Harmonic Analysis and Elliptic Homogenization Problems

本课题旨在研究一类具有非光滑边界域上的椭圆均匀化问题。主要的焦点将是二阶椭圆方程和在边界满足某些几何条件（例如Lipschitz）的域内具有快速振荡周期系数的系统。该项目的主要目标是通过在物理现实假设下建立统一估计来更好地理解解的边界正则性。近30年来，非光滑域的边值问题得到了广泛的研究。然而，对于在均匀化理论中出现的具有快速振荡周期系数的椭圆方程和系统，已知的结果很少。椭圆算子族和一类Lipschitz区域的膨胀不变性使得问题的研究变得非常有趣和具有挑战性。本文的研究是调和分析和偏微分方程的结合。在弹性力学、气动和流体力学以及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、边和顶点的区域中求解的。Lipschitz区域类是允许边界存在这种粗糙度的膨胀不变量类。该项目的研究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。该基金将部分支持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.