Nontrivial solutions to Serrin's problem in annular domains

Abstract

We construct nontrivial bounded, real analytic domains $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ of the form ${\mathrm{\Omega }}_0\setminus {\overline{\mathrm{\Omega }}}_1$, bifurcating from annuli, which admit a positive solution to the overdetermined boundary value problem

where ν stands for the inner unit normal to ∂Ω. From results by Reichel [1] and later by Sirakov [2], it was known that the condition ${\partial }_{\nu }u\le 0$ on $\partial {\mathrm{\Omega }}_1$ 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, we show that the constructed domains are self-Cheeger.