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

Abstract

Let uϵ be the solution of the Poisson equation in a domain normalΩ⊂double-struckR3 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(falsenormalΩ¯×double-struckR) 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.

Unilateral problems for thep-Laplace operator in perforated media involving large parameters

“…However, it should be noted that for p = n, due to the fact that logarithmic scale appears (cf. (7.2)), a graphic of the type of Figure 2 summarizing all the possible homogenized problem becomes more complicated (even unthinkable), and, as occurs in [18] for the Laplacian and perforation by tubes, the graphics should be performed for well defined dependence of a ε and β ε in terms of ε. In this respect, as a sample, we outline that (1.9)-(1.11) provide the homogenized problem of (1.1) when we have the following relations: 12) with k = j ≥ 0, C 2 0 > 0, α 2 > 0, and ι = n/(n − 1).…”

“…However [21,44] consider only spherical cavities while the cavities can be of different shapes in [24,25] but for relations between parameters outside the big point. See [18,20] for a long list of references on related problems.…”

“…In the most critical situation, for nonlinear Robin boundary conditions, the Laplacian and n = 2, namely, p = n = 2, we refer to [37] for general geometries of the cavities and to [18] when p = 2, n = 3 and the domain is perforated by tubes. An extension to p = n ≥ 3 can be found in [39].…”

“…(1.1)), with unilateral constraints. Also, a more restrictive σ is considered in [18,37,39]. However, it should be emphasized that, in any case, the perimeter of the perforations arises in the strange term instead of the shape of the perforations as one might expect for n ≥ 3.…”

Homogenization limit for the Diffusion Equation in a domain Perforated along (n - 1)-Dimensional Manifold with Dynamic Conditions on the Boundary of the Perforations: Critical Case

“…Усреднение краевых и начально-краевых задач с классическими краевыми условиями типа Робина изучены во многих работах, см. [3][4][5][6][7][8][9][10] и список литературы там.…”