This work uses the energetic formulation of rate-independent systems that is based on the stored-energy functionals E and the dissipation distance D. For sequences (E k ) k∈N and (D k ) k∈N we address the question under which conditions the limits q ∞ of solutions q k : [0, T ] → Q satisfy a suitable limit problem with limit functionals E ∞ and D ∞ , which are the corresponding -limits. We derive a sufficient condition, called conditional upper semi-continuity of the stable sets, which is essential to guarantee that q ∞ solves the limit problem. In particular, this condition holds if certain joint recovery sequences exist. Moreover, we show that time-incremental minimization problems can be used to approximate 123 388 A. Mielke et al. the solutions. A first example involves the numerical approximation of functionals using finite-element spaces. A second example shows that the stop and the play operator converge if the yield sets converge in the sense of Mosco. The third example deals with a problem developing microstructure in the limit k → ∞, which in the limit can be described by an effective macroscopic model. Mathematics Subject Classification (2007)

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We investigate ground state configurations for atomic potentials including both two- and three-body nearest-neighbor interaction terms. The aim is to prove that such potentials may describe crystallization in carbon nanostructures such as graphene, nanotubes, and fullerenes. We give conditions in order to prove that planar energy minimizers are necessarily honeycomb, namely graphene patches. Moreover, we provide an explicit formula for the ground state energy which exactly quantifies the lower-order surface energy contribution. This allows us to give some description of the geometry of ground states. By recasting the minimization problem in three-space dimensions, we prove that ground states are necessarily nonplanar and, in particular, rolled-up structures like nanotubes are energetically favorable. Eventually, we check that the C20 and C60 fullerenes are strict local minimizers, hence stable

This paper addresses two-dimensional crystallization in the square lattice. A suitable configurational potential featuring both two-and three-body shortranged particle interactions is considered. We prove that every ground state is a connected subset of the square lattice. Moreover, we discuss the global geometry of ground states and their optimality in terms of discrete isoperimetric inequalities on the square graph. Eventually, we study the aspect ratio of ground states and quantitatively prove the emergence of a square macroscopic Wulff shape as the number of particles grows.