A remake of Bourgain–Brezis–Mironescu characterization of Sobolev spaces

Guy Fabrice Foghem Gounoue ()
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Guy Fabrice Foghem Gounoue: Technische Universtät Dresden

Partial Differential Equations and Applications, 2023, vol. 4, issue 2, 1-36

Abstract: 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 the 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.

Keywords: Nonlocal energy forms; p-Lévy integrability; Sobolev spaces; Bounded variation spaces; Extension domains; 26B30; 46B45; 46E27; 46E30; 46E35 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s42985-023-00232-4

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