Overdetermined problems for sum Hessian equations in the space forms

Abstract

We study overdetermined problems for normalized sum Hessian equations and their quotients in the space forms. The equations and boundary conditions imply admissibility and ellipticity. For quotient equations, we prove that the domain is a geodesic ball if it is strictly star-shaped, or if the Dirichlet value satisfies an upper bound. We also prove the corresponding result for sum Hessian equations without star-shapedness. The proofs use two auxiliary functions and a Rellich--Poho\v{z}aev identity.