A compactness result for the div-curl system with inhomogeneous mixed boundary conditions for bounded Lipschitz domains and some applications

For a bounded Lipschitz domain with Lipschitz interface we show the following compactness theorem: Any \(\mathsf {L}_{}^{2}\)-bounded sequence of vector fields with \(\mathsf {L}_{}^{2}\)-bounded rotations and \(\mathsf {L}_{}^{2}\)-bounded divergences as well as \(\mathsf {L}_{}^{2}\)-bounded tangential traces on one part of the boundary and \(\mathsf {L}_{}^{2}\)-bounded normal traces on the other part of the boundary, contains a strongly \(\mathsf {L}_{}^{2}\)-convergent subsequence. This generalises recent results for homogeneous mixed boundary conditions in Bauer et al. (SIAM J Math Anal 48(4):2912-2943, 2016) Bauer et al. (in: Maxwell’s Equations: Analysis and Numerics (Radon Series on Computational and Applied Mathematics 24), De Gruyter, pp. 77-104, 2019). 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.