Isoperimetric inequality with zero magnetic field in doubly connected domains

TL;DR

This paper studies how the lowest eigenvalue of a magnetic Laplacian in doubly connected domains depends on geometry, showing maximization in annuli under certain conditions and proposing a conjecture for general optimality.

Contribution

It demonstrates that the eigenvalue is maximized by annuli for fixed area and flux, and establishes related geometric inequalities, advancing understanding of magnetic Laplacian eigenvalues.

Findings

Eigenvalue maximized by annuli for fixed area and flux

Established geometric inequalities for magnetic eigenvalues

Proved conjecture validity for large magnetic fluxes

Abstract

We investigate how the lowest eigenvalue of a magnetic Laplacian depends on the geometry of a planar domain with a disk shaped hole, where the magnetic field is generated by a singular flux. Under Dirichlet boundary conditions on the inner boundary and Neumann boundary conditions on the outer boundary, we show that this eigenvalue is maximized when the domain is an annulus, for a fixed area and magnetic flux. As consequences, we establish geometric inequalities for eigenvalues in settings with both singular and localized magnetic fields. We also propose a conjecture for a general optimality result and establish its validity for large magnetic fluxes.