Lower semicontinuity of nonlocal $L^{\infty }$ energies on $SB{V}_0(I)$

Abstract

We characterize the lower-semicontinuity of nonlocal one-dimensional energies of the type \[{\rm ess}\!\!\!\!\!\!\!\!\sup_{(s,t) \in I\times I} h([u](s), [u](t)),\] where $I$ is an open and bounded interval in the real line, $u\in SB{V}_0(I)$ and $[u](r):=u(r^{+})-u(r^{-})$, with $r\in I$.