Overdetermined problems with possibly degenerate ellipticity, a geometric approach

Given an open bounded connected subset Omega of R-n, we consider the over-determined boundary value problem obtained by adding both zero Dirichlet and constant Neumann boundary data to the elliptic equation-div(A(vertical bar del u vertical bar)del u) = 1 in Omega. We prove that, if this problem admits a solution in a suitable weak sense, then Omega is a ball. This is obtained under fairly general assumptions on Omega 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 partial derivative Omega.