Non-local, non-convex functionals converging to Sobolev norms

TL;DR

This paper investigates the convergence of non-local, non-convex functionals to Sobolev norms, extending previous work from the case p=1 to p>1, and establishes their pointwise and Gamma-convergence.

Contribution

It demonstrates the convergence of a family of non-local, non-convex functionals to Sobolev norms for p>1, generalizing earlier results for p=1.

Findings

Functionals converge to Sobolev norms in L^p

Established Gamma-convergence of the functionals

Extended previous results from p=1 to p>1

Abstract

We study the pointwise convergence and the $\Gamma$-convergence of a family of non-local, non-convex functionals $\Lambda_\delta$ in $L^p(\Omega)$ for $p>1$. We show that the limits are multiples of $\int_{\Omega} |\nabla u|^p$. This is a continuation of our previous work where the case $p=1$ was considered.