, Volume 76, Issue 1, pp 275-306
Date: 07 Dec 2007

Normal Currents: Structure, Duality Pairings and div–curl Lemmas

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Abstract.

The paper gives a decomposition of a general normal r-dimensional current [5] into the sum of three measures of which the first is an r-dimensional rectifiable measure, the second is the Cantor part of the current, and the third is Lebesgue absolutely continuous. This is analogous to the well-known decomposition of the derivative of a function of bounded variation into the jump, Cantor, and absolutely continuous parts; in fact the last is a special case of the result for (n–1)-dimensional normal currents. Further, Whitney’s cap product [15] is recast in the language of the approach to flat chains by Federer [5] and a special case (viz., currents of dimension n – 1) is shown to be closely related to the measure-valued duality pairings between vector measures with curl a measure and L vectorfields with L divergence as established by Anzellotti [2] and Kohn & Témam [6]. Finally, the cap product is shown to be jointly weak* continuous in the two factors of the product in a way similar to the compensated compactness theory; in the cases of (n – 1)-dimensional objects this reduces to results closely related to the div–curl lemmas of the standard compensated compactness theory.

Received: June 2007