Calculus of Variations and Geometric Measure Theory

A. Braides - L. D'Elia

Homogenization of discrete thin structures

created by braidesa on 17 Jul 2021
modified on 14 Apr 2023


Published Paper

Inserted: 17 jul 2021
Last Updated: 14 apr 2023

Journal: Nonlinear Analysis
Volume: 231
Pages: 112951
Year: 2023
Doi: 10.1016/
Links: journal link


We consider graphs parameterized on a portion $X\subset\mathbb Z^d\times \{1,\ldots, M\}^k$ of a cylindrical subset of the lattice $\mathbb Z^d\times \mathbb Z^k$, and perform a discrete-to-continuum dimension-reduction process for energies defined on $X$ of quadratic type. Our only assumptions are that $X$ be connected as a graph and periodic in the first $d$-directions. We show that, upon scaling of the domain and of the energies by a small parameter $\varepsilon$, the scaled energies converge to a $d$-dimensional limit energy. The main technical points are a dimension-lowering coarse-graining process and a discrete version of the $p$-connectedness approach by Zhikov.