  Let $\Omega \subset \mathbb{R}^3$ be an open and bounded set with Lipschitz
boundary and outward unit normal $\nu$. For $1<p<\infty$ we establish an
improved version of the generalized $L^p$-Korn inequality for incompatible
tensor fields $P$ in the new Banach space $$
  W^{1,\,p,\,r}_0(\operatorname{dev}\operatorname{sym}\operatorname{Curl};
\Omega,\mathbb R^{3\times3}) = \{ P \in L^p(\Omega,\mathbb R^{3\times3}) \mid
\operatorname{dev} \operatorname{sym} \operatorname{Curl} P \in
L^r(\Omega,\mathbb R^{3\times3}),\ \operatorname{dev} \operatorname{sym} (P
\times \nu) = 0 \text{ on $\partial \Omega$}\} $$ where
  $$ r \in [1, \infty), \qquad \frac1r \le \frac1p + \frac13, \qquad r >1 \quad
\text{if $p = \frac32$.}$$
  Specifically, there exists a constant $c=c(p,\Omega,r)>0$ such that the
inequality
  \[ \|P \|_{L^p}\leq c\,\left(\|\operatorname{sym} P \|_{L^p} +
\|\operatorname{dev}\operatorname{sym} \operatorname{Curl} P \|_{L^{r}}\right)
\] holds for all tensor fields $P\in W^{1,\,p, \,
r}_0(\operatorname{dev}\operatorname{sym}\operatorname{Curl})$. Here,
$\operatorname{dev} X := X -\frac13 \operatorname{tr}(X)\,\mathbb{1}$ denotes
the deviatoric (trace-free) part of a $3 \times 3$ matrix $X$ and the boundary
condition is understood in a suitable weak sense.
