TY - JOUR

T1 - On the structure of A -free measures, applications

AU - Philippis, Guido De

AU - Rindler, Filip

N1 - Publisher Copyright:
© 2016 Department of Mathematics, Princeton University.

PY - 2016

Y1 - 2016

N2 - 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.

AB - 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.

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U2 - 10.4007/annals.2016.184.3.10

DO - 10.4007/annals.2016.184.3.10

M3 - Article

AN - SCOPUS:84994170207

VL - 184

SP - 1017

EP - 1039

JO - Annals of Mathematics

JF - Annals of Mathematics

SN - 0003-486X

IS - 3

ER -