Overdetermined problems for fully nonlinear elliptic equations

We study the situation in which a solution to a fully nonlinear elliptic equation in a bounded domain with an overdetermined boundary condition prescribing both Dirichlet and Neumann constant data forces the domain to be a ball. This is a generalization of Serrin's classical result from 1971. We prove that this rigidity result holds for every fully nonlinear Hessian equation which involves a differentiable operator. We also extend the result to some equations with non differentiable operators such as Pucci operators, under the supplementary assumptions that the space dimension is two or the domain is strictly convex.