Homogenization of functional with linear growth in the context of $\mathcal{A}$-quasiconvexity

TL;DR

This paper advances the understanding of homogenization for functionals with linear growth under -quasiconvexity by establishing a new representation theorem that incorporates concentration effects through a cell problem approach.

Contribution

It extends previous homogenization results to the linear growth case, integrating the -free condition and concentration effects into the representation theorem.

Findings

Established a new homogenization representation theorem for linear growth functionals.

Solved a cell problem that captures the coupling between homogenization and -quasiconvexity.

Extended prior results to include concentration effects in the homogenization process.

Abstract

This work deals with the homogenization of functionals with linear growth in the context of $\mathcal{A}$-quasiconvexity. A representation theorem is proved, where the new integrand function is obtained by solving a cell problem where the coupling between homogenization and the $\mathcal{A}$-free condition plays a crucial role. This result extends some previous work to the linear case, thus allowing for concentration effects.