Harmonic Analysis and Elliptic Homogenization Problems

本课题旨在研究一类具有非光滑边界的区域上的椭圆齐化问题。主要的焦点将集中在边界满足某些几何条件(例如Lipschitz)的区域中具有快速振荡周期系数的二阶椭圆型方程和系统。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。在过去的30年里，非光滑区域上的边值问题得到了广泛的研究。然而，对于椭圆型方程和周期系数快速振荡的系统，在齐次化理论中所得到的结果很少。椭圆算子族和Lipschitz域的伸缩不变性使得研究问题变得非常有趣和具有挑战性。在许多应用于弹性力学、空气动力学、流体力学和电磁波散射的应用问题中，偏微分方程边值问题都是在具有面、边和顶点的区域上提出的。Lipschitz域的类是一类伸缩不变的类，它允许边界上的这种粗糙。该项目的结果将为涉及快速振荡微结构的某些计算问题提供数学基础和分析工具。这笔助学金将部分资助研究生担任研究助理。

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.