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 BoundedVariation) defined on periodically perforated domains. Given an open and bounded set Omega subset of R-n, and an open, connected, and (-1/2, 1/2)(n)-periodic set P subset of R-n, consider for any epsilon > 0 the perforated domain Omega(epsilon) := Omega boolean AND epsilon P. Let (u(epsilon)) subset of SBVp(Omega(epsilon)), p > 1, be such that integral(Omega epsilon)vertical bar del u(epsilon)vertical bar(p) dx + H-n (1)(S-u epsilon boolean AND Omega(epsilon)) + parallel to u(epsilon)parallel to(Lp)(Omega(epsilon)) is bounded. Then, we prove that, up to a subsequence, there exists u is an element of GSBV(p) boolean AND L-p(Omega) satisfying lim(epsilon) parallel to u - u(epsilon)parallel to(L1(Omega epsilon)) = 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 Poincare-Wirtinger inequality in SBV. As a possible applicationwe study the asymptotic behavior of a brittle porous material represented by the perforated domain Omega(epsilon). Finally, we slightly extend the well-known homogenization theorem for Sobolev energies on perforated domains.