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

TL;DR

This paper investigates the $ ext{BMO}$-type functional's $ ext{Gamma}$-limits under different convergence modes, revealing a nuanced limit that distinguishes between absolutely continuous, Cantor, and jump parts of BV functions.

Contribution

It establishes the $ ext{Gamma}$-limit of the $ ext{BMO}$-type functional with respect to $L^{ ext{infty}}_{ ext{loc}}$-convergence, extending previous results and clarifying the limit's structure.

Findings

The $ ext{Gamma}$-limit with $L^{ ext{infty}}_{ ext{loc}}$-convergence is a sum of measures of the different BV parts.

The limit coincides with the pointwise limit for SBV functions.

The result differentiates the behavior of the functional under various modes of convergence.

Abstract

We study the BMO-type functional $\kappa_{\varepsilon}(f,\mathbb R^n)$, which can be used to characterize BV functions $f\in BV(\mathbb R^n)$. The $\Gamma$-limit of this functional, taken with respect to $L^1_{\mathrm{loc}}$-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 SBV functions.