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 $$\mathcal A u(x)= f(x) \qquad \text{almost everywhere}.$$ 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.