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 Ω ~n, and an open, connected, and (-1/2, 1/2)~n-periodic set P ~n, consider for any {e open} > 0 the perforated domain Ω_({e open}):= Ω ∩ {e open} P. Let (u_({e open}) ? SBVp(Ω_({e open}), p > 1, be such that ∫_Ω {pipe}δu_({e open})p dx + H~(n-1)(Su_({e open}) ∩ Ω_({e open}) + {double pipe} u_({e open}){double pipe} Lp(Ω_({e open})) is bounded. Then, we prove that, up to a subsequence, there exists u {e open} GSBVp ∩ Lp(Ω)satisfying lim_({e open}) {double pipe}u - u_({e open}). 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 Focardi et al. (Math Models Methods Appl Sci 19:2065-2100, 2009) and Cagnetti and Scardia (J Math Pures Appl (9), to appear). Among the arguments we introduce, we provide a localized version of the Poincaré-Wirtinger inequality in SBV. As a possible application we study the asymptotic behavior of a brittle porous material represented by the perforated domain Ω_({e open}). Finally, we slightly extend the well-known homogenization theorem for Sobolev energies on perforated domains.
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