It is not clear to me what transmission conditions you want to set on the interface between the two regions.
If it is u_0=u_1 and \sigma_0 N=0 (\sigma_0 the stress from the region 0),
it means that you can first solve u_0 on region 0 with Neumann BC, then solve u_1 on region 1 with nonhomogeneous Dirichlet condition u_1=u_0.
If the conditions are more complicate and really coupled, an example is
The description of the problem and corresponding coupling interface conditions is in the pdf file in the beginning of that discussion.
The interface is considered either as a boundary of the domain above (boundary label 5 of Th1 int1d(Th1,5)), either as a boundary of the domain below (boundary label 3 of Th2 int1d(Th2,3)), depending if the test function corresponds to the unknown in the domain above or below.
These interface integrals can involve unknowns from both domains.