## Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity.(English)Zbl 0934.35174

Summary: We study the effects of an applied magnetic field on a superconductor and estimate the value of the upper critical magnetic field $$H_{C_3}$$ at which superconductivity can nucleate. In the case of a spatially homogeneous applied field, we show that $$H_{C_3}\simeq\kappa/\beta_0$$, the ratio of the Ginzburg-Landau parameter $$\kappa$$ and the first eigenvalue $$\beta_0$$ of a twisted Laplacian operator, and that superconductivity nucleates at the boundary when the applied field is close to $$H_{C_3}$$. In the case of a spatially non-homogeneous applied field, we give an estimate for the upper critical value and find that superconducting properties may persist only in the interior of the domain. In addition, we show that the order parameter concentrates at the minimum points of the applied field.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 82D55 Statistical mechanics of superconductors
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### References:

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