Representation of the total variation as a Gamma-limit of BMO-type seminorms
In recent years there has been a significant interest in the relations between the gradient seminorm $|Df|$ of a function of bounded variation $f: \mathbb R^n \to \mathbb R$ and certain BMO-type seminorms defined in terms of the oscillation of the function $f$ over a collection of disjoint cubes in $\mathbb R^n$. We address a question raised by Ambrosio, Bourgain, Brezis, and Figalli, proving that the Gamma-limit, with respect to the L^1_loc topology, of such BMO-type seminorms is given by 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^1_loc in terms of the boundedness of the BMO-type seminorms. This is joint work with A. Arroyo-Rabasa (UCLouvain) and G. Del Nin (Warwick).