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 Ω n , n 2 , be a bounded Lipschitz domain and 1 < q < . We prove the inequality

T L q (Ω) C DD dev T L q (Ω) + Div T L q (Ω)
being valid for tensor fields T : Ω n × n with a normal boundary condition on some open and non-empty part Γ ν of the boundary Ω. Here dev T = T - 1 n t r ( T ) · It denotes the deviatoric part of the tensor T and Div is the divergence row-wise. Furthermore, we prove
T L 2 ( Ω ) C D S C dev sym T L 2 ( Ω ) + Curl T L 2 ( Ω ) if n 3 , T L 2 ( Ω ) C D S D C dev sym T L 2 ( Ω ) + dev Curl T L 2 ( Ω ) if n = 3 ,
being valid for tensor fields T with a tangential boundary condition on some open and non-empty part Γ τ of Ω. Here, sym T = 1 2 ( T + T ) 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},
     number = {1},
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     doi = {10.1051/cocv/2014068},
     zbl = {1337.35004},
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     url = {http://www.numdam.org/articles/10.1051/cocv/2014068/}
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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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