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

TL;DR

This paper investigates the structure of divergence-free measures in the plane, demonstrating they can be decomposed into measures supported on simple curves, and extends rigidity results for divergence-free vector measures.

Contribution

It provides a decomposition theorem for divergence-free measures into measures induced by simple curves and generalizes rigidity properties to vector-valued measures.

Findings

Divergence-free measures can be decomposed into measures supported on simple curves.

If a measure has zero divergence and certain boundary conditions, it must be zero.

The results extend rigidity properties from vector fields to vector-valued measures.

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.