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.

调查员计划调查几个问题的共同主题的地方Tb定理，和奇异积分估计，泊松核估计，平方函数估计之间的相互作用，和边界的规律性。建议的主要研究方向有：1。研究层势的有界性、调和测度的性质及一致可求正性之间的关系。2.研究一致可求长集的结构。3.发展和应用“局部”Tb定理来研究自由边界的正则性和椭圆边值问题的可解性。4.发展研究复椭圆型方程或更一般的强椭圆型方程组的边值问题的可解性的技术。该项目属于调和分析及其在几何测度理论和椭圆型偏微分方程理论中的应用和相互作用。 粗略地说，在调和分析中，人们研究函数和“算子”的性质（即，将一个函数转换为另一个函数的映射），方法是将它们分解为更小的、更容易理解的组成部分，然后重新组装这些部分。 这个名字本身是通过类比音乐声音分解成各种频率成分（“谐波”）而产生的。 几何测度论涉及研究集合的几何性质和它们的“测度”（后者是长度、面积和体积概念的推广）之间的关系。 椭圆型偏微分方程和系统描述了真实的世界中的各种现象，包括静电、稳态温度分布和弹性变形。 本文的一个重点是进一步探讨几何与调和测度性质之间的关系。“后一个研究领域已经在声学工程的最新工作中得到应用，特别是在设计具有理想声学特性的房间方面。在有待审议的问题上取得进展，很可能会开辟进一步的调查渠道。所有这些进展将由调查员通过会议、研讨会和研究生课程的讲座以及通过在其网站和ArXiv网站上张贴的电子预印本进行传播。 研究者将涉及博士。学生和博士后的问题有关的拟议工作。两名前博士后已经参与了该项目。