Overdetermined problems with possibly degenerate ellipticity, a geometric approach

Given an open bounded connected subset Ω of R, we consider the overdetermined boundary value problem obtained by adding both zero Dirichlet and constant Neumann boundary data to the elliptic equation −div(A(|∇u|)∇u) = 1 in Ω. We prove that, if this problem admits a solution in a suitable weak sense, then Ω is a ball. This is obtained under fairly general assumptions on Ω and A. In particular, A may be degenerate and no growth condition is required. Our method of proof is quite simple. It relies on a maximum principle for a suitable P -function, combined with some geometric arguments involving the mean curvature of ∂Ω.