2018/03/06 by Rémy Rodiac, Rodiac, Rémy
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1803.02239
openalex publication_date 2018/03/06 · openalex created_date 2022/09/19 · openalex updated_date 2026/07/28
Let \Ω be a bounded open set in \ℝ2. The aim of this article\nis to describe the functions h in H1(\Ω) and the Radon measures \μ\nwhich satisfy -\Δ h+h=\μ and div(Th)=0 in \Ω, where Th is a\n2\× 2 matrix given by (Th)ij=2\∂ih\∂jh- (|\∇\nh|2+h2)\δij for i,j=1,2. These equations arise as equilibrium\nconditions satisfied by limiting vorticities and limiting induced magnetic\nfields of solutions of the magnetic Ginzburg-Landau equations as shown by\nSandier-Serfaty. Let us recall that they obtained that |\∇ h| is\ncontinuous in \Ω. We prove that if x0 in \Ω belongs to supp\n\μ and is such that |\∇ h(x0)|\≠ 0 then \μ is absolutely\ncontinuous with respect to the 1D-Hausdorff measure restricted to a\n\C1-curve near x0 whereas \μ_ lfloor | \∇\nh|=0 =h_| |\∇ h|=0 . We also prove that if \Ω is smooth\nbounded and star-shaped and if h=0 on \∂ \Ω then h \≡ 0 in\n\Ω. This rules out the possibility of having critical points of the\nGinzburg-Landau energy with a number of vortices much larger than the applied\nmagnetic field hex in that case.\n