  We study a fully nonlinear free transmission problem in the presence of
general degeneracy terms. Under minimal conditions on the degeneracy of the
model, we establish the existence of viscosity solutions for the associated
Dirichlet problem. Once the existence of solutions has been established, we
focus on their regularity estimates. By imposing a Dini-continuity condition on
the degeneracy laws involved in the model, we prove that viscosity solutions
are locally differentiable.
