in is a maximum at $A_n=S_{\frac23,n}$. This maximiser is unique modulo sequences of measurable sets $B_n \subset A_n,\,n\in {\mathbb{N}}$ with measure $|B_n|=0,\,n\in {\mathbb{N}}$, and sequences of measurable sets $C_n \subset \Omega_{\frac23,n}\setminus A_n,\,n\in {\mathbb{N}}$, with $\lim_{n\rightarrow 0}|C_n|=0$. This completes the proof of assertion (iv). $\square$

Proof of Theorem \[the7\]  {#sec5}
==========================

[*Proof of Theorem .*]{} The proof follows a method from [@HLP], which can be summarised as follows: construct a capacitary measure $\mu$ so that the efficiency of the eigenfunction of the first eigenvalue of $- \Delta + \mu$ almost equals $1$ and then approximate the capacitary measure in the sense of $\gamma$-convergence by a sequence of domains. [ Such an]{} approximation is possible in view of the Dal Maso-Mosco density result. (It could possibly be done locally using a Cioranescu-Murat periodic structure.) We refer the reader to [@BB05 Chapter 4] for the notions of $\gamma$-convergence, capacitary measures and approximation by sequences of domains.

Let ${\overline}B_r$ denote the closure of $B_r$. For any $0 < \varepsilon < 1$, we consider the Dirichlet-Neumann eigenvalue problem on the annulus $A_{\varepsilon}=B_1{\setminus}{\overline}B_{1-{\varepsilon}}$. Denote by $\lambda_{\varepsilon}$ the first eigenvalue and by $u_{\varepsilon}$ a corresponding eigenfunction,

$$\begin{cases}
-\Delta u_{\varepsilon}=\lambda_{\varepsilon}u_{\varepsilon},&\text{in }A_{\varepsilon},\\
{ u_{\varepsilon}}=0,&\text{on }\partial B_{1},\\
\frac{\partial { u_{\varepsilon}}}{\partial 