On critical parameters in homogenization of perforated domains by thin tubes with nonlinear flux and related spectral problems

Let u ϵ be the solution of the Poisson equation in a domain Ω ⊂ R 3perforated by thin tubes with a nonlinear Robin‐type boundary condition on the boundary of the tubes (the flux here being β ( ϵ ) σ ( x , u ϵ )), and with a Dirichlet condition on the rest of the boundary of Ω. ϵ is a small parameter that we shall make to go to zero; it denotes the period of a grid on a plane where the tubes/cylinders have their bases; the size of the transversal section of the tubes is O ( a ϵ ) with a ϵ ≪ ϵ . A certain nonperiodicity is allowed for the distribution of the thin tubes, although the perimeter is a fixed number a . Here, σ ∈ C 1 ( Ω ¯ × R ) is a strictly monotonic function of the second argument, and the adsorption parameter β ( ϵ ) > 0 can converge toward infinity. Depending on the relations between the three parameters ϵ , a ϵ , and β ( ϵ ), the effective equations in volume are obtained. Among the multiple possible relations, we provide critical relations , which imply different averages of the process ranging from linear to nonlinear. All this allows us to derive spectral convergence as ϵ →0 for the associated spectral problems in the case of σ a linear function of u ϵ . Copyright © 2014 John Wiley & Sons, Ltd.