Two subtle convex nonlocal approximations of the BVBV-norm

Inspired by the BBM formula and by work of G. Leoni and D. Spector, we analyze the asymptotic behavior of two sequences of convex nonlocal functionals (Ψn(u))(Ψn(u)) and (Φn(u))(Φn(u)) which converge formally to the BV-norm of uu. We show that pointwise convergence when uu is not smooth can be delicate; by contrast, ΓΓ-convergence to the BV-norm is a robust and very useful mode of convergence.