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Reverse Faber-Krahn inequalities for the Logarithmic potential operator

T. V. Anoop, Jiya Rose Johnson

TL;DR

This work analyzes the largest eigenvalue $\tau_1(\Omega)$ of the Logarithmic potential operator on bounded planar domains, proving reverse Faber-Krahn-type inequalities under polarization and Schwarz symmetrization for $diam(\Omega)\leq 1$ and establishing monotonicity of $\tau_1(\Omega\setminus\mathcal{O})$ under obstacle translations/rotations. It extends these ideas to the Riesz potential operator and investigates the smallest eigenvalue $\tilde{\tau}_1(\Omega)$ via the transfinite diameter, providing explicit disc-based spectral descriptions for $B_R$ (including the negative eigenvalue when $R>1$) and asymptotics as $R\to1^+$ and $R\to\infty$. A central technical tool is a Riesz-type inequality under polarization, which, together with domain rearrangement techniques, yields reversed isoperimetric-type bounds and precise eigenfunction structure. The results illuminate how geometric transformations and domain perturbations control potential-type spectra and suggest several open problems and conjectures on transfinite diameter, tilde-$\tau_1$, and extensions to the Riesz setting with broad implications for spectral geometry of nonlocal operators.

Abstract

For a bounded open set $Ω\subset \mathbb{R}^2,$ we consider the largest eigenvalue $τ_1(Ω)$ of the Logarithmic potential operator $\mathcal{L}$. If $diam(Ω)\le 1$, we prove reverse Faber-Krahn type inequalities for $τ_1(Ω)$ under polarization and Schwarz symmetrization. Further, we establish the monotonicity of $τ_1(Ω\setminus\mathcal{O})$ with respect to certain translations and rotations of the obstacle $\mathcal{O}$ within $Ω$. The analogous results are also stated for the largest eigenvalue of the Riesz potential operator. Furthermore, we investigate properties of the smallest eigenvalue $\tildeτ_1(Ω)$ for a domain whose transfinite diameter is greater than 1. Finally, we characterize the eigenvalues of $\mathcal{L}$ on $B_R$, including the $\tildeτ_1(B_R)$ when $R>1$.

Reverse Faber-Krahn inequalities for the Logarithmic potential operator

TL;DR

This work analyzes the largest eigenvalue of the Logarithmic potential operator on bounded planar domains, proving reverse Faber-Krahn-type inequalities under polarization and Schwarz symmetrization for and establishing monotonicity of under obstacle translations/rotations. It extends these ideas to the Riesz potential operator and investigates the smallest eigenvalue via the transfinite diameter, providing explicit disc-based spectral descriptions for (including the negative eigenvalue when ) and asymptotics as and . A central technical tool is a Riesz-type inequality under polarization, which, together with domain rearrangement techniques, yields reversed isoperimetric-type bounds and precise eigenfunction structure. The results illuminate how geometric transformations and domain perturbations control potential-type spectra and suggest several open problems and conjectures on transfinite diameter, tilde-, and extensions to the Riesz setting with broad implications for spectral geometry of nonlocal operators.

Abstract

For a bounded open set we consider the largest eigenvalue of the Logarithmic potential operator . If , we prove reverse Faber-Krahn type inequalities for under polarization and Schwarz symmetrization. Further, we establish the monotonicity of with respect to certain translations and rotations of the obstacle within . The analogous results are also stated for the largest eigenvalue of the Riesz potential operator. Furthermore, we investigate properties of the smallest eigenvalue for a domain whose transfinite diameter is greater than 1. Finally, we characterize the eigenvalues of on , including the when .
Paper Structure (13 sections, 26 theorems, 203 equations, 14 figures)

This paper contains 13 sections, 26 theorems, 203 equations, 14 figures.

Table of Contents

  1. Introduction
  2. The smallest eigenvalue of $\mathcal{L}$
  3. Transfinite diameter
  4. Preliminaries
  5. Proof of the Main Theorems
  6. Further properties of eigenvalues
  7. Monotonicity of eigenvalues under domain perturbations
  8. The Domain monotonicity
  9. Other remarks and open problems
  10. Riesz potential operator
  11. Some open problems
  12. Eigenvalues of Logarithmic Potential on a disc
  13. First, Second, and Third eigenvalues on BR

Key Result

Theorem 1.5

Let $\Omega\subset {\mathbb R}^2$ be a bounded open set and let $H$ be a polarizer such that an eigenfunction corresponding to $\tau_1(\Omega)$ is positive on $\Omega\setminus \sigma_H(\Omega)$. Then, The equality holds in pollog only if $P_H(\Omega)\cong\Omega$ or $P_H(\Omega)\cong\sigma_H(\Omega)$.

Figures (14)

  • Figure 1: The grey region is $\Omega$
  • Figure 2: The dark region is $P_H(\Omega)$
  • Figure 3: Shaded region is $\Omega_{t_1}$
  • Figure 4: $P_H(\Omega_{t_1})=\Omega_{t_2}$
  • Figure 5: Translations of the obstacle $\mathcal{O}$ in $h$-direction
  • ...and 9 more figures

Theorems & Definitions (63)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 53 more