Representation of the total variation as a $\Gamma$-limit 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 $L_{\mathrm{loc}}^1$ topology, of a family of $BMO$-type seminorms is given by $\frac{1}{4}$ 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 $L_{\mathrm{loc}}^1$ in terms of the boundedness of the $BMO$-type seminorms.

21 pages, 3 figures. Added some comments in the Introduction and a reference