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Zapiski Nauchnykh Seminarov POMI, 2011, Volume 397, Pages 115–125 (Mi znsl4670)  

This article is cited in 16 scientific papers (total in 16 papers)

On a canonical extension of Korn's first and Poincaré's inequality to $\mathsf H(\operatorname{Curl})$

P. Neff, D. Pauly, K.-J. Witsch

Universität Duisburg-Essen, Fakultät für Mathematik, Essen, Germany
References:
Abstract: We prove a Korn-type inequality in $\overset\circ{\mathsf H}(\operatorname{Curl};\Omega,\mathbb R^{3\times3})$ for tensor fields $P$ mapping $\Omega$ to $\mathbb R^{3\times3}$. More precisely, let $\Omega\subset\mathbb R^3$ be a bounded domain with connected Lipschitz boundary $\partial\Omega$. Then, there exists a constant $c>0$ such that
\begin{equation} c\|P\|_{\mathsf L^2(\Omega,\mathbb R^{3\times3})}\leq\|\operatorname{sym}P\|_{\mathsf L^2(\Omega,\mathbb R^{3\times3})} +\|\operatorname{Curl}P\|_{\mathsf L^2(\Omega,\mathbb R^{3\times3})} \tag{0.1} \end{equation}
holds for all tensor fields $P\in\overset\circ{\mathsf H}(\operatorname{Curl};\Omega,\mathbb R^{3\times3})$, i.e., all
$$ P\in\mathsf H(\operatorname{Curl};\Omega,\mathbb R^{3\times3}) $$
with vanishing tangential trace on $\partial\Omega$. Here, rotation and tangential trace are defined row-wise. For compatible $P$ (i.e., $P=\nabla v$), $\operatorname{Curl}P=0$, where $v\in\mathsf H^1(\Omega,\mathbb R^3)$ a vector field having components $v_n$, for which $\nabla v_n$ are normal at $\partial\Omega$, the estimate $(0.1)$ is reduced to a non-standard variant of the Korn's first inequality:
$$ c\|\nabla v\|_{\mathsf L^2(\Omega,\mathbb R^{3\times3})}\le \|\operatorname{sym}\nabla v\|_{\mathsf L^2(\Omega,\mathbb R^{3\times3})}. $$
For skew-symmetric $P$ ($\operatorname{sym}P=0$) the estimate $(0.1)$ generates a non-standard version of the Poincaré. Therefore, the estimateis a generalization of two classical inequalities of Poincaré and Korn.
Key words and phrases: Korn's inequality, gradient plasticity, theory of Maxwell's equations, Helmholtz' decomposition, Poincaré/Friedrichs type estimate.
Received: 14.12.2011
English version:
Journal of Mathematical Sciences (New York), 2012, Volume 185, Issue 5, Pages 721–727
DOI: https://doi.org/10.1007/s10958-012-0955-4
Bibliographic databases:
Document Type: Article
UDC: 517
Language: English
Citation: P. Neff, D. Pauly, K.-J. Witsch, “On a canonical extension of Korn's first and Poincaré's inequality to $\mathsf H(\operatorname{Curl})$”, Boundary-value problems of mathematical physics and related problems of function theory. Part 42, Zap. Nauchn. Sem. POMI, 397, POMI, St. Petersburg, 2011, 115–125; J. Math. Sci. (N. Y.), 185:5 (2012), 721–727
Citation in format AMSBIB
\Bibitem{NefPauWit11}
\by P.~Neff, D.~Pauly, K.-J.~Witsch
\paper On a~canonical extension of Korn's first and Poincar\'e's inequality to~$\mathsf H(\operatorname{Curl})$
\inbook Boundary-value problems of mathematical physics and related problems of function theory. Part~42
\serial Zap. Nauchn. Sem. POMI
\yr 2011
\vol 397
\pages 115--125
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl4670}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2870111}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2012
\vol 185
\issue 5
\pages 721--727
\crossref{https://doi.org/10.1007/s10958-012-0955-4}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84866950698}
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  • This publication is cited in the following 16 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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