On the Ginzburg–Landau energy with a magnetic field vanishing along a curve

Article type: Research Article

Authors: Kachmar, Ayman^a | Nasrallah, Marwa^{b; c; *}

Affiliations: [a] Department of Mathematics, Lebanese University, Nabatieh, Lebanon. E-mail: ayman.kashmar@gmail.com | [b] Lebanese International University, Beirut, Lebanon. E-mail: marwa.nasrallah@liu.edu.lb | [c] Faculty of Sciences, Section IV, Lebanese University, Bekaa, Lebanon

Correspondence: [*] Corresponding author. E-mail: marwa.nasrallah@liu.edu.lb.

Abstract: The energy of a type II superconductor placed in a strong non-uniform, smooth and signed magnetic field is displayed via a universal reference function defined by means of a simplified two dimensional Ginzburg–Landau functional. We study the asymptotic behavior of this functional in a specific asymptotic regime, thereby linking it to a one dimensional functional, using methods developed by Almog–Helffer and Fournais–Helffer devoted to the analysis of surface superconductivity in the presence of a uniform magnetic field. As a result, we obtain an asymptotic formula reminiscent of the one for the surface superconductivity regime, where the zero set of the magnetic field plays the role of the superconductor’s surface.

Keywords: Superconductivity, Ginzburg–Landau functional, ground state energy, vanishing magnetic field, Montgomery operator

DOI: 10.3233/ASY-171424

Journal: Asymptotic Analysis, vol. 103, no. 3, pp. 135-163, 2017