Unbounded periodic solutions to Serrin's overdetermined boundary value problem

Abstract

We study the existence of nontrivial unbounded domains $\Omega$ in ${\mathbb{R}}^N$ such that the overdetermined problem $$-\Delta u=1\text{in Ω},u=0,{\partial }_{\nu }u=\text{const}\text{on ∂Ω}$$ 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.

22 pages. Minor corrections have been made and the bibliographical reference is updated. To appear in Arch. Rational Mech. Anal