Local analysis of a two phase free boundary problem concerning mean curvature

We consider an overdetermined problem for a two phase elliptic operator in divergence form with piecewise constant coefficients. We look for domains such that the solution $u$ of a Dirichlet boundary value problem also satisfies the additional property that its normal derivative $\partial_n u$ is a multiple of the radius of curvature at each point on the boundary. When the coefficients satisfy some "non-criticality" condition, we construct nontrivial solutions to this overdetermined problem employing a perturbation argument relying on shape derivatives and the implicit function theorem. Moreover, in the critical case, we employ the use of the Crandall-Rabinowitz theorem to show the existence of a branch of symmetry breaking solutions bifurcating from trivial ones. Finally, some remarks on the one phase case and a similar overdetermined problem of Serrin type are given.