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 $$W^{1,\infty}(Ω;\mathbb R^d)\ni u \mapsto \sup{\rm ess}_{(x,y)\in Ω} W(x,y, \nabla u(x),\nabla u(y)),$$ where $Ω$ is a bounded open subset of $\mathbb R^N$ and $W:Ω\times Ω\times \mathbb R^{d \times N}\times \mathbb R^{d \times N} \to \mathbb R$.