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A geometric lower bound for the clamped plate eigenvalue problem in low dimensions and extensions

  • Francesca Bianchi
  • Francesco Bozzola
  • Lorenzo Brasco
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Abstract

We show that the spectrum of the Dirichlet-biLaplacian can be quantitatively bounded from below in terms of the inradius only, for a general open set in dimensions $2$ and $3$. For the same dimensions, we also extend this result to fractional powers s of the Laplacian, with N/2<s<2. The resulting estimate comes with a constant having the correct asymptotic behavior as s approaches the two endpoints N/2 and $2$. The proofs exploit capacitary techniques in the vein of Maz'ya.

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Cite

BibTeXRIS
BibTeX
@misc{bianchi2026geometric,
  title = {{A geometric lower bound for the clamped plate eigenvalue problem in low dimensions and extensions}},
  author = {Francesca Bianchi and Francesco Bozzola and Lorenzo Brasco},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16485},
  eprint = {2610.16485},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16485},
}
RIS
TY  - UNPB
TI  - A geometric lower bound for the clamped plate eigenvalue problem in low dimensions and extensions
AU  - Francesca Bianchi
AU  - Francesco Bozzola
AU  - Lorenzo Brasco
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16485
UR  - https://arxiv.org/abs/2610.16485
AB  - We show that the spectrum of the Dirichlet-biLaplacian can be quantitatively bounded from below in terms of the inradius only, for a general open set in dimensions $2$ and $3$. For the same dimensions, we also extend this result to fractional powers s of the Laplacian, with N/2<s<2. The resulting estimate comes with a constant having the correct asymptotic behavior as s approaches the two endpoints N/2 and $2$. The proofs exploit capacitary techniques in the vein of Maz'ya.
ER  -