A symmetry problem with applications to finance

In this paper we will consider an overdetermined problem in an unknown ring-shaped domain, and we prove that the domain has to be spherical ring. We will apply this problem to an \(n\) -dimensional toy model in the mean field game theory introduced by Lasry-Lions too. We show that if in Lasry-Lions model, the additional extra data ”the boundary is a level set” is assumed, then the region, where the solution is harmonic, has to have spherical symmetry.