We study the Föppl-von Kármán theory for isotropically compressed thin plates in a geometrically linear setting, which is commonly used to model weak buckling of thin films. We consider generic smooth domains with clamped boundary conditions, and obtain rigorous upper and lower bounds on the minimum energy linear in the plate thickness σ. This energy is much lower than previous estimates based on certain dimensional reductions of the problem, which had lead to energies of order 1 + σ (scalar approximation) or σ 2/3 (two-component approximation).
