Nontrivial solutions to the relative overdetermined torsion problem in a cylinder

Given a bounded regular domain $$\omega \subset \mathbb {R}^{N-1}$$ ω ⊂ R N - 1 and the half-cylinder $$\Sigma = \omega \times (0,+\infty )$$ Σ = ω × ( 0 , + ∞ ) , we consider the relative overdetermined torsion problem in $$\Sigma $$ Σ , i.e. $$\begin{aligned} {\left\{ \begin{array}{ll} \Delta {u}+1=0 & \hbox { in}\ \Omega ,\\ \partial _\eta u = 0 & \hbox { on}\ {\widetilde{\Gamma }}_\Omega ,\\ u=0 & \hbox { on}\ \Gamma _\Omega ,\\ \partial _{\nu }u =c & \hbox { on}\ \Gamma _\Omega , \end{array}\right. } \end{aligned}$$ Δ u + 1 = 0 in Ω , ∂ η u = 0 on Γ ~ Ω , u = 0 on Γ Ω , ∂ ν u = c on Γ Ω , where $$\Omega \subset \Sigma $$ Ω ⊂ Σ , $$\Gamma _\Omega = \partial \Omega \cap \Sigma $$ Γ Ω = ∂ Ω ∩ Σ , $${\widetilde{\Gamma }}_\Omega = \partial \Omega {\setminus } \Gamma _\Omega $$ Γ ~ Ω = ∂ Ω \ Γ Ω , $$\nu $$ ν is the outer unit normal vector on $$\Gamma _\Omega $$ Γ Ω and $$\eta $$ η is the outer unit normal vector on $${\widetilde{\Gamma }}_\Omega $$ Γ ~ Ω . We build nontrivial solutions to this problem in domains $$\Omega $$ Ω that are the hypograph of certain nonconstant functions $$v: {\overline{\omega }} \rightarrow (0, + \infty )$$ v : ω ¯ → ( 0 , + ∞ ) . Such solutions can be reflected with respect to $$\omega $$ ω , 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 $$\omega $$ ω .