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We propose a homogenized supremal functional rigorously derived via power-lawapproximation by functionals of the type $\esssup f\left(\frac{x}{\varepsilon},Du\right)$, when $\Omega$ is a bounded open set of $\mathbb R^n$ and $u\inW^{1,\infty}(\Omega;\mathbb R^d)$. The homogenized functional is also deduceddirectly in the case where the sublevel sets of $f(x,\cdot)$ satisfy suitableconvexity properties, as a corollary of homogenization results dealing withpointwise gradient constrained integral functionals.

Homogenization of supremal functionals in vectorial setting (via power-law approximation)