Upper critical field and location of surface nucleation of superconductivity
type: Item
instance of: scholarly article
title: Upper critical field and location of surface nucleation of superconductivity
zbMATH Open document ID: 1060.35132
DOI: 10.1016/S0294-1449(02)00005-7
author: Bernard Helffer
author: Xingbin Pan
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2003-04-27T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78571
review text: The two-dimensional Landau-Ginzburg functional as a function of the order parameter \(\psi\) and the magnetic vector potential \(A\) as well as the corresponding Euler-Lagrange equations are considered on a domain \(\Omega\). In this article, the authors consider only Neumann boundary conditions. The external magnetic field \({\mathcal H} = (0,0,\sigma)\) is assumed to be a constant orthogonal to the two-dimensional plane considered. The behavior of the minimizers as the Ginzburg-Landau parameter \(\kappa\) becomes large is investigated. The paper has three major results that are motivated by results by \textit{K. Lu} and \textit{X. B. Pan} [J. Math. Phys. 40, No. 6, 2647--2670 (1999; Zbl 0943.35058)]. The first concerns the critical external magnetic field: It is well known that there exists a unique divergence free vector field \(F\) such that \(\nabla \times F= 1\) on \(\Omega\) and \(F\) is orthogonal to the boundary on \(\partial \Omega\). Let \({\mathcal H}_C(\kappa) = \inf\{\sigma>0\mid (0,\sigma F)\) is the only minimizer\} be the critical external magnetic field. The first result gives an asymptotic formula for \({\mathcal H}_C(\kappa)\) for large \(\kappa\). The second and third result give results on the concentration of \(\psi\). In particular, if the domain is non-degenerate, the minimizer \(\psi\) concentrates near the boundary and there in particular at the points of highest curvature.
reviewed by: Heinz Siedentop
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 82D55
Mathematics Subject Classification ID: 47N55
Mathematics Subject Classification ID: 58E50
zbMATH DE Number: 1901030
zbMATH Keywords: critical field
zbMATH Keywords: location of the superconductivity
zbMATH Keywords: magnetic vector potential
zbMATH Keywords: Euler-Lagrange equations
zbMATH Keywords: minimizers
MaRDI profile type: Publication
cites work: Q3969238
cites work: Stable nucleation for the Ginzburg-Landau system with an applied magnetic field
cites work: Onset of superconductivity in decreasing fields for general domains
cites work: Q5285802
cites work: Q4764824
cites work: Macroscopic Models for Superconductivity
cites work: Eigenvalues variation. I: Neumann problem for Sturm--Liouville operators
cites work: Q5512940
cites work: Boundary concentration for eigenvalue problems related to the onset of superconductivity
cites work: Analysis and Approximation of the Ginzburg–Landau Model of Superconductivity
cites work: The Breakdown of Superconductivity Due to Strong Fields for the Ginzburg--Landau Model
cites work: Semi-classical analysis for the Schrödinger operator and applications
cites work: Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells
cites work: Magnetic bottles in connection with superconductivity
cites work: Q4260816
cites work: Gauge Invariant Eigenvalue Problems in ℝⁿ and in ℝⁿ₊
cites work: Eigenvalue problems of Ginzburg–Landau operator in bounded domains
cites work: Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity
cites work: Surface nucleation of superconductivity in 3-dimensions
cites work: Ginzburg-Landau system and surface nucleation of superconductivity
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description: scientific article; zbMATH DE number 1901030