Weck's Selection Theorem: The Maxwell Compactness Property for Bounded Weak Lipschitz Domains with Mixed Boundary Conditions in Arbitrary Dimensions

Bauer, Sebastian; Pauly, Dirk; Schomburg, Michael

doi:10.48550/arxiv.1809.01192

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“…Moreover, let the boundary of be decomposed into two relatively open and strong Lipschitz subsets 0 and 1 := \ 0 forming the interface 0 ∩ 1 for the mixed boundary conditions. See [2][3][4] for exact definitions. We call ( , 0 ) a bounded strong Lipschitz pair.…”

Section: Notationsmentioning

confidence: 99%

“…In [4, Theorem 5.5], see [3,Theorem 7.4] for more details and compare to [2], the following theorem about the existence of regular potentials for the rotation with homogeneous mixed boundary conditions has been shown.…”

Section: Preliminariesmentioning

confidence: 99%

“…The main result (compact embedding) is formulated in Theorem 4.1. Note that while Theorem 1.1 even holds for bounded weak 3 Lipschitz pairs ( , 0 ), cf. [2, Theorem 4.7], Theorem 4.1 is only shown for bounded strong Lipschitz pairs.…”

Section: Introductionmentioning

confidence: 99%

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

Pauly

Skrepek

2022

Ann Univ Ferrara

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For a bounded Lipschitz domain with Lipschitz interface we show the following compactness theorem: Any $$\mathsf {L}_{}^{2}$$ L 2 -bounded sequence of vector fields with $$\mathsf {L}_{}^{2}$$ L 2 -bounded rotations and $$\mathsf {L}_{}^{2}$$ L 2 -bounded divergences as well as $$\mathsf {L}_{}^{2}$$ L 2 -bounded tangential traces on one part of the boundary and $$\mathsf {L}_{}^{2}$$ L 2 -bounded normal traces on the other part of the boundary, contains a strongly $$\mathsf {L}_{}^{2}$$ 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.

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Section: Notationsmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%