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Harmonic Analysis, Geometric Measure Theory and Partial Differential Equations

调和分析、几何测度论和偏微分方程

基本信息

批准号：
0653180
负责人：
Marius Mitrea
金额：
$ 15.94万
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：
2007
资助国家：
美国
起止时间：
2007-06-01 至 2011-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0653180&HistoricalAwards=false
关键词：
Harmonic Analysis Geometric Measure Theory

项目摘要

HARMONIC ANALYSIS, GEOMETRIC MEASURE THEORY AND PARTIAL DIFFERENTIAL EQUATIONSAbstract of Proposed ResearchMarius Mitrea This project will develop the systematic use of singular integral operators (SIO) under more general hypotheses than has been previously attained. The goal is to display the effectiveness of SIO-based methods in circumstances traditionally handled by other, more specialized, tools (such as variational methods, harmonic measure techniques, etc).It is well-recognized that there are subtle connections between the boundedness of singular integral operators and the geometric measure-theoretic properties of sets. A fundamental result in this direction is the boundedness of SIO with reasonable kernels on surfaces which are Ahlfors regular (i.e., behave like n-1 dimensional at all scales), and contain ``big pieces of Lipschitz surfaces'' in a uniform fashion (one calls such surfaces uniformly rectifiable). This earlier work involved geometric measure theory, but has not yet been applied to problems in Partial Differential Equations (PDE). The ultimate goal of this proposal is to explore the role that SIO may play in the treatment of boundary value problems under sharp geometric measure theoretic assumptions on the domain and its boundary. In particular, this work will develop the analysis of SIO on uniformly rectifiable surfaces for applications to problems in PDE, such as boundary problems for the Laplace operator, other second order elliptic operators and systems (such as the Lame, Stokes and Maxwell systems).

调和分析、几何测度理论与偏微分方程--拟研究摘要本计画将在比先前更一般的假设下，发展奇异积分算子（SIO）的系统应用。我们的目标是显示基于SIO的方法在传统上由其他更专业的工具（如变分方法，调和测度技术等）处理的情况下的有效性。众所周知，奇异积分算子的有界性与集合的几何测度论性质之间存在着微妙的联系。在这个方向上的一个基本结果是SIO在Ahlfors正则表面上具有合理内核的有界性（即，在所有尺度上都表现得像n-1维），并且以统一的方式包含“大片的Lipschitz曲面”（人们称这种曲面为一致可求长的）。这一早期的工作涉及几何测度理论，但尚未应用于偏微分方程（PDE）的问题。这个建议的最终目标是探讨的作用，SIO可能发挥在尖锐的几何测度理论假设下的域及其边界的边值问题的治疗。特别是，这项工作将开发的分析SIO的一致可求长的表面上的应用程序的问题，在PDE，如边界问题的拉普拉斯运营商，其他二阶椭圆运营商和系统（如拉梅，斯托克斯和麦克斯韦系统）。