the first eigenvalue $\lambda_{\varepsilon}$ is simple.

In view of the Dal Maso-Mosco density result [@DMM87], there exists a sequence ${\Omega}_n{\subseteq}B_1$, $n \ge 1$, such that ${\Omega}_n $ $\gamma$-converges to $\mu$. We can assume that the boundary $\partial {\Omega}_n$ of ${\Omega}_n$ is smooth. Furthermore, we can also assume that ${\Omega}_n$ is connected. For otherwise we keep from ${\Omega}_n$ only a connected component $\Omega_n'$ with the largest $L^1-$norm of its torsion function among all the connected components of $\Omega_n$. Then $ \Omega_n \setminus \Omega_n'$ $\gamma$-converges to the empty set as $n \to \infty$ and the removal of $ \Omega_n \setminus \Omega_n'$ from $\Omega_n$ does not affect the first Dirichlet eigenvalue of $-\Delta$ on $\Omega_n$, while $|\Omega_n'| \le |\Omega_n|$. It is possible to explicitly construct such a sequence $(\Omega_n)$ in the spirit of Cioranescu-Murat [@CM], but such a construction is not needed for the rest of the proof. The only fact we need to keep in mind is that ${\Omega}_n {\subseteq}B_1$, so that $|{\Omega}_n| \le |B_1|$. Denote by $\lambda_1(\Omega_n)$ the first Dirichlet eigenvalue of $-\Delta$ on $\Omega_n$ and by $u_n$ the $L^2-$normalized, positive eigenfunction corresponding to $\lambda_1(\Omega_n)$. We extend $u_n$ to $ B_1$ by setting it to $0$ on $B_1 \setminus {\overline}\Omega_n$ and 