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

Abstract

For a bounded Lipschitz domain with Lipschitz interface we show the following compactness theorem: Any L 2 -bounded sequence of vector fields with L 2 -bounded rotations and L 2 -bounded divergences as well as L 2 -bounded tangential traces on one part of the boundary and L 2 -bounded normal traces on the other part of the boundary, contains a strongly L 2 -convergent subsequence. This generalises recent results for homogeneous mixed boundary conditions in [2,4]. As applications we present a related Friedrichs/Poincaré type estimate, a div-curl lemma, and show that the Maxwell operator with mixed tangential and impedance boundary conditions (Robin type boundary conditions) has compact resolvents.