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

Abstract

We introduce a class of concentrated p-Lévy integrable functions approximating the unity, which serves as the core tool to characterize the Sobolev spaces and the space of functions of bounded variation in the spirit of Bourgain-Brezis-Mironescu. We provide this characterization for a class of unbounded domains satisfying the extension property. We also examine the situation where the extension property fails.

Nonlocal complement value problem for a global in time parabolic equation

“…We have shown that v = V(u 0 − u T ) = ∫ T 0 udt. Therefore, we obtain which, according to the relation (25), implies that u is a weak solution to the problem (1). ◻…”

“…The aforementioned spaces are Hilbert spaces. Additional, recent finds about these function spaces and their relations with classical Sobolev spaces can be found in [23][24][25]. Let (V (Ω|ℝ N )) * and (𝕏 (Ω|ℝ N )) * be the dual spaces of V (Ω|ℝ N ) and 𝕏 (Ω|ℝ N ) respectively.…”

“…as s → 1 − . More generally, we have L u(x) → −Δu(x) as → 0 (see [21,25]), where L u is defined as in (2) with replaced by satisfying If we assume that ≥ 0 is radial and satisfies ∫ ℝ N (1 ∧ |h| 2 ) (h)dh = 1 N then a remarkable example of family ( ) satisfying (3), is obtained from the rescaled version as follows In [49], the weak solvability of the problem is proven for the case where u is a positive bounded function and is the so called Flory-Huggins potential, i.e., is a convex increasing function that tends to infinity as its argument approaches a certain positive value. The positiveness is a natural requirement since u is a density probability.…”