Strong diamagnetism for general domains and application

Fournais, Soeren; Helffer, Bernard

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

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 ${\lambda }_1\left(B\right)$ 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.

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 ${\lambda }_1\left(B\right)$ be the first eigenvalue of this Laplacian. It is proved that $B\mapsto {\lambda }_1\left(B\right)$ 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.

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

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

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

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

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

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

[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., Volume 38 (1997) no. 3, pp. 1289-1317 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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