Nontrivial solutions to Serrin's problem in annular domains

We construct nontrivial smooth bounded domains $\Omega \subseteq \mathbb{R}^n$ of the form $\Omega_0 \setminus \overline{\Omega}_1$, bifurcating from annuli, for which there exists a positive solution to the overdetermined boundary value problem $$-\Delta u = 1, \; u>0 \quad \text{in } \Omega, \qquad u = 0 ,\; \partial_\nu u = \text{const} \quad \text{on } \partial\Omega_0, \qquad u = \text{const} ,\; \partial_\nu u = \text{const} \quad \text{on } \partial \Omega_1,$$ where $\nu$ stands for the inner unit normal to $\partial\Omega$. From results by Reichel and later by Sirakov, it was known that the condition $\partial_\nu u \leq 0$ on $\partial\Omega_1$ is sufficient for rigidity to hold, namely, the only domains which admit such a solution are annuli and solutions are radially symmetric. Our construction shows that the condition is also necessary. In addition, the constructed domains are shown to be self-Cheeger.