H-convergence and homogenization of non-local elliptic operators in both perforated and non-perforated domains

Abstract

In this paper, we focus on the homogenization process of the non-local elliptic boundary value problem $$\mathcal{L}_\varepsilon^s u_\varepsilon =(-\nabla\cdot (A_\varepsilon(x)\nabla))^{s}u_\varepsilon=f \mbox{ in } \mathcal O, $$ with $0<s<1$, considering non-homogeneous Dirichlet type condition outside of the bounded domain $\mathcal O\subseteq \mathbb{R}^n$. We find the homogenized problem by using the $H$-convergence method, as $\varepsilon\to 0$, under standard uniform ellipticity, boundedness and symmetry assumptions on coefficients $A_\varepsilon(x)$, with the homogenized coefficients as the standard $H$-limit (cf. \cite{MT1}) of the sequence $\{A_\varepsilon\}_{\varepsilon>0}$. We also prove that the commonly referred to as \textit{the strange term} in the literature (see \cite[Chapter 4]{MT}) does not appear in the homogenized problem associated with the fractional Laplace operator $(-\Delta)^s$ in a perforated domain. Both of these results have been obtained in the class of general microstructures. Consequently, we could certify that the homogenization process, as $\varepsilon\to 0$, is stable under $s\to 1^{-}$ in the non-perforated domains, but not necessarily in the case of perforated domains.