Uniform Rectifiability and Elliptic Equations

一致可整流性和椭圆方程

• 批准号: 1361701
• 负责人: Steven Hofmann
• 金额: $ 24万
• 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: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1361701&HistoricalAwards=false
• 关键词: Uniform; Rectifiability; Elliptic; Equations

This 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 that 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 with 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, steady-state temperature distributions, and elastic deformations. A particular focus of the present project is to explore further the relationship between geometry and the behavior of solutions to these differential equations that arise in the physical world.The project has two main areas of focus. First, the principal investigator plans to characterize quantitative rectifiability properties of a closed set E, of codimension one in d-dimensional Euclidean space, in terms of the behavior of harmonic functions (and of solutions of linear and quasi-linear elliptic equations more generally) in the complement of E. In particular, a primary goal of the proposal is to prove theorems of F. and M. Riesz type, and converses, without imposing any connectivity assumptions on either the set E or its complement. In the classical F. and M. Riesz Theorem, and its modern descendants, one obtains existence of, and in some cases quantitative estimates for, the Poisson kernel (i.e., the Radon-Nykodym derivative of harmonic measure for a domain D, with respect to arclength or surface measure on the boundary of D), as a consequence of rectifiability properties of the boundary. In the presence of suitable connectivity hypotheses, say when the set E is the boundary of a domain D enjoying an appropriate quantitative version of path connectedness, the geometry of E can be characterized in terms of the behavior of the Poisson kernel associated to D. On the other hand, a counterexample of Bishop and Jones precludes such results in the absence of connectivity. Thus, to prove theorems of F. and M. Riesz type without connectivity hypotheses will entail finding estimates for solutions of elliptic equations, estimates that serve as appropriate substitutes for Poisson kernel regularity. In the second area of focus of this project, the principal investigator plans to continue to investigate solvability of elliptic boundary value problems, in particular, the Neumann problem, for divergence form, second-order elliptic equations with "radially"-independent coefficients in the half-space, without assuming self-adjointness of the coefficient matrix. Previous work of the principal investigaor and his coauthors has treated the Dirichlet and regularity problems in this setting.

该项目属于调和分析及其应用领域，并与几何测量理论和椭圆偏微分方程理论相互作用。 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。本项目的一个特别重点是进一步探索几何和物理世界中出现的这些微分方程的解的行为之间的关系。该项目有两个重点领域。首先，主要研究者计划描述d维欧氏空间中余维为1的闭集E的定量求长性质，根据调和函数（以及更一般的线性和准线性椭圆方程的解）在E的补空间中的行为。特别地，该提案的主要目标是证明F的定理。和M. Riesz类型，并转换，而不强加任何连通性的假设，无论是集E或其补充。在经典的F.和M. Riesz定理及其现代后代，人们获得了泊松核的存在性，并且在某些情况下获得了泊松核的定量估计（即，作为边界可求长性质的结果，给出了区域D的调和测度对边界上弧长或面测度的Radon-Nykodym导数。在存在适当的连通性假设的情况下，比如说当集合E是具有适当的路径连通性定量版本的域D的边界时，E的几何可以根据与D相关联的泊松核的行为来表征。另一方面，Bishop和Jones的反例排除了在缺乏连通性的情况下的这种结果。因此，为了证明F.和M.没有连通性假设的Riesz类型将需要找到椭圆方程解的估计，估计可以作为泊松核正则性的适当替代品。在该项目的第二个重点领域，首席研究员计划继续研究椭圆边值问题的可解性，特别是Neumann问题，对于发散形式，具有“径向”独立系数的二阶椭圆方程，在半空间中，不假设系数矩阵的自伴性。以前的工作的主要investigaor和他的合著者已经处理的狄利克雷和规律性问题在此设置。