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Isoperimetric inequality with zero magnetic field in doubly connected domains

Mrityunjoy Ghosh, Ayman Kachmar

TL;DR

The paper studies how the lowest eigenvalue of a magnetic Laplacian with a singular flux, in planar doubly connected domains with a disk-shaped hole and mixed Dirichlet/Neumann boundaries, depends on geometry under fixed area and flux. It proves a reverse Faber-Krahn-type isoperimetric inequality: among such domains, the annulus maximizes the first eigenvalue $\lambda^{\Omega_0}(\Phi)$; the proof first handles the annulus via separation of variables in polar coordinates and then extends to general domains using a radial trial function that leads to a strict comparison unless the domain is already annular. Corollaries for localized magnetic fields and a general optimality conjecture (verified for large flux) are derived, and the results are extended to the Neumann problem, confirming that the annulus remains optimal in that setting. The work clarifies the influence of topology (hole) and magnetic flux on spectral optimization and connects to classical isoperimetric results in the magnetic context.

Abstract

We investigate how the lowest eigenvalue of a magnetic Laplacian depends on the geometry of a planar domain with a disk shaped hole, where the magnetic field is generated by a singular flux. Under Dirichlet boundary conditions on the inner boundary and Neumann boundary conditions on the outer boundary, we show that this eigenvalue is maximized when the domain is an annulus, for a fixed area and magnetic flux. As consequences, we establish geometric inequalities for eigenvalues in settings with both singular and localized magnetic fields. We also propose a conjecture for a general optimality result and establish its validity for large magnetic fluxes.

Isoperimetric inequality with zero magnetic field in doubly connected domains

TL;DR

The paper studies how the lowest eigenvalue of a magnetic Laplacian with a singular flux, in planar doubly connected domains with a disk-shaped hole and mixed Dirichlet/Neumann boundaries, depends on geometry under fixed area and flux. It proves a reverse Faber-Krahn-type isoperimetric inequality: among such domains, the annulus maximizes the first eigenvalue ; the proof first handles the annulus via separation of variables in polar coordinates and then extends to general domains using a radial trial function that leads to a strict comparison unless the domain is already annular. Corollaries for localized magnetic fields and a general optimality conjecture (verified for large flux) are derived, and the results are extended to the Neumann problem, confirming that the annulus remains optimal in that setting. The work clarifies the influence of topology (hole) and magnetic flux on spectral optimization and connects to classical isoperimetric results in the magnetic context.

Abstract

We investigate how the lowest eigenvalue of a magnetic Laplacian depends on the geometry of a planar domain with a disk shaped hole, where the magnetic field is generated by a singular flux. Under Dirichlet boundary conditions on the inner boundary and Neumann boundary conditions on the outer boundary, we show that this eigenvalue is maximized when the domain is an annulus, for a fixed area and magnetic flux. As consequences, we establish geometric inequalities for eigenvalues in settings with both singular and localized magnetic fields. We also propose a conjecture for a general optimality result and establish its validity for large magnetic fluxes.
Paper Structure (4 sections, 12 theorems, 53 equations)

This paper contains 4 sections, 12 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. The case of the annulus
  3. Proof of Theorem \ref{['theo:FB']}
  4. The Neumann problem

Key Result

Theorem 1.1

Suppose that eq:constraint and eq:cond-geom hold. If $\Omega_0$ is not an annulus, then we have where $\Omega_0^*$ is introduced in eq:def-opt-dom.

Theorems & Definitions (23)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Conjecture 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • ...and 13 more