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 by the first author and collaborators. As applications we present a related Friedrichs/Poincare 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.

Keywords: compact embeddings, div-curl system, mixed boundary conditions, inhomogeneous boundary conditions

References in corpus (1)

• Complexes from complexes

Cited by in corpus (3)

• Semi-uniform stabilization of anisotropic Maxwell's equations via boundary feedback on split boundary
• Homogenization and nonselfadjoint spectral optimization for dissipative Maxwell eigenproblems
• Weak equals strong L2 regularity for partial tangential traces on Lipschitz domains