Rectifiability and Elliptic Partial Differential Equations

这个项目位于几何测度论和偏微分方程的交界处，它将利用调和分析的技术。在几何测度论中，人们通过一些测量在集合上的行为来研究集合的几何性质(“测量”的概念推广了长度、面积和体积的概念)。在这个项目中，所考虑的这类集合通常是空间中某一区域的边界(即，包围的周长)，为此，一个具体的度量--即“调和度量”--发挥着中心作用。调和测度及其推广可用于构造具有给定边界值的所谓“椭圆型”偏微分方程解。这些方程控制着现实世界中的各种稳态现象，包括静电学和稳态温度分布。例如，在给定某一区域周边已知温度分布的情况下，可以使用调和测量来确定该区域内的稳态温度分布。毫不奇怪，该区域的几何形状及其边界在很大程度上影响了一个人在实践中执行这样一个计划的能力。这个项目的一个主要目标是在自然意义上量化几何和调和测量行为及其推广之间的联系。该项目有两个主要的重点领域。首先，主要研究人员计划在d维欧氏空间中的开集边界上找到调和测度相对于曲面测度的定量绝对连续性的内在几何刻画。他预计这样的刻画应该包括两个部分：边界的定量可纠正性，以及对边界的大量部分的比例不变的非切向可达性。与这项工作相关的是，主要研究者还计划研究与更一般的二阶椭圆算子有关的椭圆调和测度的定量绝对连续性的类似问题。第二，主要研究者将在几种不同的环境下研究椭圆型方程的其他边值问题的可解性。更准确地说，他打算在比Lipschitz域更一般的区域和非对称散度形式的算子中更好地理解Neumann问题。作为第一步，他计划考虑一类特定的传输问题，诺伊曼问题是该问题的一个终点。他还计划处理某些高阶散度型椭圆算子的边值问题。

This project lies at the interface of geometric measure theory and partial differential equations, and it will utilize techniques from harmonic analysis. In geometric measure theory, one studies geometric properties of sets via the behavior of some measure on them (the concept of "measure" generalizes the notions of length, area, and volume). In this project, the sort of set under consideration is typically the boundary (i.e., the enclosing perimeter) of some region in space, and for this a specific measure -- namely, "harmonic measure"-- plays a central role. Harmonic measure and its generalizations may be used to construct solutions in some region, with prescribed boundary values, to partial differential equations of so-called "elliptic" type. Such equations govern various steady-state phenomena in the real world, including electrostatics and steady-state temperature distributions. For example, harmonic measure may be used to determine the steady-state temperature distribution within some region, given a known temperature distribution on the perimeter of the region. Unsurprisingly, the geometry of the region and its boundary greatly affect one's ability to carry out such a program in practice. A principal goal of this project is to quantify, in a natural sense, the connection between geometry and the behavior of harmonic measure and its generalizations.The project has two main areas of focus. First, the principal investigator plans to find an intrinsically geometric characterization of quantitative absolute continuity of harmonic measure with respect to surface measure, on the boundary of an open set in d-dimensional Euclidean space. He expects that such a characterization should comprise two parts: a quantitative rectifiablity property of the boundary, plus scale invariant nontangential accessibility to ample portions of the boundary. Related to this work, the principal investigator also plans to investigate the analogous question concerning quantitative absolute continuity of elliptic-harmonic measure associated to a more general second-order elliptic operator. Second, the principal investigator will study the solvability of other boundary value problems for elliptic equations, in several different settings. More precisely, he intends to work toward an improved understanding of the Neumann problem in domains more general than Lipschitz domains and for nonsymmetric divergence form operators. As a first step, he plans to consider a certain family of transmission problems, for which the Neumann problem is an endpoint case. He also plans to treat boundary-value problems for certain higher-order divergence-form elliptic operators.