Regularity properties of stationary harmonic functions whose Laplacian is a Radon measure

Abstract

We study the regularity of Radon measures $\mu$ which satisfy that there exists a function $h_{\mu }$ in $H^1(\Omega )$, stationary harmonic such that $\Delta h_{\mu }=\mu$ in $\Omega$ (here $\Omega$ is an open set of ${\mathbb{R}}^2$). Such conditions appear in physical contexts such as the study of a limiting vorticity measure associated to a family $(u_{\epsilon }{)}_{\epsilon }$ of solutions of the Ginzburg-Landau system without magnetic field. Under these conditions we prove that locally there exists a harmonic function $H$ such that the support of the measure is contained in the set of zeros of $H$. Using the local structure of the set of zeros of harmonic functions we can thus obtain that locally the support of $\mu$ is a union of smooth simple

arXiv comment35 pages, 6 figures