Calculus of Variations and Geometric Measure Theory

A. Chambolle - L. Kreutz

Crystallinity of the homogenized energy density of periodic lattice systems

created by kreutz on 15 Jun 2021
modified on 25 Feb 2023


Published Paper

Inserted: 15 jun 2021
Last Updated: 25 feb 2023

Journal: Multiscale Modeling & Simulation
Pages: 37
Year: 2021

ArXiv: 2106.08111 PDF


We study the homogenized energy densities of periodic ferromagnetic Ising systems. We prove that, for finite range interactions, the homogenized energy density, identifying the effective limit, is crystalline, i.e. its Wulff crystal is a polytope, for which we can (exponentially) bound the number of vertices. This is achieved by deriving a dual representation of the energy density through a finite cell formula. This formula also allows easy numerical computations: we show a few experiments where we compute periodic patterns which minimize the anisotropy of the surface tension.