On the structure of A -free measures, applications

Guido De Philippis, Filip Rindler

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)1017-1039
Number of pages23
JournalAnnals of Mathematics
Issue number3
StatePublished - 2016

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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