Scale limit of a variational inequality modeling diffusive flux in a domain with small holes and strong adsorption in case of a critical scaling
Willi Jäger, Maria Neuss‐Radu, Т. А. Шапошникова · Doklady Mathematics · 2011
In this paper we study the asymptotic behavior of solutions u ɛ of the elliptic variational inequality for the Laplace operator in domains periodically perforated by balls with radius of size C 0ɛα, C 0 > 0, α = n/n−2, and distributed with period ɛ. On the boundary of balls, we have the following nonlinear restrictions u ɛ ≥ 0, ∂ν u ɛ ≥ −ɛ−ασ(x, u ɛ), u ɛ(∂ν u ɛ + ɛ−ασ(x, u ɛ)) = 0. The weak convergence of the solutions u ɛ 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.