Preprint

Inserted: 12 oct 2026
Last Updated: 12 oct 2026

Pages: 48
Year: 2026

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.

Keywords: capacity, fractional Sobolev spaces, Inradius, clamped plate, eigenvalue problems

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• biabozbra_clamped_final.pdf