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A global higher regularity result for the static relaxed micromorphic model on smooth domains

Part of: Partial differential equations Miscellaneous topics in calculus of variations and optimal control Equations of mathematical physics and other areas of application Generalities, axiomatics, foundations of continuum mechanics of solids Equilibrium (steady-state) problems

Published online by Cambridge University Press:  20 May 2024

Dorothee Knees Open the ORCID record for Dorothee Knees [Opens in a new window] ,
Sebastian Owczarek Open the ORCID record for Sebastian Owczarek [Opens in a new window]  and
Patrizio Neff Open the ORCID record for Patrizio Neff [Opens in a new window]
Dorothee Knees
Affiliation:
Institute of Mathematics, University of Kassel, Heinrich-Plett Str. 40, 34132 Kassel, Germany (dknees@mathematik.uni-kassel.de)
Sebastian Owczarek
Affiliation:
Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland (sebastian.owczarek@pw.edu.pl)
Patrizio Neff
Affiliation:
Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg-Essen, Campus Essen, Thea-Leymann Str. 9, 45127 Essen, Germany (patrizio.neff@uni-due.de)
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Abstract

We derive a global higher regularity result for weak solutions of the linear relaxed micromorphic model on smooth domains. The governing equations consist of a linear elliptic system of partial differential equations that is coupled with a system of Maxwell-type. The result is obtained by combining a Helmholtz decomposition argument with regularity results for linear elliptic systems and the classical embedding of $H(\operatorname {div};\Omega )\cap H_0(\operatorname {curl};\Omega )$ into $H^1(\Omega )$.

Keywords

global regularitysmooth domainrelaxed micromorphic modelelasticity coupled with Maxwell systemHelmholtz decompositiongeneralized continuadislocation model

MSC classification

Primary: 35Q74: PDEs in connection with mechanics of deformable solids
Secondary: 35B65: Smoothness and regularity of solutions 49N60: Regularity of solutions 74A35: Polar materials 74G40: Regularity of solutions

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Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 15
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Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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