*Published Paper*

**Inserted:** 9 jul 2010

**Last Updated:** 7 dec 2013

**Journal:** Advances in Calculus of Variations

**Volume:** Volume 6

**Number:** 3

**Pages:** 299–319

**Year:** 2013

**Doi:** 10.1515/acv-2012-0102

**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 of a map $u\in W^{1,p}(\Omega,S^1)$. Such result extends to exponents $p>1$ the result on distributional Jacobians of Alberti, Baldo, Orlandi.

**Keywords:**
distributional jacobian, Topological singularities, vector fields with integer fluxes, Sobolev maps between manifolds

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