Does first eigenfunction of Laplacian with mixed Dirichlet-Neumann condition attains maxima only on the Neumann boundary?

Question

Consider a doubly connected bounded domain of the form $\Omega=\Omega_0 \setminus \overline\Omega_1$, where $\Omega_0,\Omega_1$ are bounded open sets and consider the following equation on $\Omega$: \begin{align} -\Delta u&=\lambda_1u , \;\text{in}\; \Omega\\ u&=0,\; \text{on}\; \partial\Omega_1\\ \frac{\partial u}{\partial \eta}&=0,\; \text{on}\; \partial\Omega_0 \end{align} where $\eta$ is the outer unit normal to $\partial \Omega_0$ and $\lambda_1$ is the first eigenvalue. Note that first eigenfunction $u>0$ inside $\Omega$ and hence attains minima on $\partial\Omega_1$ only. Whether maxima attains only on the Neumann boundary $\partial\Omega_0$ ? For example when $\Omega$ is concentric annulus it happens. Does it hold for any general set or is there any counterexample?