Homogenization of a class of Neumann problems in perforated domains

摘要

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 e 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 e (the last ones corresponding to the classical homogenization situation). Our study is performed by the periodic unfolding method from C. R. hcad. 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 Model. Math. Anal. Numer. 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.