  For $1<p<\infty$ we prove an $L^p$-version of the generalized Korn inequality
for incompatible tensor fields $P$ in $ W^{1,\,p}_0(\operatorname{Curl};
\Omega,\mathbb{R}^{3\times3})$. More precisely, let $\Omega\subset\mathbb{R}^3$
be a bounded Lipschitz domain. Then there exists a constant $c>0$ such that
\begin{equation*} \| P\|_{L^p(\Omega,\mathbb{R}^{3\times3})}\leq c\,\left(
\|\operatorname{sym} P\|_{L^p(\Omega,\mathbb{R}^{3\times3})} + \|
\operatorname{Curl}P \|_{L^p(\Omega,
\mathbb{R}^{3\times3})}\right)\end{equation*} holds for all tensor fields $P\in
W^{1,\,p}_0(\operatorname{Curl}; \Omega,\mathbb{R}^{3\times3})$, i.e., for all
$P\in W^{1,\,p}(\operatorname{Curl}; \Omega,\mathbb{R}^{3\times3})$ with
vanishing tangential trace $ P\times \nu=0 $ on $ \partial\Omega$ where $\nu$
denotes the outward unit normal vector field to $\partial\Omega$. For
compatible $P=D u$ this recovers an $L^p$-version of the classical Korn's first
inequality $$ \|D u \|_{L^p(\Omega,\mathbb{R}^{3\times 3})} \le c\,
\|\operatorname{sym}D u\|_{L^p(\Omega,\mathbb{R}^{3\times3})} \quad \text{with
}D u \times \nu = 0 \quad \text{on $\partial \Omega$}, $$ and for
skew-symmetric $P=A\in\mathfrak{so}(3)$ an $L^p$-version of the Poincar\'{e}
inequality $$ \|A\|_{L^p(\Omega,\mathfrak{so}(3))}\le c\, \|\operatorname{Curl}
A\|_{L^p(\Omega,\mathbb{R}^{3\times3})} \quad \text{with } A \times \nu = 0 \
\Leftrightarrow \ A=0 \quad \text{on $\partial \Omega$}. $$

Analysis of PDEs