An integrability result for Lp -vector fields in the plane

摘要

We prove that if p > 1, then the divergence of an L~p-vector field V on a 2-di-mensional domain Ω is the boundary of an integral 1-current if and only if V can be represented as the rotated gradient ▽⊥u for a W~(1,p)-map u: Ω → S~1. Such a result extends to exponents p > 1 the result on distributional Jacobians of Alberti, Baldo, Orlandi (2003).