A periodic homogenization problem with defects rare at infinity

Abstract

<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>

Defects in homogenization theory

“…Another option is to study periodic coefficients that are perturbed by defects that are not vanishing at infinity but that are only "rare" at infinity. This is the case of the work [32] by R. Goudey.…”