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

TL;DR

This paper characterizes conditions on the density function W to ensure the lower semicontinuity of a nonlocal supremal functional in Sobolev spaces, and describes its relaxation via level convex envelopes in one dimension.

Contribution

It provides necessary and sufficient conditions for lower semicontinuity of nonlocal supremal functionals and describes their relaxation in the one-dimensional case.

Findings

Characterizes conditions for weak* lower semicontinuity of the functional.

Shows the relaxation can be obtained by replacing W with its level convex envelope in 1D.

Provides a complete description of the relaxation process for these functionals.

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.