An overdetermined problem in a weakly star shaped annular domain in space forms

We obtain symmetry results of the solution to some overdetermined problems related to the eigenvalue equation on a weakly star shaped annular domain in space forms. If the associated P-function is constant on the boundary, then we get a Serrin type symmetry. To justify the assumption, we obtain a radial function satisfying the eigenvalue equation, which is different from the well-known Serrin type solution. On the other hand, we prove the same result in aweakly star shaped annular domain with a spherical boundary component without the condition on P.