Isoperimetric inequalities for the magnetic Neumann and Steklov problems with Aharonov-Bohm magnetic potential

TL;DR

This paper establishes isoperimetric inequalities for the lowest eigenvalues of magnetic Laplacians with Aharonov-Bohm potential on planar domains, extending classical results to magnetic and curved geometries.

Contribution

It introduces new isoperimetric inequalities for magnetic Laplacian eigenvalues with Aharonov-Bohm potential, including non-integer flux cases and curved geometries.

Findings

Inequalities hold for non-integer flux around the pole.

Results extend classical inequalities to magnetic and curved settings.

Disk with pole at center optimizes the eigenvalue bounds.

Abstract

We discuss isoperimetric inequalities for the magnetic Laplacian on bounded domains of $\mathbb R^2$ endowed with an Aharonov-Bohm potential. When the flux of the potential around the pole is not an integer, the lowest eigenvalue for the Neumann and the Steklov problems is positive. We establish isoperimetric inequalitites for the lowest eigenvalue in the spirit of the classical inequalities of Szeg\"o-Weinberger, Brock and Weinstock, the model domain being a disk with the pole at its center. We consider more generally domains in the plane endowed with a rotationally invariant metric, which include the spherical and the hyperbolic case.