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Rectifiability and Elliptic Partial Differential Equations

可修正性和椭圆偏微分方程

基本信息

批准号：
1664047
负责人：
Steven Hofmann
金额：
$ 21.9万
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：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1664047&HistoricalAwards=false
关键词：
Rectifiability Elliptic Partial Differential Equations

项目摘要

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.

该项目处于几何测量理论和偏微分方程的接口，它将利用调和分析技术。在几何测度论中，人们通过一些测度在集合上的行为来研究集合的几何性质（“测度”的概念推广了长度、面积和体积的概念）。在这个项目中，所考虑的集合类型通常是边界（即，空间中某个区域的围界），为此，一个特定的测度--即“调和测度”--起着核心作用。调和测度及其推广可以用来构造所谓“椭圆”型偏微分方程在某些区域上的解，并具有给定的边值。这些方程支配着真实的世界中的各种稳态现象，包括静电和稳态温度分布。例如，谐波测量可以用于确定某个区域内的稳态温度分布，给定该区域周边上的已知温度分布。毫不奇怪，该区域的几何形状及其边界极大地影响了人们在实践中执行这样一个程序的能力。该项目的主要目标是在自然意义上量化几何与调和测度及其推广行为之间的联系。该项目有两个主要关注领域。首先，主要研究者计划在d维欧氏空间的开集的边界上，找到调和测度相对于表面测度的定量绝对连续性的内在几何表征。他预计，这样的表征应该包括两个部分：一个定量的rectifiablity属性的边界，加上规模不变的非切向访问充足的部分边界。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的可解性，在几个不同的设置。更确切地说，他打算努力提高对诺依曼问题的理解，在比Lipschitz域更一般的域和非对称发散形式算子。作为第一步，他计划考虑一个特定的家庭的传输问题，其中诺依曼问题是一个端点的情况。他还计划处理某些高阶发散型椭圆算子的边值问题。