BMO-type seminorms from Escher-type tessellations

The paper is about a representation formula introduced by Fusco, Moscariello, and Sbordone in [14]. The formula permits to characterize the gradient norm of a Sobolev function, defined on the whole space R-n, as the limit of non-local energies (BMO-type seminorms) defined on tessellations of R-n generated by cubic cells with arbitrary orientation. We improve the main result in [14] in three different regards: we give a new concise proof of the representation formula, we analyze the case of a generic open subset Omega subset of R-n and consider general tessellations of Omega by means of cells more general than cubes, again arbitrarily-oriented, inspired by the creative mind of the graphic artist M.C. Escher. (C) 2020 Elsevier Inc. All rights reserved.