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

TL;DR

This paper investigates the regularity of Radon measures linked to stationary harmonic functions with Laplacian equal to the measure, revealing that their support locally lies within zeros of harmonic functions, with applications in physics such as Ginzburg-Landau vortices.

Contribution

It establishes the local structure of measures satisfying certain harmonicity conditions, connecting their support to zeros of harmonic functions, and applies this to physical models.

Findings

Support of measures is contained in zeros of harmonic functions

Locally, the support is a union of smooth simple curves

Results apply to vorticity measures in Ginzburg-Landau systems

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 $(u_\varepsilon)_\varepsilon$ 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 $\mu$ is a union of smooth simple