On the structure of divergence-free measures on $\mathbb R^2$

Paolo Bonicatto; Nikolay A. Gusev

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arXiv · 1912.10936

On the structure of divergence-free measures on $\mathbb R^2$

Paolo Bonicatto·Nikolay A. Gusev

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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.

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Paolo Bonicatto, Nikolay A. Gusev. 2019-12-23. On the structure of divergence-free measures on $\mathbb R^2$. https://arxiv.org/abs/1912.10936

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