Let $\Omega\subset \mathbb R^N$ be a open bounded set with smooth boundary. Then we can prove that for any $u\in BV(\Omega)$ and $\omega\in \operatorname{ker}\mathcal E $, where $\mathcal E$ denotes the distributional symmetrize derivative $\mathcal E\omega = \frac{1}{2}(\nabla w+\nabla w^T)$ from $\mathbb R^N$ to $\mathbb R^N$, there exists a constant $C>0$ independent of $u$ and $\omega$ such that $$ \|Du\|_{\mathcal M(\Omega)}\leq C(\|Du-w\|_{\mathcal M(\Omega)}+\|u\|_{L^1}) $$ The proof is not long and can be found here, Theorem 3.3, equation $(4)$.
It can also be shown that $\omega\in \operatorname{ker}\mathcal E$ iff $\omega = Ax+b$ where $A=-A^T$, $A\in\mathbb R^{N\times N}$ and $b\in \mathbb R^N$
However, the proof I know is done by using contradiction, and while the argument is short and simple it cannot give any information on the constant $C$.
I am sure that this constant only depends on $\Omega$ but I really want to know its dependence explicitly. What is the best value for the constant $C$ and can we make it larger or smaller by changing $\Omega$? Also, if $\Omega:=[0,1]\times[0,1]$ in $\mathbb R^2$, can we explicitly compute this constant?
Thank you!