Harmonic Analysis and Elliptic PDE

9706641 Pipher这个项目是关于调和分析技术在线性椭圆理论中各种问题的应用。利用Littlewood-Paley理论研究二阶散度型方程的边值问题，特别是具有非光滑系数和/或其矩阵不是实值或对称的方程的边值问题。非光滑区域中的线性问题与线性问题有着密切的联系。因此，这个项目包括对具有Lipschitz边界的区域中的高阶椭圆算子的研究。特别是，它处理Sobolev空间或Hardy空间中的数据的边界问题。在数学、物理、工程和制造的各个领域中出现了许多问题，这些问题归结为能够在有角和边的区域上近似求解线性偏微分方程组。这种现象，即角和边的存在，在获得良好的近似值时会导致固有的数学困难。尤其重要的是，能够根据数据(已知量)来衡量误差有多小。本项目致力于阐明这些数值近似和应用所基于的理论。

9706641 Pipher This project is concerned with the applications of techniques of harmonic analysis to various problems in linear elliptic theory. It uses Littlewood-Paley theory to study boundary value problems for second order divergence form equations, in particular for equations with non-smooth coefficients and/or whose matrix is not assumed to be real-valued or symmetric. There are close connections to linear problems in non-smooth domains. Thus, included in this project is the study of higher order elliptic operators in domains with Lipschitz boundary. In particular, it treats boundary problems with data in Sobolev spaces or Hardy spaces. There are many problems arising in various areas of mathematics, physics, engineering, and manufacturing which reduce to being able to approximate solutions of linear partial differential equations in regions which have corners and edges. This phenomenon, the presence of corners and edges, causes inherent mathematical difficulties in obtaining good approximations. In particular, it is important to be able to measure how small the error will be in terms of the data (the known quantities). This project is concerned with formulating the theory upon which these numerical approximations and applications are based.