On the behavior of clamped plates under large compression

We determine the asymptotic behavior of eigenvalues of clamped plates under large compression by relating this problem to eigenvalues of the Laplacian with Robin boundary conditions. Using the method of fundamental solutions, we then carry out a numerical study of the extremal domains for the first eigenvalue, from which we see that these depend on the value of the compression, and start developing a boundary structure as this parameter is increased. The corresponding number of nodal domains of the first eigenfunction of the extremal domain also increases with the compression.