Accepted Paper

Inserted: 30 jan 2023
Last Updated: 6 aug 2024

Journal: Proceedings of the Royal Society of Edinburgh: Sect. A Mathematics
Year: 2023

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 $\{\rho_\varepsilon\}$, $\varepsilon\in(0,1)$, we devise sufficient and necessary conditions on $\{\rho_\varepsilon\}$ for the associated nonlocal functionals to converge as $\varepsilon \to 0$ to a variant of the Dirichlet integral. Finally, some comparison between our result and the existing literature is provided.

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