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For a bounded Lipschitz domain with Lipschitz interface we show the followingcompactness theorem: Any $L^2$-bounded sequence of vector fields with$L^2$-bounded rotations and $L^2$-bounded divergences as well as $L^2$-boundedtangential traces on one part of the boundary and $L^2$-bounded normal traceson the other part of the boundary, contains a strongly $L^2$-convergentsubsequence. This generalises recent results for homogeneous mixed boundaryconditions by the first author and collaborators. As applications we present arelated Friedrichs/Poincare type estimate, a div-curl lemma, and show that theMaxwell operator with mixed tangential and impedance boundary conditions (Robintype boundary conditions) has compact resolvents.

A Compactness Result for the div-curl System with Inhomogeneous Mixed Boundary Conditions for Bounded Lipschitz Domains and Some Applications