Let $BV(\mathbb R^n; \mathbb R^n)$ be the space of (vector-valued) [functions of bounded variation](https://www.encyclopediaofmath.org/index.php/Function_of_bounded_variation) and let $BD(\mathbb R^n;\mathbb R^n)$ the space of [functions with bounded deformation](https://en.wikipedia.org/wiki/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? 



Bounded deformation vs bounded variation
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. 


