Unbounded periodic solutions to Serrin's overdetermined boundary value problem

We study the existence of nontrivial unbounded domains $Ω$ in $\mathbb{R}^N$ such that the overdetermined problem $-Δu = 1 \quad \text{in $Ω$}, \qquad u=0, \quad \partial_νu=\textrm{const} \qquad \text{on $\partial Ω$}$ 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.

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