A Serrin-type symmetry result on model manifolds: An extension of the Weinberger argument

We consider the classical ‘‘Serrin's symmetry result” for the overdetermined boundary value problem related to the equation $\mathrm{\Delta }u=?1$ in a model manifold of non-negative Ricci curvature. Using an extension of the Weinberger classical argument we prove a Euclidean symmetry result under a suitable ‘‘compatibility” assumption between the solution and the geometry of the model.