  We introduce Robin boundary conditions for biharmonic operators, which are a
model for elastically supported plates and are closely related to the study of
spaces of traces of Sobolev functions. We study the dependence of the operator,
its eigenvalues, and eigenfunctions on the Robin parameters. We show in
particular that when the parameters go to plus infinity the Robin problem
converges to other biharmonic problems, and obtain estimates on the rate of
divergence when the parameters go to minus infinity. We also analyse the
dependence of the operator on smooth perturbations of the domain, computing the
shape derivatives of the eigenvalues and giving a characterisation for critical
domains under volume and perimeter constraints. We include a number of open
problems arising in the context of our results.
