Asymptotic analysis of a class of minimization problems in a thin multidomain

Abstract

We consider a quasilinear Neumann problem with exponent p ∈]1, +∞[, in a multidomain of R N , N ≥ 2, consisting of two vertical cylinders, one placed upon the other: the first one with given height and small cross section, the other one with small height and given cross section. Assuming that the volumes of the two cylinders tend to zero with same rate, we prove that the limit problem is well posed in the union of the limit domains, with respective dimension 1 and N − 1. Moreover, this limit problem is coupled if p > N − 1 and uncoupled if 1 < p ≤ N − 1. IntroductionLet N ≥ 2, let ω ⊂ R N −1 be a bounded open connected set with a smooth boundary such that the origin in R N −1 , denoted by 0 , belongs to ω, and let {r n } n∈N , {h n } n∈N be two sequences of positive numbers converging to 0. For every n ∈ N, consider the thin multidomain Ω n = Ω 1 n ∪ Ω 2 n , the union of two vertical cylinders with small volumes: Ω 1 n = r n ω × [0, 1[ with small cross section r n ω and constant height, Ω 2 n = ω×] − h n , 0[ with small height h n and constant cross section (see figure next page). This paper arises from the desire of studying the asymptotic behaviour, as n → +∞, of the following model problem:

Asymptotic analysis of the eigenvalues of a Laplacian problem in a thin multidomain

“…Ciarlet and P. Destuynder in [3], and having introduced suitable weighted inner products, by using the min-max Principle we obtain a priori estimates (with respect to n) for the sequences {λ n,k } k∈N (see Proposition 2.1). Then, by making use of the method of oscillating test functions, introduced by L. Tartar in [17], by applying some results obtained by A. Gaudiello, B. Gustafsson, C. Lefter and J. Mossino in [6] and [7] and by adapting the techniques used by M. Vanninathan in [16], we derive the limit eigenvalue problem and the limit of the rescaled basis, as n → +∞, in the case h n ≃ r For the study of thin multi-structures we refer to [2], [3], [4], [11], [12], [13], [14] and the references quoted therein. For a thin multi-structure as considered in this paper, we refer to [5], [6], [7], [8], [9] and [10].…”

“…Then, the Dirichlet version of Theorem 1.1 in [7] (see also Remark 1.4 in [7]), with f = λu, entails the existence of ϕ ∈ V such that…”

Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem

“…To study the asymptotic behaviour of {U ε } ε , as ε → 0, we introduce the classical transformation mapping Ω − ε onto the fixed domain [9,14,16] and [17]) and we set, for every ε,…”

Dimension reduction and homogenization of random degenerate operators. Part I

“…As usual in the study of structures with thin thickness, it is better to work in a domain with fixed thickness by rescaling the original problem (1.3) (see for example [7,17]), which permits us to simplify the derivation of a priori estimates. The resulting problem takes the form…”

“…Moreover, we obtain an auxiliary problem in which the probability space plays the role of the cell (see equation (3.10)). For various dimension reduction problems, see for example [6,7,[16][17][18], and for the homogenization of partial differential equations in the random context, see for example [4,5,11,12,27,28] and the references therein.…”