An introduction to the qualitative and quantitative theory of homogenization

TL;DR

This paper introduces the fundamental concepts of homogenization theory for elliptic PDEs, covering qualitative and quantitative aspects, with a focus on periodic and stochastic cases, and provides elementary methods and recent quantitative results.

Contribution

It offers a unified, elementary approach to qualitative homogenization and discusses recent advances in quantitative stochastic homogenization, including error representation and ergodicity quantification.

Findings

Elementary construction of the sublinear corrector for stochastic homogenization

Representation of homogenization error via two-scale expansion

Quantification of ergodicity and error growth using concentration inequalities

Abstract

We present an introduction to periodic and stochastic homogenization of ellip- tic partial differential equations. The first part is concerned with the qualitative theory, which we present for equations with periodic and random coefficients in a unified approach based on Tartar's method of oscillating test functions. In partic- ular, we present a self-contained and elementary argument for the construction of the sublinear corrector of stochastic homogenization. (The argument also applies to elliptic systems and in particular to linear elasticity). In the second part we briefly discuss the representation of the homogenization error by means of a two- scale expansion. In the last part we discuss some results of quantitative stochastic homogenization in a discrete setting. In particular, we discuss the quantification of ergodicity via concentration inequalities, and we illustrate that the latter in combi- nation with elliptic regularity theory leads to a quantification of the growth of the sublinear corrector and the homogenization error.