On the structure of ${\mathscr A}$-free measures and applications

TL;DR

This paper provides a general structure theorem for the singular parts of ${\mathscr A}$-free measures, extending classical results and applying to functions of bounded deformation and normal currents.

Contribution

It introduces a unifying structure theorem for ${\mathscr A}$-free measures and extends Alberti's rank-one theorem to new contexts like functions of bounded deformation.

Findings

Established a general structure theorem for ${\mathscr A}$-free Radon measures.

Extended Alberti's rank-one theorem to functions of bounded deformation.

Proved that Lipschitz functions are differentiable only almost everywhere with respect to absolutely continuous measures.

Abstract

We establish a general structure theorem for the singular part of ${\mathscr A}$-free Radon measures, where ${\mathscr A}$ is a linear PDE operator. By applying the theorem to suitably chosen differential operators ${\mathscr A}$, we obtain a simple proof of Alberti's rank-one theorem and, for the first time, its extensions to functions of bounded deformation (BD). We also prove a structure theorem for the singular part of a finite family of normal currents. The latter result implies that the Rademacher theorem on the differentiability of Lipschitz functions can hold only for absolutely continuous measures and that every top-dimensional Ambrosio--Kirchheim metric current in $\mathbb R^d$ is a Federer-Fleming flat chain.