Uniform Rectifiability and Elliptic Equations

该项目属于调和分析及其在几何度量理论和椭圆型偏微分方程组理论中的应用及其相互作用的领域。粗略地说，在调和分析中，人们通过将它们分解成更小的、更容易理解的组成部分，然后重新组装，来研究函数和“运算符”(即，将一个函数转换为另一个函数的映射)的性质。这个名字本身就是与音乐声音分解成各种频率成分(“谐波”)的类比而产生的。几何测度论涉及研究集合的几何性质和它们的“度量”(后者是长度、面积和体积概念的推广)之间的关系。椭圆型偏微分方程组描述了现实世界中各种各样的现象，包括静电学、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与物理世界中出现的这些微分方程解的行为之间的关系。该项目有两个主要的重点领域。首先，主要研究人员计划用E的补集中的调和函数(以及更一般的线性和拟线性椭圆方程解)的行为来刻画d维欧氏空间中余维为1的闭集E的定量可整性性质。特别地，该建议的一个主要目标是证明F.和M.Riesz型定理，以及逆集E，而不对集合E或其补集强加任何连通性假设。在经典的F.和M.Riesz定理及其现代派生定理中，由于边界的可正性，人们得到了Poisson核(即区域D的调和测度关于D的边界上的弧长或曲面测度的Radon-Nykodym导数)的存在性，并且在某些情况下得到了定量的估计。在存在适当的连通性假设的情况下，例如当集合E是享受路径连通性的适当量化版本的域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的反例在没有连通性的情况下排除了这样的结果。因此，要在没有连通性假设的情况下证明F.和M.Riesz型定理，就需要找到椭圆方程解的估计，这些估计可以作为Poisson核正则性的适当替代品。在本项目的第二个重点领域，主要研究者计划继续研究散度形式的椭圆边值问题的可解性，特别是Neumann问题，即在半空间中具有径向无关系数的二阶椭圆型方程，而不假设系数矩阵的自伴性。主要研究者和他的合著者以前的工作已经在这种背景下处理过狄利克莱特和正则性问题。

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.