Scale limit of a variational inequality modeling diffusive flux in a domain with small holes and strong adsorption in case of a critical scaling

摘要

In this paper we study the asymptotic behavior of solutions u ε of the elliptic variational inequality for the Laplace operator in domains periodically perforated by balls with radius of size C 0εα, C 0 > 0, α = n-2, and distributed with period ε. On the boundary of balls, we have the following nonlinear restrictions u ε > 0, αν u ε > -ε -ασ(x, u ε), u ε(α ν u ε + ε-ασ(x, u ε)) = 0. The weak convergence of the solutions u ε to the solution of an effective variational equality is proved. In this case, the effective equation contains a nonlinear term which has to be determined as solution of a functional equation. Furthermore, a corrector result with respect to the energy norm is given.