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Existence of steady stable solutions for the Ginzburg-Landau equation in a domain with nontrivial topology

Author: Norimichi Hirano
Journal: Proc. Amer. Math. Soc. 138 (2010), 1701-1710
MSC (2000): Primary 35J50, 35Q80
Published electronically: January 6, 2010
MathSciNet review: 2587455
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Abstract: Let $ N\geq2$ and $ \Omega$ $ \subset\mathbb{R}^{N}$ be a bounded domain with boundary $ \partial\Omega$. Let $ \Gamma\subset\partial\Omega$ be closed. Our purpose in this paper is to consider the existence of stable solutions $ u\in H^{1}(\Omega,\mathcal{\mathbb{C}})$ of the Ginzburg-Landau equation

\begin{displaymath}\left\{ \begin{array}[c]{rlll} -\Delta u(x) & = & \lambda(w_{... ...\nu} & = & 0 & \qquad\qquad\text{on }\Gamma \end{array}\right. \end{displaymath}

where $ \lambda>0,$ $ w_{0}\in C^{2}(\overline{\Omega},\mathbb{\mathbb{R}^{+}})$ and $ g\in C^{2}(\partial\Omega\backslash\Gamma)$ such that $ \left\vert g(x)\right\vert =w_{0}(x)$ on $ \partial\Omega\backslash\Gamma$.

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Additional Information

Norimichi Hirano
Affiliation: Department of Mathematics, Graduate School of Environment and Information Sciences, Yokohama National University, Tokiwadai, Hodogayaku, Yokohama, Japan

Keywords: Ginzburg-Landau equation, nontrivial topology of domain
Received by editor(s): June 28, 2009
Received by editor(s) in revised form: July 3, 2009, and August 20, 2009
Published electronically: January 6, 2010
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.