In the context of a variational model for the epitaxial growth of strained elastic films,we study the effects of the presence of anisotropic surface energies in the determinationof equilibrium configurations. We show that the threshold effect that describes thestability of flat morphologies in the isotropic case remains valid for weak anisotropies,but is no longer present in the case of highly anisotropic surface energies, where we showthat the flat configuration is always a local minimizer of the total energy. Following theapproach of [N. Fusco and M. Morini, Equilibrium configurations of epitaxially strainedelastic films: second order minimality conditions and qualitative properties of solutions.Preprint], we obtain these results by means of a minimality criterion based on thepositivity of the second variation.
