Strong diamagnetism for general domains and applications

TL;DR

This paper investigates the behavior of the first eigenvalue of the magnetic Neumann Laplacian on regular domains under strong magnetic fields, revealing monotonicity and implications for superconductivity critical fields.

Contribution

It proves the monotonic increase of the first eigenvalue with magnetic field strength for large B, linking spectral properties to superconductivity phenomena.

Findings

Eigenvalue $ ightarrow$ monotone increasing for large B

All third critical fields for strongly Type II superconductors coincide

Results extend previous spectral and superconductivity analyses

Abstract

We consider the Neumann Laplacian with constant magnetic field on a regular domain. Let $B$ be the strength of the magnetic field, and let $\lambda_1(B)$ be the first eigenvalue of the magnetic Neumann Laplacian on the domain. It is proved that $B \mapsto \lambda_1(B)$ is monotone increasing for large $B$. Combined with the results of \cite{FournaisHelffer3}, this implies that all the `third' critical fields for strongly Type II superconductors coincide.