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

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.