$L^p$-trace-free generalized Korn inequalities for incompatible tensor fields in three space dimensions

For $1<p<\mathrm{\infty }$ we prove an $L^p$-version of the generalized trace-free Korn inequality for incompatible tensor fields $P$ in $W_0^{1,p}(\mathrm{Curl};\mathrm{\Omega },{\mathbb{R}}^{3\times 3})$. More precisely, let $\mathrm{\Omega }\subset {\mathbb{R}}^3$ be a bounded Lipschitz domain. Then there exists a constant $c>0$ such that $$\mathrm{\parallel }P{\mathrm{\parallel }}_{L^p(\mathrm{\Omega },{\mathbb{R}}^{3\times 3})}\le c(\mathrm{\parallel }\mathrm{dev}\mathrm{sym}P{\mathrm{\parallel }}_{L^p(\mathrm{\Omega },{\mathbb{R}}^{3\times 3})}+\mathrm{\parallel }\mathrm{dev}\mathrm{Curl}P{\mathrm{\parallel }}_{L^p(\mathrm{\Omega },{\mathbb{R}}^{3\times 3})})$$ holds for all tensor fields $P\in W_0^{1,p}(\mathrm{Curl};\mathrm{\Omega },{\mathbb{R}}^{3\times 3})$, i.e., for all $P\in W^{1,p}(\mathrm{Curl};\mathrm{\Omega },{\mathbb{R}}^{3\times 3})$ with vanishing tangential trace $P\times \nu =0$ on $\mathrm{\partial }\mathrm{\Omega }$ where $\nu$ denotes the outward unit normal vector field to $\mathrm{\partial }\mathrm{\Omega }$ and $\mathrm{dev}P:=P-\frac{1}{3}\mathrm{tr}(P){\mathbb{1}}_3$ denotes the deviatoric (trace-free) part of $P$. We also show the norm equivalence $$\mathrm{\parallel }P{\mathrm{\parallel }}_{L^p(\mathrm{\Omega },{\mathbb{R}}^{3\times 3})}+\mathrm{\parallel }\mathrm{Curl}P{\mathrm{\parallel }}_{L^p(\mathrm{\Omega },{\mathbb{R}}^{3\times 3})}\le c(\mathrm{\parallel }\mathrm{dev}\mathrm{sym}P{\mathrm{\parallel }}_{L^p(\mathrm{\Omega },{\mathbb{R}}^{3\times 3})}+\mathrm{\parallel }\mathrm{dev}\mathrm{Curl}P{\mathrm{\parallel }}_{L^p(\mathrm{\Omega },{\mathbb{R}}^{3\times 3})})$$ for tensor fields $P\in W^{1,,p}_0(\operatorname{Curl}; \Omega,$\mathbb{R}$^{3\times3})$. These estimates also hold true for tensor fields with vanishing tangential trace only on a relatively open (non-empty) subset $\mathrm{\Gamma }\subseteq \mathrm{\partial }\mathrm{\Omega }$ of the boundary.