Averaging of variational inequalities for the Laplacian with nonlinear restrictions along manifolds
Delfina Gómez, Miguel Lobo, María‐Eugenia Pérez‐Martínez, Т. А. Шапошникова · Applicable Analysis · 2011
In this article, we consider variational inequalities arising, e.g., in modelling diffusion of substances in porous media. We assume that the media fills a domain Ωϵ of ℝ n with n ≥ 3. We study the case where the number of cavities is large and they are periodically distributed along a (n − 1)-dimensional manifold. ϵ is the period while ϵα is the size of each cavity with α ≥ 1; ϵ is a parameter that converges towards zero. Moreover, we also assume that the nonlinear process involves a large parameter ϵ−κ with κ = (α − 1)(n − 1). Passing to the scale limit, and depending on the value of α, the effective equation or variational inequality is obtained. In particular, we find a critical size of the cavities when α = κ = (n − 1)/(n − 2). We also construct correctors which improve convergence for α ≥ (n − 1)/(n − 2).