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

Abstract

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

to appear in European Journal of Mathematics