BMO-type seminorms and local Poincaré constants for BV functions

Seminario periodico del Dipartimento di Matematica
7 maggio 2024
Orario di inizio 
PovoZero - Via Sommarive 14, Povo (Trento)
Aula Seminari Dipartimento di Fisica
Comunità universitaria
Comunità studentesca UniTrento
Ingresso libero
Andrea Pinamonti
Università degli Studi Trento 38123 Povo (TN) - Staff Dipartimento di Matematica
+39 04 61/281508-1625-1701-3786-1980
Giacomo Del Nin (Max Planck Institute)


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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