Faber-Krahn inequality for the first Dirichlet eigenvalues of the logarithmic Hardy operator

The purpose of this paper is to study Faber–Krahn inequality of Dirichlet eigenvalues $\{{\lambda }_k^{\ln ,\mu }(\mathrm{\Omega }){\}}_{k\in \mathbb{N}}$ for the logarithmic Hardy operator $${\mathcal{L}}_{\mu }=(-\mathrm{\Delta }{)}^{\ln }+2\mu \log \frac{1}{\mathrm{\mid }x\mathrm{\mid }}$$ on a bounded Lipschitz domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$ containing the origin, with $\mu >-1$. We establish a Faber–Krahn inequality for $\mu \in (-1,0)$, the ball centered at the origin minimizes the first eigenvalue at fixed volume, with equality only for that ball. Finally, we show this inequality fails for $\mu >0$ via an explicit two-ball-and-tube construction.