A note on an overdetermined system involving mean curvature

Let ΩΩ be a bounded simply connected domain in RnRn with connected boundary of class C2C2. We show that, if there exist nn functions satisfying some overdetermined system with boundary conditions involving mean curvature of ∂Ω∂Ω, then ΩΩ is a ball. The proof is based on the maximum principle and Serrin’s theorem.