<i>L</i>^<i>p</i>-trace-free generalized Korn inequalities for incompatible tensor fields in three space dimensions

For $10$ such that \[ \|{ P }\|_{L^p(\Omega,\mathbb{R}^{3\times3})}\leq c\,\left(\|{\operatorname{dev} \operatorname{sym} }\|_{L^p(\Omega,\mathbb{R}^{3\times3})} + \operatorname{dev} \operatorname{Curl} }\|_{L^p(\Omega,\mathbb{R}^{3\times3})}\right) \] holds all $P\in \Omega,\mathbb{R}^{3\times3})$, i.e., W^{1,\,p}(\operatorname{Curl}; \Omega,\mathbb{R}^{3\times3})$ with vanishing tangential trace P\times \nu=0 on \partial\Omega$ where $\nu$ denotes outward unit normal vector field to $\partial\Omega$ and $\operatorname{dev} := -\frac13 \operatorname{tr}(P)\,\mathbb{1}_3$ deviatoric (trace-free) part $P$. We also show norm equivalence }\|_{L^p(\Omega,\mathbb{R}^{3\times3})}+\|{\operatorname{Curl} \operatorname{dev}\operatorname{Curl} These estimates hold true only relatively open (non-empty) subset $\Gamma \subseteq boundary.