  We consider a transmission problem in which the interior domain has
infinitely ramified structures. Transmission between the interior and exterior
domains occurs only at the fractal component of the interface between the
interior and exterior domains. We also consider the sequence of the
transmission problems in which the interior domain is obtained by stopping the
self-similar construction after a finite number of steps; the transmission
condition is then posed on a prefractal approximation of the fractal interface.
We prove the convergence in the sense of Mosco of the energy forms associated
with these problems to the energy form of the limit problem. In particular,
this implies the convergence of the solutions of the approximated problems to
the solution of the problem with fractal interface. The proof relies in
particular on an extension property. Emphasis is put on the geometry of the
ramified domain. The convergence result is obtained when the fractal interface
has no self-contact, and in a particular geometry with self-contacts, for which
an extension result is proved.
