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 ∂f ∂ν ∈ Y⊥, where Y is a closed subspace of L(∂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 .