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 u(epsilon) of the elliptic variational inequality for the Laplace operator in domains periodically perforated by balls with radius of size C-0 epsilon(alpha), C-0 > 0, alpha is an element of (1, n/n-2] and distributed with period epsilon. On the boundary of the balls, we have the following nonlinear restrictions u(epsilon) >= 0, partial derivative(nu)u(epsilon) >= -epsilon(-gamma) sigma (x, u(epsilon)), u(epsilon)(partial derivative(nu)u(epsilon) + epsilon(-gamma) sigma(x, u(epsilon)) = 0, gamma = alpha (n - 1) - n. The weak convergence of the solutions s to the solution of an effective problem is given. In the critical case alpha = n/n-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. (C) 2012 Elsevier Ltd. All rights reserved.