Homogenization of a class of Neumann problems in perforated domains

Abstract

We consider elliptic problems in periodically perforated domains in R^N, N≥3, with nonhomogeneous Neumann conditions on the boundary of the holes. The aim is to give the asymptotic behavior of the solutions as the period ε goes to zero. Two geometries are considered. In the first one, all the holes are “small”, i.e., their size is of order of εr(ε) with r(ε)→0. The second geometry is more general, there are small holes as before but also holes of size of the order of ε (the last ones corresponding to the classical homogenization situation). Our study is performed by the periodic unfolding method from C. R. Acad. Sci. Paris Ser. I 335 (2002), 99–104, adapted to the case of holes of size εr(ε) (see J. Math. Pures Appl. 89 (2008), 248–277). The use of this method allows us to study second-order operators with highly oscillating coefficients and so, to generalize here the results of RAIRO Modél. Math. Anal. Numér. 4(22) (1988), 561–608. In both cases, if r(ε)=exp (N/N−1), an additional term appears in the right-hand side of the limit equation.