
Journal of Convex Analysis 30 (2023), No. 1, 217247 Copyright Heldermann Verlag 2023 External Forces in the Continuum Limit of Discrete Systems with NonConvex Interaction Potentials: Compactness for a ΓDevelopment Marcello Carioni Institute of Mathematics, University of Wuerzburg, Germany and: Dept. of Applied Mathematics, University of Twente, Enschede, The Netherlands m.c.carioni@utwente.nl Julian Fischer Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany and: Institute of Science and Technology, Klosterneuburg, Austria julian.fischer@ist.ac.at Anja Schloemerkemper Institute of Mathematics, University of Wuerzburg, Germany anja.schloemerkemper@mathematik.uniwuerzburg.de This paper is concerned with equilibrium configurations of onedimensional particle systems with nonconvex nearestneighbour and nexttonearestneighbour interactions and its passage to the continuum. The goal is to derive compactness results for a Γdevelopment of the energy with the novelty that external forces are allowed. In particular, the forces may depend on Lagrangian or Eulerian coordinates and thus may model dead as well as live loads. Our result is based on a new technique for deriving compactness results which are required for calculating the firstorder Γlimit in the presence of external forces: instead of comparing a configuration of n atoms to a global minimizer of the Γlimit, we compare the configuration to a minimizer in some subclass of functions which in some sense are "close to" the configuration. The paper is complemented with the study of the minimizers of the Γlimit. Keywords: Discretetocontinuum limit, fracture, external force, LennardJones potential. MSC: 74R10, 49J45, 74Q05. [ Fulltextpdf (222 KB)] for subscribers only. 