On the clamped buckling eigenvalues
Let νj be the eigenvalues of the problem ∆2 u + ν∆u = 0 in a Eucledean domain, with the condition
u(x) = ∇u(x) = 0 on the boundary, and let λj be the eigenvalues of the Dirichlet Laplacian in the same
domain. Payne conjectured that, for planar domains, ν1 ≥ λ3 . I will show that this is not the case.
Moreover, if a domain is close to a disk then most likely ν1 < λ3 . On the other hand, for domains that
are invariant under rotation by the angle π/2, the Payne conjecture holds. I will also review the status of
another conjecture of Payne: νk ≥ λk+1 . This conjecture remains open.
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