Let $\Omega \subset \mathbb{R}^n$ be a smooth bounded domain and $B \subset \Omega$ be an open ball, for a given $\epsilon>0$ consider
$$a_\epsilon(x) = \begin{cases} 1, \qquad x \in \Omega \backslash B \\ \epsilon, \qquad x \in B \end{cases}$$
Let $f \in H^{1/2}(\partial \Omega)$, i.e. $f$ is boundary data of some $H^1(\Omega)$ function, in the sense of traces, and let $u_\epsilon \in H^1(\Omega)$ be the unique weak solution of the problem
$$\begin{cases} div(a_\epsilon \nabla u_\epsilon) = 0, \qquad &\text{in} \ \Omega \\ u_\epsilon=f, \qquad &\text{on} \ \partial \Omega \end{cases}$$
Let also $u_0 \in H^1(\Omega \backslash \overline{B})$ be the unique weak solution of
$$\begin{cases} \Delta u_0 = 0, \qquad &\text{in} \ \Omega \backslash \overline{B} \\ \frac{\partial u_0}{\partial n} = 0 \qquad &\text{on} \ \partial B \\ u_0=f, \qquad &\text{on} \ \partial \Omega \end{cases}$$
How can we show that $u_\epsilon \to u_0$ in $H^1(\Omega \backslash \overline{B})$ as $\epsilon \to 0$?
There are the associated Dirichlet energies:
$$E_\epsilon(u_\epsilon) := \int_\Omega a_\epsilon |\nabla u_\epsilon|^2 dx, \qquad E_0(u_0) = \int_{\Omega \backslash \overline{B}} |\nabla u_0|^2 dx$$
We then have,
$$E_\epsilon(u_\epsilon) = \int_{\Omega \backslash \overline{B}} |\nabla u_\epsilon|^2 dx + \int_B \epsilon |\nabla u_\epsilon|^2 dx $$
The idea should be to prove that $\|u_\epsilon\|_{H^1(\Omega \backslash \overline{B})}$ is bounded uniformly in $\epsilon$, so that we can extract weakly convergent subsequence with a limit $u_0 \in H^1(\Omega \backslash \overline{B})$. This limit clearly satisfies $\Delta u_0 = 0$ in $\Omega \backslash \overline{B}$, by just using the weak convergence and the weak formulations of PDEs. I don't know how to obtain the Neumann boundary condition for $u_0$ on $\partial B$. Then if we show the convergence of energies $E_\epsilon(u_\epsilon) \to E_0(u_0)$ we can conclude that $u_\epsilon \to u_0$ strongly in $H^1(\Omega \backslash \overline{B})$.
We clearly have that $\|\nabla u_\epsilon\|_{L^2(\Omega \backslash \overline{B})}$ is bounded uniformly in $\epsilon$, for $\epsilon \leq 1$, using that the energy $E_\epsilon(v)$ is minimized for $v=u_\epsilon$. But I don't know how to conclude the boundedness of $\|u_\epsilon\|_{L^2(\Omega \backslash \overline{B})}$.
Can one also consider a limit $\epsilon \to \infty$? If so what would be the limiting problem? Thank you.