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.
UR - http://www.scopus.com/inward/record.url?scp=84994170207&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84994170207&partnerID=8YFLogxK
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 -