We consider a homogenization problem for the diffusion equation − div (a ε ∇u ε ) = f when the coefficient a ε is a non-local perturbation of a periodic coefficient. The perturbation does not vanish but becomes rare at infinity in a sense made precise in the text. We prove the existence of a corrector, identify the homogenized limit and study the convergence rates of u ε to its homogenized limit.

We consider an homogenization problem for the second order elliptic equation −∆u ε + 1 ε V (./ε)u ε + νu ε = f when the highly oscillatory potential V belongs to a particular class of non-periodic potentials. We show the existence of an adapted corrector and prove the convergence of u ε to its homogenized limit.

We consider an homogenization problem for the second order elliptic equation − div ( a ( · / ε ) ∇ u ε ) = 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 ( R d ) for p ∈ [ 1 , + ∞ [. 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 ε . When p ⩾ d, we exhibit admissible coefficients a such that u ε possesses different subsequences that converge to different limits in L 2 .

<p style='text-indent:20px;'>We consider a homogenization problem for the diffusion equation <inline-formula><tex-math id="M1">\begin{document}$ -\operatorname{div}\left(a_{\varepsilon} \nabla u_{\varepsilon} \right) = f $\end{document}</tex-math></inline-formula> when the coefficient <inline-formula><tex-math id="M2">\begin{document}$ a_{\varepsilon} $\end{document}</tex-math></inline-formula> is a non-local perturbation of a periodic coefficient. The perturbation does not vanish but becomes rare at infinity in a sense made precise in the text. We prove the existence of a corrector, identify the homogenized limit and study the convergence rates of <inline-formula><tex-math id="M3">\begin{document}$ u_{\varepsilon} $\end{document}</tex-math></inline-formula> to its homogenized limit.</p>

We consider an homogenization problem for the second order elliptic equation − div (a(./ε)∇u ε ) = 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 (R d ) for p ∈ [1, +∞[. 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 ε . When p ≥ d, we exhibit admissible coefficients a such that u ε possesses different subsequences that converge to different limits in L 2 .