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

Let $\mathrm{\Omega }\subset {\mathbb{R}}^3$ be an open and bounded set with Lipschitz boundary and outward unit normal $\nu$. For $1<p<\mathrm{\infty }$ we establish an improved version of the generalized $L^p$-Korn inequality for incompatible tensor fields $P$ in the new Banach space $$W_0^{1,p,r}(\mathrm{dev}\mathrm{sym}\mathrm{Curl};\mathrm{\Omega },{\mathbb{R}}^{3\times 3})=\{P\in L^p(\mathrm{\Omega },{\mathbb{R}}^{3\times 3})\mid \mathrm{dev}\mathrm{sym}\mathrm{Curl}P\in L^r(\mathrm{\Omega },{\mathbb{R}}^{3\times 3}),\mathrm{dev}\mathrm{sym}(P\times \nu )=0\text{ on }\mathrm{\partial }\mathrm{\Omega }\}$$ where $$r\in [1,\mathrm{\infty }),\frac{1}{r}\le \frac{1}{p}+\frac{1}{3},r>1\text{if }p=\frac{3}{2}\text{.}$$ Specifically, there exists a constant $c=c(p,\mathrm{\Omega },r)>0$ such that the inequality $$\mathrm{\parallel }P{\mathrm{\parallel }}_{L^p}\le c(\mathrm{\parallel }\mathrm{sym}P{\mathrm{\parallel }}_{L^p}+\mathrm{\parallel }\mathrm{dev}\mathrm{sym}\mathrm{Curl}P{\mathrm{\parallel }}_{L^r})$$ holds for all tensor fields $P\in W^{1,,p, , r}_0(\operatorname{dev}\operatorname{sym}\operatorname{Curl})$\mathrm{.}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.