On the structure of divergence-free measures on $\mathbb R^2$
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Publication:6331693
DOI10.1515/ACV-2020-0066arXiv1912.10936MaRDI QIDQ6331693
Paolo Bonicatto, Nikolay A. G. Gusev
Publication date: 23 December 2019
Abstract: We consider the structure of divergence-free vector measures on the plane. We show that such measures can be decomposed into measures induced by closed simple curves. More generally, we show that if the divergence of a planar vector-valued measure is a signed measure, then the vector-valued measure can be decomposed into measures induced by simple curves (not necessarily closed). As an application we generalize certain rigidity properties of divergence-free vector fields to vector-valued measures. Namely, we show that if a locally finite vector-valued measure has zero divergence, vanishes in the lower half-space and the normal component of the unit tangent vector of the measure is bounded from below (in the upper half-plane), then the measure is identically zero.
Geometric measure and integration theory, integral and normal currents in optimization (49Q15) Length, area, volume, other geometric measure theory (28A75) Currents in global analysis (58A25)
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