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{\bf Homogenization of the Neumann Problem in Preforated Domains:  Al
  Alternative Approach}

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Marco Barchiesi\\
BCAM\\
Derio, Spain\\
barchiesi@bcamath.org 

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M. Focardi\\
Universit\`a di Firenze\\
Firenze, Italy\\
focardi@math.unifi.it
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\noindent {\bf 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 Poincar\'{e}-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 Poincar\'{e}-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.

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