Traces of Differential Forms on Lipschitz Boundaries

Weck, Norbert

doi:10.1524/anly.2004.24.14.147

Cited by 14 publications

(29 citation statements)

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“…The aim of this section is to transfer Lemma 4.2 to arbitrary weak Lipschitz pairs (Ω, Γ τ ). To this end we will employ a technical lemma, whose proof is sketched in[23, Section 3] and[30, Remark 2]. We give a detailed proof in the appendix.…”

mentioning

confidence: 99%

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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mentioning

confidence: 99%

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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show abstract

“…is well defined, linear and continuous. Moreover [35,Thm. 4], it is surjective and hence admits a continuous right inverse t −1 .…”

Section: For Smooth Boundaries the Spaces Hmentioning

confidence: 99%

“…Letω = t ω denote the tangential trace according to (19). It is shown in [35,Sec. 2] that the tangential trace can be extended to H 1 Λ p (Ω).…”

Section: Sobolev Spaces On the Boundarymentioning

confidence: 99%

“…The domain integral on the right-hand side vanishes due to the gauge condition (35). With (55) and (56) we can write down a first version of the representation formula for ω ∈ Y p (Ω) ∩ F p (Ω),…”

Section: Representation Formula For Maxwell Solutionsmentioning

confidence: 99%

“…There exists a Hodge decomposition [35, Thm. 11] of the spaces H It has been shown in[35, Thm. 8] that the L 2 inner product on the bound-…”

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confidence: 99%

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Differential Forms and Boundary Integral Equations for Maxwell-Type Problems

Kurz

Auchmann

2012

Fast Boundary Element Methods in Engineering and Industrial Applications

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We present boundary-integral equations for Maxwell-type problems in a differential-form setting. Maxwell-type problems are governed by the differential equation (δd − k 2 )ω = 0, where k ∈ C holds, subject to some restrictions. This problem class generalizes curl curland div grad-types of problems in three dimensions. The goal of the paper is threefold: 1) Establish the Sobolev-space framework in the full generality of differential-form calculus on a smooth manifold of arbitrary dimension and with Lipschitz boundary. 2) Introduce integral transformations and fundamental solutions, and derive a representation formula for Maxwell-type problems. 3) Leverage the power of differential-form calculus to gain insight into properties and inherent symmetries of boundary-integral equations of Maxwell-type.Acknowledgements The authors would like to thank Ralf Hiptmair for feedback and valuable suggestions.

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Dirac Integral Equations for Dielectric and Plasmonic Scattering

Helsing

Rosén

2021

Integr. Equ. Oper. Theory

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A new integral equation formulation is presented for the Maxwell transmission problem in Lipschitz domains. It builds on the Cauchy integral for the Dirac equation, is free from false eigenwavenumbers for a wider range of permittivities than other known formulations, can be used for magnetic materials, is applicable in both two and three dimensions, and does not suffer from any low-frequency breakdown. Numerical results for the two-dimensional version of the formulation, including examples featuring surface plasmon waves, demonstrate competitiveness relative to state-of-the-art integral formulations that are constrained to two dimensions. However, our Dirac integral equation performs equally well in three dimensions, as demonstrated in a companion paper.

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Traces of Differential Forms on Lipschitz Boundaries

Cited by 14 publications

References 0 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

Differential Forms and Boundary Integral Equations for Maxwell-Type Problems

Dirac Integral Equations for Dielectric and Plasmonic Scattering

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