Sharp Asymptotics for the Neumann Laplacian with Variable Magnetic Field: Case of Dimension 2

The aim of this paper is to establish estimates of the lowest eigen-value of the Neumann realization of (i del + BA)(2) on an open bounded subset Omega subset of R(2) with smooth boundary as B tends to infinity. We introduce a "magnetic" curvature mixing the curvature of partial derivative Omega 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.