On some anisotropic singular perturbation problems

We investigate the asymptotic behavior of some anisotropic diffusion problems and give some estimates on the rate of convergence of the solution toward its limit. We also relate this type of elliptic problems to problems set in cylinder becoming unbounded in some directions and show how some information on one type leads to information for the other type and conversely. 1. A model problem The goal of this note is to study diffusion problems for which the diffusion in some directions is very small. More precisely we would like to find the limit behavior of the solution of such a problem when the small diffusion parameters approach zero. First we would like to explain the issues on a very simple example in two dimensions. Let us denote by ω the interval (−1, 1) and for a > 0 by Ωa the rectangle (−a, a)×ω. The points in Ω1 will be labeled by x = (x1, x2). We denote by H1 0 (ω) the usual Sobolev space of functions vanishing on ∂ω and by H −1(ω) its dual (see [6]). The duality bracket between H−1(ω) and H1 0 (ω) will be denoted by 〈 , 〉ω or simply 〈 , 〉. Let us consider f such that f ∈W 1,2(ω;H−1(ω)) (1.1) with an obvious notation for this space (see [1], [6]). Note that clearly W 1,2(ω;H−1(ω)) ⊂ C0(ω;H−1(ω)), (1.2) where C0(ω;H−1(ω)) denotes the space of continuous functions from ω into H−1(ω). We define a continuous linear form onH1 0 (Ω1) – i.e. an element ofH (Ω1) by setting 〈f, v〉Ω1 = ∫ ω 〈f, v(x1, ·)〉ω dx1, ∀ v ∈ H 0 (Ω1). (1.3) Since there is no ambiguity this linear form will be also denoted by f . Then we would like to consider for ε > 0, uε the weak solution to { −ε2∂2 x1uε − ∂ 2 x2uε = f in Ω1, uε = 0 on ∂Ω1, (1.4)