$\Gamma$-convergence of non-local, non-convex functionals in one dimension

TL;DR

This paper investigates the $3$-convergence of non-local, non-convex functionals in one dimension, establishing the limit as a multiple of the Sobolev or BV semi-norms, extending previous results without monotonicity constraints.

Contribution

It extends $33$-convergence results for non-local, non-convex functionals in one dimension by removing the monotonicity condition.

Findings

Limit functional is a multiple of the $W^{1,p}$ semi-norm for $p>1$.

Limit functional is the $BV$ semi-norm for $p=1$.

Extends earlier results to non-monotone cases.

Abstract

We study the $\Gamma$-convergence of a family of non-local, non-convex functionals in $L^p(I)$ for $p \ge 1$, where $I$ is an open interval. We show that the limit is a multiple of the $W^{1, p}(I)$ semi-norm to the power $p$ when $p>1$ (resp. the $BV(I)$ semi-norm when $p=1$). In dimension one, this extends earlier results which required a monotonicity condition.