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

Abstract

We provide necessary and sufficient conditions on the density $W:{\mathbb{R}}^d\times {\mathbb{R}}^d\to \mathbb{R}$ 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 $I$ in general can be obtained by replacing $W$ by its separately level convex envelope.