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 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.

On the structure of A-free measures and applications / De Philippis, Guido; Rindler, F.. - In: ANNALS OF MATHEMATICS. - ISSN 0003-486X. - 184:3(2016), pp. 1017-1039. [10.4007/annals.2016.184.3.10]

On the structure of A-free measures and applications

De Philippis, Guido;
2016-01-01

Abstract

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 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.
2016
184
3
1017
1039
https://arxiv.org/abs/1601.06543
De Philippis, Guido; Rindler, F.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/15970
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