We prove a Korn-type inequality in
$\overset{\circ}{\text{H}}(\text{Curl};\Omega,\mathbb{R}^{3\times 3})$ for
tensor fields $P$ mapping $\Omega$ to $\mathbb{R}^{3\times 3}$. More
precisely, let $\Omega\subset \mathbb{R}^3$ be a bounded domain with
connected Lipschitz boundary $\partial\Omega$. Then, there exists a
constant $c > 0$ such that
\begin{equation}\label{01}
c\|P\|_{L^2(\Omega,\mathbb{R}^{3\times 3})}\leq \|\operatorname{sym}
P\|_{L^2(\Omega,\mathbb{R}^{3\times 3})}+\|\operatorname{Curl} P\|_{L^2(\Omega,\mathbb{R}^{3\times 3})}
\end{equation}
holds for all tensor fields
$P\in\overset{\cdot}{\text{H}}(\text{Curl};\Omega,\mathbb{R}^{3\times 3})$,
i.e., all $P\in
\overset{\cdot}{\text{H}}(\text{Curl};\Omega,\mathbb{R}^{3\times 3})$ with
vanishing tangential trace on $\partial\Omega$. Here, rotation and
tangential trace are defined row-wise. For compatible $P$, i.e., $P
= \nabla v$ and thus $\operatorname{Curl} P = 0$, where $v\in
\text{H}^1(\Omega,\mathbb{R}^3)$ are vector fields having components
$v_n$, for which $\nabla v_n$ are normal at $\partial \Omega$, the
presented estimate \eqref{01} reduces to a non-standard variant of
Korn's first inequality, i.e.,
\[
c\|\nabla v\|_{L^2(\Omega,\mathbb{R}^{3\times 3})}\leq
\|\operatorname{sym}\nabla v\|_{L^2(\Omega,\mathbb{R}^{3\times 3})}.
\]
On the other hand, for skew-symmetric $P$, i.e., sym $P = 0$,
\eqref{01} reduces to a non-standard version of Poincaré's
estimate. Therefore, since \eqref{01} admits the classical boundary
conditions our result is a common generalization of the two
classical estimates, namely Poincaré's resp. Korn's first
inequality. Applications to infinitesimal gradient plasticity with
plastic spin are given.
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