Strong diamagnetism for general domains and application

Fournais, Soeren; Helffer, Bernard

doi:10.5802/aif.2337

Strong diamagnetism for general domains and application
[Diamagnétisme fort pour des domaines généraux et applications]

Fournais, Soeren ¹ ; Helffer, Bernard ²

¹ Université Paris-Sud Laboratoire de Mathématiques UMR CNRS 8628 Bât 425 91405 Orsay Cedex (France) and University of Aarhus Department of Mathematical Sciences Ny Munkegade, Building 1530 8000 Aarhus C (Denmark)
² Université Paris-Sud Laboratoire de Mathématiques UMR CNRS 8628 Bât 425 91405 Orsay Cedex (France)

Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2389-2400.

Résumé
Abstract

Nous considérons le Laplacien de Neumann avec champ magnétique constant dans un domaine régulier de $ℝ^{2}$ . Si $B$ désigne l’intensité de ce champ et si $λ_{1} (B)$ désigne la première valeur propre de ce Laplacien, il est démontré que $λ_{1}$ 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.

We consider the Neumann Laplacian with constant magnetic field on a regular domain in $ℝ^{2}$ . Let $B$ be the strength of the magnetic field and let $λ_{1} (B)$ be the first eigenvalue of this Laplacian. It is proved that $B \mapsto λ_{1} (B)$ 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.

MR Zbl

DOI : 10.5802/aif.2337

Classification : 35P15, 35J55, 82D55
Keywords: Spectral theory, bottom of the spectrum, Neumann condition, superconductivity
Mot clés : théorie spectrale, bas du spectre, condition de Neumann, supraconductivité

Affiliations des auteurs :

Fournais, Soeren ¹ ; Helffer, Bernard ²

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     title = {Strong diamagnetism for general domains and application},
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     pages = {2389--2400},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Fournais, Soeren; Helffer, Bernard. Strong diamagnetism for general domains and application. Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2389-2400. doi : 10.5802/aif.2337. http://www.numdam.org/articles/10.5802/aif.2337/

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