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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2501.13569 |
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| _version_ | 1866929684829175808 |
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| author | Anoop, T. V. Johnson, Jiya Rose |
| author_facet | Anoop, T. V. Johnson, Jiya Rose |
| contents | 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 $R>1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13569 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Reverse Faber-Krahn inequalities for the Logarithmic potential operator Anoop, T. V. Johnson, Jiya Rose Analysis of PDEs 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 $R>1$. |
| title | Reverse Faber-Krahn inequalities for the Logarithmic potential operator |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.13569 |