On the weighted logarithmic potential operator
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866915809444495360 |
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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 $Ω\subset \mathbb{R}^N$ with $N\geq 2$, and for positive continuous functions $w,g$ on $\overlineΩ$, we consider the weighted eigenvalue problem
\begin{equation*}
\mathcal{L}_{w} u =τgu,
\end{equation*}
where $\mathcal{L}_{w}$ is the weighted logarithmic potential operator on $L^2(Ω)$ as defined below:
\begin{equation*}
\mathcal{L}_{w} u(x)=\int_Ω\log\left(\frac{w(x)w(y)}{|x-y|}\right)u(y)dy.
\end{equation*}
We study the monotonicity and continuity of the largest positive eigenvalue $τ_{w,g}^+(Ω)$ with respect to $Ω$, $w$, and $g$. We also establish that $τ_{w,g}^+(Ω)$ satisfies a reverse Faber Krahn inequality under polarization. We provide a sufficient condition for the existence of a negative eigenvalue in terms of the weighted transfinite diameter of $Ω$, under the assumption that $\log w$ is superharmonic. For $Ω\subset \mathbb{R}^2$, if $Δ\log w $ is a constant $C$, we show that 0 can be an eigenvalue of $\mathcal{L}_{w}$ only when $C=\frac{2π}{|Ω|}$. For such domains, if $\log w$ is a harmonic function on $Ω$, we provide a representation formula for the eigenfunctions. Using this representation, we establish variants of the maximum principles that give some insight into the geometry of these eigenfunctions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_18138 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the weighted logarithmic potential operator Anoop, T. V. Johnson, Jiya Rose Analysis of PDEs For a bounded open set $Ω\subset \mathbb{R}^N$ with $N\geq 2$, and for positive continuous functions $w,g$ on $\overlineΩ$, we consider the weighted eigenvalue problem \begin{equation*} \mathcal{L}_{w} u =τgu, \end{equation*} where $\mathcal{L}_{w}$ is the weighted logarithmic potential operator on $L^2(Ω)$ as defined below: \begin{equation*} \mathcal{L}_{w} u(x)=\int_Ω\log\left(\frac{w(x)w(y)}{|x-y|}\right)u(y)dy. \end{equation*} We study the monotonicity and continuity of the largest positive eigenvalue $τ_{w,g}^+(Ω)$ with respect to $Ω$, $w$, and $g$. We also establish that $τ_{w,g}^+(Ω)$ satisfies a reverse Faber Krahn inequality under polarization. We provide a sufficient condition for the existence of a negative eigenvalue in terms of the weighted transfinite diameter of $Ω$, under the assumption that $\log w$ is superharmonic. For $Ω\subset \mathbb{R}^2$, if $Δ\log w $ is a constant $C$, we show that 0 can be an eigenvalue of $\mathcal{L}_{w}$ only when $C=\frac{2π}{|Ω|}$. For such domains, if $\log w$ is a harmonic function on $Ω$, we provide a representation formula for the eigenfunctions. Using this representation, we establish variants of the maximum principles that give some insight into the geometry of these eigenfunctions. |
| title | On the weighted logarithmic potential operator |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.18138 |