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|∂Ω∈Y and , where Y is a closed subspace of L2(∂Ω;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 Y.