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

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