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Let $BV(\mathbb R^n; \mathbb R^n)$ be the space of (vector-valued) functions of bounded variation and let $BD(\mathbb R^n;\mathbb R^n)$ the space of functions with bounded deformation. They are made up respectively of functions $u$ for which the full distributional derivative $$ Du \in \mathcal M(\mathbb R^n) $$ is represented by a measure with finite total variation and of the functions for which the symmetric part of the distributional derivative $$ Eu := \frac{Du+(Du)^t}{2} \in \mathcal M(\mathbb R^n) $$ is represented by a measure with finite total variation.

If $n=1$ of course the two definitions coincide. For $n\ge 2$ they are different, but I do not find an explicit example.

Q. Let $n\ge 2$. Find an element in $BD \setminus BV$.

Is a characterization of such functions available somewhere in the literature?

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Example 7.7 in

L. Ambrosio, A. Coscia, Alessandra, G. Dal Maso, Fine properties of functions with bounded deformation. Arch. Rational Mech. Anal. 139 (1997), no. 3, 201–238.

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  • $\begingroup$ Thanks a lot, this is indeed what I was looking for. I did go - quite quick - through that paper but did not find that example. $\endgroup$ – user111164 Oct 9 '18 at 19:19
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This paper discusses counterexamples to Korn's inequality in $L^1$ spaces: https://www.mis.mpg.de/preprints/2003/preprint2003_93.pdf

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