Serrin’s Overdetermined Problem on S^N x R

We show the existence of non-trivial domains Ω of SN× R (SN being the N-dimensional unit sphere) which support the solution to the Serrin’s overdetermined boundary value problem {ΔSN×Ru=-1inΩ,u=0on∂Ω,∂u∂ν=const.on∂Ω. Here ΔSN×R denotes the Laplace–Beltrami operator on SN× R and ∂∂ν denotes the derivative in the direction of the outer unit normal vector to ∂Ω. These domains are obtained by bifurcation of symmetric straight tubular neighborhoods of SN× { 0 } and they are not bounded by geodesic spheres.