Multiscale homogenization of convex functionals with discontinuous integrand

TL;DR

This paper derives the effective limit behavior of a family of multiscale convex functionals with discontinuous integrands using Young measures, advancing the understanding of homogenization in complex multiscale settings.

Contribution

It introduces a novel multiscale Young measure approach to homogenize convex functionals with discontinuous, multiscale integrands under weak regularity assumptions.

Findings

Derived the $ ext{Gamma}$-limit of multiscale convex functionals.

Extended homogenization techniques to discontinuous integrands.

Provided a framework for weak regularity assumptions in multiscale analysis.

Abstract

This article is devoted to obtain the $\Gamma$-limit, as $\epsilon$ tends to zero, of the family of functionals $$F_{\epsilon}(u)=\int_{\Omega}f\Bigl(x,\frac{x}{\epsilon},..., \frac{x}{\epsilon^n},\nabla u(x)\Bigr)dx$$, where $f=f(x,y^1,...,y^n,z)$ is periodic in $y^1,...,y^n$, convex in $z$ and satisfies a very weak regularity assumption with respect to $x,y^1,...,y^n$. We approach the problem using the multiscale Young measures.