Year: 2026
Type: preprint
Abstract: In this paper, we study the Dirichlet eigenvalue problem for the logarithmic Hardy operator, defined as the logarithmic Laplacian with a critical logarithmic potential, in a bounded Lipschitz domain containing the origin. We first establish a logarithmic Hardy inequality and use it to identify the associated energy space, showing that the embedding into \(L^2\) is compact when the perturbation parameter exceeds the endpoint value \(-1\), but fails to be compact at that endpoint. For parameters above \(-1\), we develop a complete variational spectral theory, prove that the problem admits a discrete sequence of eigenvalues tending to infinity, characterized by successive minimizations over orthogonal complements, whose eigenfunctions form a complete orthonormal basis of \(L^2\). We further establish a uniform lower bound for the first eigenvalue, a scaling property under domain dilation that determines a critical radius for the sign of the first eigenvalue and leaves the eigenvalue gaps invariant.
Source: arXiv (Cornell University)
Authors: Liao, Tangrui, Wang, Ying, Yang, Minghua
Institution: Jiangxi University of Finance and Economics
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1193735095, G3048397482, G7754512266
OpenAlex ID: W7214047751