  In this paper we prove uniform a priori estimates for transmission problems
with constant coefficients on two subdomains, with a special emphasis for the
case when the ratio between these coefficients is large. In the most part of
the work, the interface between the two subdomains is supposed to be Lipschitz.
We first study a scalar transmission problem which is handled through a
converging asymptotic series. Then we derive uniform a priori estimates for
Maxwell transmission problem set on a domain made up of a dielectric and a
highly conducting material. The technique is based on an appropriate
decomposition of the electric field, whose gradient part is estimated thanks to
the first part. As an application, we develop an argument for the convergence
of an asymptotic expansion as the conductivity tends to infinity.
