Lectures on Periodic Homogenization of Elliptic Systems

TL;DR

This paper surveys the theory and recent advances in quantitative homogenization of second-order elliptic systems with periodic coefficients, focusing on convergence rates, regularity estimates, and asymptotic expansions in bounded domains.

Contribution

It provides a comprehensive overview of quantitative homogenization, including new interior and boundary regularity estimates and convergence rate results for elliptic systems with periodic coefficients.

Findings

Uniform regularity estimates for solutions in various norms.

Convergence rates for solutions and eigenvalues.

Asymptotic expansions of fundamental solutions and Green functions.

Abstract

In recent years considerable advances have been made in quantitative homogenization of partial differential equations in the periodic and non-periodic settings. This monograph surveys the theory of quantitative homogenization for second-order linear elliptic systems in divergence form with rapidly oscillating periodic coefficients, $$ \mathcal{L}_\e =-\text{\rm div} \big (A(x/\e)\nabla \big), $$ in a bounded domain $\Omega$ in $\br^d$. It begins with a review of the classical qualitative homogenization theory, and addresses the problem of convergence rates of solutions. The main body of the monograph investigates various interior and boundary regularity estimates (H\"older, Lipschitz, $W^{1, p}$, nontangnetial-maximal-function) that are uniform in the small parameter $\e>0$. Additional topics include convergence rates for Dirichlet eigenvalues and asymptotic expansions of fundamental solutions, Green functions, and Neumann functions. Part of this monograph is based on lecture notes for courses the author taught at several summer schools and at the University of Kentucky. Much of material in Chapters 6 and 7 is taken from his joint papers \cite{KLS-2013, KLS-2014} with Carlos Kenig and Fang-Hua Lin, and from \cite{KS-2011-L} with Carlos Kenig.