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 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 er(e) with r(e) → 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. Acad. Sci. Paris Ser. I 335 (2002), 99-104, adapted to the case of holes of size er(e )( seeJ. 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(e) = exp(N/N − 1), an additional term appears in the right-hand side of the limit equation.