Γ-convergence, Sobolev norms, and BV functions

We prove that the family of functionals (Iδ) defined by Iδ(g)=∫∫RN×RN|g(x)-g(y)|>δδp|x-y|N+pdxdy, ∀g∈Lp(RN), for p≥1 and δ>0, Γ-converges in Lp(RN), as δ goes to zero, when p≥1. Hereafter | | denotes the Euclidean norm of RN. We also introduce a characterization for bounded variation (BV) functions which has some advantages in comparison with the classic one based on the notion of essential variation on almost every line.