Elliptic homogenization with almost translation-invariant coefficients

Abstract

We consider an homogenization problem for the second order elliptic equation $-\mathrm{div}\left(a(./\epsilon )\nabla u^{\epsilon }\right)=f$ 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 $u^{\epsilon }$. When $p\ge d$, we exhibit admissible coefficients $a$ such that $u^{\epsilon }$ possesses different subsequences that converge to different limits in $L^2$.

33 pages, 3 figures