Uniform Rectifiability and Elliptic Equations

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。本项目的一个特别重点是进一步探索几何与这些在物理世界中出现的微分方程的解的行为之间的关系。该项目主要关注两个领域。首先，主要研究者计划描述d维欧几里德空间中余维为1的闭集E的定量可纠偏性性质，根据E的补中的调和函数（以及更一般的线性和拟线性椭圆方程的解）的行为。特别是，该建议的一个主要目标是证明f和M. Riesz型定理，以及反转。没有对集合E或它的补集施加任何连通性假设。在经典的Riesz定理和Riesz定理及其现代派生定理中，我们得到了泊松核（即D域的调和测度的Radon-Nykodym导数，相对于D边界上的弧长或曲面测度）的存在性，并在某些情况下给出了定量估计，这是边界的可纠偏性的结果。在适当的连通性假设存在的情况下，比如当集合E是具有适当的路径连通性定量版本的域D的边界时，E的几何形状可以用与D相关的泊松核的行为来表征。另一方面，Bishop和Jones的一个反例排除了在没有连通性的情况下的这种结果。因此，为了证明没有连通性假设的F.和M. Riesz型定理，将需要找到椭圆方程解的估计，这些估计可以作为泊松核正则性的适当替代品。在这个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，特别是诺伊曼问题，对于发散形式，二阶椭圆方程在半空间中具有“径向”独立的系数，不假设系数矩阵的自伴随性。首席研究员和他的合作者以前的工作已经处理了狄利克雷和规则问题在这种情况下。

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.