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 W Í mathbbRn{Omegasubseteqmathbb{R}^n}, and an open, connected, and (−1/2, 1/2) n -periodic set P Í mathbbRn{Psubseteqmathbb{R}^n}, consider for any ε > 0 the perforated domain Ω ε := Ω ∩ ε P. Let (ue) Ì SBVp(We){(u_varepsilon)subset SBV^p(Omega_{varepsilon})}, p > 1, be such that òWe|Ñue|pdx+Hn-1(Sue Ç We) +||ue||Lp(We){int_{Omega_{varepsilon}}left|{nabla{u}_varepsilon}right|^pdx+mathcal{H}^{n-1}(S_{u_varepsilon},cap,Omega_{varepsilon}) +leftVert{u_varepsilon}rightVert_{L^p(Omega_{varepsilon})}} is bounded. Then, we prove that, up to a subsequence, there exists u Î GSBVp Ç Lp(W){uin GSBV^p,cap, L^p(Omega)} satisfying lime||u-ue||L1(We)=0{lim_varepsilonleftVert{u-u_varepsilon}rightVert_{L^1(Omega_{varepsilon})}=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 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 ε . Finally, we slightly extend the well-known homogenization theorem for Sobolev energies on perforated domains.
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机译：本文的主要结果是在周期穿孔域上定义的空间SBV（有界变分的特殊函数）中的函数族的紧致性定理。给定一个开放有界集合WÍmathbbR n {Omegasubseteqmathbb {R} ^ n}，并得到一个开放的，连通的和（−1/2，1/2） n -周期集PÍmathbbR n {Psubseteqmathbb {R} ^ n}，对于任何ε> 0，考虑穿孔区域Ωε：=Ω∩εP。 （u e ）ÌSBV p （W e ）{（u_varepsilon）子集SBV ^ p（Omega_ {varepsilon}）}，p> 1，所以ò W e |Ñu e | p dx + H n-1 （S u e ÇW e ）+ || u e || L p （W e ） {int_ {Omega_ {varepsilon}}左| {nabla {u} _varepsilon}右| ^ pdx + mathcal {H} ^ {n-1}（S_ {u_varepsilon}，cap，Omega_ {varepsilon}）+ leftVert {u_varepsilon} rightVert_ {L ^ p（Omega_ {varepsilon}）}}是有界的。然后，我们证明，直到一个子序列，都存在uγGSBV p ÇL p （W）{uin GSBV ^ p，cap，L ^ p（Omega ）}满足lim e || uu e || L 1 （W e ） = 0 {lim_varepsilonleftVert {u-u_varepsilon} rightVert_ {L ^ 1（Omega_ {varepsilon}）} = 0}。我们的分析避免了在SBV中使用任何扩展程序，将P上的假设减弱为最小假设，并简化了Focardi等人最近获得的结果的证明。 （Math Models Methods Appl Sci 19：2065-2100，2009）和Cagnetti and Scardia（J Math Pures Appl（9）出现）。在我们介绍的论点中，我们提供了SBV中庞加莱-维特林不等式的本地化版本。作为一种可能的应用，我们研究了以多孔区域Ωε表示的脆性多孔材料的渐近行为。最后，我们略微扩展了穿孔区域上Sobolev能量的众所周知的均化定理。
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