Harmonic Analysis and Elliptic Homogenization Problems

本计画旨在研究一类具非光滑边界区域的椭圆均匀化问题。主要重点将放在二阶椭圆方程和系统的快速振荡周期系数的域，其边界满足一些几何条件（例如，Lipschitz）。这个项目的主要目标是通过在物理上现实的假设下建立统一的估计来更好地理解解的边界正则性。非光滑区域上的边值问题在过去的30年里得到了广泛的研究。然而，很少有结果是已知的椭圆方程和系统的快速振荡的周期系数，这是在理论的均匀化。椭圆算子族和Lipschitz域类的伸缩不变性质使得研究的问题非常有趣和具有挑战性。在弹性力学、空气动力学、流体动力学和电磁波散射等许多应用问题中，偏微分方程的边值问题都是在边界具有面、边和顶点的区域中提出的。类Lipschitz域是一个膨胀不变类，允许这种粗糙的边界。本计画之研究结果将提供快速振荡微结构之数学基础与分析工具。该补助金将部分支持研究生作为研究助理。

This project aims to study a class of elliptic homogenization problems in domains with non-smooth boundaries. The main focus will be on second order elliptic equations and systems with rapidly oscillating periodic coefficients in domains whose boundaries satisfy some geometric conditions (for example, Lipschitz). The primary objective of this project is to gain a better understanding of the boundary regularity properties of solutions by establishing uniform estimates under physically realistic assumptions. Boundary value problems in non-smooth domains have received extensive study in the last 30 years. However, very few results are known for elliptic equations and systems with rapidly oscillating periodic coefficients, which arise in the theory of homogenization. The dilation-invariant property of the family of elliptic operators and that of the class of Lipschitz domains make problems to be investigated very interesting and challenging. The proposed research lies at the interface of harmonic analysis and partial differential equations.In many applied problems of elasticity, aero- and hydrodynamics, and electro-magnetic wave scattering, the boundary value problems for the partial differential equations are posed in domains whose boundaries have faces, edges, and vertices. The class of Lipschitz domains is a dilation invariant class which allows such roughness on the boundaries. The results of this project will provide the mathematical foundation and analytical tools for certain computational problems which involve rapid oscillating microstructures. The grant will partially support graduate students as research assistants.