A Lebesgue-Lusin property for linear operators of first and second order

We prove that for a homogeneous linear partial differential operator $\mathcal{A}$ of order $k\le 2$ and an integrable map $f$ taking values in the essential range of that operator, there exists a function $u$ of special bounded variation satisfying $$\mathcal{A}u(x)=f(x)\text{almost everywhere}\mathrm{.}$$ This extends a result of G. Alberti for gradients on ${\mathbf{R}}^N$. In particular, for $0\le m<N$, it is shown that every integrable $m$-vector field is the absolutely continuous part of the boundary of a normal $(m+1)$-current.