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

Abstract

We study the BMO-type functional ${\kappa }_{\epsilon }(f,{\mathbb{R}}^n)$, which can be used to characterize BV functions $f\in BV({\mathbb{R}}^n)$. The $\Gamma$-limit of this functional, taken with respect to $L_{\mathrm{loc}}^1$-convergence, is known to be $\frac{1}{4}|Df|({\mathbb{R}}^n)$. We show that the $\Gamma$-limit with respect to $L_{\mathrm{loc}}^{\infty }$-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 SBV functions.