Harmonic Analysis and Elliptic Homogenization Problems

University Of Kentucky Research Foundation, Lexington KY

Investigators

Abstract

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.