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

Abstract

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.