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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.

Motivated by the theory of superconductivity and more precisely by the problem of the onset of superconductivity in dimension two, many papers devoted to the analysis in a semi-classical regime of the lowest eigenvalue of the Schrödinger operator with magnetic field have appeared recently. Here we would like to mention the works by Bernoff-Sternberg, Lu-Pan, Del Pino-Felmer-Sternberg and Helffer-Morame and also Bauman-Phillips-Tang for the case of a disc. In the present paper we settle one important...