\(\Gamma\)-expansion for a 1D confined Lennard-Jones model with point defect (Q1953044)

The very important and useful paper studies the relationship between atomistic and continuum models relying upon high level of symmetry which is maintained after deforming a crystal. The author applies \(\Gamma\)-convergence to a sequence of atomistic energy functionals that depend on the parameter \(\varepsilon\), which is the inverse of the number of atoms per unit volume. The confined Lennard-Jones model for interatomic interactions is proposed. Here it is assumed that nearest neighbour potentials are globally convex and second neighbour potentials are globally concave. A model for interatomic interactions in a 1D chain including a point defect is presented. Next the author makes a formal analysis of the model to motivate this study. The \(\Gamma\)-limit for the energy functional as the number of atoms per period tends to infinity and an explicit form for the first order term in a \(\Gamma\)-expansion in terms of an infinite problem is derived. Some results about the minimum problem for the 0-th order \(\Gamma\)-limit of the atomistic energies, including existence, uniqueness and regularity for minimizers, are proved. Finally a first order \(\Gamma\)-limit, expressing it in terms of a minimization problem in an infinite cell, is derived and some properties of this problem are proved along the way.