Quantitative homogenization for the case of an interface between two heterogeneous media

TL;DR

This paper develops quantitative homogenization results for linear elliptic equations with an interface between two heterogeneous media, extending previous work to more general media types and providing optimal convergence rates.

Contribution

It introduces a new quantitative analysis of homogenization near interfaces for both periodic and random media, improving upon existing methods.

Findings

Quantification of the sublinearity of the interface-adapted homogenization corrector

Extension of homogenization results to more general media types

Derivation of almost-optimal convergence rates

Abstract

In this article we are interested in quantitative homogenization results for linear elliptic equations in the non-stationary situation of a straight interface between two heterogenous media. This extends the previous work [Josien, 2019] to a substantially more general setting, in which the surrounding heterogeneous media may be periodic or random stationary and ergodic. Our main result is a quantification of the sublinearity of a homogenization corrector adapted to the interface, which we construct using an improved version of the method developed in [Fischer and Raithel, 2017]. This quantification is optimal up to a logarithmic loss and allows to derive almost-optimal convergence rates.