# Bounded deformation vs bounded variation

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?

This paper discusses counterexamples to Korn's inequality in $$L^1$$ spaces: https://www.mis.mpg.de/preprints/2003/preprint2003_93.pdf