Sharp conditions for the validity of the Bourgain-Brezis-Mironescu formula

TL;DR

This paper establishes precise conditions on weight functions ensuring the convergence of nonlocal functionals to the Dirichlet integral, extending the understanding of the Bourgain-Brezis-Mironescu formula.

Contribution

It provides necessary and sufficient conditions on weights for the convergence of nonlocal functionals to local Dirichlet integrals, refining previous results.

Findings

Derived conditions on weights for convergence.

Compared new results with existing literature.

Extended the applicability of the Bourgain-Brezis-Mironescu formula.

Abstract

Following the seminal paper by Bourgain, Brezis and Mironescu, we focus on the asymptotic behavior of some nonlocal functionals that, for each $u\in L^2(\mathbb{R}^N)$, are defined as the double integrals of weighted, squared difference quotients of $u$. Given a family of weights $\{\rho_\epsilon\}$, $\epsilon\in (0,1)$, we devise sufficient and necessary conditions on $\{\rho_\epsilon\}$ for the associated nonlocal functionals to converge as $\epsilon\to 0$ to a variant of the Dirichlet integral. Finally, some comparison between our result and the existing literature is provided.