Given a bounded regular domain \(\omega \subset \mathbb {R}^{N-1}\) and the half-cylinder \(\Sigma = \omega \times (0,+\infty )\) , we consider the relative overdetermined torsion problem in \(\Sigma \) , i.e. \(\begin{aligned} {\left\{ \begin{array}{ll} \Delta {u}+1=0 & \hbox { in}\ \Omega ,\\ \partial _\eta u = 0 & \hbox { on}\ {\widetilde{\Gamma }}_\Omega ,\\ u=0 & \hbox { on}\ \Gamma _\Omega ,\\ \partial _{\nu }u =c & \hbox { on}\ \Gamma _\Omega , \end{array}\right. } \end{aligned}\) where \(\Omega \subset \Sigma \) , \(\Gamma _\Omega = \partial \Omega \cap \Sigma \) , \({\widetilde{\Gamma }}_\Omega = \partial \Omega {\setminus } \Gamma _\Omega \) , \(\nu \) is the outer unit normal vector on \(\Gamma _\Omega \) and \(\eta \) is the outer unit normal vector on \({\widetilde{\Gamma }}_\Omega \) . We build nontrivial solutions to this problem in domains \(\Omega \) that are the hypograph of certain nonconstant functions \(v: {\overline{\omega }} \rightarrow (0, + \infty )\) . Such solutions can be reflected with respect to \(\omega \) , giving nontrivial solutions to the relative overdetermined torsion problem in a cylinder. The proof uses a local bifurcation argument which, quite remarkably, works for most smooth domains \(\omega \) .
