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This paper investigates the Faber-Krahn inequality for the first Dirichlet eigenvalues of the logarithmic Hardy operator $\mathcal{L}_{\mu}=(-\Delta)^{\ln}+2\mu\log\frac{1}{|x|}$ on bounded Lipschitz domains. For $\mu \in (-1, 0)$, the authors prove that the ball centered at the origin uniquely minimizes the first eigenvalue for a fixed volume. However, for $\mu > 0$, this inequality is shown to fail through a specific construction involving two balls and a tube.
The Faber-Krahn inequality for the logarithmic Hardy operator holds for $\mu \in (-1, 0)$ but breaks down for $\mu > 0$, revealing a critical phase transition in spectral geometric inequalities.
The purpose of this paper is to study Faber--Krahn inequality of Dirichlet eigenvalues $\{ \lambda^{\ln,\mu}_{k}(\Omega)\}_{k\in \N}$ for the logarithmic Hardy operator \[ \mathcal{L}_{\mu}=(-\Delta)^{\ln}+2\mu\log\frac{1}{|x|} \] on a bounded Lipschitz domain $\Omega\subset\mathbb{R}^{N}$ 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.
