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

We study the regularity of Radon measures $μ$ which satisfy that there exists a function $h_μ$ in $H^1(Ω)$, stationary harmonic such that $Δh_μ=μ$ in $Ω$ (here $Ω$ 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 $(u_\varepsilon)_\varepsilon$ 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 $μ$ is a union of smooth simple