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Surface nucleation of superconductivity in 3-dimensions. (English) Zbl 0972.35152
The authors study the surface nucleation of superconductivity and estimate the value of the upper critical field \(H_{C_3}\) for superconductors occupying an arbitrary smooth domain in \(\mathbb{R}^3\). It is proved that \(H_{C_3}\simeq K/\beta_0\), the ratio of the Ginzburg-Landau parameter \(K\) and the first eigenvalue \(\beta_0\) for the Schrödinger operator with unit magnetic field on the half plane. When the applied magnetic field is spatially homogeneous and close to \(H_{C_3}\), a superconducting layer nucleates on a portion of the surface at which the applied field is tangential to the surface. Nucleation under nonhomogeneous applied fields is also discussed.

35Q55 NLS equations (nonlinear Schrödinger equations)
82D55 Statistical mechanical studies of superconductors
Full Text: DOI
[1] Bethuel, F.; Brézis, H.; Hélein, E., Ginzburg – landau vortices, (1994), Birkhäuser Boston/Basel/Berlin
[2] 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) · Zbl 0922.35157
[3] Bethuel, F.; Riviere, T., Vortices for a variational problem related to superconductivity, Ann. inst. H. poincare, anal. non lineaire, 12, 243-303, (1995) · Zbl 0842.35119
[4] Bernoff, A.; Sternberg, P., Onset of superconductivity in decreasing fields for general domains, J. math. phys., 39, 1272-1284, (1998) · Zbl 1056.82523
[5] Chapman, S.J., Nucleation of superconductivity in decreasing fields, European J. appl. math., 5, 449-468, (1994) · Zbl 0820.35124
[6] Chapman, S.J.; Howison, S.D.; Ockendon, J.R., Macroscopic models for superconductivity, SIAM rev., 34, 529-560, (1992) · Zbl 0769.73068
[7] Chanillo, S.; Kiessling, M., Symmetry of solutions of ginzburg – landau equations, C. R. acad. sci. Paris ser. I, 321, 1023-1026, (1995) · Zbl 0843.35004
[8] Du, Q.; Gunzburger, M.; Peterson, J., Analysis and approximation of the ginzburg – landau model of superconductivity, SIAM rev., 34, 4-81, (1992) · Zbl 0787.65091
[9] De Gennes, P.G., Superconductivity of metals and alloys, (1966), Benjamin New York · Zbl 0138.22801
[10] E, W., Dynamics of vortices in ginzburg – landau theories and applications to superconductivity, Physica D, 77, 383-404, (1994) · Zbl 0814.34039
[11] Ginzburg, V.; Landau, L., On the theory of superconductivity, Soviet phys. JETP, 20, 1064-1082, (1950)
[12] Gunzburger, M.; Ockendon, J., Mathematical models in superconductivity, SIAM news, (November and December (1994))
[13] 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) · Zbl 0920.35058
[14] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1983), Springer-Verlag Berlin/Heidelberg/New York/Tokyo · Zbl 0691.35001
[15] Helffer, B., Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells, J. funct. anal., 138, 40-81, (1996) · Zbl 0851.58046
[16] Rose-Innes, A.C.; Rhoderick, E.H., Introduction to superconductivity, International series in solid state physics, (1978), Pergamon Press Oxford/New York
[17] Jaffe, A.; Taubes, C., Vortices and monopoles, (1980), Birkhäuser Boston/Basel · Zbl 0457.53034
[18] Lin, F.-H., Solutions of ginzburg – landau equations and critical points of the renormalized energy, Ann. inst. H. poincare, anal. non lineaire, 12, 599-622, (1995) · Zbl 0845.35052
[19] Lin, F.-H.; Du, Q., Ginzburg – landau vortices: dynamics, pinning, and hysteresis, SIAM J. math. anal., 28, 1265-1293, (1997) · Zbl 0888.35054
[20] Lu, Kening; Pan, Xing-Bin, Ginzburg – landau equation with de Gennes boundary condition, J. differential equations, 129, 136-165, (1996) · Zbl 0873.35088
[21] Lu, Kening; Pan, Xing-Bin, Gauge invariant eigenvalue problems in \(R\)^2 and in \(R\)2+, Trans. amer. math. soc., 352, 1247-1276, (2000) · Zbl 1053.35124
[22] Lu, Kening; Pan, Xing-Bin, Eigenvalue problems of ginzburg – landau operator in bounded domains, J. math. phys., 40, 2647-2670, (1999) · Zbl 0943.35058
[23] Lu, Kening; Pan, Xing-Bin, Estimates of the upper critical field for the ginzburg – landau equations of superconductivity, Physica D, 127, 73-104, (1999) · Zbl 0934.35174
[24] Mironescu, P., On the stability of radial solutions of the ginzburg – landau equations, J. funct. anal., 130, 334-347, (1995) · Zbl 0839.35011
[25] Neu, John, Vortices in complex scalar fields, Physica D, 43, 385-406, (1990) · Zbl 0711.35024
[26] Struwer, M., On the asymptotical behavior of minimizers of the ginzburg – landau model in 2-dimensions, J. differential integral equations, 7, 1613-1627, (1994)
[27] Saint-James, D.; De Gennes, P., Onset of superconductivity in decreasing fields, Phys. lett., 6, 306-308, (1963)
[28] Teman, R., Navier – stokes equations, theory and numerical analysis, Elsevier science, (1987), North-Holland Amsterdam/New York/Oxford
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