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

Abstract

The purpose of this paper is to study Faber--Krahn inequality of Dirichlet eigenvalues $\{{\lambda }_k^{\ln ,\mu }(\Omega ){\}}_{k\in \mathbb{N}}$ for the logarithmic Hardy operator \[ \mathcal{L}_μ=(-Δ)^{\ln}+2μ\log\frac{1}{|x|} \] 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.

arXiv comment14 pages