Faber-Krahn inequality for the first Dirichlet eigenvalues of the logarithmic Hardy operator
Year: 2026
Type: preprint
Abstract: The purpose of this paper is to study Faber--Krahn inequality of Dirichlet eigenvalues $\{ λ^{\ln,μ}_{k}(Ω)\}_{k\in \N}$ for the logarithmic Hardy operator \[ \mathcal{L}_μ=(-Δ)^{\ln}+2μ\log\frac{1}{|x|} \] on a bounded Lipschitz domain $Ω\subset\mathbb{R}^{N}$ containing the origin, with $μ>-1$. We establish a Faber--Krahn inequality for $μ\in(-1,0)$, 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 $μ>0$ via an explicit two-ball-and-tube construction.
Source: arXiv (Cornell University)
Authors: Wang, Ying, Zhang, Tao, Wang, Wanwan
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1193735095, G1200305164, G8846185683
OpenAlex ID: W7221017814