Nontrivial solutions to the relative overdetermined torsion problem in a cylinder

Given a bounded regular domain $\omega \subset {\mathbb{R}}^{N-1}$ and the half-cylinder $\mathrm{\Sigma }=\omega \times (0,+\mathrm{\infty })$, we consider the relative overdetermined torsion problem in $\mathrm{\Sigma }$, i.e. \left\{ \begin{array}{ll} Δ{u}+1=0 &in $Ω$,\newline \partial_ηu = 0 &on $\widetilde Γ_Ω$,\newline u=0 &on $Γ_Ω$,\newline \partial_νu =c &on $Γ_Ω$. \end{array} \right. where $\mathrm{\Omega }\subset \mathrm{\Sigma }$, ${\mathrm{\Gamma }}_{\mathrm{\Omega }}=\mathrm{\partial }\mathrm{\Omega }\cap \mathrm{\Sigma }$, ${\tilde{\mathrm{\Gamma }}}_{\mathrm{\Omega }}=\mathrm{\partial }\mathrm{\Omega }\setminus {\mathrm{\Gamma }}_{\mathrm{\Omega }}$, $\nu$ is the outer unit normal vector on ${\mathrm{\Gamma }}_{\mathrm{\Omega }}$ and $\eta$ is the outer unit normal vector on ${\tilde{\mathrm{\Gamma }}}_{\mathrm{\Omega }}$. We build nontrivial solutions to this problem in domains $\mathrm{\Omega }$ that are the hypograph of certain nonconstant functions $v:\bar{\omega }\to (0,+\mathrm{\infty })$. 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 any generic base $\omega$.