On homogenization of nonlinear Robin type boundary conditions for cavities along manifolds and associated spectral problems

Abstract

Abstract. Let ue be the solution of the Poisson equation in a domain periodically perforated along a manifold γ = Ω ∩ {x1 = 0}, with a nonlinear Robin type boundary condition on the perforations (the flux here being O(e−κ)σ(x,ue)), and with a Dirichlet condition on ∂Ω. Ω is a domain of R with n 3, the small parameter e, that we shall make to go to zero, denotes the period, and the size of each cavity is O(e) with α 1. The function σ involving the nonlinear process is a C1(Ω × R) function and the parameter κ ∈ R. Depending on the values of α and κ, the effective equations on γ are obtained; we provide a critical relation between both parameters which implies a different average of the process on γ ranging from linear to nonlinear. For each fixed κ a critical size of the cavities which depends on n is found. As e → 0, we show the convergence of ue in the weak topology of H1 and construct correctors which provide estimates for convergence rates of solutions. All this allows us to derive convergence for the eigenelements of the associated spectral problems in the case of σ a linear function.