Nontrivial solutions to the relative overdetermined torsion problem in a cylinder

Given a bounded regular domain ω⊂RN-1 and the half-cylinder Σ=ω×(0,+∞), we consider the relative overdetermined torsion problem in Σ, i.e. (Formula presented.) where Ω⊂Σ, ΓΩ=∂Ω∩Σ, Γ~Ω=∂Ω\ΓΩ, ν is the outer unit normal vector on ΓΩ and η is the outer unit normal vector on Γ~Ω. We build nontrivial solutions to this problem in domains Ω that are the hypograph of certain nonconstant functions v:ω¯→(0,+∞). Such solutions can be reflected with respect to 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 ω.