Equilibrium configurations of epitaxially strained elastic films: existence, regularity, and qualitative properties of solutions.

We consider a variational model introduced in the physical literature to describe the epitaxial growth of an elastic lm over a thick at substrate when a lat- tice mismatch between the two materials is present. We prove existence of minimizing congurations, study their regularity properties, and establish several quantitative and qualitative properties of local and global minimizers of the free-energy functional. Among the other results, we determine analytically the critical threshold for the local minimality of the at conguration, we investigate also its global minimality, and we provide some conditions under which the non occurrence of singularities in non at global minimizers is guaranteed. One of the main tools is a new second order sucient condition for lo- cal minimality, which provides the rst extension of the classical criteria based on the positivity of second variation to the context of functionals with bulk and surface energies.