Periodic unfolding and nonhomogeneous Neumann problems in domains with small holes

Abstract

We consider elliptic problems in periodically perforated domains in R N , with nonhomogeneous Neumann conditions on the boundary of the holes. We are interested in the asymptotic behavior of the solutions as the period ε goes to zero. In a first case all the holes are “small”, i.e., are of size r ( ε ) with r ( ε ) / ε → 0 . In the second case, there are again small holes but also holes of size ε. We use the periodic unfolding method introduced in Cioranescu et al. (2002), which allows us to study second order operators with highly oscillating coefficients and so, to generalize here the results of Conca and Donato (1988). In both cases, if r ( ε ) = exp ( N / N − 1 ) , an additional term appears in the right-hand side of the limit equation. To cite this article: A. Ould Hammouda, C. R. Acad. Sci. Paris, Ser. I 346 (2008).