We prove a Korn-type inequality in \( \mathop {\text{H}}\limits^\circ \left( {{\text{Curl}};\Omega, {\mathbb{R}^{3 \times 3}}} \right) \) for tensor fields P mapping Ω to \( {\mathbb{R}^{3 \times 3}} \). More precisely, let \( \Omega \subset {\mathbb{R}^3} \) be a bounded domain with connected Lipschitz boundary ∂Ω. Then there exists a constant c > 0 such that
for all tensor fields P ∈ \( \mathop {\text{H}}\limits^\circ \left( {{\text{Curl}};\Omega, {\mathbb{R}^{3 \times 3}}} \right) \), i.e., all P ∈ \( {\text{H}}\left( {{\text{Curl}};\Omega, {\mathbb{R}^{3 \times 3}}} \right) \) with vanishing tangential trace on ∂Ω. Here the rotation and tangential trace are defined row-wise. For compatible P of form P = ∇v, Curl P = 0, where \( v \in {{\text{H}}^1}\left( {\Omega, {\mathbb{R}^3}} \right) \) is a vector field with components v n for which ∇v n are normal at ∂Ω, estimates (0.1) is reduced to a non standard variant of Korn’s first inequality:
For skew-symmetric P (with sym P = 0), estimates (0.1) generates a nonstandard version of Poincaré’s inequality. Therefore, the estimate is a generalization of two classical inequalities of Poincaré and Korn. Bibliography: 24 titles.
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 397, 2011, pp. 115–125.
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Neff, P., Pauly, D. & Witsch, KJ. On a canonical extension of Korn’s first and Poincaré’s inequalities to H(CURL). J Math Sci 185, 721–727 (2012). https://doi.org/10.1007/s10958-012-0955-4
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DOI: https://doi.org/10.1007/s10958-012-0955-4