# On Shape Optimization with Large Magnetic Fields in Two Dimensions

## TL;DR

This paper shows that shapes optimizing magnetic eigenvalues become symmetric under strong magnetic fields.

## Contribution

The paper introduces new asymptotic bounds and symmetry results for magnetic eigenvalues in strong field limits.

## Key findings

- Optimal domains for magnetic eigenvalues tend to be symmetric in strong magnetic fields.
- A new estimate for the torsion function on rectangles is derived.
- Results are extended to magnetic Dirac operators with infinite mass boundary conditions.

## Abstract

This paper aims to show that, in the limit of strong magnetic fields, the optimal domains for eigenvalues of magnetic Laplacians tend to exhibit symmetry. We establish several asymptotic bounds on magnetic eigenvalues to support this conclusion. Our main result implies that if, for a bounded simply-connected planar domain, the n-th eigenvalue of the magnetic Dirichlet Laplacian with uniform magnetic field is smaller than the corresponding eigenvalue for a disk of the same area, then the Fraenkel asymmetry of that domain tends to zero in the strong magnetic field limit. Comparable results are also derived for the magnetic Dirichlet Laplacian on rectangles, as well as the magnetic Dirac operator with infinite mass boundary conditions on smooth domains. As part of our analysis, we additionally provide a new estimate for the torsion function on rectangles.