preprint

Inserted: 29 sep 2022

Year: 2022

Abstract:

We prove that for a first-order homogeneous linear partial differential operator $\mathcal A$ and a map $f$ taking values in the essential range of the 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 $\mathbb R^N$. In particular, for $m < N$, it is shown that every integrable $m$-form field on $\mathbb R^N$ is the absolutely continuous part of the boundary of some locally normal current with finite mass.