  We consider elliptic operators with operator-valued coefficients and discuss
the associated parabolic problems. The unknowns are functions with values in a
Hilbert space $W$. The system is equipped with a general class of coupled
boundary conditions of the form $f_{|\partial\Omega}\in \mathcal Y$ and
$\frac{\partial f}{\partial \nu}\in {\mathcal Y}^\perp$, where $\mathcal Y$ is
a closed subspace of $L^2(\partial\Omega;W)$. We discuss well-posedness and
further qualitative properties, systematically reducing features of the
parabolic system to operator-theoretical properties of the orthogonal
projection onto $\mathcal Y$.
