ON THE STRUCTURE OF A -FREE MEASURES AND APPLICATIONS

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. MSC (2010): 35D30 (primary); 28B05, 42B37 (secondary).
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Guido de Philippis, Filip Rindler. ON THE STRUCTURE OF A -FREE MEASURES AND APPLICATIONS. Annals of Mathematics, Princeton University, Department of Mathematics, 2016, 184, ⟨10.4007/annals.2016.184.3.10⟩. ⟨ensl-01413644⟩

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