Inserted: 15 feb 2010
Last Updated: 26 jul 2011
Journal: Calc. Var. Partial Differential Equations
The main goal of this paper is a compactness result for families of functions in the space $SBV$ (Special functions of Bounded Variation) defined on periodically perforated domains.
Our analysis avoids the use of any extension procedure in $SBV$, weakens the hypothesis on the reference perforation to minimal ones and simplifies the proof of the results recently obtained in Refs 19, 15. Among the arguments we introduce, we provide a localized version of the Poincaré-Wirtinger inequality in $SBV$.
As an application we study the asymptotic behavior of a variational model for brittle porous materials.
Finally, we slightly extend the well known homogenization theorem for Sobolev energies on perforated domains.