On the structure of A-free measures and applications

• 成果类型: Article
• 署名作者: De Philippis, Guido; Rindler, Filip
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2016.184.3.10
• 发表日期: 2016
• 页码: 1017-1039
• 关键词: lower semicontinuity; lipschitz functions; transport-equation; young measures; cauchy-problem; bv; differentiability; relaxation

摘要

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.