Journal
ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS
Volume 22, Issue 1, Pages 112-133Publisher
EDP SCIENCES S A
DOI: 10.1051/cocv/2014068
Keywords
Korn's inequality; Lie-algebra decomposition; Poincare's inequality; Maxwell estimates; relaxed micromorphic model
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Let Omega subset of R-n, n >= 2, be a bounded Lipschitz domain and 1 < q < infinity. We prove the inequality parallel to T parallel to(Lq(Omega)) <= C-DD (parallel to dev T parallel to(Lq(Omega)) + parallel to Div T parallel to(Lq(Omega))) being valid for tensor fields T : Omega --> R-nxn with a normal boundary condition on some open and non- empty part Gamma(nu) of the boundary partial derivative Omega. Here dev T = T - 1/n tr (T) center dot Id denotes the deviatoric part of the tensor T and Div is the divergence row-wise. Furthermore, we prove parallel to T parallel to(L2(Omega)) <= C-DSC (parallel to dev sym T parallel to(L2(Omega)) + parallel to Curl T parallel to(Lq(Omega))) if n >= 3, parallel to T parallel to(L2(Omega)) <= C-DSDC (parallel to dev sym T parallel to(L2(Omega)) + parallel to dev Curl T parallel to(L2(Omega))) if n = 3, being valid for tensor fields T with a tangential boundary condition on some open and non- empty part Gamma(tau) of partial derivative Omega. Here, sym T = 1/2 (T + T-inverted perpendicular) denotes the symmetric part of T and Curl is the rotation row-wise.
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