The energy of a type II superconductor placed in a strong non-uniform, smooth and signed magnetic field is displayed via a universal reference function defined by means of a simplified two dimensional Ginzburg–Landau functional. We study the asymptotic behavior of this functional in a specific asymptotic regime, thereby linking it to a one dimensional functional, using methods developed by Almog–Helffer and Fournais–Helffer devoted to the analysis of surface superconductivity in the presence of a uniform magnetic field. As a result, we obtain an asymptotic formula reminiscent of the one for the surface superconductivity regime, where the zero set of the magnetic field plays the role of the superconductor’s surface.
Y.Almog and B.Helffer, The distribution of surface superconductivity along the boundary: On a conjecture of X. B. Pan, SIAM J. Math. Anal.38 (2007), 1715–1732. doi:10.1137/050636796.
2.
Y.Almog, B.Helffer and X.B.Pan, Mixed normal-superconducting states in the presence of strong electric currents, Arch. Rational Mech. Anal.223 (2017), 419–462. doi:10.1007/s00205-016-1037-4.
3.
W.Assaad and A.Kachmar, The influence of magnetic steps on bulk superconductivity, Discrete and Continuous Dynamical Systems (A)36(12) (2016), 6623–6643. doi:10.3934/dcds.2016087.
4.
K.Attar, The ground state energy of the two dimensional Ginzburg–Landau functional with variable magnetic field, Annales de l’Institut Henri Poincaré – Analyse Non-Linéaire32 (2015), 325–345. doi:10.1016/j.anihpc.2013.12.002.
5.
K.Attar, Energy and vorticity of the Ginzburg–Landau model with variable magnetic field, Asymptot. Anal.93 (2015), 75–114. doi:10.3233/ASY-151286.
6.
K.Attar, Pinning with a variable magnetic field of the two-dimensional Ginzburg–Landau model, Non-Linear Analysis: TMA139 (2016), 1–54. doi:10.1016/j.na.2016.02.002.
7.
A.Contreras and X.Lamy, Persistence of superconductivity in thin shells beyond , Commun. Contemp. Math.18 (2016), article no. 1550047.
8.
M.Correggi and N.Rougerie, On the Ginzburg–Landau functional in the surface superconductivity regime, Comm. Math. Phys.332 (2014), 1297–1343. doi:10.1007/s00220-014-2095-9.
9.
M.Correggi and N.Rougerie, Boundary behavior of the Ginzburg–Landau order parameter in the surface superconductivity regime, Arch. Rational Mech. Anal.219 (2015), 553–606. doi:10.1007/s00205-015-0900-z.
10.
M.Correggi and N.Rougerie, Effects of boundary curvature on surface superconductivity, Letters in Mathematical Physics106 (2016), 445–467. doi:10.1007/s11005-016-0824-z.
11.
P.G.de Gennes, Superconductivity of Metals and Alloys, Benjamin, Amsterdam, 1996.
12.
S.Fournais and B.Helffer, Energy asymptotics for type II superconductors, Calc. Var. Partial Differential Equations24(3) (2005), 341–376. doi:10.1007/s00526-005-0333-x.
13.
S.Fournais and B.Helffer, Spectral Methods in Surface Superconductivity, Progress in Nonlinear Differential Equations and Their Applications, Vol. 77, Birkhäuser Boston Inc., Boston, MA, 2010.
14.
S.Fournais and A.Kachmar, Nucleation of bulk superconductivity close to critical magnetic field, Advan. Math.226(2) (2011), 1213–1258. doi:10.1016/j.aim.2010.08.004.
15.
B.Helffer, The Montgomery operator revisited, Colloquium Mathematicum118(2) (2011), 391–400. doi:10.4064/cm118-2-3.
16.
B.Helffer and A.Kachmar, The Ginzburg–Landau functional with vanishing magnetic field, Arch. Rational Mech. Anal.218 (2015), 55–122. doi:10.1007/s00205-015-0856-z.
17.
B.Helffer and A.Kachmar, Decay of superconductivity away from the magnetic zero set, 2016, available at arXiv:1604.02402v1.
18.
B.Helffer and A.Kachmar, From constant to non-degenerately vanishing magnetic fields in superconductivity, Annales de l’Institut Henri Poincaré – Analyse Non-Linéaire34 (2017), 423–438. doi:10.1016/j.anihpc.2015.12.008.
19.
B.Helffer, Y.Kordyukov, N.Raymond and S.Vũ Ngo̧c, Magnetic wells in dimension three, Analysis and PDE9(7) (2016), 1575–1608. doi:10.2140/apde.2016.9.1575.
20.
B.Helffer and A.Mohamed, Semi classical analysis for the ground state energy of a Schrödinger operator with magnetic wells, J. Funct. Anal138(1) (1996), 40–81. doi:10.1006/jfan.1996.0056.
21.
R.Montgomery, Hearing the zero locus of a magnetic field, Commun. Math. Phys.168(3) (1995), 651–675. doi:10.1007/BF02101848.
22.
X.B.Pan and H.Kwek, Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains, Trans. Am. Math. Soc.354(10) (2002), 4201–4227. doi:10.1090/S0002-9947-02-03033-7.
23.
E.Sandier and S.Serfaty, Vortices for the Magnetic Ginzburg–Landau Model, Progress in Nonlinear Differential Equations and Their Applications, Vol. 70, Birkhäuser, Basel, 2007.
24.
E.Sandier and S.Serfaty, From the Ginzburg–Landau model to vortex lattice problems, Commun. Math. Phys.313(3) (2012), 635–741. doi:10.1007/s00220-012-1508-x.
