Calculus of Variations and Geometric Measure Theory

A. Braides - L. Kreutz

An integral-representation result for continuum limits of discrete energies with multi-body interactions

created by kreutz on 06 Mar 2017
modified by braidesa on 16 Jun 2018


Published Paper

Inserted: 6 mar 2017
Last Updated: 16 jun 2018

Journal: SIAM J. Math. Anal.
Volume: 50
Pages: 1485-1520
Year: 2018
Doi: 10.1137/17M1121433


We prove a compactness and integral-representation theorem for sequences of families of lattice energies describing atomistic interactions defined on lattices with vanishing lattice spacing. The densities of these energies may depend on interactions between all points of the corresponding lattice contained in a reference set. We give conditions that ensure that the limit is an integral defined on a Sobolev space. A homogenization theorem is also proved. The result is applied to multibody interactions corresponding to discrete Jacobian determinants and to linearizations of Lennard- Jones energies with mixtures of convex and concave quadratic pair-potentials.