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PreprintCC BY-SA 4.0arXiv

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 L2 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 L2. 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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Cite

BibTeXRIS
BibTeX
@misc{liao2026dirichlet,
  title = {{Dirichlet Eigenvalues for the Logarithmic-Hardy operator}},
  author = {Tangrui Liao and Ying Wang and Minghua Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23593},
  eprint = {2609.23593},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23593},
  note = {CC BY-SA 4.0},
}
RIS
TY  - UNPB
TI  - Dirichlet Eigenvalues for the Logarithmic-Hardy operator
AU  - Tangrui Liao
AU  - Ying Wang
AU  - Minghua Yang
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.23593
UR  - https://arxiv.org/abs/2609.23593
AB  - 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.
ER  -