Interaction of a Bulk and a Surface Energy with a Geometrical Constraint

This study is an attempt to generalize in dimension higher than two the mathematical results in [8] (Computing the equilibrium con guration of epitaxially strained crystalline lms, SIAM J. Appl. Math. 62 (2002), no. 4, 1093 1121) by E. Bonnetier and the rst author. It is the study of a physical system whose equilibrium is the result of a competition between an elastic energy inside a domain and a surface tension, proportional to the perimeter of the domain. The domain is constrained to remain a subgraph. It is shown in [8] that several phenomenon appear at various scales as a result of this competition. In this paper, we focus on establishing a sound mathematical framework for this problem in higher dimension. We also provide an approximation, based on a phaseeld representation of the domain.