BMO-type seminorms and local Poincaré constants for BV functions

Abstract

In 2015 Bourgain, Brezis, and Mironescu introduced a class of BMO-type functionals that measure the oscillation of a function on a family of disjoint $\epsilon$-cubes. These functionals turned out to be related to the total variation of the function, and over the years several authors have addressed the problem of finding an expression for their limit as $\epsilon$ goes to zero. Thanks to the work of many, we now know that for SBV functions the limit exists and coincides with 1/2 times the jump variation plus 1/4 times the absolutely continuous variation. However, for BV functions with a non-trivial Cantor part, the limit might not exist. In this talk I will present a natural relaxation of these functionals that enforces the existence of the limit for any BV function. I will show that this limit is related to a quantity that we introduce, the local Poincare' constant of the function, and I will discuss some challenging open questions. This result is based on a project with Adolfo Arroyo-Rabasa (Bonn) and Paolo Bonicatto (Trento).

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