From $\mathrm{BV}^\mathcal A$ to $\mathrm{BV}$: An endpoint Korn estimate

In this short note we prove that, given a $\mathbb{C}$-elliptic operator $\mathcal{A}$ and a map $u\in \mathrm{BV}^{\mathcal{A}}(Ω;V)$ satisfying \( \nabla_{\mathrm{ap}}u\in L^1(Ω;V\otimes\mathbb{R}^d), \) then $u\in \mathrm{BV}(Ω;V)$. The result is quantitative and follows from the Korn-type estima