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.

该项目位于几何测量理论和部分微分方程的界面，它将利用谐波分析的技术。在几何测量理论中，一个人通过某种度量的行为研究集合的几何特性（“测量”的概念概括了长度，面积和体积的概念）。在该项目中，所考虑的设置通常是空间中某些区域的边界（即封闭的周长），而对于这一特定措施（即“谐波度量”）起着核心作用。谐波度量及其概括可用于在某些区域构建解决方案，并具有规定的边界值，用于所谓的“椭圆”类型的部分微分方程。这种方程控制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，给定该区域周长的已知温度分布，可以使用谐波度量来确定某些区域内的稳态温度分布。毫不奇怪，该地区及其边界的几何形状极大地影响了一个人在实践中执行这样的程序的能力。该项目的主要目标是从自然意义上量化几何形状与谐波测量及其概括之间的联系。该项目具有两个主要重点领域。首先，主要研究者计划在d维欧几里得空间中的开放式集合的边界上找到谐波测量的定量绝对连续性的本质几何表征。他期望这样的特征应包括两个部分：边界的定量整流性质，以及对边界充足部分的规模不变的非倾斜度可访问性。与这项工作相关的是，首席研究人员还计划研究与更一般的二阶椭圆运算符相关的椭圆谐波措施的定量绝对连续性的类似问题。其次，主要研究者将在几种不同的环境中研究椭圆方程其他边界价值问题的可溶性。更确切地说，他打算与Lipschitz域更笼统地对诺伊曼问题和非对称差异形式的运算符进行更笼统的诺曼问题的理解。作为第一步，他计划考虑某种传播问题，而诺伊曼问题是终点案例。他还计划治疗某些高阶差异椭圆算子的边界值问题。