Reverse Faber-Krahn inequalities for the logarithmic potential operator

T. V. Anoop; Jiya Rose Johnson

doi:10.3934/dcds.2026012

Article Contents

2026, Volume 50: 321-357. Doi: 10.3934/dcds.2026012

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Reverse Faber-Krahn inequalities for the logarithmic potential operator

T. V. Anoop^, and
Jiya Rose Johnson

Department of Mathematics, Indian Institute of Technology Madras, India

^*Corresponding author: T. V. Anoop
^*Corresponding author: T. V. Anoop

Received: May 2025

Revised: November 2025

Early access: December 8, 2025

Published online: December 8, 2025

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Abstract

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

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Mathematics Subject Classification: Primary: 35P05, 47G40, 47A75; Secondary: 35P15.

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Figure 1. The grey region is $ \Omega$

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Figure 2. The dark region is $ P_H(\Omega)$

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Figure 3. Shaded region is $ \Omega_{t_1}$

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Figure 4. $ P_H(\Omega_{t_1}) = \Omega_{t_2}$

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Figure 5. Graph of $ g$

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Figure 6. Graph of $ h$

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Figure 7. Graph of $ 1h$ and $ -\frac{t^2}{2}$

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Figure 8. Graph of $ J_0$ and $ h$

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Figure 9. Graph of $ J_0, \, J_1$ and $ f$

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Figure 10. Translations of the obstacle $ \mathcal{O} $ in $ h $-direction

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Figure 11. Rotations with respect to $ a$

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Figure 12. $ \Omega_{d} = B_1\cup B_1(d\mathbf{e_1})$

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Figure 13. $ \Omega_{d} = S_1\cup P_{d}\cup S_d$

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Figure 14. $ B_{R_1}(x)\subseteq \Omega \subseteq B_{R_2}(y)$

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