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

2016/01/25 by De Philippis, Guido, Rindler, Filip
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1601.06543

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.

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