This study is an attempt to generalize in dimension higher than two the mathematical results in [8] (Computing the equilibrium conguration of epitaxially strained crystalline lms, SIAM J. Appl. Math. 62 (2002), no. 4, 10931121) 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 phase-eld representation of the domain.

Pair-interaction atomistic energies may give rise, in the framework of the passage from discrete systems to continuous variational problems, to nonlinear energies with genuinely quasiconvex integrands. This phenomenon takes place even for simple harmonic interactions as shown by an example by Friesecke and Theil [19]. On the other hand, a rigorous derivation of linearly elastic energies from energies with quasiconvex integrands can be obtained by Gamma-convergence following the method by Dal Maso, Negri and Percivale [14]. We show that the derivation of linear theories by Gamma-convergence can be obtained directly from lattice interactions in the regime of small deformations. Our proof relies on a lower bound by comparison with the continuous result, and on a direct Taylor expansion for the upper bound. The computation is carried over for a family of lattice energies comprising interactions on the triangular lattice in dimension two

Abstract. We consider, in an open subset Ω of R N , energies depending on the perimeter of a subset E ⊂ Ω (or some equivalent surface integral) and on a function u which is defined only on Ω \ E. We compute the lower semicontinuous envelope of such energies. This relaxation has to take into account the fact that in the limit, the "holes" E may collapse into a discontinuity of u, whose surface will be counted twice in the relaxed energy. We discuss some situations where such energies appear, and give, as an application, a new proof of convergence for an extension of Ambrosio-Tortorelli's approximation to the Mumford-Shah functional.Mathematics Subject Classification. 49J45, 49Q20, 49Q10.

We consider a one-dimensional system of Lennard-Jones nearest-and next-to-nearest-neighbour interactions. It is known that if a monotone parameterization is assumed then the limit of such a system can be interpreted as a Griffith fracture energy with an increasing condition on the jumps. In view of possible applications to a higher-dimensional setting, where an analogous parameterization does not always seem reasonable, we remove the monotonicity assumption and describe the limit as a Griffith fracture energy where the increasing condition on the jumps is removed and is substituted by an energy that accounts for changes in orientation ('creases'). In addition, fracture may be generated by 'macroscopic' or 'microscopic' cracks.

The Sardinia Radio Telescope (SRT), located near Cagliari (Italy), is the world’s second largest fully steerable radio telescope endowed with an active-surface system. Its primary mirror has a quasi-parabolic shape with a diameter of 64 m. The configuration of the primary mirror surface can be modified by means of electro-mechanical actuators. This capability ensures, within a fixed range, the balancing of the deformation caused, for example, by loads such as self-weight, thermal effects and wind pressure. In this way, the difference between the ideal shape of the mirror (which maximizes its performances) and the actual surface can be reduced. In this paper the authors describe the characteristics of the SRT, the close-range photogrammetry (CRP) survey developed in order to set up the actuator displacements, and a finite element model capable of accurately estimating the structural deformations. Numerical results are compared with CRP measurements in order to test the accuracy of the model.