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Stable nucleation for the Ginzburg-Landau system with an applied magnetic field. (English) Zbl 0922.35157
The authors deal with the boundary value problem \[ \Biggl({i\over\kappa}+ A\Biggr)^2\psi- \psi+| \psi|^2\psi= 0,\;(\text{curl})^2A+{i\over 2\kappa} (\psi^*\nabla\psi- \psi\nabla\psi^*)+ |\psi|^2 A=0\quad\text{in }\Omega, \] \[ n\Biggl({i\over\kappa} \nabla+ A\Biggr)\psi= 0,\quad \text{curl }A= he_3\quad \text{on }\partial\Omega \] describing a superconducting material placed in a vacuum subject to an applied magnetic field. The domain \(\Omega\) is a cylinder and the applied field \(he_3\) is parallel to the axis. Then \(|\psi|^2\) represents the density of the superconducting electrons and \(\text{curl }A\) (where \(A\) is a two-dimensional vector field) represents the induced magnetic field.
If \(h\) is large, then the normal (nonsuperconducting) state (with \(\psi= 0\)) is stable. As \(h\) decreases, the normal state becomes unstable. However, if the constant \(\kappa\) is large, then, excepting a discrete set of radii of \(\Omega\), there exists precisely one superconducing solution which is stable. Then \(|\psi|\) becomes uniformly positive in a strip near \(\partial\Omega\) and rapidly decreases toward the interior of \(\Omega\). This is the surface nucleation phenomenon thoroughly analyzed in the article.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
82D55 Statistical mechanical studies of superconductors
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