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

10 Mar 2021  ·  Dirk Pauly, Nathanael Skrepek ·

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. read more

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Analysis of PDEs Functional Analysis