Local Poincaré constants and mean oscillation functionals for BV functions

Abstract

We introduce the concept of local Poincaré constant of a BV function as a tool to understand the relation between its mean oscillation and its total variation at small scales. This enables us to study a variant of the BMO-type seminorms on $\epsilon$-size cubes introduced by Ambrosio, Bourgain, Brezis, and Figalli. More precisely,we relax the size constraint by considering a family of functionals that allow cubes of sidelength smaller than or equal to $\epsilon$. These new functionals converge, as $\epsilon$ tends to zero, to a local functional defined on BV, which can be represented by integration in terms of the local Poincaré constant and the total variation. This contrasts with the original functionals, whose limit is defined on SBV and may not exist for functions with a non-trivial Cantor part. Moreover, we characterize the local Poincaré constant of a function with a cell-formula given by the maximum mean oscillation of its BV blow-ups. As a corollary of this characterization, we show that the new limit functional extends the original one to all BV functions. Finally, we discuss rigidity properties and other challenging questions relating the local Poincaré constant of a function to its fine properties.