A scaling law for a model of epitaxially strained elastic films with dislocations

A static variational model for shape formation in heteroepitaxial crystal growth is considered. The energy functional takes into account surface energy, elastic misfit-energy and nucleation energy of dislocations. A scaling law for the infimal energy is proven. The results quantify the expectation that in certain parameter regimes, island formation or topological defects are favorable. This generalizes results in the purely elastic setting from [Goldman and Zwicknagl (2014)]. To handle dislocations in the lower bound, a new variant of a ball-construction combined with thorough local estimates is presented.