A Remake of Bourgain-Brezis-Mironescu Characterization of Sobolev Spaces

Abstract

We introduce a large class of concentrated $p$-Lévy integrable functions approximating the unity, which serves as the core tool from which we provide a nonlocal characterization of Sobolev spaces and the space of functions of bounded variation via nonlocal energies forms. It turns out that this nonlocal characterization is a necessary and sufficient criterion to define Sobolev spaces on domains satisfying the extension property. We also examine the general case where the extension property does not necessarily hold. In the latter case, we establish weak convergence of the nonlocal Radon measures involved to the local Radon measures induced by the distributional gradient.

Published in: Partial Differential Equations and Applications. https://doi.org/10.1007/s42985-023-00232-4

References in corpus (2)

• Bourgain-Brezis-Mironescu Domains
• Nonlocal complement value problem for a global in time parabolic equation

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• Convergence rates of the fractional to the local Dirichlet problem
• Robust nonlocal trace and extension theorems