Overdetermined boundary value problems and torsion

We discuss overdetermined elliptic boundary value problems in the Euclidean space.  We will focus in particular on the torsion problem, which consists in the study of pairs (Ω,u), where Ω⊂R^n is a bounded domain and u: Ω→R is a function with constant nonzero laplacian and such that u= 0 on the boundary ∂Ω. A celebrated result, proven by Serrin, states that, if the normal derivative ∂u/∂ν is constant on ∂Ω, then Ω is a ball and u is rotationally symmetric.  After reviewing Serrin’s work, we will introduce and apply some recent geometric techniques to study the case where ∂Ω is disconnected and ∂u/∂ν is allowed to assume distinct values on different boundary components.