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

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 $N\mathrm{/}2<s<2$. The resulting estimate comes with a constant having the correct asymptotic behavior as $s$ approaches the two endpoints $N\mathrm{/}2$ and $2$. The proofs exploit capacitary techniques in the vein of Maz’ya.