Unbounded periodic solutions to Serrin's overdetermined boundary value problem

We study the existence of nontrivial unbounded domains $\mathrm{\Omega }$ in ${\mathbb{R}}^N$ such that the overdetermined problem $$-\mathrm{\Delta }u=1\text{in }\mathrm{\Omega },u=0,{\mathrm{\partial }}_{\nu }u=\text{const}\text{on }\mathrm{\partial }\mathrm{\Omega }$$ admits a solution $u$. By this, we complement Serrin’s classification result from 1971 which yields that every bounded domain admitting a solution of the above problem is a ball in ${\mathbb{R}}^N$. The domains we construct are periodic in some variables and radial in the other variables, and they bifurcate from a straight (generalized) cylinder or slab. We also show that these domains are uniquely self Cheeger relative to a period cell for the problem.