On the asymptotic behaviour of the discrete spectrum in buckling problems for thin plates

We consider the buckling problem for a family of thin plates with thickness parameter (epsilon). This involves finding the least positive multiple (lambda)[stack min(epsilon) ] of the load that makes the plate buckle, a value that can be expressed in terms of an eigenvalue problem involving a non-compact operator. We show that under certain assumptions on the load, we have (lambda)[[stack min(epsilon) ] = (oscr)((epsilon)2). This guarantees that provided the plate is thin enough, this minimum value can be numerically approximated without the spectral pollution that is possible due to the presence of the non-compact operator. We provide numerical computations illustrating some of our theoretical results.