Homogenization effects on non-local functionals

TL;DR

This paper investigates how non-local functionals with oscillating weights behave under homogenization, revealing complex microstructures and the failure of standard integral representations in the effective limit.

Contribution

It explicitly evaluates the mma-limit for non-local functionals with periodic weights, highlighting the impact of oscillations on homogenization and functional representation.

Findings

Explicit mma-limit for the class of non-local functionals.

Oscillating microstructures emerge in minimizing sequences.

Effective functional cannot be represented as a standard double integral.

Abstract

We study the homogenization of a class of non-local functionals featuring a rapidly oscillating periodic weight. By means of two-scale convergence, we explicitly evaluate the {\Gamma}-limit for constant target functions, revealing how the interplay between periodicity and non-locality forces the minimizing sequences to develop highly oscillating microstructures. As a natural consequence, we establish that the effective macroscopic functional fails to admit a standard double-integral representation.