I am trying to give meaning to the notion of second differential of total variation. 

For sufficiently regular $u:\Omega \subset \mathbb{R}^2 \to \mathbb{R}$ let the total variation be given by 
$$TV(u)=\int_{\Omega} |Du| dx.$$ 
Assuming that $u$ is sufficiently smooth and $|Du|\neq 0$ everywhere on $\Omega$ we have
$$ \partial TV(u)(v):=\frac{d}{d \epsilon} TV(u+\epsilon v)|_{\epsilon=0}=\int_{\Omega} Dv \cdot \frac{Du}{|Du|} dx=-\int_\Omega \text{div}\left(\frac{Du}{|Du|}\right) v~ dx, ~~ \forall v \in C^\infty_c(\Omega).$$
Analogously
$$\partial^2 TV(u)(v,w):=\frac{d}{d \epsilon} \partial TV(u+\epsilon w)(v)=\int_\Omega \frac{((Du)^\perp \cdot Dv) ((Du)^\perp \cdot Dw) }{|Du|^3} dx, ~~ \forall v,w \in C^\infty_c(\Omega),$$
where $(Du)^\perp=(\partial u/\partial y, - \partial u/\partial x)$.

I would like to write the above integral in terms of $v,w$, without using their derivatives, something like it was done for the first differential. Is this possible, at least for $\Omega=[0,1]^2$?

Furthermore, does anyone know any results in direction of understanding the second differential of total variation?