A unified framework for parabolic equations with mixed boundary conditions and diffusion on interfaces

In this paper we consider scalar parabolic equations in a general non-smooth setting emphasizing interface conditions and mixed boundary conditions. In particular, we study dynamics and diffusion on a Lipschitz interface and on the boundary, where the diffusion coefficients are only assumed to be bounded, measurable and positive semidefinite. In the bulk, we consider diffusion coefficients which may degenerate towards a Lipschitz surface. For this problem class, we introduce a unified functional analytic framework based on sesquilinear forms and show maximal LpLp-regularity and bounded H∞H∞-calculus for the corresponding operator, providing well-posedness for a large class of initial conditions and external forces.