t)$ when $\epsilon$ is close to $0$ and positive?

3.  Let $j \in \{1,2\}$. Let $t$ be a fixed point of $\mathbb{R}^n\setminus\Omega^i_j$ such that $p^j +r_\ast t \not \in (p^l+r_\ast \mathrm{cl}\Omega_l)$ if $l \neq j$. What can be said of the map $\epsilon\mapsto u_{\epsilon}(\eta(\epsilon)p^j+\epsilon t)$ when $\epsilon$ is close to $0$ and positive?

In a sense, question (i) concerns the ‘macroscopic’ behavior of $u_{\epsilon}$ far from the holes $\eta(\epsilon)p^1+\epsilon \Omega^i_1$ and $\eta(\epsilon)p^2+\epsilon \Omega^i_2$, whereas question (ii) concerns the ‘microscopic’ behavior of $u_{\epsilon}$ in proximity of centers of the perforations, and question (iii) concerns the ‘microscopic’ behavior of $u_{\epsilon}$ in proximity of the boundary of one of the perforations.

Boundary value problems in domains with small holes are typical in the frame of asymptotic analysis and are usually investigated by means of asymptotic expansion methods. As an example, we mention the method of matching outer and inner asymptotic expansions proposed by Il’in (see [@Il78], [@Il92], and [@Il99]) and the compound asymptotic expansion method of Maz’ya, Nazarov, and Plamenevskij, which allows the treatment of general Douglis–Nirenberg elliptic boundary value problems in domains with perforations and corners (cf. [@MaNaPl00]). Moreover, in Kozlov, Maz’ya, and Movchan [@KoMaMo99] one can find the study of boundary 