Scaling law and reduced models for epitaxially strained crystalline films

In this talk I will present some results obtained with B. Zwicknagl on a variational model for the epitaxial deposition of a film on a rigid substrate in the presence of a crystallographic misfit. I will show the scaling behavior of the minimal energy in terms of the volume of the film and the size of the misfit, and derive the reduced models in the various regimes by $\Gamma$-convergence methods. Depending on the relation between the thickness of the film and the size of the misfit, the surface or the elastic energy contribution dominate, and in the critical case, the two contributions balance. In particular, the formation of islands is proven if the misfit is large compared to the volume of the film.