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Zapiski Nauchnykh Seminarov POMI, 2011, Volume 397, Pages 115–125
(Mi znsl4670)
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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
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
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
Linking options:
https://www.mathnet.ru/eng/znsl4670 https://www.mathnet.ru/eng/znsl/v397/p115
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