Dev-Div- and DevSym-DevCurl-inequalities for incompatible square tensor fields with mixed boundary conditions
Fakultät für Mathematik, Universität Duisburg-Essen, Campus
Revised: 29 July 2014
Let Ω ⊂ ℝn, n ≥ 2, be a bounded Lipschitz domain and 1 <q< ∞. We prove the inequality being valid for tensor fields T:Ω → ℝn × n with a normal boundary condition on some open and non-empty part Γν of the boundary ∂Ω. Here dev T = T - 1/n tr(T)·Id denotes the deviatoric part of the tensor T and Div is the divergence row-wise. Furthermore, we prove being valid for tensor fields T with a tangential boundary condition on some open and non-empty part Γτ of ∂Ω. Here, sym T = 1/2(T + T⊤) denotes the symmetric part of T and Curl is the rotation row-wise.
Mathematics Subject Classification: 35A23 / 35Q61 / 74C05 / 78A25 / 78A30
Key words: Korn’s inequality / Lie-algebra decomposition / Poincaré’s inequality / Maxwell estimates / relaxed micromorphic model
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