BMO-type functionals, total variation, and $\Gamma$-convergence

Abstract:

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