Rank one property for derivatives of functions with bounded variation. (English) Zbl 0791.26008

Alberti, Giovanni

Proc. R. Soc. Edinb., Sect. A 123, No. 2, 239-274 (1993).

The basic result of this well organized and clearly written paper is contained in the following Theorem: Let \(\Omega\) be an open subset of \(\mathbb{R}^ n\), let \(u\) be a function of bounded variation of \(\Omega\) into \(\mathbb{R}^ m\) and denote by \(D_ s u\) the part of \(Du\) which is singular with respect to Lebesgue measure. Then \(D_ s u\) is a rank one measure and this means that for \(| D_ s u|\) almost all \(x\) the matrix \([dD_ s u/d| D_ s u|](x)\) has rank one.

Reviewer: J.Albrycht (Poznań)

MSC:

26B30

Absolutely continuous real functions of several variables, functions of bounded variation

Keywords:

derivative; function of bounded variation; rank one measure

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