Rigidity results for Serrin's overdetermined problems in Riemannian manifolds

In this work, we are interested in studying Serrin's overdetermined problems in Riemannian manifolds. For manifolds endowed with a conformal vector field, we prove a Pohozoaev-type identity to show a Serrin's type rigidity result using the P-function approach introduced by Weinberger. We proceed with a conformal change to achieve this goal, starting from a geometric Pohozaev identity due to Schoen. Moreover, we obtain a symmetry result for the associated Dirichlet problem by using a generalized normalized wall shear stress bound.