Averaging of variational inequalities for the Laplacian with nonlinear restrictions along manifolds

Abstract

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 Ω_ϵ 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 ϵ^α 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).

Keywords:

• porous media
• boundary homogenization
• variational inequalities
• nonlinear flux

AMS Subject Classifications::

• 35B27
• 35B40
• 47J20
• 76R50

Acknowledgement

The work has been partially supported by the Spanish MICINN: MTM2009-12628.