Abstract.
The aim of this paper is to establish estimates of the lowest eigenvalue of the Neumann realization of \((i\nabla + B{\bf{A}})^2\) on an open bounded subset \(\Omega \subset \mathbb(R)^2 \) with smooth boundary as B tends to infinity. We introduce a “magnetic” curvature mixing the curvature of ∂Ω and the normal derivative of the magnetic field and obtain an estimate analogous with the one of constant case. Actually, we give a precise estimate of the lowest eigenvalue in the case where the restriction of magnetic field to the boundary admits a unique minimum which is non degenerate. We also give an estimate of the third critical field in Ginzburg–Landau theory in the variable magnetic field case.
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Communicated by Christian Gérard.
Submitted: June 26, 2008., Accepted: November 28, 2008.
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Raymond, N. Sharp Asymptotics for the Neumann Laplacian with Variable Magnetic Field: Case of Dimension 2. Ann. Henri Poincaré 10, 95–122 (2009). https://doi.org/10.1007/s00023-009-0405-0
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DOI: https://doi.org/10.1007/s00023-009-0405-0