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

M. Fuchs   ; S. Repin

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 41, Tome 385 (2010), pp. 224-233

Résumé

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.

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