Some Poincaré-type inequalities for functions of bounded deformation involving the deviatoric part of the symmetric gradient

If $\Omega\subset\mathbb R^n$ is a bounded Lipschitz domain, we prove the inequality $\|u\|_1\le c(n)\operatorname{diam}(\Omega)\int_\Omega|\varepsilon^D(u)|$ being valid for functions of bounded deformation vanishing on $\partial\Omega$. Here $\varepsilon^D(u)$ denotes the deviatoric part of the symmetric gradient and $\int_\Omega|\varepsilon^D(u)|$ stands for the total variation of the tensor-valued measure $\varepsilon^D(u)$. Further results concern possible extensions of this Poincaré-type inequality. Bibl. 27 titles.