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 [E. Bonnetier and A. Chambolle, SIAM J. Appl. Math., 62 (2002), pp. 1093-1121]. 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 by Bonnetier and Chambolle that several phenomena 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 a higher dimension. We also provide an approximation, based on a phase-field representation of the domain.