A Korn's inequality for incompatible tensor fields

Abstract

Abstract We prove a Korn‐type inequality for bounded Lipschitz domains in $\Omega {\rm ~in~}{\rm I\!R}^3$ and non‐symmetric square integrable tensor fields $P : \Omega \to {\rm I\!R}^{3\times 3}$ having square integrable rotation ${\rm Curl~}P : \Omega \to {\rm I\!R}^{3\times 3}$ . For skew‐symmetric P or compatible $P =\nabla\;v$ our estimate reduces to non‐standard variants of Poincaré's or Korn's first inequality, respectively, for which our new estimate can be viewed as a common generalized version. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)