Harmonic Analysis, Geometric Measure Theory and Partial Differential Equations

调和分析，几何测量理论和偏微分方程研究摘要marius Mitrea本项目将在比以前更一般的假设下发展奇异积分算子（SIO）的系统使用。目标是展示基于sio的方法在传统上由其他更专业的工具（如变分方法、谐波测量技术等）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有着微妙的联系。在这个方向上的一个基本结果是SIO在具有合理核的表面上的有界性，这些表面是Ahlfors正则（即，在所有尺度上都表现得像n-1维），并且以均匀的方式包含“大块Lipschitz表面”（人们称这种表面为均匀可整流的）。这项早期的工作涉及几何测量理论，但尚未应用于偏微分方程（PDE）的问题。本提案的最终目的是探讨SIO在处理在尖锐几何测量理论假设下的域及其边界的边值问题中可能发挥的作用。特别是，这项工作将发展均匀可整流表面上的SIO分析，用于PDE中的问题，例如拉普拉斯算子的边界问题，其他二阶椭圆算子和系统（如Lame， Stokes和Maxwell系统）。

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).