Dirichlet Eigenvalues for the Logarithmic-Hardy operator

Tangrui Liao; Ying Wang; Minghua Yang

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arXiv · 2609.23593

Dirichlet Eigenvalues for the Logarithmic-Hardy operator

Tangrui Liao·Ying Wang·Minghua Yang

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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.

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BibTeX RIS

Tangrui Liao, Ying Wang, Minghua Yang. 2026-09-20. Dirichlet Eigenvalues for the Logarithmic-Hardy operator. https://arxiv.org/abs/2609.23593

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