Dev-Div- and DevSym-DevCurl-inequalities for incompatible square tensor fields with mixed boundary conditions
ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 1, pp. 112-133.

Let $\Omega \subset {ℝ}^{n},\phantom{\rule{4pt}{0ex}}n\ge 2$, be a bounded Lipschitz domain and $1. We prove the inequality

 $\begin{array}{c}\hfill {\parallel T\parallel }_{{L}^{q}\left(\Omega \right)}\le {C}_{D\phantom{\rule{-0.166667em}{0ex}}D}\left({\parallel \mathrm{dev}T\parallel }_{{L}^{q}\left(\Omega \right)}+{\parallel \mathrm{Div}T\parallel }_{{L}^{q}\left(\Omega \right)}\right)\end{array}$
being valid for tensor fields $T:\Omega \to {ℝ}^{n×n}$ with a normal boundary condition on some open and non-empty part $\Gamma {}_{\nu }$ of the boundary $\partial \Omega$. Here $\mathrm{dev}T=T-\frac{1}{n}tr\left(T\right)\phantom{\rule{4pt}{0ex}}·$ It denotes the deviatoric part of the tensor $T$ and Div is the divergence row-wise. Furthermore, we prove
 $\begin{array}{ccccc}\hfill {\parallel T\parallel }_{{L}^{2}\left(\Omega \right)}& & \le {C}_{D\phantom{\rule{-0.166667em}{0ex}}S\phantom{\rule{-0.166667em}{0ex}}C\phantom{\rule{-0.166667em}{0ex}}}\left({\parallel \mathrm{dev}\phantom{\rule{0.166667em}{0ex}}\mathrm{sym}\phantom{\rule{0.166667em}{0ex}}T\parallel }_{{L}^{2}\left(\Omega \right)}+{\parallel \mathrm{Curl}\phantom{\rule{0.166667em}{0ex}}T\parallel }_{{L}^{2}\left(\Omega \right)}\right)\phantom{\rule{1em}{0ex}}\hfill & & \mathrm{if}\phantom{\rule{0.166667em}{0ex}}n\ge 3,\\ \hfill {\parallel T\parallel }_{{L}^{2}\left(\Omega \right)}& & \le {C}_{D\phantom{\rule{-0.166667em}{0ex}}S\phantom{\rule{-0.166667em}{0ex}}D\phantom{\rule{-0.166667em}{0ex}}C\phantom{\rule{-0.166667em}{0ex}}}\left({\parallel \mathrm{dev}\phantom{\rule{0.166667em}{0ex}}\mathrm{sym}\phantom{\rule{0.166667em}{0ex}}T\parallel }_{{L}^{2}\left(\Omega \right)}+{\parallel \mathrm{dev}\phantom{\rule{0.166667em}{0ex}}\mathrm{Curl}\phantom{\rule{0.166667em}{0ex}}T\parallel }_{{L}^{2}\left(\Omega \right)}\right)\phantom{\rule{1em}{0ex}}\hfill & & \mathrm{if}\phantom{\rule{0.166667em}{0ex}}n=3,\end{array}$
being valid for tensor fields $T$ with a tangential boundary condition on some open and non-empty part $\Gamma {}_{\tau }$ of $\partial \Omega$. Here, $\mathrm{sym}T=\frac{1}{2}\left(T+{T}^{\top }\right)$denotes the symmetric part of denotes the symmetric part of $T$ and Curl is the rotation row-wise.

DOI: 10.1051/cocv/2014068
Classification: 35A23, 35Q61, 74C05, 78A25, 78A30
Keywords: Korn’s inequality, Lie-algebra decomposition, Poincaré’s inequality, Maxwell estimates, relaxed micromorphic model
Bauer, Sebastian 1; Neff, Patrizio 1; Pauly, Dirk 1; Starke, Gerhard 1

1 Fakultät für Mathematik, Universität Duisburg-Essen, Campus Essen, Thea-Leymann-Str. 9, 45127 Essen, Germany.
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author = {Bauer, Sebastian and Neff, Patrizio and Pauly, Dirk and Starke, Gerhard},
title = {Dev-Div- and {DevSym-DevCurl-inequalities} for incompatible square tensor fields with mixed boundary conditions},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {112--133},
publisher = {EDP-Sciences},
volume = {22},
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Bauer, Sebastian; Neff, Patrizio; Pauly, Dirk; Starke, Gerhard. Dev-Div- and DevSym-DevCurl-inequalities for incompatible square tensor fields with mixed boundary conditions. ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 1, pp. 112-133. doi : 10.1051/cocv/2014068. http://www.numdam.org/articles/10.1051/cocv/2014068/

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