Berestycki–Caffarelli–Nirenberg classification conjecture for Serrin domains

Classify every noncompact domain \(\Omega\subset\mathbb{R}^n\) admitting a solution of Serrin’s overdetermined problem \(-\Delta u=n\) in \(\Omega\), \(u>0\) in \(\Omega\), \(u=0\) on \(\partial\Omega\), and \(|Du|=1\) on \(\partial\Omega\), by proving that \(\Omega\) is either a radius-one ball or is congruent to one of the cylinders \(\{y\in\mathbb{R}^m:|y|<m/n\}\times\mathbb{R}^{n-m}\) for some integer \(1\leq m\leq n-1\).

Background

The paper recalls a conjecture of Berestycki, Caffarelli, and Nirenberg concerning the classification of noncompact solutions to Serrin’s overdetermined problem. Cylinders with cross-sectional radius $m/n$ provide explicit noncompact solutions, while the classical compact-domain theorem identifies the radius-one ball as the only compact solution. The present paper proves the ball conclusion under the additional assumption of weak stability, so the unrestricted classification conjecture remains unresolved.