Existence of steady stable solutions for the Ginzburg-Landau equation in a domain with nontrivial topology
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- by Norimichi Hirano PDF
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Abstract:
Let $N\geq 2$ 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 \[ \left \{ \begin {array} [c]{rlll} -\Delta u(x) & = & \lambda (w_{0}^{2}(x)-\left \vert u\right \vert ^{2})u & \qquad \qquad \text {in }\Omega ,\\ u & = & g & \qquad \qquad \text {on }\partial \Omega \backslash \Gamma ,\\ \frac {\partial u}{\partial \nu } & = & 0 & \qquad \qquad \text {on }\Gamma \end {array} \right . \] 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 $\partial \Omega \backslash \Gamma$.References
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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
- Email: hirano@math.sci.ynu.ac.jp
- 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
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 1701-1710
- MSC (2000): Primary 35J50, 35Q80
- DOI: https://doi.org/10.1090/S0002-9939-10-10225-1
- MathSciNet review: 2587455