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PreprintCC BY 4.0arXiv

Analytical properties of the principal clamped transmission eigenpair for a planar region

  • Haoran Qi
  • Isaac Harris
  • Andreas Kleefeld
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Abstract

This paper investigates the analytical properties of the principal eigenpair for the clamped transmission eigenvalue problem, which arises in the context of biharmonic (i.e. flexural wave) scattering by a clamped planar obstacle. By deriving a suitable Rayleigh quotient, we provide rigorous proofs for several fundamental properties that were previously only observed through numerical experiments. One of the results we have shown here is that the first eigenvalue is monotone decreasing with respect to the domain. Furthermore, we prove that the corresponding eigenfunction is of a single sign, a property analogous to the principal eigenfunction of the Dirichlet eigenvalues of the Laplacian. Another key result is a Faber--Krahn type inequality, establishing that the disk minimizes the principal eigenvalue among all domains of a given measure. The theoretical results are supported by a comprehensive numerical investigation.

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Cite

BibTeXRIS
BibTeX
@misc{qi2026analytical,
  title = {{Analytical properties of the principal clamped transmission eigenpair for a planar region}},
  author = {Haoran Qi and Isaac Harris and Andreas Kleefeld},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.04741},
  eprint = {2611.04741},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.04741},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Analytical properties of the principal clamped transmission eigenpair for a planar region
AU  - Haoran Qi
AU  - Isaac Harris
AU  - Andreas Kleefeld
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.04741
UR  - https://arxiv.org/abs/2611.04741
AB  - This paper investigates the analytical properties of the principal eigenpair for the clamped transmission eigenvalue problem, which arises in the context of biharmonic (i.e. flexural wave) scattering by a clamped planar obstacle. By deriving a suitable Rayleigh quotient, we provide rigorous proofs for several fundamental properties that were previously only observed through numerical experiments. One of the results we have shown here is that the first eigenvalue is monotone decreasing with respect to the domain. Furthermore, we prove that the corresponding eigenfunction is of a single sign, a property analogous to the principal eigenfunction of the Dirichlet eigenvalues of the Laplacian. Another key result is a Faber--Krahn type inequality, establishing that the disk minimizes the principal eigenvalue among all domains of a given measure. The theoretical results are supported by a comprehensive numerical investigation.
ER  -