  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.
