Strong diamagnetism for general domains and application

[Diamagnétisme fort pour des domaines généraux et applications]

Fournais, Soeren ; Helffer, Bernard

Annales de l'Institut Fourier, Tome 57 (2007), p. 2389-2400 / Harvested from Numdam

Résumé

Nous considérons le Laplacien de Neumann avec champ magnétique constant dans un domaine régulier de $\mathbb{R}^2$. Si $B$ désigne l’intensité de ce champ et si $\lambda _1(B)$ désigne la première valeur propre de ce Laplacien, il est démontré que $\lambda _1$ est une fonction monotone croissante de $B$ pour $B$ grand. En combinant avec des résultats antérieurs des auteurs, ceci implique la coïncidence de toutes les définitions raisonables du troisième champ critique pour les matériaux supraconducteurs de type II.

Abstract

We consider the Neumann Laplacian with constant magnetic field on a regular domain in $\mathbb{R}^2$. Let $B$ be the strength of the magnetic field and let $\lambda _1(B)$ be the first eigenvalue of this Laplacian. It is proved that $B \mapsto \lambda _1(B)$ is monotone increasing for large $B$. Together with previous results of the authors, this implies the coincidence of all the “third” critical fields for strongly type 2 superconductors.

Détail

Publié le : 2007-01-01

MR 2394546 | Zbl 1133.35073

DOI : https://doi.org/10.5802/aif.2337
Classification:  35P15,  35J55,  82D55
Mots clés: théorie spectrale, bas du spectre, condition de Neumann, supraconductivité

Bibliographie

[1] Agmon, Shmuel Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of $N$-body Schrödinger operators, Princeton University Press, Princeton, NJ, Mathematical Notes, Tome 29 (1982) | MR 745286 | Zbl 0503.35001

[2] Bauman, P.; Phillips, D.; Tang, Q. Stable nucleation for the Ginzburg-Landau system with an applied magnetic field, Arch. Rational Mech. Anal., Tome 142 (1998) no. 1, pp. 1-43 | Article | MR 1629119 | Zbl 0922.35157

[3] Bernoff, Andrew; Sternberg, Peter Onset of superconductivity in decreasing fields for general domains, J. Math. Phys., Tome 39 (1998) no. 3, pp. 1272-1284 | Article | MR 1608449 | Zbl 1056.82523

[4] Bonnaillie, Virginie On the fundamental state for a Schrödinger operator with magnetic field in a domain with corners, Asymptotic Anal, Tome 41 (2005) no. 3-4, pp. 215-258 | MR 2127997 | Zbl 1067.35054

[5] Bonnaillie-Noël, Virginie; Dauge, Monique Asymptotics for the low-lying eigenstates of the Schrödinger operator with magnetic field near corners, Ann. Henri Poincaré, Tome 7 (2006) no. 5, pp. 899-931 | Article | MR 2254755 | Zbl 1134.81021

[6] Bonnaillie-Noël, Virginie; Soeren, Fournais Superconductivity in domains with corners (In preparation)

[7] Erdős, László Dia- and paramagnetism for nonhomogeneous magnetic fields, J. Math. Phys., Tome 38 (1997) no. 3, pp. 1289-1317 | Article | MR 1435670 | Zbl 0875.81047

[8] Erdős, László Spectral shift and multiplicity of the first eigenvalue of the magnetic Schrödinger operator in two dimensions, Ann. Inst. Fourier (Grenoble), Tome 52 (2002) no. 6, pp. 1833-1874 | Article | Numdam | MR 1954326 | Zbl 1106.35039

[9] Fournais, S.; Helffer, B. On the third critical field in Ginzburg-Landau theory, Comm. Math. Phys., Tome 266 (2006) no. 1, pp. 153-196 | Article | MR 2231969 | Zbl 1107.58009

[10] Giorgi, T.; Phillips, D. The breakdown of superconductivity due to strong fields for the Ginzburg-Landau model, SIAM Rev., Tome 44 (2002) no. 2, p. 237-256 (electronic) (Reprinted from SIAM J. Math. Anal. 30 (1999), no. 2, 341–359 [MR 2002b:35235]) | Article | MR 1926099 | Zbl 1094.82021

[11] Helffer, Bernard; Morame, Abderemane Magnetic bottles in connection with superconductivity, J. Funct. Anal., Tome 185 (2001) no. 2, pp. 604-680 | Article | MR 1856278 | Zbl 1078.81023

[12] Helffer, Bernard; Pan, Xing-Bin Upper critical field and location of surface nucleation of superconductivity, Ann. Inst. H. Poincaré Anal. Non Linéaire, Tome 20 (2003) no. 1, pp. 145-181 | Article | Numdam | MR 1958165 | Zbl 1060.35132

[13] Loss, Michael; Thaller, Bernd Optimal heat kernel estimates for Schrödinger operators with magnetic fields in two dimensions, Comm. Math. Phys., Tome 186 (1997) no. 1, pp. 95-107 | Article | MR 1462758 | Zbl 0873.35078

[14] Lu, Kening; Pan, Xing-Bin Eigenvalue problems of Ginzburg-Landau operator in bounded domains, J. Math. Phys., Tome 40 (1999) no. 6, pp. 2647-2670 | Article | MR 1694223 | Zbl 0943.35058

[15] Lu, Kening; Pan, Xing-Bin Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity, Phys. D, Tome 127 (1999) no. 1-2, pp. 73-104 | Article | MR 1678383 | Zbl 0934.35174

[16] Lu, Kening; Pan, Xing-Bin Gauge invariant eigenvalue problems in ${\bf R}^2$ and in ${\bf R}^2_+$, Trans. Amer. Math. Soc., Tome 352 (2000) no. 3, pp. 1247-1276 | Article | MR 1675206 | Zbl 1053.35124

[17] Pan, Xing-Bin Superconductivity near critical temperature, J. Math. Phys., Tome 44 (2003) no. 6, pp. 2639-2678 | Article | MR 1979105 | Zbl 1062.82057

[18] Del Pino, Manuel; Felmer, Patricio L.; Sternberg, Peter Boundary concentration for eigenvalue problems related to the onset of superconductivity, Comm. Math. Phys., Tome 210 (2000) no. 2, pp. 413-446 | Article | MR 1776839 | Zbl 0982.35077

[19] Soeren, Fournais; Helffer, Bernard Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian, Ann. Inst. Fourier (Grenoble), Tome 56 (2006) no. 1, pp. 1-67 | Article | Numdam | MR 2228679 | Zbl 1097.47020