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

Abstract

Let u_ε 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(ε^−κ)σ(x,u_ε)), and with a Dirichlet condition on ∂Ω. Ω is a domain of Rⁿ with n≥3, the small parameter ε, that we shall make to go to zero, denotes the period, and the size of each cavity is O(ε^α) 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 ε→0, we show the convergence of u_ε 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.