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.

研究者计划研究与局部Tb定理的共同主题相关的几个问题，以及奇异积分估计、泊松核估计、平方函数估计和边界规则之间的相互作用。本文拟研究的主要方向有：1、研究方向：探讨层势的有界性、谐波测度性质与均匀整流性之间的关系。2. 研究一致可整集的结构。3. 发展和应用“局部”Tb定理，研究自由边界的正则性和椭圆型边值问题的可解性。4. 发展研究复杂椭圆方程边值问题的可解性的技术，或者更一般地说，研究具有有界可测系数的强椭圆系统边值问题的可解性。本课题的研究方向是谐波分析及其在几何测度理论和椭圆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，人们通过将函数和“算子”（即将一个函数转换为另一个函数的映射）分解成更容易理解的更小的组成部分，然后重新组合这些部分来研究函数和“算子”的性质。这个名字本身是通过类比将音乐声音分解成各种频率成分（“谐波”）而产生的。几何度量理论涉及对集合的几何性质和它们的“度量”（后者是长度、面积和体积概念的概括）之间关系的研究。偏微分方程和椭圆型系统描述了现实世界中各种各样的现象，包括静电、稳态温度分布和弹性变形。本提案的一个特别焦点是进一步探索几何与“调和测度”性质之间的关系。后一个领域的研究已经在最近的声学工程工作中得到了应用，特别是在设计具有理想声学性能的房间时。待审议问题的进展极有可能开辟进一步的调查途径。研究者将通过在会议、研讨会和研究生课程上的演讲，以及在其网站和ArXiv网站上发布的电子预印本来传播所有这些进展。研究人员将包括一名博士生和一名博士后，研究与拟议工作相关的问题。两位前博士后已经参与了这个项目。