  We consider the situation when an elliptic problem in a subdomain $\Omega_1$
of an $n$-dimensional bounded domain $\Omega$ is coupled via inhomogeneous
canonical transmission conditions to a parabolic problem in
$\Omega\setminus\Omega_1$. In particular, we can treat elliptic-parabolic
equations in bounded domains with discontinuous coefficients. Using Fourier
multiplier techniques, we prove an a priori estimate for strong solutions to
the equations in $L^p$-Sobolev spaces.
