PreprintCC BY-SA 4.0arXiv
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 { λln,μk(Ω)}k∈ N for the logarithmic Hardy operator ℒμ=(−Δ)ln+2μlog1/|x| on a bounded Lipschitz domain Ω⊂ℝN containing the origin, with μ>−1. We establish a Faber--Krahn inequality for μ∈(−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.
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BibTeX
@misc{wang2026faberkrahn,
title = {{Faber-Krahn inequality for the first Dirichlet eigenvalues of the logarithmic Hardy operator}},
author = {Ying Wang and Tao Zhang and Wanwan Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09578},
eprint = {2610.09578},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09578},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Faber-Krahn inequality for the first Dirichlet eigenvalues of the logarithmic Hardy operator
AU - Ying Wang
AU - Tao Zhang
AU - Wanwan Wang
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09578
UR - https://arxiv.org/abs/2610.09578
AB - The purpose of this paper is to study Faber--Krahn inequality of Dirichlet eigenvalues { λ^ln,μ_k(Ω)}_k∈ N for the logarithmic Hardy operator ℒ_μ=(−Δ)^ln+2μlog1/|x| on a bounded Lipschitz domain Ω⊂ℝ^N containing the origin, with μ>−1. We establish a Faber--Krahn inequality for μ∈(−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.
ER -