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

Abstract

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 + Hn-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.