Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy

Let $\Omega \subset \mathbb{R}^3$ be an open and bounded set with Lipschitz boundary outward unit normal $\nu$. For $11 \quad \text{if $p \frac32$.}$$ Specifically, there exists a constant $c=c(p,\Omega,r)>0$ such that \[ \|P \|_{L^p}\leq c\,\left(\|\operatorname{sym} \|_{L^p} \|\operatorname{dev}\operatorname{sym} \|_{L^{r}}\right) \] holds all $P\in W^{1,\,p, \, r}_0(\operatorname{dev}\operatorname{sym}\operatorname{Curl})$. Here, $\operatorname{dev} X := -\frac13 \operatorname{tr}(X)\,\mathbb{1}$ denotes deviatoric (trace-free) part $3 3$ matrix $X$ condition is understood suitable weak sense.