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A global div-curl-lemma for mixed boundary conditions in weak Lipschitz domains and a corresponding generalized A0*-A1-lemma in Hilbert spaces

  • Dirk Pauly EMAIL logo
From the journal Analysis

Abstract

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.

Acknowledgements

The author is grateful to Sören Bartels for bringing up the topic of the ÷-curl-lemma, and especially to Marcus Waurick for lots of inspiring discussions on the ÷-curl-lemma and for his substantial contributions to the Special Semester at RICAM in Linz late 2016.

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Received: 2018-04-19
Revised: 2018-11-29
Accepted: 2019-02-10
Published Online: 2019-04-17
Published in Print: 2019-08-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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