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We provide necessary and sufficient conditions on the density $W:\mathbbR^d\times\mathbb R ^d\to\mathbb R$ in order to ensure the sequential weak*lower semicontinuity of the functional $J: W^{1,\infty}(I;\mathbb R^d)\to\mathbb R$, defined as \begin{align*} J(u):=ess\,sup_{I\times I}W(u'(x),u'(y)), \end{align*} when $I$ is an open and bounded interval of $\mathbb R$.We also show that, when $d=1$, the lower semicontinuous envelope of $I$ ingeneral can be obtained by replacing $W$ by its separately level convexenvelope.

Relaxation of one-dimensional nonlocal supremal functionals in the Sobolev setting