Lower semicontinuity of nonlocal $L^\infty$ energies on $SBV_0(I)$
Jose Matias, Pedro M. Santos, Elvira Zappale
TL;DR
The paper addresses the $L^1$-lower semicontinuity of a nonlocal supremal energy $H(u)=\mathrm{ess}\sup_{(s,t)\in S(u)\times S(u)} h([u](s),[u](t))$ defined on $SBV_0(I)$ for piecewise constant functions, focusing on the jump data $[u](t)$. It proves that $H$ is $L^1(I)$-lower semicontinuous if and only if the density $h:(\mathbb{R}\setminus\{0\})\times(\mathbb{R}\setminus\{0\})\to\mathbb{R}$ is lower semicontinuous and Cartesian submaximal after a diagonal symmetric extension $\hat{h}$. The key tool is the $\hat{h}$ representation, $\hat{h}(\xi,\eta)=\max\{ h(\xi,\xi), h(\eta,\eta), h(\eta,\xi), h(\xi,\eta)\}$, along with constructive $L^1$-convergent sequences of piecewise-constant approximants that concentrate jumps at finite points. These results characterize when supremal, nonlocal energies are stable under $L^1$-limits and link the theory to Cartesian submaximality and the geometry of jump sets in one dimension.
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 $r\in I$.