Homogenization of supremal functionals in the vectorial case (via $L^p$-approximation)

TL;DR

This paper develops a rigorous method to derive homogenized supremal functionals in the vectorial setting using $L^p$-approximation, extending previous scalar results and considering convexity properties of the sublevel sets.

Contribution

It introduces a new homogenization approach for supremal functionals in the vectorial case via $L^p$-approximation, including cases with convex sublevel sets.

Findings

Homogenized functional derived through $L^p$-approximation.

Extension to cases with convex sublevel sets.

Connection to gradient constrained integral functionals.

Abstract

We propose a homogenized supremal functional rigorously derived via $L^p$-approximation by functionals of the type $\underset{x\in\Omega}{\mbox{ess-sup}}\hspace{0.03cm} f\left(\frac{x}{\varepsilon}, Du\right)$, when $\Omega$ is a bounded open set of $\mathbb R^n$ and $u\in W^{1,\infty}(\Omega;\mathbb R^d)$. The homogenized functional is also deduced directly in the case where the sublevel sets of $f(x,\cdot)$ satisfy suitable convexity properties, as a corollary of homogenization results dealing with pointwise gradient constrained integral functionals.