Parabolic systems with coupled boundary conditions

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 $f_{\mathrm{\mid }\mathrm{\partial }\mathrm{\Omega }}\in \mathcal{Y}$ and $\frac{\mathrm{\partial }f}{\mathrm{\partial }\nu }\in {\mathcal{Y}}^{\perp }$, where $\mathcal{Y}$ is a closed subspace of $L^2(\mathrm{\partial }\mathrm{\Omega };W)$. We discuss well-posedness and further qualitative properties, systematically reducing features of the parabolic system to operator-theoretical properties of the orthogonal projection onto $\mathcal{Y}$.