Gamma-liminf estimate for a class of non-local approximations of Sobolev and BV norms

TL;DR

This paper establishes a lower bound for a class of non-local, non-convex functionals, linking their Gamma-liminf to Sobolev and BV norms, and addresses open questions about their limiting behavior.

Contribution

It provides a Gamma-liminf estimate for non-local functionals approximating Sobolev and BV norms, advancing understanding of their asymptotic properties.

Findings

Gamma-liminf is bounded below by Sobolev or BV norms

Answers open questions on the limiting behavior of these functionals

Uses discretized analysis to establish bounds

Abstract

We consider a family of non-local and non-convex functionals, and we prove that their Gamma-liminf is bounded from below by a positive multiple of the Sobolev norm or the total variation. As a by-product, we answer some open questions concerning the limiting behavior of these functionals. The proof relies on the analysis of a discretized version of these functionals.