An integrability result for $L^p$-vectorfields in the plane

TL;DR

This paper establishes a characterization of divergence-free $L^p$-vector fields in the plane, linking their divergence to boundary currents and representing them as rotated gradients of $W^{1,p}$ maps into the circle.

Contribution

It extends the understanding of $L^p$-vector fields by characterizing divergence as boundary currents and representing fields as rotated gradients, generalizing previous Jacobian results.

Findings

Divergence of $L^p$-vector fields corresponds to boundary of an integral 1-current.

Such vector fields can be represented as rotated gradients of $W^{1,p}$ functions into $S^1$.

The result extends known distributional Jacobian theorems to all $p>1$.

Abstract

We prove that if $p>1$ then the divergence of a $L^p$-vectorfield $V$ on a 2-dimensional domain $\Omega$ is the boundary of an integral 1-current, if and only if $V$ can be represented as the rotated gradient $\nabla^\perp u$ for a $W^{1,p}$-map $u:\Omega\to S^1$. Such result extends to exponents $p>1$ the result on distributional Jacobians of Alberti, Baldo, Orlandi.