Isoperimetric inequalities for the lowest magnetic Steklov eigenvalue

TL;DR

This paper proves that among smooth simply-connected planar domains, the disk maximizes the lowest magnetic Steklov eigenvalue for moderate magnetic field strengths, extending to exterior domains under certain conditions.

Contribution

It establishes new isoperimetric inequalities for the magnetic Steklov eigenvalue in both bounded and exterior domains, using novel trial function methods.

Findings

Disk maximizes eigenvalue among smooth simply-connected domains

Isoperimetric inequality holds for exterior domains with symmetry

Results apply for magnetic fields of moderate strength

Abstract

This paper studies the optimization of the lowest eigenvalue of the magnetic Steklov problem on planar domains. In the bounded domain setting and for magnetic fields of moderate strengths, we prove that among all simply-connected smooth domains of given area, the disk maximises the lowest magnetic Steklov eigenvalue. For exterior domains, we establish a similar isoperimetric inequality for magnetic fields of moderate strength under fixed perimeter constraint and additional geometric and symmetry assumptions. The proofs rely on the method of torsion-type trial functions in the bounded domain case and on the method of trial functions dependent only on the distance to the boundary in the exterior domain case.