Regularity Properties of Stationary Harmonic Functions Whose Laplacian is a Radon Measure
Rémy Rodiac · SIAM Journal on Mathematical Analysis · 2016
We study the regularity of Radon measures $\mu$ which satisfy that there exists a function $h_{\mu}$ in $H^1$, stationary harmonic such that $\Delta h_{\mu} =\mu$. These conditions appear in physical contexts such as the study of a limiting vorticity measure associated with 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 zero set of $H$. By using the local structure of the zero set of harmonic functions we can thus obtain that locally the support of $\mu$ is a union of smooth simple curves.