Energy asymptotics for type II superconductors. (English) Zbl 1160.82365
Summary: We study the Ginzburg-Landau functional in the parameter regime describing ‘Type II superconductors’. In the exact regime considered minimizers are localized to the boundary-i.e. the sample is only superconducting in the boundary region. Depending on the relative size of different parameters we describe the concentration behavior and give leading order energy asymptotics. This generalizes previous results by K. Lu and X.-B. Pan [Physica D 127, No. 1–2, 73–104 (1999; Zbl 0934.35174)], the second author and Pan [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20, No. 1, 145–181 (2003; Zbl 1060.35132)], and Pan [Calc. Var. Partial Differ. Equ. 14, No. 4, 447–482 (2002; Zbl 1006.35090)].
MSC:
| 82D55 | Statistical mechanics of superconductors |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35J50 | Variational methods for elliptic systems |
| 58E50 | Applications of variational problems in infinite-dimensional spaces to the sciences |
References:
| [1] | Adams, R.A.: Sobolev spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, Pure Appl. Math., vol. 65 [MR 56 #9247] (1975) · Zbl 0314.46030 |
| [2] | Avron, J., Herbst, I., Simon, B.: Schrödinger operators with magnetic fields. I. General interactions. Duke Math. J. 45(4), 847–883 (1978) [MR 80k:35054] · Zbl 0399.35029 |
| [3] | Bolley, C., Helffer, B.: An application of semi-classical analysis to the asymptotic study of the supercooling field of a superconducting material. Ann. Inst. H. Poincaré Phys. Théor. 58, 189–233 (1993) [MR 94k:82120] · Zbl 0779.35104 |
| [4] | Bonnaillie, V.: On the fundamental state for a Schrödinger operator with magnetic field in a domain with corners. C. R. Math. Acad. Sci. Paris 336, 135–140 (2003) [MR 1969567] · Zbl 1038.35043 |
| [5] | Bonnaillie, V.: On the fundamental state energy for a Schrödinger operator with magnetic field in a domain with corners. Asympt. Analysis (2004) (in press). · Zbl 1061.65114 |
| [6] | Bauman, P., Phillips, D., Tang, Q.: Stable nucleation for the Ginzburg-Landau system with an applied magnetic field. Arch. Rational Mech. Anal. 142, 1–43 (1998) [MR 99g:58040] · Zbl 0922.35157 · doi:10.1007/s002050050082 |
| [7] | Bernoff, A., Sternberg, P.: Onset of superconductivity in decreasing fields for general domains. J. Math. Phys. 39, 1272–1284 (1998) [MR 99a:82099] · Zbl 1056.82523 · doi:10.1063/1.532379 |
| [8] | Dauge, M., Helffer, B.: Eigenvalues variation. I. Neumann problem for Sturm-Liouville operators. J. Differential Equations 104, 243–262 (1993) [MR 94j:47097] · Zbl 0784.34021 |
| [9] | del Pino, M., Felmer, P.L., Sternberg, P.: Boundary concentration for eigenvalue problems related to the onset of superconductivity. Commun. Math. Phys. 210, 413–446 (2000) [MR 2001k:35231] · Zbl 0982.35077 · doi:10.1007/s002200050786 |
| [10] | Giorgi, T., Phillips, D.: The breakdown of superconductivity due to strong fields for the Ginzburg-Landau model. SIAM J. Math. Anal. 30, 341–359 (1999) [MR 2000b:35235] · Zbl 0920.35058 · doi:10.1137/S0036141097323163 |
| [11] | Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin Heidelberg New York, Reprint of the 1998 edition (2001) [MR 2001k:35004] · Zbl 1042.35002 |
| [12] | Helffer, B., Morame, A.: Magnetic bottles in connection with superconductivity. J. Funct. Anal. 185, 604–680 (2001) [MR 2002m:81051] · Zbl 1078.81023 · doi:10.1006/jfan.2001.3773 |
| [13] | Helffer, B., Pan, X.: Upper critical field and location of surface nucleation for superconductivity. Ann. I.H. Poincaré 20, 145–181 (2003) · Zbl 1060.35132 · doi:10.1016/S0294-1449(02)00005-7 |
| [14] | Lieb, E.H., Loss, M.: Analysis, American Mathematical Society. Providence, RI (1997) [MR 98b:00004] |
| [15] | Lu, K., Pan, X.-B.: Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity. Physics. D 127, 73–104 (1999) [MR 2000a:82075] · Zbl 0934.35174 · doi:10.1016/S0167-2789(98)00246-2 |
| [16] | Marcinkiewicz, J.: Sur les multiplicateurs des séries de Fourier. Stud. Math. 8, 78–91 (French) (1939) · Zbl 0020.35403 |
| [17] | Pan, X.-B.: Surface superconductivity in applied magnetic fields above HC 2. Commun. Math. Phys. 228, 327–370 (2002) [MR 2003i:82094] · Zbl 1004.82020 · doi:10.1007/s002200200641 |
| [18] | Pan, X.-B.: Upper critical field for superconductors with edges and corners. Calc. Var. Partial Differential Equations 14, 447–482 (2002) [MR 2003f:82105] · Zbl 1006.35090 · doi:10.1007/s005260100111 |
| [19] | Sandier, E., Serfaty, S.: On the energy of type-II superconductors in the mixed phase. Rev. Math. Phys. 12, 1219–1257 (2000) [MR 2002f:58023] · Zbl 0964.49006 · doi:10.1142/S0129055X00000411 |
| [20] | Sandier, E., Serfaty, S.: The decrease of bulk-superconductivity close to the second critical field in the Ginzburg-Landau model. SIAM J. Math. Anal. 34 939–956 (electronic) (2003) [MR 1 969 609] · Zbl 1030.82015 |
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