The Maxwell Compactness Property in Bounded Weak Lipschitz Domains with Mixed Boundary Conditions

Bauer, Sebastian; Pauly, Dirk; Schomburg, Michael

doi:10.1137/16m1065951

Cited by 59 publications

(136 citation statements)

References 20 publications

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“…The main result is given by Theorem 4.8. Here N ≥ 2 and 0 ≤ q ≤ N are natural numbers, the dimension of the domain Ω and the rank of the differential forms, respectively. This generalises the results from [1], where bounded weak Lipschitz domains in the classical setting of R 3 were considered. In fact, the results from [1] can be recovered by setting N = 3 and q = 1 or q = 2.…”

Section: Qsupporting

confidence: 81%

“…This generalises the results from [1], where bounded weak Lipschitz domains in the classical setting of R 3 were considered. In fact, the results from [1] can be recovered by setting N = 3 and q = 1 or q = 2.…”

Section: Qsupporting

confidence: 81%

“…The results of this section immediately follow in the framework of a general functional analytic toolbox, see [20,19,21]. For details, see also the proofs in [1] for the classical case of vector analysis.…”

Section: Applicationsmentioning

confidence: 80%

“…we have ||M −1 0 || ≤ (c 2 m + 1) 1/2 . For details and a proof of Theorem 5.4 in the classical setting of vector analysis see [1].…”

Section: The Maxwell Estimatementioning

confidence: 99%

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3. Weck’s selection theorem: The Maxwell compactness property for bounded weak Lipschitz domains with mixed boundary conditions in arbitrary dimensions

Bauer¹,

Pauly²,

Schomburg³

2019

Maxwell’s Equations

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It is proved that the space of differential forms with weak exterior-and co-derivative, is compactly embedded into the space of square integrable differential forms. Mixed boundary conditions on weak Lipschitz domains are considered. Furthermore, canonical applications such as Maxwell estimates, Helmholtz decompositions and a static solution theory are proved. As a side product and crucial tool for our proofs we show the existence of regular potentials and regular decompositions as well.

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Section: Qsupporting

confidence: 81%

Section: Qsupporting

confidence: 81%

Section: Applicationsmentioning

confidence: 80%

“…we have ||M −1 0 || ≤ (c 2 m + 1) 1/2 . For details and a proof of Theorem 5.4 in the classical setting of vector analysis see [1].…”

Section: The Maxwell Estimatementioning

confidence: 99%

See 2 more Smart Citations

3. Weck’s selection theorem: The Maxwell compactness property for bounded weak Lipschitz domains with mixed boundary conditions in arbitrary dimensions

Bauer¹,

Pauly²,

Schomburg³

2019

Maxwell’s Equations

Self Cite

View full text Add to dashboard Cite

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“…For a proof see [3,Theorem 4.7]. A short historical overview of Weck's selection theorem is given in the introduction of [3], see also the original paper [38] and [27,37,9,39,14,16] for simpler proofs and generalizations.…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

A global div-curl-lemma for mixed boundary conditions in weak Lipschitz domains and a corresponding generalized A 0 * \mathrm{A}_{0}^{*} - A 1 \mathrm{A}_{1} -lemma in Hilbert spaces

Pauly

2019

Analysis

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We prove global and local versions of the so-called div-curl-lemma, a crucial result in the homogenization theory of partial differential equations, for mixed boundary conditions on bounded weak Lipschitz domains in 3D with weak Lipschitz interfaces. We will generalize our results using an abstract Hilbert space setting, which shows corresponding results to hold in arbitrary dimensions as well as for various differential operators. The crucial tools and the core of our arguments are Hilbert complexes and related compact embeddings.We will prove a global version of the div-curl-lemma stating that under certain (mixed tangential and normal) boundary conditions and (very weak) regularity assumptions on a domain Ω ⊂ R 3 , see Section 2, the following holds:Theorem II (global div-curl-lemma). Let Ω ⊂ R 3 be a bounded weak Lipschitz domain with boundary Γ and weak Lipschitz boundary parts Γ t and Γ n . Let (E n ) and (H n ) be two sequences bounded in L 2 (Ω), such that (curl E n ) and (div H n ) are also bounded in L 2 (Ω) and ν × E n = 0 on Γ t and ν · H n = 0 on Γ n . Then there exist subsequences, again denoted by (E n ) and (H n ), such that (E n ), (curl E n ) and (H n ), (div H n ) converge weakly to E, curl E and H, div H in L 2 (Ω), respectively, and the inner products converge as well, i.e.,A local version similar to the classical div-curl-lemma from Theorem I (distributional like convergence for arbitrary domains and no boundary conditions needed) is then immediately implied.Corollary III (local div-curl-lemma). Let Ω ⊂ R 3 be an open set. Let (E n ) and (H n ) be two sequences bounded in L 2 (Ω), such that (curl E n ) and (div H n ) are also bounded in L 2 (Ω). Then there exist subsequences, again denoted by (E n ) and (H n ), such that (E n ), (curl E n ) and (H n ), (div H n ) converge weakly to E, curl E and H, div H in L 2 (Ω), respectively, and the inner products converge in the distributional sense as well, i.e., for all ϕ ∈C ∞ (Ω) it holds

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Nonlocal homogenisation theory for curl‐div‐systems

Nicaise

Waurick

2022

Mathematische Nachrichten

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We study the curl‐div‐system with variable coefficients and a nonlocal homogenisation problem associated with it. Using, in part refining, techniques from nonlocal H‐convergence for closed Hilbert complexes, we define the appropriate topology for possibly nonlocal and non‐periodic coefficients in curl‐div systems to model highly oscillatory behaviour of the coefficients on small scales. We address curl‐div systems under various boundary conditions and analyse the limit of the ratio of small scale over large scale tending to zero. Already for standard Dirichlet boundary conditions and local coefficients the limit system is nontrivial and unexpected. Furthermore, we provide an analysis of highly oscillatory local coefficients for a curl‐div system with impedance type boundary conditions relevant in scattering theory for Maxwell's equations and relate the abstract findings to local H‐convergence and weak*‐convergence of the coefficients.

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The Maxwell Compactness Property in Bounded Weak Lipschitz Domains with Mixed Boundary Conditions

Cited by 59 publications

References 20 publications

3. Weck’s selection theorem: The Maxwell compactness property for bounded weak Lipschitz domains with mixed boundary conditions in arbitrary dimensions

3. Weck’s selection theorem: The Maxwell compactness property for bounded weak Lipschitz domains with mixed boundary conditions in arbitrary dimensions

A global div-curl-lemma for mixed boundary conditions in weak Lipschitz domains and a corresponding generalized A 0 * \mathrm{A}_{0}^{*} - A 1 \mathrm{A}_{1} -lemma in Hilbert spaces

Nonlocal homogenisation theory for curl‐div‐systems

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