Homogenization of the Neumann Problem in Preforated Domains: Al Alternative Approach

Marco Barchiesi BCAM Derio, Spain barchiesi@bcamath.org

M. Focardi Università di Firenze Firenze, Italy focardi@math.unifi.it

Abstract: 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 18, 14. 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.

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 18, 14. 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.