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Communications in Mathematical Physics

, Volume 210, Issue 2, pp 413–446 | Cite as

Boundary Concentration for Eigenvalue Problems Related to the Onset of Superconductivity

  • Manuel del Pino
  • Patricio L. Felmer
  • Peter Sternberg

Abstract:

We examine the asymptotic behavior of the eigenvalue μ(h) and corresponding eigenfunction associated with the variational problem
$$$$
in the regime h>>1. Here A is any vector field with curl equal to 1. The problem arises within the Ginzburg–Landau model for superconductivity with the function μ(h) yielding the relationship between the critical temperature vs. applied magnetic field strength in the transition from normal to superconducting state in a thin mesoscopic sample with cross-section \(\Omega\subset\R^{2}\). We first carry out a rigorous analysis of the associated problem on a half-plane and then rigorously justify some of the formal arguments of [BS], obtaining an expansion for μ while also proving that the first eigenfunction decays to zero somewhere along the sample boundary \(\) when Ω is not a disc. For interior decay, we demonstrate that the rate is exponential.

Keywords

Eigenvalue Problem Boundary Concentration 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Manuel del Pino
    • 1
  • Patricio L. Felmer
    • 1
  • Peter Sternberg
    • 2
  1. 1.Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile. E-mail: delpino@dim.uchile.cl; pfelmer@dim.uchile.clCL
  2. 2.Department of Mathematics, Indiana University, Bloomington, IN 47405, USA.¶E-mail: sternber@indiana.eduUS

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