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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 mechanics of superconductors
##### Keywords:
superconduction; Ginzburg-Landau system; nucleation
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