## 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

Zbl 0864.35097
Full Text:

### References:

 [1] Bolley, C.; Helffer, B.; Bolley, C.; Helffer, B., Rigorous results on the Ginzburg-Landau models in a film submitted to an exterior parallel magnetic field, Nonlinear studies, Nonlinear studies, Vol. 3, n^{o} 2, 1-32, (1996), Part II · Zbl 0857.34006 [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) [5] C. BOLLEY and B. HELFFER, In preparation. [6] Chapman, S.J., Asymptotic analysis of the Ginzburg-Landau model of superconductivity: reduction to a free boundary model, (1992), Preprint · Zbl 0848.35130 [7] Galaiko, V.P., Superheating critical field for superconductors of the first kind, Soviet physics JETP, Vol. 27, n^{o} 1, (July 1968) [8] de Gennes, P.G., Superconductivity, selected topics in solid state physics and theoretical physics, () · Zbl 0138.22801 [9] Ginzburg, V.L., On the theory of superconductivity, Nuovo cimento, Vol. 2, 1234, (1955) · Zbl 0067.23504 [10] Ginzburg, V.L., On the destruction and the onset of superconductivity in a magnetic field, Soviet physics JETP, Vol. 7, 78, (1958) · Zbl 0099.44703 [11] Ginzburg, V.L.; Landau, L.D.; Landau, L.D., On the theory of superconductivity, (), Vol. 20, 138-167, (1950), English translation [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 [13] Saint James, D.; de Gennes, P.G., Onset of superconductivity in decreasing fields, Phys. lett., Vol. 7, 306, (1963) [14] Saint James, D.; Sarma, G.; Thomas, E.J., Type II superconductivity, (1969), Pergamon Press [15] Parr, H., Superconductive superheating field for finite κ, Z. physik, Vol. B25, 359-361, (1976)
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