Parabolic systems with coupled boundary conditions

Abstract

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 $\mathcal{Y}$.

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