*preprint*

**Inserted:** 4 oct 2023

**Last Updated:** 4 oct 2023

**Year:** 2023

**Abstract:**

We propose a homogenized supremal functional rigorously derived via power-law approximation by functionals of the type ${\rm esssup} f\left(\frac{x}{\varepsilon}, Du\right)$, 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.