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

Abstract

We establish a general structure theorem for the singular part of Afree 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 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 R d is a Federer-Fleming flat chain.

Locations

• arXiv (Cornell University) - View - PDF
• Warwick Research Archive Portal (University of Warwick) - View - PDF
• HAL (Le Centre pour la Communication Scientifique Directe) - View - PDF
• Annals of Mathematics - View