Calculus of Variations and Geometric Measure Theory

G. De Philippis - F. Rindler

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

created by dephilipp on 25 Jan 2016
modified on 30 Oct 2017


Accepted Paper

Inserted: 25 jan 2016
Last Updated: 30 oct 2017

Journal: Ann. of Math.
Year: 2016

ArXiv: 1601.06543 PDF


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.