Representation of the total variation as a $\Gamma$-limit of $BMO$-type seminorms

TL;DR

This paper proves that the total variation can be represented as a $$-scaled $$-limit of $BMO$-type seminorms, providing new insights into their relationship and compactness properties.

Contribution

It establishes that the $$-limit of $BMO$-type seminorms equals a scaled total variation, offering an alternative proof and compactness results.

Findings

The $$-limit of $BMO$-type seminorms equals one-fourth of the total variation.

Provides an alternative proof for known lower bounds of the pointwise limit.

Establishes a compactness criterion in $L^1_{ m loc}$ based on boundedness of $BMO$-type seminorms.

Abstract

We address a question raised by Ambrosio, Bourgain, Brezis, and Figalli, proving that the $\Gamma$-limit, with respect to the $L^1_{\rm loc}$ topology, of a family of $BMO$-type seminorms is given by $\tfrac14$ times the total variation seminorm. Our method also yields an alternative proof of previously known lower bounds for the pointwise limit and conveys a compactness result in $L^1_{\rm loc}$ in terms of the boundedness of the $BMO$-type seminorms.