Unbounded periodic solutions to Serrin's overdetermined boundary value problem

TL;DR

This paper constructs unbounded, periodic domains in Euclidean space where a specific overdetermined boundary value problem admits solutions, extending Serrin's classical result from bounded to certain unbounded domains.

Contribution

It introduces new unbounded, periodic domains that admit solutions to Serrin's overdetermined problem, expanding the classification beyond bounded domains.

Findings

Constructed unbounded periodic domains with solutions

Domains bifurcate from cylinders or slabs

Domains are uniquely self Cheeger relative to a period cell

Abstract

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