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 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.

A general framework for nonlocal Neumann problems

“…The proof is analogous to the one of Theorem 2.1. See also [46,Proposition 2.14] or [47,Proposition 3.46] for a general setting.…”

“…The latter embedding may fail when Ω is not an extension domain (see [46,Counterexample 3.8] or [68,Example 9.1]). Note that H 1 (Ω) can be viewed as limiting space of a sequence of nonlocal spaces of type H ν (Ω) and V ν (Ω|R d ) see [44][45][46] for additional results.…”

The divergence theorem and nonlocal counterparts

“…In order to prove the convergence of the nonlocal operators to their local counterparts, we generate whole families of nonlocal vector fields from the same vector field 𝐹 ∶ Ω → ℝ 𝑑 , using families of radial functions (𝛼 𝜀 ) 𝜀 and symmetric measures (𝜇 𝜀 (ℎ)dℎ) 𝜀 . These families are closely related to the Lévy measures introduced, for example, in [19,20]. Specifically, we consider radial functions (𝛼 𝜀 ∶ ℝ 𝑑 ⧵ {0} → ℝ) 𝜀∈(0,1) and symmetric measures 𝜇 𝜀 (ℎ)dℎ (where the functions 𝜇 𝜀 are also radial) that satisfy the following properties:…”