Calculus of Variations and Geometric Measure Theory

M. Petrache

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

created by petrache on 09 Jul 2010
modified on 07 Dec 2013


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


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