Lower semicontinuity of nonlocal $L^\infty$ energies on $SBV_0(I)$

TL;DR

This paper characterizes the lower semicontinuity of certain nonlocal supremal energies defined on SBV functions in one dimension, focusing on the behavior of jump discontinuities and their influence on energy limits.

Contribution

It provides a characterization of lower semicontinuity for nonlocal supremal energies involving jumps in SBV functions, a novel analysis in this context.

Findings

Identifies conditions for lower semicontinuity of nonlocal supremal energies.

Establishes a link between jump discontinuities and energy lower bounds.

Provides a framework for analyzing nonlocal energies in one-dimensional SBV spaces.

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, $u \in SBV_0(I)$ and $[u](r):= u(r^+)- u(r^-)$, with $r\in I$.