The ground state energy of the two dimensional Ginzburg–Landau functional with variable magnetic field
Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 2, pp. 325-345.

We consider the Ginzburg–Landau functional with a variable applied magnetic field in a bounded and smooth two dimensional domain. We determine an accurate asymptotic formula for the minimizing energy when the Ginzburg–Landau parameter and the magnetic field are large and of the same order. As a consequence, it is shown how bulk superconductivity decreases in average as the applied magnetic field increases.

DOI : https://doi.org/10.1016/j.anihpc.2013.12.002
Classification : 82D55
Mots clés : Superconductivity, Ginzburg–Landau, Variable magnetic field
@article{AIHPC_2015__32_2_325_0,
author = {Attar, K.},
title = {The ground state energy of the two dimensional Ginzburg--Landau functional with variable magnetic field},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {325--345},
publisher = {Elsevier},
volume = {32},
number = {2},
year = {2015},
doi = {10.1016/j.anihpc.2013.12.002},
zbl = {1320.82071},
mrnumber = {3325240},
language = {en},
url = {www.numdam.org/item/AIHPC_2015__32_2_325_0/}
}
Attar, K. The ground state energy of the two dimensional Ginzburg–Landau functional with variable magnetic field. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 2, pp. 325-345. doi : 10.1016/j.anihpc.2013.12.002. http://www.numdam.org/item/AIHPC_2015__32_2_325_0/

[1] A. Aftalion, S. Serfaty, Lowest Landau level approach in superconductivity for the Abrikosov lattice close to ${H}_{{c}_{2}}$ , Sel. Math. New Ser. 13 no. 2 (2007), 183 -202 | MR 2361092 | Zbl 1138.82034

[2] S. Fournais, B. Helffer, Optimal uniform elliptic estimates for the Ginzburg–Landau system, Adventures in Mathematical Physics, Contemp. Math. vol. 447 , Amer. Math. Soc. (2007), 83 -102 | MR 2423573 | Zbl 1161.35042

[3] S. Fournais, B. Helffer, Spectral Methods in Surface Superconductivity, Prog. Nonlinear Differ. Equ. Appl. vol. 77 , Birkhäuser, Boston (2010) | MR 2662319 | Zbl 1256.35001

[4] S. Fournais, A. Kachmar, The ground state energy of the three dimensional Ginzburg–Landau functional part I: Bulk regime, Commun. Partial Differ. Equ. 38 no. 2 (2013), 339 -383 | MR 3009083 | Zbl 1267.82140

[5] V. Girault, P.-A. Raviart, Finite Elements Methods for Navier–Stokes Equations, Springer (1986) | MR 851383 | Zbl 0413.65081

[6] X.B. Pan, Surface superconductivity in applied magnetic fields above ${\mathrm{𝐻𝐶}}_{2}$ , Commun. Math. Phys. 228 no. 2 (2002), 327 -370 | MR 1911738 | Zbl 1004.82020

[7] X.B. Pan, Surface superconductivity in 3 dimensions, Trans. Am. Math. Soc. 356 no. 10 (2004), 3899 -3937 | MR 2058511 | Zbl 1051.35090

[8] X.B. Pan, K.H. Kwek, Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains, Trans. Am. Math. Soc. 354 no. 10 (2002), 4201 -4227 | MR 1926871 | Zbl 1004.35110

[9] S. Sandier, S. Serfaty, The decrease of bulk-superconductivity close to the second critical field in the Ginzburg–Landau model, SIAM J. Math. Anal. 34 no. 4 (2003), 939 -956 | MR 1969609 | Zbl 1030.82015