×
Helffer, Bernard; Morame, Abderemane

Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 (general case). (English) Zbl 1057.35061

Ann. Sci. Éc. Norm. Supér. (4) 37, No. 1, 105-170 (2004).
Authors’ summary: In a recent paper Lu and Pan have analyzed the asymptotic behavior, in the semi-classical regime, of the ground state energy of the Neumann realization of the Schrödinger operator in the case of dimension 3. Although these results are rather satisfactory when the magnetic field is non-constant and satisfies some generic conditions, they are not sufficient in the case of a constant magnetic field for understanding phenomena like the onset of superconductivity and more accurate results should be obtained. In the two-dimensional case, the effects due to the curvature of the boundary were predicted by a formal analysis of Bernoff-Sternberg and finally proved by the joint efforts of Lu-Pan, Del Pino-Felmer-Sternberg and Helffer-Morame. Our aim is to analyze similar effects in dimension 3. As known from physicists and roughly analyzed by Lu-Pan, it turns out that the results depend on the geometry of the boundary especially at the points where the magnetic field is tangent at the boundary. We present here the analog of the Bernoff-Sternberg conjecture (also formulated in a different form by Pan) and prove it in the generic situation.
Reviewer: Luis Vazquez (Madrid)

Cited in 25 Documents

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
82D55 Statistical mechanics of superconductors
78A30 Electro- and magnetostatics
58E99 Variational problems in infinite-dimensional spaces

Keywords:

magnetic bottles; Ginzburg-Landau functional; Neumann condition; superconductivity
Cite Review PDF
Full Text: DOI Numdam Numdam EuDML

References:

[1] Agmon S. , Lectures on Exponential Decay of Solutions of Second Order Elliptic Equations , Math. Notes , vol. 29 , Princeton University Press , 1982 . MR 745286 | Zbl 0503.35001 · Zbl 0503.35001
[2] Bauman P. , Phillips D. , Tang Q. , Stable nucleation for the Ginzburg-Landau system with an applied magnetic field , IMA Preprint Series 1416 (1996) , Arch. Rational Mech. Anal. 142 ( 1998 ) 1 - 43 . MR 1629119 | Zbl 0922.35157 · Zbl 0922.35157 · doi:10.1007/s002050050082
[3] Bernoff A. , Sternberg P. , Onset of superconductivity in decreasing fields for general domains , J. Math. Phys. 39 ( 1998 ) 1272 - 1284 . MR 1608449 | Zbl 1056.82523 · Zbl 1056.82523 · doi:10.1063/1.532379
[4] Bolley C. , Modélisation du champ de retard à la condensation d’un supraconducteur par un problème de bifurcation , M 26 ( 2 ) ( 1992 ) 235 - 287 . Numdam | MR 1153002 | Zbl 0741.35085 · Zbl 0741.35085
[5] E.B. Davies , Spectral Theory and Differential Operators , in: Cambridge Studies in Advanced Mathematics. Zbl 0893.47004 · Zbl 0893.47004
[6] Dauge M. , Helffer B. , Eigenvalues variation I, Neumann problem for Sturm-Liouville operators , J. Differential Equations 104 ( 2 ) ( 1993 ) 243 - 262 . MR 1231468 | Zbl 0784.34021 · Zbl 0784.34021 · doi:10.1006/jdeq.1993.1071
[7] Dierkes U. , Hildebrandt S. , Kuster A. , Wohlrab O. , Minimal Surfaces I , Springer-Verlag , Berlin , 1992 . MR 1215267 | Zbl 0777.53012 · Zbl 0777.53012
[8] Helffer B. , Introduction to the Semiclassical Analysis for the Schrödinger Operator and Applications , Springer Lecture Notes in Math. , vol. 1336 , 1988 . MR 960278 | Zbl 0647.35002 · Zbl 0647.35002
[9] Helffer B. , Semi-classical analysis for the Schrödinger operator with magnetic wells (after R. Montgomery, B. Helffer and A. Mohamed) , in: Proceedings of the Conference in Minneapolis , IMA Volumes in Mathematics and its Applications , vol. 95 , Springer-Verlag , 1997 , pp. 99 - 114 . MR 1477211 | Zbl 0887.35131 · Zbl 0887.35131
[10] Helffer B. , Bouteilles magnétiques et supraconductivité (d’après Helffer-Morame, Lu-Pan et Helffer-Pan), Séminaire EDP de l’École Polytechnique 2000-2001 , Reprint , http://www.math.polytechnique.fr . Numdam | MR 1860683 | Zbl 02124161 · Zbl 1222.82086
[11] Helffer B. , Mohamed A. , Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells , J. Funct. Anal. 138 ( 1 ) ( 1996 ) 40 - 81 . MR 1391630 | Zbl 0851.58046 · Zbl 0851.58046 · doi:10.1006/jfan.1996.0056
[12] Helffer B. , Morame A. , Magnetic bottles in connection with superconductivity , J. Funct. Anal. 185 ( 2 ) ( 2001 ) 604 - 680 . MR 1856278 | Zbl 1078.81023 · Zbl 1078.81023 · doi:10.1006/jfan.2001.3773
[13] Helffer B. , Morame A. , Magnetic bottles in connection with superconductivity: Case of dimension 3 , Proc. Indian Acad. Sci. (Math. Sci.) 112 ( 1 ) ( 2002 ). MR 1894543 · Zbl 1199.81016
[14] B. Helffer , A. Morame , Magnetic bottles for the Neumann problem: Curvature effects in the case of dimension 3, Preprint mp_arc 01-362, 2001. MR 1856278 · Zbl 1057.35061
[15] B. Helffer , A. Morame , Magnetic bottles for the Neumann problem: Curvature effects in the case of dimension 3, general case, Preprint mp_arc 02-145, 2002. · Zbl 1199.81016
[16] Helffer B. , Nourrigat J. , Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs , Progress in Mathematics , vol. 58 , Birkhäuser , 1985 . MR 897103 | Zbl 0568.35003 · Zbl 0568.35003
[17] Helffer B. , Pan X.-B. , Upper critical field and location of surface nucleation of superconductivity , Annales de l’Institut Henri Poincaré (Analyse Non Linéaire) 20 ( 1 ) ( 2003 ) 145 - 181 . Numdam | MR 1958165 | Zbl 1060.35132 · Zbl 1060.35132 · doi:10.1016/S0294-1449(02)00005-7
[18] Helffer B. , Sjöstrand J. , Multiple wells in the semiclassical limit I , Comm. PDE 9 ( 4 ) ( 1984 ) 337 - 408 . MR 740094 | Zbl 0546.35053 · Zbl 0546.35053 · doi:10.1080/03605308408820335
[19] Lu K. , Pan X.-B. , Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity , Phys. D 127 ( 1999 ) 73 - 104 . MR 1678383 | Zbl 0934.35174 · Zbl 0934.35174 · doi:10.1016/S0167-2789(98)00246-2
[20] Lu K. , Pan X.-B. , Eigenvalue problems of Ginzburg-Landau operator in bounded domains , J. Math. Phys. 40 ( 6 ) ( 1999 ) 2647 - 2670 . MR 1694223 | Zbl 0943.35058 · Zbl 0943.35058 · doi:10.1063/1.532721
[21] Lu K. , Pan X.-B. , Gauge invariant eigenvalue problems on R 2 and R 2 + , Trans. Amer. Math. Soc. 352 ( 3 ) ( 2000 ) 1247 - 1276 . MR 1675206 | Zbl 1053.35124 · Zbl 1053.35124 · doi:10.1090/S0002-9947-99-02516-7
[22] Lu K. , Pan X.-B. , Ginzburg-Landau system and surface nucleation of superconductivity , in: Proceeding of the IMS Workshop on Reaction-Diffusion systems, Chinese University of Hong-Kong , 1999 .
[23] Lu K. , Pan X.-B. , Surface nucleation of superconductivity in 3-dimension , J. Differential Equations 168 ( 2 ) ( 2000 ) 386 - 452 . MR 1808455 | Zbl 0972.35152 · Zbl 0972.35152 · doi:10.1006/jdeq.2000.3892
[24] Montgomery R. , Hearing the zerolocus of a magnetic field , Comm. Math. Phys. 168 ( 1995 ) 651 - 675 . Article | MR 1328258 | Zbl 0827.58076 · Zbl 0827.58076 · doi:10.1007/BF02101848
[25] Pan X.-B. , Upper critical field for superconductors with edges and corners , Calc. Var. PDE 14 ( 2002 ) 447 - 482 . MR 1911825 | Zbl 1006.35090 · Zbl 1006.35090 · doi:10.1007/s005260100111
[26] X.-B. Pan , Surface conductivity in 3 dimensions, Personal communication, October 2001, Unpublished.
[27] Pan X.-B. , Kwek K.-H. , Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domain , Trans. Amer. Math. Soc. 354 ( 10 ) ( 2002 ) 4201 - 4227 . MR 1926871 | Zbl 1004.35110 · Zbl 1004.35110 · doi:10.1090/S0002-9947-02-03033-7
[28] del Pino M. , Felmer P.L. , Sternberg P. , Boundary concentration for eigenvalue problems related to the onset of superconductivity , Comm. Math. Phys. 210 ( 2000 ) 413 - 446 . MR 1776839 | Zbl 0982.35077 · Zbl 0982.35077 · doi:10.1007/s002200050786
[29] Reed M. , Simon B. , Methods of modern Mathematical Physics, IV: Analysis of Operators , Academic Press , New York , 1978 . MR 493421 | Zbl 0401.47001 · Zbl 0401.47001
[30] Saint-James D. , Sarma G. , Thomas E.J. , Type II Superconductivity , Pergamon , Oxford , 1969 .
[31] Spivak M. , Differential Geometry , Publish or Perish , Houston , 1975 .
[32] Thaller B. , The Dirac Equation , Texts and Monographs in Physics , Springer-Verlag , 1992 . MR 1219537 | Zbl 0765.47023 · Zbl 0765.47023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.