Homogenization of a variational inequality for the Laplace operator with nonlinear restriction for the flux on the interior boundary of a perforated domain

In this paper, we study the asymptotic behavior of solutions uεuε of the elliptic variational inequality for the Laplace operator in domains periodically perforated by balls with radius of size C0εαC0εα, C0>0C0>0, α∈(1,nn−2], and distributed with period εε. On the boundary of the balls, we have the following nonlinear restrictions uε≥0uε≥0, ∂νuε≥−ε−γσ(x,uε)∂νuε≥−ε−γσ(x,uε), uε(∂νuε+ε−γσ(x,uε))=0uε(∂νuε+ε−γσ(x,uε))=0, γ=α(n−1)−nγ=α(n−1)−n. The weak convergence of the solutions uεuε to the solution of an effective problem is given. In the critical case α=nn−2, the effective equation contains a nonlinear term which has to be determined as a solution of a functional equation. Furthermore, a corrector result with respect to the energy norm is proved.