Characterization of Sobolev and BV spaces

Abstract

The main results of this paper are new characterizations of W 1,p (Ω), 1 < p < ∞, and BV (Ω) for Ω ⊂ R N an arbitrary open set. Using these results, we answer some open questions of Brezis [11] and Ponce [32].

Characterization of Orlicz Sobolev Spaces

“…..,N form a linearly independet set, (see Remark 1.4 of (Leoni & Spector, 2011)), we have a contradiction.…”

“…In (Leoni & Spector, 2011), Leoni and Spector have given an alternative characteization of the Sobolev spaces using the not-local semi-norm …”

Estimates for the topological degree and related topics

“…Further studies of their characterization are given in [12,37,40,49,56,75,76] and the references therein. Theorem 3.1 (Bourgain, Brezis, and Mironescu [15]…”

Two subtle convex nonlocal approximations of the BV-norm

“…Let (ρ n ) be a sequence of radial mollifiers in the sense that ρ n ∈ L It has also been established by A. Ponce [23] that I n → I as n → +∞ in the sense of Γ -convergence in L 1 (Ω ). For works related to the BBM formula, see [5][6][7]15,16]. Other functionals converging to the BV-norm are considered in [3,8,9,[17][18][19][20][21][22].…”