Analytical Validation of a Continuum Model for Epitaxial Growth with Elasticity on Vicinal Surfaces

Within the context of heteroepitaxial growth of a film onto a substrate, terraces and steps self-organize according to misfit elasticity forces. Discrete models of this behavior were developed by Duport et al. (J Phys I 5:1317-1350, 1995) and Tersoff et al. (Phys Rev Lett 75:2730-2733, 1995). A continuum limit of these was in turn derived by Xiang (SIAM J Appl Math 63:241-258, 2002) (see also the work of Xiang and Weinan Phys Rev B 69:035409-1-035409-16, 2004; Xu and Xiang SIAM J Appl Math 69:1393-1414, 2009). In this paper we formulate a notion of weak solution to Xiang's continuum model in terms of a variational inequality that is satisfied by strong solutions. Then we prove the existence of a weak solution.