Ginzburg-Landau equation with DeGennes boundary condition
type: Item
instance of: scholarly article
title: Ginzburg-Landau equation with DeGennes boundary condition
zbMATH Open document ID: 0873.35088
author: Kening Lu
author: Xingbin Pan
published in: Journal of Differential Equations
publication date: 1996-11-22T00:00:00Z
review text: Let \(\Omega\) be a bounded domain in \(\mathbb{R}^2\) and \(\varepsilon>0\), \(\gamma(\varepsilon)\) be parameters. The authors study the following semilinear elliptic boundary value problem \[ \varepsilon^2\Delta u+(1- u^2)u\quad\text{in }\Omega,\quad {\partial u\over\partial\nu}+ \gamma(\varepsilon)u=0\quad\text{on }\partial\Omega\tag{1} \] and its relation to the minimizing problem of the Ginzburg-Landau type energy functional \[ L(\psi,A)= \int_\Omega\Biggl(|(\nabla- iA)\psi|^2+ |\text{rot }A- H|^2+ {1\over2\varepsilon} (|\psi|^2-1)^2\Biggr) dx+\int_{\partial\Omega} |\psi|^2 dS.\tag{2} \] It is proved that if \(\Omega\) is simply-connected, \(H\equiv 0\) and \(\varepsilon\) is small, the global minimizer \((\psi,A)\) of (2) is expressed by the aid of the unique positive solution of (1), that is \((\psi,A)\equiv(u_\varepsilon e^{\kappa i},\nabla\kappa)\). The behavior of \(u_\varepsilon\) is investigated for small \(\varepsilon>0\).
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35A15
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 933272
zbMATH Keywords: DeGennes boundary condition
zbMATH Keywords: Ginzburg-Landau type energy functional
zbMATH Keywords: global minimizer
zbMATH Keywords: unique positive solution
reviewed by: Shuichi Jimbo
MaRDI profile type: Publication
full work available at URL: jdeq.1996.0114
OpenAlex ID: W1972220685
DOI: 10.1006/JDEQ.1996.0114
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description: scientific article; zbMATH DE number 933272