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We study the regularity of Radon measures $\mu$ which satisfy that thereexists 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 studyof a limiting vorticity measure associated to a family$(u_\varepsilon)_\varepsilon$ of solutions of the Ginzburg-Landau systemwithout magnetic field. Under these conditions we prove that locally thereexists a harmonic function $H$ such that the support of the measure iscontained in the set of zeros of $H$. Using the local structure of the set ofzeros of harmonic functions we can thus obtain that locally the support of$\mu$ is a union of smooth simple

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