Calculus of Variations and Geometric Measure Theory

M. Barchiesi - M. Focardi

Homogenization of the Neumann problem in perforated domains: an alternative approach

created by focardi on 15 Feb 2010
modified by barchiesi on 26 Jul 2011


Published Paper

Inserted: 15 feb 2010
Last Updated: 26 jul 2011

Journal: Calc. Var. Partial Differential Equations
Volume: 42
Pages: 257-288
Year: 2011


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.

Keywords: Homogenization, perforated domains, free-discontinuity, compactness in SBV