BMO-type functionals, total variation, and Γ-convergence

We study the BMO-type functional κ ε ( f , R n ) \kappa _{\varepsilon }(f,\mathbb {R}^n) , which can be used to characterize bounded variation functions f ∈ B V ( R n ) f\in \mathrm {BV}(\mathbb {R}^n) . The Γ \Gamma -limit of this functional, taken with respect to L l o c 1 L^1_{\mathrm {loc}} -convergence, is known to be 1 4 | D f | ( R n ) \tfrac 14 |Df|(\mathbb {R}^n) . We show that the Γ \Gamma -limit with respect to L l o c ∞ L^{\infty }_{\mathrm {loc}} -convergence is \[ 1 4 | D a f | ( R n ) + 1 4 | D c f | ( R n ) + 1 2 | D j f | ( R n ) , \tfrac 14 |D^a f|(\mathbb {R}^n)+\tfrac 14 |D^c f|(\mathbb {R}^n)+\tfrac 12 |D^j f|(\mathbb {R}^n), \] which agrees with the “pointwise” limit in the case of special functions of bounded varation.