Rectifiability and Elliptic Partial Differential Equations

本课题是几何测量理论与偏微分方程的结合，将运用调和分析的技术。在几何测度理论中，人们通过测度在集合上的行为来研究集合的几何性质（测度的概念概括了长度、面积和体积的概念）。在这个项目中，考虑的集合类型通常是空间中某些区域的边界（即封闭的周长），为此，一个特定的度量——即“谐波度量”——起着核心作用。调和测度及其推广可用于构造所谓“椭圆”型偏微分方程在给定边值区域内的解。这些方程控制着现实世界中的各种稳态现象，包括静电和稳态温度分布。例如，谐波测量可用于确定某区域内的稳态温度分布，给定该区域周长上的已知温度分布。不出所料，该地区的几何形状及其边界极大地影响了人们在实践中执行此类计划的能力。这个项目的一个主要目标是量化，在自然的意义上，几何和谐波测度及其推广之间的联系。该项目主要关注两个领域。首先，在d维欧几里德空间的开集边界上，寻找调和测度相对于曲面测度的定量绝对连续性的内在几何特征。他期望这样的表征应该包括两个部分：边界的定量可纠偏性，加上边界足够部分的尺度不变非切向可达性。在此基础上，主要研究者还计划研究与更一般的二阶椭圆算子相关的椭圆调和测度的定量绝对连续性的类似问题。其次，主要研究者将研究其他椭圆方程边值问题在几种不同情况下的可解性。更确切地说，他打算在比Lipschitz域更一般的域和非对称散度形式算子中改进对诺伊曼问题的理解。作为第一步，他计划考虑某一类传输问题，其中诺伊曼问题是一个端点情况。他还打算处理某些高阶发散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.