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

The main result of this paper is a compactness theorem for families of functions in the space SBV (Special functions of Bounded Variation) defined on periodically perforated domains. Given an open and bounded set $\Om\subseteq\R^n$, and an open, connected, and $(-1/2,1/2)^n$-periodic set $P\subseteq\R^n$, consider for any $\e>0$ the perforated domain $\Om_\e:=\Om\cap\e P$. Let $(u_\e)\subset SBV^p(\Om_{\e})$, $p>1$, be such that $\int_{\Om_{\e}}\snorm{\g u_\e}^pdx+\HH^{n-1}(S_{u_\e}\cap\Om_{\e}) +\norm{u_\e}_{L^p(\Om_{\e})}$ is bounded. Then, we prove that, up to a subsequence, there exists $u\in GSBV^p\cap L^p(\Om)$ satisfying $\lim_\e\norm{u-u_\e}_{L^1(\Om_{\e})}=0$. Our analysis avoids the use of any extension procedure in $SBV$, weakens the hypotheses on $P$ to the minimal ones and simplifies the proof of the results recently obtained in \cite{FGP07,CS07}. Among the arguments we introduce, we provide a localized version of the Poincar\'{e}-Wirtinger inequality in $SBV$. As a possible application we study the asymptotic behavior of a brittle porous material represented by the perforated domain $\Om_{\e}$. Finally, we slightly extend the well-known homogenization theorem for Sobolev energies on perforated domains.