Nečas-Lions lemma revisited: An $L^p$-version of the generalized Korn inequality for incompatible tensor fields

Abstract

For $1<p<\infty$ we prove an $L^p$-version of the generalized Korn inequality for incompatible tensor fields $P$ in $W_0^{1,p}(\mathrm{Curl};\hat{I}\text{c◯},{\mathbb{R}}^{3\times 3})$. More precisely, let $\hat{I}\text{c◯}\subset {\mathbb{R}}^3$ be a bounded Lipschitz domain. Then there exists a constant $c>0$ such that \begin{equation*} \| P\|_{L^p(Ω,\mathbb{R}^{3\times3})}\leq c\,\left( \|\operatorname{sym} P\|_{L^p(Ω,\mathbb{R}^{3\times3})} + \| \operatorname{Curl}P \|_{L^p(Ω, \mathbb{R}^{3\times3})}\right)\end{equation*} holds for all tensor fields $P\in W_0^{1,p}(\mathrm{Curl};\hat{I}\text{c◯},{\mathbb{R}}^{3\times 3})$, i.e., for all $P\in W^{1,p}(\mathrm{Curl};\hat{I}\text{c◯},{\mathbb{R}}^{3\times 3})$ with vanishing tangential trace $P\times \hat{I}½=0$ on $\partial \hat{I}\text{c◯}$ where $\hat{I}½$ denotes the outward unit normal vector field to $\partial \hat{I}\text{c◯}$. For compatible $P=Du$ this recovers an $L^p$-version of the classical Korn's first inequality $$\Vert Du{\Vert }_{L^p(\hat{I}\text{c◯},{\mathbb{R}}^{3\times 3})}\le c\Vert \mathrm{sym}Du{\Vert }_{L^p(\hat{I}\text{c◯},{\mathbb{R}}^{3\times 3})}\text{with }Du\times \hat{I}½=0\text{on ∂I^c◯},$$ and for skew-symmetric $P=A\in \mathfrak{so}(3)$ an $L^p$-version of the Poincaré inequality $$\Vert A{\Vert }_{L^p(\hat{I}\text{c◯},\mathfrak{so}(3))}\le c\Vert \mathrm{Curl}A{\Vert }_{L^p(\hat{I}\text{c◯},{\mathbb{R}}^{3\times 3})}\text{with }A\times \hat{I}½=0\iff A=0\text{on ∂I^c◯}.$$