  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.
