探索几何形状和“谐波测量”的性质之间的关系。后一个调查领域已经发现了最近在声学工程中的工作中的应用，尤其是在具有理想声学特性的房间的设计中。在所有可能性的问题上，要考虑的问题的进展将开辟进一步的调查途径。调查员将通过会议，研讨会和研究生课程的讲座以及通过其网站和ARXIV网站上发布的电子预印本来传播所有这些进展。 研究人员将涉及博士学位。学生和与拟议工作有关的问题的博士后。该项目已经参与了两个以前的博士后。