Elliptic homogenization with almost translation-invariant coefficients

TL;DR

This paper studies homogenization of elliptic equations with coefficients that are nearly translation-invariant, establishing conditions under which the coefficients are close to periodic with local defects and analyzing the limits of solutions.

Contribution

It introduces a new framework for elliptic homogenization with almost translation-invariant coefficients, including a discrete Gagliardo-Nirenberg-Sobolev inequality and analysis of solution limits.

Findings

Coefficients belong to a class of periodic plus local defect models.

Existence of a corrector and identification of the homogenized limit.

Different subsequences of solutions can converge to different limits when p ≥ d.

Abstract

We consider an homogenization problem for the second order elliptic equation $-\operatorname{div}\left(a(./\varepsilon) \nabla u^{\varepsilon} \right)=f$ when the coefficient $a$ is almost translation-invariant at infinity and models a geometry close to a periodic geometry. This geometry is characterized by a particular discrete gradient of the coefficient $a$ that belongs to a Lebesgue space $L^p(\mathbb{R}^d)$ for $p\in[1,+\infty[$. When $p<d$, we establish a discrete adaptation of the Gagliardo-Nirenberg-Sobolev inequality in order to show that the coefficient $a$ actually belongs to a certain class of periodic coefficients perturbed by a local defect. We next prove the existence of a corrector and we identify the homogenized limit of $u^{\varepsilon}$. When $p\geq d$, we exhibit admissible coefficients $a$ such that $u^{\varepsilon}$ possesses different subsequences that converge to different limits in $L^2$.