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

2015/04/28 by Rémy Rodiac, Rodiac, Rémy
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AP #math.CV #math.MP

paper · pdf · doi:10.48550/arxiv.1504.07457

35 pages, 6 figures

arxiv created 2015/04/28 · arxiv updated 2015/04/29

Abstract

We study the regularity of Radon measures μ which satisfy that there exists a function h_μ in H¹(Ω), stationary harmonic such that Δh_μ=μ in Ω (here Ω is an open set of ℝ²). Such conditions appear in physical contexts such as the study of a limiting vorticity measure associated to a family (u_ε)_ε 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

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