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

TL;DR

This paper proves a compactness theorem for vector fields in Lipschitz domains with mixed boundary conditions, extending previous results, and applies it to derive estimates, a div-curl lemma, and properties of the Maxwell operator.

Contribution

It generalizes compactness results for the div-curl system to inhomogeneous mixed boundary conditions on Lipschitz domains and explores applications to Maxwell's equations.

Findings

Established a compactness theorem for vector fields with mixed boundary conditions.

Derived a Friedrichs/Poincare type estimate and a div-curl lemma.

Proved the Maxwell operator with mixed boundary conditions has compact resolvents.

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.