A buckling eigenvalue 
of a smooth bounded domain 
is an eigenvalue
of the clamped boundary-value problem
 |
with 
and 
on the boundary, where 
is the Laplacian and 
is the outward unit normal (Ashbaugh
1999). The positive eigenvalues are ordered 
, counting multiplicity.
This differs from the vibrating clamped-plate equation 
.
Ashbaugh (1999) conjectured that a ball maximizes the ratio 
. Dai et al.
(2026) report a counterexample in every dimension 
. Their domains are bounded,
connected, have smooth boundaries, and contain finitely
many interior holes. The construction does not settle the comparison for convex
or simply connected domains. The Pólya-Szegő
buckling conjecture instead compares the first eigenvalue
at fixed volume, so it is a distinct problem.
