  We study the spectral stability of two fourth order Steklov problems upon
domain perturbation. One of the two problems is the classical DBS - Dirichlet
Biharmonic Steklov - problem, the other one is a variant. Under a comparatively
weak condition on the convergence of the domains, we prove the stability of the
resolvent operators for both problems, which implies the stability of
eigenvalues and eigenfunctions. The stability estimates for the eigenfunctions
are expressed in terms of the strong $H^2$-norms. The analysis is carried out
without assuming that the domains are star-shaped. Our condition turns out to
be sharp at least for the variant of the DBS problem. In the case of the DBS
problem, we prove stability of a suitable Dirichlet-to-Neumann type map under
very weak conditions on the convergence of the domains and we formulate an open
problem. As bypass product of our analysis, we provide some stability and
instability results for Navier and Navier-type boundary value problems for the
biharmonic operator.
