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**Proof of the De Gennes formula for the superheating field in the weak \(\kappa\) limit.**
*(English)*
Zbl 0889.34010

Summary: In continuation of our preceding paper [Rev. Math. Phys. 8, No. 1, 43-83 (1996; Zbl 0864.35097)] concerning superconducting films, we present in this article new estimates for the superheating field in the weak \(\kappa\) limit. The principal result is the proof of the existence of a finite superheating field \(h^{sh,+}(\kappa)\) (obtained by restricting the usual definition of the superheating field to solutions of the Ginzburg-Landau system \((f, A)\) with \(f\) positive) in the case of a semi-infinite interval. The bound is optimal in the limit \(\kappa\to 0\) and permits to prove (combining with our previous results) the De Gennes formula
\[
2^{-{3\over 4}}= \lim_{\kappa\to 0} \kappa^{{1\over 2}} h^{sh,+}(\kappa).
\]
The proof is obtained by improving slightly the estimates given in the paper of the authors, where an upper bound was found but under the additional condition that the function \(f\) was bounded from below by some fixed constant \(\rho>0\).

### MSC:

34A34 | Nonlinear ordinary differential equations and systems |

82D55 | Statistical mechanics of superconductors |

### Keywords:

Ginzburg-Landau system### Citations:

Zbl 0864.35097
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\textit{C. Bolley} and \textit{B. Helffer}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 14, No. 5, 597--613 (1997; Zbl 0889.34010)

### References:

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[2] | Bolley, C.; Helffer, B., Sur LES asymptotiques des champs critiques pour l’équation de Ginzburg-Landau, () · Zbl 0877.35120 |

[3] | Bolley, C.; Helffer, B., Rigorous results for the Ginzburg-Landau equations associated to a superconducting film in the weak κ-limit, Reviews in math. physics, Vol. 8, n^{o} 1, 43-83, (1996) · Zbl 0864.35097 |

[4] | Bolley, C.; Helffer, B., Superheating in a film in the weak κ limit: numerical results and approximate models, (Dec. 1994), (Part I to appear in M2AN) |

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[12] | Hastings, S.P.; Kwong, M.K.; Troy, W.C., The existence of multiple solutions for a Ginzburg-Landau type model of superconductivity, (May 1995), Preprint |

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[15] | Parr, H., Superconductive superheating field for finite κ, Z. physik, Vol. B25, 359-361, (1976) |

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