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Ginzburg-Landau equation with magnetic effect in a thin domain

  • Shuichi Jimbo
  • Yoshihisa Morita
Original article

Abstract.

We study the Ginzburg-Landau equation with magnetic effect in a thin domain \(\Omega(\epsilon)\) in \({\mathbb R}^3\), where the thickness of the domain is controlled by a parameter \(\epsilon>0\). This equation is an Euler equation of a free energy functional and it has trivial solutions that are minimizers of the functional. In this article we look for a nontrivial stable solution to the equation, that is, a local minimizer of the energy functional. To prove the existence of such a stable solution in \(\Omega(\epsilon)\), we consider a reduced problem as \(\epsilon\to 0\) and a nondegenerate stable solution to the reduced equation. Applying the standard variational argument, we show that there exists a stable solution in \(\Omega(\epsilon)\) near the solution to the reduced equation if \(\epsilon>0\) is sufficiently small. We also present a specific example of a domain which allows a stable vortex solution, that is, a stable solution with zeros.

Mathematics Subject Classification (2000): 35J20, 35J50, 35Q60, 74K35 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Shuichi Jimbo
    • 1
  • Yoshihisa Morita
    • 2
  1. 1.Department of Mathematics, Hokkaido University, Sapporo 060-0810 Japan (e-mail: jimbo@math.sci.hokudai.ac.jp) JP
  2. 2.Department of Applied Mathematics and Informatics, Ryukoku University, Seta Otsu 520-2194 Japan (e-mail: morita@rins.ryukoku.ac.jp) JP

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