Elliptic homogenization with almost translation-invariant coefficients
Résumé
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 $L^2$.