# Inequalities for the lowest magnetic Neumann eigenvalue

## TL;DR

This paper investigates bounds on the lowest magnetic Neumann eigenvalue for planar domains, focusing on whether the disk maximizes this eigenvalue under fixed area and exploring related inequalities.

## Contribution

It provides new and existing bounds on the magnetic Neumann eigenvalue and examines the extremal properties of the disk in this context.

## Key findings

- The disk's role as a potential maximizer of the eigenvalue is analyzed.
- New bounds on the magnetic Neumann eigenvalue are discussed.
- The paper compares old and new inequalities related to the problem.

## Abstract

We study the ground state energy of the Neumann magnetic Laplacian on planar domains. For a constant magnetic field we consider the question whether, under an assumption of fixed area, the disc maximizes this eigenvalue. More generally, we discuss old and new bounds obtained on this problem.