Calculus of Variations and Geometric Measure Theory
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M. Carioni - J. Fischer - A. Schlömerkemper

External forces in the continuum limit of discrete systems with non-convex interaction potentials: Compactness for a Γ-development

created by carioni on 13 Apr 2019



Inserted: 13 apr 2019
Last Updated: 13 apr 2019

Year: 2018


This paper is concerned with equilibrium configurations of one-dimensional particle system with non-convex nearest-neighbour and next-to-nearest-neighbour interactions and its passage to the continuum. The goal is to derive compactness results for a Γ-development of the energy with the novelty that external forces are allowed. In particular, the forces may depend on Lagrangian or Eulerian coordinates. Our result is based on a new technique for deriving compactness results which are required for calculating the first-order Γ-limit: instead of comparing a configuration of n atoms to a global minimizer of the Γ-limit, we compare the configuration to a minimizer in some subclass of functions which in some sense are “close to” the configuration. This new technique is required due to the additional presence of forces with non-convex potentials. The paper is complemented with the study of the minimizers of the Γ–limit.


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