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.MSC (2010): 35D30 (primary); 28B05, 42B37 (secondary).

We thank Emil Wiedemann for pointing out that in the proof of the sufficiency part of Theorem 9 (p. 584) in [2] the reduction to the case with underlying deformation zero does not work. This is so, because in the definition of μ it might happen that μ ∞ x is not a positive measure. The current proof works only if one restricts attention to the subclass of Young measures μ with underlying deformation u ∈ BV( ; R m ) that admit a generating sequence of the special form (u + ϕ j ), where (ϕ j ) ⊂ W 1,1 ( ; R m ) with ϕ j * 0. Instead, to prove Theorem 9 in the stated generality, we extend Step 1 in the proof to the case of Young measures with arbitrary underlying deformation.For u ∈ BV( ; R m ) we letthat is, the set of gradient Young measures that can be generated by sequences converging weakly* to u in BV( ; R m ). First we need the following approximation lemma in BV( ; R m ).Lemma 1 Let u ∈ BV( ; R m ) and ε > 0. Then, there exists a countable family Q of (rotated) rectangles Q ⊂ and a mapping v ∈ BV( ; R m ) with the following properties:The online version of the original article can be found under doi:10.1007/s00205-009-0287-9.1 Here and in the following the numbers refer to those in [2], and we also use the notation introduced there.

of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.