We establish a general structure theorem for the singular part of A -free Radon measures, where A is a linear PDE operator. By applying the theorem to suitably chosen differential operators A , we obtain a simple proof of Alberti's rank-one theorem, 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, that every top-dimensional Ambrosio-Kirchheim metric current in Rd is a Federer-Fleming at chain.
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty