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We construct nontrivial smooth bounded domains $\Omega \subseteq\mathbb{R}^n$ of the form $\Omega_0 \setminus \overline{\Omega}_1$, bifurcatingfrom annuli, for which there exists a positive solution to the overdeterminedboundary 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 tohold, namely, the only domains which admit such a solution are annuli andsolutions are radially symmetric. Our construction shows that the condition isalso necessary. In addition, the constructed domains are shown to beself-Cheeger.

Nontrivial solutions to Serrin's problem in annular domains