Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Isoperimetric inequalities for the lowest magnetic Steklov eigenvalue

Ayman Kachmar, Vladimir Lotoreichik

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.

Isoperimetric inequalities for the lowest magnetic Steklov eigenvalue

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.
Paper Structure (14 sections, 10 theorems, 97 equations)

This paper contains 14 sections, 10 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Bounded domains
  3. The lowest magnetic Steklov eigenvalue
  4. The case of the disk
  5. Method of torsion-type trial functions
  6. Auxiliary problem
  7. Isoperimetric inequality
  8. Exterior domains
  9. The lowest magnetic Steklov eigenvalue
  10. Isoperimetric inequality
  11. Existence of ground states
  12. Existence of a ground state for the magnetic Steklov problem on a bounded domain
  13. Existence of a ground state for the lowest fiber problem
  14. Variational characterisation of the lowest eigenvalue of the magnetic Dirichlet-to-Neumann map

Key Result

Proposition 2.2

Let ${\mathcal{B}}\subset{\mathbb R}^2$ be the disk of radius $R > 0$. If $b|{\mathcal{B}}| < \pi$, then the infimum in eq:def-ev for $\Omega ={\mathcal{B}}$ is attained on the radial function where $f_\star\in C^\infty([0,R])$ satisfies the system In particular, it holds (for some $A\in{\mathbb C}\setminus\{0\}$)

Theorems & Definitions (26)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5: CLPS
  • Proposition 2.6
  • proof
  • Remark 2.7
  • Proposition 2.8
  • ...and 16 more