On the weighted logarithmic potential operator

Abstract

For a bounded open set $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$ with $N\ge 2$, and for positive continuous functions $w,g$ on $\overline{\hat{I}}\text{c◯}$, 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 $L^2(\hat{I}\text{c◯})$ 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 $\ddot{I}{„}_{w,g}^{+}(\hat{I}\text{c◯})$ with respect to $\hat{I}\text{c◯}$, $w$, and $g$. We also establish that $\ddot{I}{„}_{w,g}^{+}(\hat{I}\text{c◯})$ 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 $\hat{I}\text{c◯}$, under the assumption that $\log w$ is superharmonic. For $\hat{I}\text{c◯}\subset {\mathbb{R}}^2$, if $\hat{I}”\log w$ is a constant $C$, we show that 0 can be an eigenvalue of ${\mathcal{L}}_w$ only when $C=\frac{2\ddot{I}€}{|\hat{I}\text{c◯}|}$. For such domains, if $\log w$ is a harmonic function on $\hat{I}\text{c◯}$, 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.

28 pages