On a~canonical extension of Korn's first and Poincar\'e's inequality to~$\mathsf H(\operatorname{Curl})$

P. Neff ; D. Pauly ; K.-J. Witsch

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 42, Tome 397 (2011), pp. 115-125

Résumé

We prove a Korn-type inequality in $\overset\circ{\mathsf H}(\operatorname{Curl};\Omega,\mathbb R^{3\times3})$ for tensor fields $P$ mapping $\Omega$ to $\mathbb R^{3\times3}$. More precisely, let $\Omega\subset\mathbb R^3$ be a bounded domain with connected Lipschitz boundary $\partial\Omega$. Then, there exists a constant $c>0$ such that \begin{equation} c\|P\|_{\mathsf L^2(\Omega,\mathbb R^{3\times3})}\leq\|\operatorname{sym}P\|_{\mathsf L^2(\Omega,\mathbb R^{3\times3})} +\|\operatorname{Curl}P\|_{\mathsf L^2(\Omega,\mathbb R^{3\times3})} \tag{0.1} \end{equation} holds for all tensor fields $P\in\overset\circ{\mathsf H}(\operatorname{Curl};\Omega,\mathbb R^{3\times3})$, i.e., all $$ P\in\mathsf H(\operatorname{Curl};\Omega,\mathbb R^{3\times3}) $$ with vanishing tangential trace on $\partial\Omega$. Here, rotation and tangential trace are defined row-wise. For compatible $P$ (i.e., $P=\nabla v$), $\operatorname{Curl}P=0$, where $v\in\mathsf H^1(\Omega,\mathbb R^3)$ a vector field having components $v_n$, for which $\nabla v_n$ are normal at $\partial\Omega$, the estimate $(0.1)$ is reduced to a non-standard variant of the Korn's first inequality: $$ c\|\nabla v\|_{\mathsf L^2(\Omega,\mathbb R^{3\times3})}\le \|\operatorname{sym}\nabla v\|_{\mathsf L^2(\Omega,\mathbb R^{3\times3})}. $$ For skew-symmetric $P$ ($\operatorname{sym}P=0$) the estimate $(0.1)$ generates a non-standard version of the Poincaré. Therefore, the estimateis a generalization of two classical inequalities of Poincaré and Korn.