On the weighted logarithmic potential operator

TL;DR

This paper investigates the spectral properties of a weighted logarithmic potential operator on bounded domains, analyzing eigenvalues, their monotonicity, and inequalities, with applications to domain geometry and potential theory.

Contribution

It introduces new results on the eigenvalues' behavior, including a reverse Faber Krahn inequality and conditions for negative eigenvalues, extending potential theory in weighted settings.

Findings

Largest eigenvalue is monotonic with respect to domain and weights.

Established a reverse Faber Krahn inequality under polarization.

Provided conditions for the existence of negative eigenvalues based on weighted transfinite diameter.

Abstract

For a bounded open set $\Omega \subset \mathbb{R}^N$ with $N\geq 2$, and for positive continuous functions $w,g$ on $\overline{\Omega}$, we consider the weighted eigenvalue problem \begin{equation*} \mathcal{L}_{w} u =\tau gu, \end{equation*} where $\mathcal{L}_{w}$ is the weighted logarithmic potential operator on $L^2(\Omega)$ as defined below: \begin{equation*} \mathcal{L}_{w} u(x)=\int_\Omega \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 $\tau_{w,g}^+(\Omega)$ with respect to $\Omega$, $w$, and $g$. We also establish that $\tau_{w,g}^+(\Omega)$ 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 $\Omega$, under the assumption that $\log w$ is superharmonic. For $\Omega\subset \mathbb{R}^2$, if $\Delta\log w $ is a constant $C$, we show that 0 can be an eigenvalue of $\mathcal{L}_{w}$ only when $C=\frac{2\pi}{|\Omega|}$. For such domains, if $\log w$ is a harmonic function on $\Omega$, 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.