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

• Published in: Mathematical methods in the applied sciences Vol. 38; no. 12; pp. 2606 - 2629
• Main Authors: Gómez, D., Lobo, M., Pérez, M. E., Shaposhnikova, T. A., Zubova, M. N.
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 01.08.2015; Wiley Subscription Services, Inc
• Subjects: Boundaries; Dirichlet problem; Flux; Functions (mathematics); homogenization; Homogenizing; Mathematical analysis; nonlinear flux; Nonlinearity; perforated media; spectral analysis; Tubes
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.3246

Let uϵ be the solution of the Poisson equation in a domain Ω⊂R3 perforated by thin tubes with a nonlinear Robin‐type boundary condition on the boundary of the tubes (the flux here being β(ϵ)σ(x,uϵ)), 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(aϵ) with aϵ≪ϵ. A certain nonperiodicity is allowed for the distribution of the thin tubes, although the perimeter is a fixed number a. Here, σ∈C1(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, aϵ, 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 uϵ. Copyright © 2014 John Wiley & Sons, Ltd.