A sufficient condition for the lower semicontinuity of nonlocal supremal functionals in the vectorial case

In this note we present a sufficient condition ensuring lower semicontinuity for nonlocal supremal functionals of the type $$W^{1,\mathrm{\infty }}(\mathrm{\Omega };{\mathbb{R}}^d)\ni u\mapsto \sup {\mathrm{e}\mathrm{s}\mathrm{s}}_{(x,y)\in \mathrm{\Omega }}W(x,y,\mathrm{\nabla }u(x),\mathrm{\nabla }u(y)),$$ where $\mathrm{\Omega }$ is a bounded open subset of ${\mathbb{R}}^N$ and $W:\mathrm{\Omega }\times \mathrm{\Omega }\times {\mathbb{R}}^{d\times N}\times {\mathbb{R}}^{d\times N}\to \mathbb{R}$.