Asymptotic development of anisotropic singular perturbation problems

Abstract

In this work we construct an asymptotic expansion to the weak solution of anisotropic singular perturbation problems of elliptic type. The paper aims to go deep in the study of the rate of convergence. In fact the analysis of the asymptotic expansion strongly helps in understanding what is, and what is not, needed to improve the rate and the type of convergence. In order to define the coefficients of the development some smoothness basic properties (with respect to the parameters) of elliptic problems are established.

Keywords

Singular perturbations asymptotic development anisotropic elliptic problems regularity of solutions

1. Introduction

In order to describe the class of problems that we would like to address, we first introduce some basic notation and hypotheses. Let Ω be a bounded open cylinder in $R^{n}$ , i.e. $\begin{matrix} Ω = ω_{1} \times ω_{2}, \end{matrix}$ where $ω_{1}$ , $ω_{2}$ are bounded Lipschitz domains of $R^{p}$ and $R^{n - p}$ respectively ( $n > p ⩾ 1$ ). We denote by $x = (x_{1}, \dots, x_{n}) = (X_{1}, X_{2})$ the points of $R^{n}$ where $\begin{matrix} X_{1} = (x_{1}, \dots, x_{p}), X_{2} = (x_{p + 1}, \dots, x_{n}), \end{matrix}$ i.e. we split the coordinates of a point $x \in Ω$ into two parts $X_{1} \in ω_{1}$ and $X_{2} \in ω_{2}$ . Throughout this manuscript $\partial_{x_{i}}$ denotes the partial derivative in the $x_{i}$ -direction and $D^{α} u = \partial_{x_{1}}^{α_{1}} \dots \partial_{x_{n}}^{α_{n}} u$ is the αth-weak derivative for a multi-index $α = (α_{1}, \dots, α_{n}) \in N^{n}$ . With these notation we set $\begin{matrix} \nabla u = {(\partial_{x_{1}} u, \dots, \partial_{x_{n}} u)}^{T} = (\binom{\nabla_{X_{1}} u}{\nabla_{X_{2}} u}), \end{matrix}$ where $\begin{matrix} \nabla_{X_{1}} u = {(\partial_{x_{1}} u, \dots, \partial_{x_{p}} u)}^{T}, \nabla_{X_{2}} u = {(\partial_{x_{p + 1}} u, \dots, \partial_{x_{n}} u)}^{T} . \end{matrix}$ Let $A = (a_{i j} (x))$ be a $n \times n$ matrix such that $\begin{matrix} (1.1) & a_{i j} \in L^{\infty} (Ω), \forall i, j = 1, \dots, n \end{matrix}$ and for some constant $λ > 0$ , it satisfies the ellipticity condition $\begin{matrix} (1.2) & A ζ \cdot ζ ⩾ λ | ζ |^{2} \forall ζ \in R^{n}, a.e. x \in Ω . \end{matrix}$ (“·” denotes the canonical scalar product in $R^{n}$ .) We decompose A into four blocks by writing $\begin{matrix} A = (\begin{matrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix}), \end{matrix}$ where $A_{11}$ , $A_{22}$ are respectively $p \times p$ and $(n - p) \times (n - p)$ matrices. We then define for $ε > 0$ , the perturbed matrix, $\begin{matrix} A_{ε} = (\begin{matrix} ε^{2} A_{11} & ε A_{12} \\ ε A_{21} & A_{22} \end{matrix}) . \end{matrix}$ Therefore we have, for a.e. $x \in Ω$ and every $ζ \in R^{n}$ $\begin{matrix} A_{ε} ζ \cdot ζ = A ζ_{ε} \cdot ζ_{ε} ⩾ λ | ζ_{ε} |^{2} = λ {ε^{2} | {\bar{ζ}}_{1} |^{2} + | {\bar{ζ}}_{2} |^{2}}, \end{matrix}$ where $ζ = (\binom{{\bar{ζ}}_{1}}{{\bar{ζ}}_{2}})$ , ${\bar{ζ}}_{1} = {(ζ_{1}, \dots, ζ_{p})}^{T}$ , ${\bar{ζ}}_{2} = {(ζ_{p + 1}, \dots, ζ_{n})}^{T}$ and $ζ_{ε} = (\binom{ε {\bar{ζ}}_{1}}{{\bar{ζ}}_{2}})$ . Thus we have $\begin{array}{l} A_{ε} ζ \cdot ζ ⩾ λ (ε^{2} \land 1) | ζ |^{2} \forall ζ \in R^{n}, a.e. x \in Ω, \\ (1.3) & A_{22} {\bar{ζ}}_{2} \cdot {\bar{ζ}}_{2} ⩾ λ | {\bar{ζ}}_{2} |^{2} \forall {\bar{ζ}}_{2} \in R^{n - p}, a.e. x \in Ω . \end{array}$ (“∧” denotes the minimum of two numbers.) It follows that $A_{ε}$ and $A_{22}$ are positive definite matrices. For a function $\begin{matrix} f \in L^{2} (Ω), \end{matrix}$ we ensure the existence and the uniqueness of a weak solution $u_{ε}$ to $\begin{matrix} (1.4) & \{\begin{matrix} - \nabla \cdot (A_{ε} \nabla u_{ε}) = f & in Ω, \\ u_{ε} = 0 & on \partial Ω, \end{matrix} \end{matrix}$ in the following sense $\begin{matrix} (1.5) & \{\begin{matrix} u_{ε} \in H_{0}^{1} (Ω), \\ ε^{2} \int_{Ω} A_{11} \nabla_{X_{1}} u_{ε} \cdot \nabla_{X_{1}} v d x + ε \int_{Ω} A_{12} \nabla_{X_{2}} u_{ε} \cdot \nabla_{X_{1}} v d x \\ + ε \int_{Ω} A_{21} \nabla_{X_{1}} u_{ε} \cdot \nabla_{X_{2}} v d x \\ + \int_{Ω} A_{22} \nabla_{X_{2}} u_{ε} \cdot \nabla_{X_{2}} v d x = \int_{Ω} f v d x \forall v \in H_{0}^{1} (Ω) . \end{matrix} \end{matrix}$ The main idea of the present work aims to go beyond the limit in describing the asymptotic behavior of $u_{ε}$ . It focuses on finding an upgraded approximation simply depending on ε. It is useful and common to approach $u_{ε}$ by a finite partial sum of terms of its possible series expansion and insure a sharper quantitative estimate of the error. This allows to go deeper in the analysis of the asymptotic behavior of $u_{ε}$ and to understand what happens if we want to improve the rate of convergence and under which optimal conditions these improvements take place. For example we will see that the difference between the rates of convergence of $\nabla_{X_{2}} u_{ε}$ and $\nabla_{X_{1}} u_{ε}$ , established in [5] for the limit as an approximation, is more attached to a smoothness statement of the approximation. Another more interesting example is dealt with concerning the exponential rate of convergence that is always limited to some symmetries as sufficient conditions (see [5]). Here we will see sufficient and necessary conditions that guarantee an exponential error of the asymptotic development and give a concrete example without any symmetry. Note that the convergence results are ensured in a Sobolev type space on subdomains located far away from the boundary layer. For more details about the anisotropic singular perturbations we refer the reader to the recent papers [2,4–9,13] and the references therein. See also [11,14] for the isotropic singular perturbation theory.

In the next section, we define the coefficients of the development as solutions of an iterative sequence of elliptic problems and since the perturbation is only taken in some directions we have to study the smoothness of the coefficients in order to complete the definition of the sequence of these elliptic problems. The third section is devoted to show the basic convergence theorem and in the last section we deal with different upgraded results.

2. Formal asymptotic expansion and some regularity results

As it is shown in [5] the limit $u_{0}$ of $u_{ε}$ is the unique solution to the following lower dimension problem, for a.e. $X_{1} \in ω_{1}$ , $\begin{matrix} (2.1) & \{\begin{matrix} - \nabla_{X_{2}} \cdot (A_{22} \nabla_{X_{2}} u_{0} (X_{1}, \cdot)) = f (X_{1}, \cdot) & in ω_{2}, \\ u_{0} (X_{1}, \cdot) = 0 & on \partial ω_{2}, \end{matrix} \end{matrix}$ in the following sense, for a.e. $X_{1} \in ω_{1}$ , $\begin{matrix} (2.2) & \{\begin{matrix} u_{0} (X_{1}, \cdot) \in H_{0}^{1} (ω_{2}), \\ \int_{ω_{2}} A_{22} \nabla_{X_{2}} u_{0} (X_{1}, \cdot) \cdot \nabla_{X_{2}} v d X_{2} = \int_{ω_{2}} f (X_{1}, \cdot) v d X_{2}, & \forall v \in H_{0}^{1} (ω_{2}) . \end{matrix} \end{matrix}$ The existence and the uniqueness of $u_{0}$ is followed from the Lax-Milgram theorem since we have (1.1), (1.3) and $f (X_{1}, \cdot) \in L^{2} (ω_{2})$ for a.e. $X_{1} \in ω_{1}$ . The convergence holds out to the boundary of Ω, but with respect to topologies weaker than those of the existence space of $u_{ε}$ ; $H_{0}^{1} (Ω)$ . We mean in the following functional space $\begin{matrix} V (Ω) = {u \in L^{2} (Ω) ∣ \nabla_{X_{2}} u \in L^{2} (Ω)}, \end{matrix}$ equipped with the norm $\begin{matrix} | u |_{V (Ω)}^{2} = \int_{Ω} {| u (x) |}^{2} + {| \nabla_{X_{2}} u (x) |}^{2} d x . \end{matrix}$ Note that $u_{0}$ is also the unique solution to (2.1) in $V_{0} (Ω)$ , the closure of $D (Ω)$ in $V (Ω)$ , and there is an equivalent weak formulation to (2.2) defined on $V_{0} (Ω)$ (see [5]). The improvements related to the convergence $u_{ε} \to u_{0}$ investigated on one hand the topology type by considering the standard Sobolev space on domains located far away from the boundary layer $\partial ω_{1} \times ω_{2}$ and on the other hand the rate of convergence. Unfortunately, these improvements are limited by the nature of the problem. It can go until an exponential rate of convergence if $A_{12}$ , $A_{22}$ and f are independent of $X_{1}$ , however, a rate of convergence as $\begin{matrix} u_{ε} - u_{0} = o (ε) in L^{2} (ω_{1}^{'} \times ω_{2}), ω_{1}^{'} \subset \subset ω_{1} \end{matrix}$ cannot take place if $\begin{matrix} \nabla_{X_{1}} \cdot (A_{12} \nabla_{X_{2}} u_{0}) + \nabla_{X_{2}} \cdot (A_{21} \nabla_{X_{1}} u_{0}) \neq 0 in Ω . \end{matrix}$ Now, in order to reduce the approximation error we have no choice than to consider an approximation $U_{ε}$ of $u_{ε}$ depending on ε in a simple way. Arguing as in many singular perturbation (isotropic) works, we can propose an asymptotic development of $u_{ε}$ , i.e. it should be expressed as a power series of ε in the form $\begin{matrix} (2.3) & u_{ε} = u_{0} + ε u_{1} + \dots . \end{matrix}$ Consequently, this allows to chose $U_{ε}$ as a polynomial in ε, i.e. $\begin{matrix} (2.4) & U_{ε}^{N} = u_{0} + ε u_{1} + \dots + ε^{N} u_{N}, \end{matrix}$ where the integer N is the order of the development. However the main goal of this work is to deal with the limit behavior of the error $\begin{matrix} R_{ε}^{N} = u_{ε} - U_{ε}^{N} \end{matrix}$ and evaluate its rate of convergence with respect to different norms.

Formally, if we substitute the asymptotic expansion of (2.3) into (1.4) and expand the left-hand side in powers of ε, we then deduce, after equating the coefficients of equal powers of ε, that the coefficients $u_{N}$ are solutions of the following system of boundary value problems, defined on the section $ω_{2}$ for a.e. $X_{1} \in ω_{1}$ , $\begin{array}{l} (2.5) & \{\begin{matrix} - \nabla_{X_{2}} \cdot (A_{22} \nabla_{X_{2}} u_{0} (X_{1}, \cdot)) = f (X_{1}, \cdot) & in ω_{2}, \\ u_{0} (X_{1}, \cdot) \in H_{0}^{1} (ω_{2}), \end{matrix} \\ (2.6) & \{\begin{matrix} - \nabla_{X_{2}} \cdot (A_{22} \nabla_{X_{2}} u_{1} (X_{1}, \cdot)) = \nabla_{X_{1}} \cdot (A_{12} \nabla_{X_{2}} u_{0} (X_{1}, \cdot)) + \nabla_{X_{2}} \cdot (A_{21} \nabla_{X_{1}} u_{0} (X_{1}, \cdot)) & in ω_{2}, \\ u_{1} (X_{1}, \cdot) \in H_{0}^{1} (ω_{2}), \end{matrix} \end{array}$ and for $N ⩾ 2$ , $\begin{matrix} (2.7) & \{\begin{matrix} - \nabla_{X_{2}} \cdot (A_{22} \nabla_{X_{2}} u_{N} (X_{1}, \cdot)) = \nabla_{X_{1}} \cdot (A_{11} \nabla_{X_{1}} u_{N - 2} (X_{1}, \cdot)) \\ + \nabla_{X_{1}} \cdot (A_{12} \nabla_{X_{2}} u_{N - 1} (X_{1}, \cdot)) \\ + \nabla_{X_{2}} \cdot (A_{21} \nabla_{X_{1}} u_{N - 1} (X_{1}, \cdot)) & in ω_{2}, \\ u_{N} (X_{1}, \cdot) \in H_{0}^{1} (ω_{2}) . \end{matrix} \end{matrix}$ Our perturbed problem is now reduced to the sequence of the elliptic boundary value problems (2.5), (2.6) and (2.7) which can be easily solved iteratively once the solution of (2.5) has been constructed and has, in addition to $A_{11}$ , $A_{12}$ and $A_{21}$ , the necessary smoothness. In fact, since the solution $u_{ε}$ remains in the Sobolev space $H_{0}^{1} (Ω)$ , its approximation $U_{ε}^{N}$ is envisaged at least in a Sobolev type space as $V_{0} (Ω)$ . Moreover, according to the expressions of the left-hand side in the equations (2.5), (2.6) and (2.7), reliable hypotheses have to be ensured on the data to make the suitable purpose workable.

Let us now deal with the regularity of the solutions to problems occurred above in a general context.

Theorem 1.
Under the above assumptions on $A_{22}$ , let u be a solution (in the weak sense) to the following problem, for a.e. $X_{1} \in ω_{1}$ , $\begin{matrix} (2.8) & \{\begin{matrix} - \nabla_{X_{2}} \cdot (A_{22} \nabla_{X_{2}} u (X_{1}, \cdot)) = g (X_{1}, \cdot) + \nabla_{X_{2}} \cdot G (X_{1}, \cdot) & in ω_{2}, \\ u (X_{1}, \cdot) \in H_{0}^{1} (ω_{2}), \end{matrix} \end{matrix}$ where $g \in L^{2} (Ω)$ , $G \in {[L^{2} (Ω)]}^{n - p}$ . In addition, we assume, for a multi-index $α \in N^{p} \times {0}^{n - p}$ , that $\begin{matrix} (2.9) & D^{β} a_{i j} \in L^{\infty} (Ω), \forall i, j = p + 1, \dots, n \end{matrix}$ and $\begin{matrix} (2.10) & D^{β} g, D^{β} G \in L^{2} (Ω), \end{matrix}$ for all $β \in N^{p} \times {0}^{n - p}$ with $β ⩽ α$ . Then the solution u has the maximal regularity in the $X_{1}$ -direction according to that of g, G and $A_{22}$ , i.e. we have $\begin{matrix} (2.11) & D^{β} u, D^{β} (\nabla_{X_{2}} u) \in L^{2} (Ω) for β ⩽ α . \end{matrix}$ (The comparison “⩽” between the multi-indices is meant component by component.)

To show this and more let us introduce the higher order difference quotients which are the main tool to deal with the smoothness of the solution. For a multi-index $α = (α_{1}, \dots, α_{n}) \in N^{n}$ and a step $h > 0$ , we define the difference quotients of order 1, i.e. $α = e_{i}$ where $e_{i}$ denotes the unit coordinate vector in the $x_{i}$ -direction, by $\begin{matrix} △_{h}^{α} v (x) : = △_{i, h} v (x) = \frac{v (x + h e_{i}) - v (x)}{h}, \end{matrix}$ and the higher order differences by $\begin{matrix} △_{h}^{α} v = △_{h}^{α_{1}} △_{h}^{α_{2}} \dots △_{h}^{α_{n}} v, \end{matrix}$ where $\begin{matrix} △_{h}^{α_{k}} v = △_{k, h} △_{h}^{α_{k} - 1} v . \end{matrix}$ The following lemma shows the link between the weak derivatives and the difference quotients (see also [3,12] for more properties). Lemma 1.

Let $v \in W^{m, p} (Ω)$ , $m \in N$ , $1 ⩽ p ⩽ \infty$ , and $α \in N^{n}$ be a multi-index such that $| α | ⩽ m$ . Then $△_{h}^{α} v \in L^{p} (Ω^{'})$ for any $Ω^{'} \subset \subset Ω$ satisfying $0 < h < \frac{1}{| α |} dist (Ω^{'}, \partial Ω)$ , and we have $\begin{matrix} {| △_{h}^{α} v |}_{p, Ω^{'}} ⩽ {| D^{α} v |}_{p, Ω} . \end{matrix}$ ( $| \cdot |_{p, Ω}$ denotes the norm in the space $L^{p} (Ω)$ .)

Let $v \in L^{p} (Ω)$ with $1 < p ⩽ \infty$ , and we suppose that for a multi-index $α \in N^{n}$ , there exists a constant $C > 0$ such that $△_{h}^{α} v \in L^{p} (Ω^{'})$ and $| △_{h}^{α} v |_{p, Ω^{'}} ⩽ C$ for any $Ω^{'} \subset \subset Ω$ and all h satisfying $0 < h < \frac{1}{| α |} dist (Ω^{'}, \partial Ω)$ . Then the αth-weak derivative $D^{α} v$ exists in $L^{p} (Ω)$ and also satisfies $\begin{matrix} {| D^{α} v |}_{p, Ω} ⩽ C . \end{matrix}$

Proof.
In fact this lemma is an extension of [1, Proposition IX.3] shown for $m = 1$ . For the first assertion, it is enough to apply [1, Proposition IX.3] $| α |$ times. It then holds, for any choice $β^{1}, β^{2}, \dots, β^{| α |}$ with $β^{1} ⩽ β^{2} ⩽ \dots ⩽ β^{| α |} : = α$ and $| β^{i} | + 1 = | β^{i + 1} |$ , $i = 1, \dots, | α | - 1$ , $\begin{array}{rcl} {| △_{h}^{α} v |}_{p, Ω^{'}} & ⩽ & {| △_{h}^{α - β^{1}} D^{β^{1}} v |}_{p, Ω_{h}^{'}} ⩽ {| △_{h}^{α - β^{2}} D^{β^{2}} v |}_{p, Ω_{2 h}^{'}} \\ ⩽ & \dots \\ ⩽ & {| △_{h}^{α - β^{| α |}} D^{β^{| α |}} v |}_{p, Ω_{| α | h}^{'}} ⩽ {| D^{α} v |}_{p, Ω}, \end{array}$ where $Ω_{τ}^{'}$ denotes the τ-neighborhood of $Ω^{'}$ for a positive number τ.

Now we pass to the second assertion. Assume, for some constant C, that $| △_{h}^{α} v |_{p, Ω^{'}} ⩽ C$ holds for any $Ω^{'} \subset \subset Ω$ and all $0 < h < \frac{1}{| α |} dist (Ω^{'}, \partial Ω)$ . Therefore, there exist a function $w \in L^{p} (Ω^{'})$ and a subsequence $h_{k} ⟶ 0$ such that $\begin{array}{l} △_{h_{k}}^{α} v ⇀ w in L^{p} (Ω^{'}) for 1 < p < \infty, \\ △_{h_{k}}^{α} v \overset{*}{⇀} w in L^{\infty} (Ω^{'}) if p = \infty . \end{array}$ Then, it follows that for each $φ \in D (Ω^{'})$ and $h_{k}$ small enough $\begin{array}{rcl} \int_{Ω^{'}} v D^{α} φ d x & = & \int_{Ω} v D^{α} φ d x \\ = & lim_{h_{k} ⟶ 0} \int_{Ω} v △_{- h_{k}}^{α} φ d x \\ = & {(- 1)}^{| α |} lim_{h_{k} ⟶ 0} \int_{Ω} △_{h_{k}}^{α} v φ d x \\ = & {(- 1)}^{| α |} \int_{Ω} w φ d x . \end{array}$ This means that $w = D^{α} v$ on $Ω^{'}$ and since the norm is lower semi-continuous we derive $\begin{matrix} {| D^{α} v |}_{p, Ω^{'}} ⩽ \underset{h_{k} ⟶ 0}{lim inf} {| △_{h_{k}}^{α} v |}_{p, Ω^{'}} ⩽ C . \end{matrix}$ Of course the above inequality holds on the whole domain Ω since $Ω^{'}$ is arbitrary in Ω. □
Proof of Theorem 1.
The Theorem 1 is an extension of [5, Proposition 1], where it was shown for $| α | = 1$ . We then proceed by induction on α with $| α | ⩾ 2$ . Suppose that (2.11) holds for all $β \in N^{n}$ with $β < α$ . Let $ω_{1}^{'}$ be a bounded subset such that $ω_{1}^{'} \subset \subset ω_{1}$ and h be a positive constant such that $0 < | α | h < dist (ω_{1}^{'}, \partial ω_{1})$ . For a.e. $X_{1} \in ω_{1}^{'}$ , if we apply the difference quotient operator $△_{h}^{α}$ , with $α = (α_{1}, \dots, α_{p}, 0, \dots, 0) \in N^{p} \times {0}^{n - p}$ , to the integral formulation of problem (2.8), $\begin{matrix} \int_{ω_{2}} A_{22} \nabla_{X_{2}} u (X_{1}, \cdot) \cdot \nabla_{X_{2}} v d X_{2} = \int_{ω_{2}} g (X_{1}, \cdot) v d X_{2} - \int_{ω_{2}} G (X_{1}, \cdot) \cdot \nabla_{X_{2}} v d X_{2}, \forall v \in H_{0}^{1} (ω_{2}), \end{matrix}$ we get, using [3, Lemma 3.3] to develop the difference quotient of a product, $\begin{matrix} \int_{ω_{2}} \sum_{i, j = p + 1}^{n} \sum_{β ⩽ α} (\binom{α}{β}) △_{h}^{β} a_{i j} (x + (α - β) h) △_{h}^{α - β} (\partial_{x_{j}} u) \partial_{x_{i}} v d X_{2} = \int_{ω_{2}} △_{h}^{α} g v - △_{h}^{α} G \cdot \nabla_{X_{2}} v d X_{2}, \end{matrix}$ where $(\binom{α}{β}) = \frac{α!}{β! (α - β)!}$ . Permuting the derivatives and the difference quotient and taking, for a.e. $X_{1} \in ω_{1}^{'}$ , $v = △_{h}^{α} u (X_{1}, \cdot) \in H_{0}^{1} (ω_{2})$ as a test function yield $\begin{array}{l} \int_{ω_{2}} \sum_{i, j = p + 1}^{n} a_{i j} (x + α h) \partial_{x_{j}} (△_{h}^{α} u) \partial_{x_{i}} (△_{h}^{α} u) d X_{2} \\ = \int_{ω_{2}} (△_{h}^{α} g) (△_{h}^{α} u) - △_{h}^{α} G \cdot \nabla_{X_{2}} (△_{h}^{α} u) d X_{2} \\ - \int_{ω_{2}} \sum_{i, j = p + 1}^{n} \sum_{0 < β ⩽ α} (\binom{α}{β}) △_{h}^{β} a_{i j} (x + (α - β) h) \partial_{x_{j}} (△_{h}^{α - β} u) \partial_{x_{i}} (△_{h}^{α} u) d X_{2} . \end{array}$ Integrating now on $ω_{1}^{'}$ and using the ellipticity assumption, we derive $\begin{array}{l} λ {| \nabla_{X_{2}} (△_{h}^{α} u) |}_{2, ω_{1}^{'} \times ω_{2}}^{2} \\ ⩽ {| △_{h}^{α} g |}_{2, ω_{1}^{'} \times ω_{2}} {| △_{h}^{α} u |}_{2, ω_{1}^{'} \times ω_{2}} + {| △_{h}^{α} G |}_{2, ω_{1}^{'} \times ω_{2}} {| △_{h}^{α} (\nabla_{X_{2}} u) |}_{2, ω_{1}^{'} \times ω_{2}} \\ + \sum_{0 < β ⩽ α} (\binom{α}{β}) {| △_{h}^{β} A_{22} |}_{\infty, ω_{1}^{'} \times ω_{2}} {| △_{h}^{α - β} (\nabla_{X_{2}} u) |}_{2, ω_{1}^{'} \times ω_{2}} {| \nabla_{X_{2}} (△_{h}^{α} u) |}_{2, ω_{1}^{'} \times ω_{2}} . \end{array}$ Applying the Poincaré inequality in the $X_{2}$ -direction on the norm $| △_{h}^{α} u |_{2, ω_{1}^{'} \times ω_{2}}$ , then Young’s inequality, we end up with $\begin{matrix} {| \nabla_{X_{2}} (△_{h}^{α} u) |}_{2, ω_{1}^{'} \times ω_{2}}^{2} ⩽ C {| △_{h}^{α} g |}_{2, ω_{1}^{'} \times ω_{2}}^{2} + C {| △_{h}^{α} G |}_{2, ω_{1}^{'} \times ω_{2}}^{2} + C \sum_{0 < β ⩽ α} {| △_{h}^{α - β} (\nabla_{X_{2}} u) |}_{2, ω_{1}^{'} \times ω_{2}}^{2} . \end{matrix}$ Of course, thanks to Lemma 1(i) with (2.9), (2.10) and the inductive hypotheses, we can easily deduce that the right-hand side of the above inequality is bounded independently of h, i.e. $\begin{matrix} (2.12) & {| △_{h}^{α} u |}_{2, ω_{1}^{'} \times ω_{2}}, {| \nabla_{X_{2}} (△_{h}^{α} u) |}_{2, ω_{1}^{'} \times ω_{2}} ⩽ C . \end{matrix}$ We applied the Poincaré inequality in the $X_{2}$ -direction to get the first above estimate from the second one. Finally, (2.11) follows from (2.12) and Lemma 1(ii). □
Remark 1.
Theorem 1 remains valid if we suppose that the domain $ω_{2}$ is only bounded in some directions (not all) to apply the Poincaré inequality.

3. Convergence results

As it is mentioned above, the problem (2.7) will be solved iteratively. That is to say, in order to define $u_{N}$ , for $N ⩾ 2$ , as a solution of (2.7) in the following weak sense, for a.e. $X_{1} \in ω_{1}$ , $\begin{matrix} (3.1) & \{\begin{matrix} u_{N} (X_{1}, \cdot) \in H_{0}^{1} (ω_{2}), \\ \int_{ω_{2}} A_{22} \nabla_{X_{2}} u_{N} (X_{1}, \cdot) \cdot \nabla_{X_{2}} v d X_{2} \\ = \int_{ω_{2}} \nabla_{X_{1}} \cdot (A_{11} \nabla_{X_{1}} u_{N - 2} (X_{1}, \cdot)) v d X_{2} \\ + \int_{ω_{2}} \nabla_{X_{1}} \cdot (A_{12} \nabla_{X_{2}} u_{N - 1} (X_{1}, \cdot)) v d X_{2} \\ - \int_{ω_{2}} (A_{21} \nabla_{X_{1}} u_{N - 1} (X_{1}, \cdot)) \cdot \nabla_{X_{2}} v d X_{2} & \forall v \in H_{0}^{1} (ω_{2}), \end{matrix} \end{matrix}$ we need to ensure the smoothness of $u_{N - 1}$ , $u_{N - 2}$ , $A_{11}$ and $A_{12}$ in the following sense $\begin{matrix} (3.2) & D_{X_{1}}^{1} A_{11}, D_{X_{1}}^{1} A_{12} \in L^{\infty} (Ω), D_{X_{1}}^{2} u_{N - 2}, D_{X_{1}}^{1} u_{N - 1}, D_{X_{1}}^{1} \nabla_{X_{2}} u_{N - 1} \in L^{2} (Ω) . \end{matrix}$ ( $D_{X_{i}}^{m}$ denotes the derivatives in the direction $X_{i}$ of order up to m.) Again we have to ensure the existence and the uniqueness of $u_{N - 1}$ , $u_{N - 2}$ , of course as solutions of the same problem (3.1) replacing N by $N - 1$ , $N - 2$ respectively, as well as their smoothness in (3.2). So on, this leads to require, taking into account Theorem 1, the following hypotheses on $u_{0}$ , $u_{1}$ that appear in the left-hand side of the equation defining $u_{2}$ $\begin{array}{l} (3.3) & \begin{matrix} D_{X_{1}}^{N - 1} A_{11}, D_{X_{1}}^{N - 1} A_{12}, D_{X_{1}}^{N - 2} A_{21}, D_{X_{1}}^{N - 2} A_{22} \in L^{\infty} (Ω), \\ D_{X_{1}}^{N} u_{0}, D_{X_{1}}^{N - 1} u_{1}, D_{X_{1}}^{N - 1} \nabla_{X_{2}} u_{1} \in L^{2} (Ω) . \end{matrix} \end{array}$ Now, if we require a solution $u_{1}$ satisfying $D_{X_{1}}^{N - 1} u_{1}, D_{X_{1}}^{N - 1} \nabla_{X_{2}} u_{1} \in L^{2} (Ω)$ we have to assume, taking into account (3.3) and (2.6), that $\begin{array}{l} (3.4) & \begin{matrix} D_{X_{1}}^{N - 1} A_{11}, D_{X_{1}}^{N} A_{12}, D_{X_{1}}^{N - 1} A_{21}, D_{X_{1}}^{N - 1} A_{22} \in L^{\infty} (Ω), \\ D_{X_{1}}^{N} u_{0}, D_{X_{1}}^{N} \nabla_{X_{2}} u_{0} \in L^{2} (Ω), \end{matrix} \end{array}$ of course this is always according to Theorem 1. Note that we can also consider that (3.1) is the weak formulation of (2.6) with $N = 1$ and $u_{- 1} = 0$ . Finally, the smoothness requested on $u_{0}$ solution to (2.5) with (3.4) leads to state, using again Theorem 1, the following result.

Corollary 1.
The existence and the uniqueness of $u_{1}, \dots, u_{N} \in V_{0} (Ω)$ as solutions of the elliptic system ( 2.6 ), ( 2.7 ) are ensured if we assume that $\begin{matrix} (3.5) & D_{X_{1}}^{N - 1} A_{11}, D_{X_{1}}^{N} A_{12}, D_{X_{1}}^{N - 1} A_{21}, D_{X_{1}}^{N} A_{22} \in L^{\infty} (Ω), D_{X_{1}}^{N} f \in L^{2} (Ω) . \end{matrix}$
Remark 2.
If we assume that the smoothness hypotheses (3.5) are satisfied for $K + N$ instead of N, where K is a positive integer, we have $\begin{matrix} (3.6) & u_{N} \in H^{K} (ω_{1}; H_{0}^{1} (ω_{2})) . \end{matrix}$

The above remark shows the regularity of the coefficients which allowed to consider $U_{ε}^{N}$ as an approximation in a standard Sobolev space. In fact if we take $K = 1$ and $N = 0$ , we may covert the hypotheses assumed in [5, Theorem 3] to get $\begin{matrix} (3.7) & u_{ε} - u_{0}, \nabla_{X_{2}} (u_{ε} - u_{0}) = O (ε), \nabla_{X_{1}} u_{ε} ⇀ \nabla_{X_{1}} u_{0} in L^{2} (ω_{1}^{'} \times ω_{2}), \end{matrix}$ where $ω_{1}^{'} \subset \subset ω_{1}$ . Now we pass to the main outer asymptotic expansion result improving (3.7) and we state.
Theorem 2 (Outer Asymptotic Expansion up to Order N).

Under the assumptions ( 1.1 ), ( 1.2 ) and ( 3.5 ) satisfied for $N + 1$ , it holds that, when $ε \to 0$ , $\begin{array}{l} (3.8) & \begin{matrix} u_{ε} - (u_{0} + ε u_{1} + \dots + ε^{N} u_{N}) = O (ε^{N + 1}) in V (ω_{1}^{'} \times ω_{2}), \\ \frac{1}{ε^{N}} \nabla_{X_{1}} [u_{ε} - (u_{0} + ε u_{1} + \dots + ε^{N} u_{N})] ⇀ 0 weakly in L^{2} (ω_{1}^{'} \times ω_{2}), \end{matrix} \end{array}$ for any $ω_{1}^{'} \subset \subset ω_{1}$ . (The vectorial convergence is meant component by component.)

Proof.
First, we notice that under the above assumptions and according to Remark 2, we have $\begin{matrix} u_{k} \in H^{1} (Ω), for k = 0, \dots, N . \end{matrix}$ We can also see that for $N = 0$ , the same results of this theorem are shown in [5, Theorem 3]. For $N > 0$ , we proceed by induction on N and suppose that the estimates (3.8) take place for any $N^{'} < N$ , and we will prove this result for N. Starting from (3.1) written for $k = 1, \dots, N$ and multiplying each k-identity by $ε^{k}$ then summing up over $k = 1, \dots, N$ , we obtain, for $v \in H_{0}^{1} (Ω)$ $\begin{array}{l} ε^{2} \int_{Ω} A_{11} \nabla_{X_{1}} U_{ε}^{N - 2} \cdot \nabla_{X_{1}} v d x + ε \int_{Ω} A_{12} \nabla_{X_{2}} U_{ε}^{N - 1} \cdot \nabla_{X_{1}} v d x \\ + ε \int_{Ω} A_{21} \nabla_{X_{1}} U_{ε}^{N - 1} \cdot \nabla_{X_{2}} v d x + \int_{Ω} A_{22} \nabla_{X_{2}} U_{ε}^{N} \cdot \nabla_{X_{2}} v d x = \int_{Ω} f v d x, \end{array}$ where $U_{ε}^{- 1} = 0$ . Of course the identity (2.2) is added and we integrated over $ω_{1}$ . Then subtracting the above identity from (1.5), we get $\begin{array}{l} ε^{2} \int_{Ω} A_{11} \nabla_{X_{1}} R_{ε}^{N - 2} \cdot \nabla_{X_{1}} v d x + ε \int_{Ω} A_{12} \nabla_{X_{2}} R_{ε}^{N - 1} \cdot \nabla_{X_{1}} v d x \\ + ε \int_{Ω} A_{21} \nabla_{X_{1}} R_{ε}^{N - 1} \cdot \nabla_{X_{2}} v d x + \int_{Ω} A_{22} \nabla_{X_{2}} R_{ε}^{N} \cdot \nabla_{X_{2}} v d x = 0 . \end{array}$ Thus $\begin{array}{l} ε^{2} \int_{Ω} A_{11} \nabla_{X_{1}} R_{ε}^{N} \cdot \nabla_{X_{1}} v d x + ε \int_{Ω} A_{12} \nabla_{X_{2}} R_{ε}^{N} \cdot \nabla_{X_{1}} v d x \\ + ε \int_{Ω} A_{21} \nabla_{X_{1}} R_{ε}^{N} \cdot \nabla_{X_{2}} v d x + \int_{Ω} A_{22} \nabla_{X_{2}} R_{ε}^{N} \cdot \nabla_{X_{2}} v d x \\ = - ε^{N + 1} \int_{Ω} A_{11} \nabla_{X_{1}} u_{N - 1} \cdot \nabla_{X_{1}} v d x - ε^{N + 2} \int_{Ω} A_{11} \nabla_{X_{1}} u_{N} \cdot \nabla_{X_{1}} v d x \\ (3.9) & - ε^{N + 1} \int_{Ω} A_{12} \nabla_{X_{2}} u_{N} \cdot \nabla_{X_{1}} v d x - ε^{N + 1} \int_{Ω} A_{21} \nabla_{X_{1}} u_{N} \cdot \nabla_{X_{2}} v d x . \end{array}$ Next, we consider a smooth function $ρ = ρ (X_{1})$ supported in $ω_{1}^{″}$ such that $ω_{1}^{'} \subset \subset ω_{1}^{″} \subset \subset ω_{1}$ , $ρ = 1$ on $ω_{1}^{'}$ and $0 ⩽ ρ ⩽ 1$ . This allows to take $v = R_{ε}^{N} (\cdot) ρ^{2} (X_{1}) \in H_{0}^{1} (Ω)$ as a test function in (3.9) that leads to $\begin{array}{l} \int_{Ω} ρ^{2} A_{ε} \nabla R_{ε}^{N} \cdot \nabla R_{ε}^{N} d x \\ = - 2 ε^{2} \int_{Ω} A_{11} \nabla_{X_{1}} R_{ε}^{N - 1} \cdot \nabla_{X_{1}} ρ ρ R_{ε}^{N} d x - 2 ε^{N + 1} \int_{Ω} A_{11} \nabla_{X_{1}} u_{N - 1} \cdot \nabla_{X_{1}} ρ ρ R_{ε}^{N} d x \\ - 2 ε \int_{Ω} A_{12} \nabla_{X_{2}} R_{ε}^{N - 1} \cdot \nabla_{X_{1}} ρ ρ R_{ε}^{N} d x - ε^{N + 2} \int_{Ω} ρ^{2} A_{11} \nabla_{X_{1}} u_{N} \cdot \nabla_{X_{1}} R_{ε}^{N} d x \\ - ε^{N + 1} \int_{Ω} ρ^{2} A_{21} \nabla_{X_{1}} u_{N} \cdot \nabla_{X_{2}} R_{ε}^{N} d x - ε^{N + 1} \int_{Ω} ρ^{2} A_{11} \nabla_{X_{1}} u_{N - 1} \cdot \nabla_{X_{1}} R_{ε}^{N} d x \\ (3.10) & - ε^{N + 1} \int_{Ω} ρ^{2} A_{12} \nabla_{X_{2}} u_{N} \cdot \nabla_{X_{1}} R_{ε}^{N} d x . \end{array}$ Next applying the ellipticity assumption, the Poincaré and the Cauchy–Schwarz inequalities for the first five terms of the right-hand side of the above inequality, we get $\begin{array}{l} ε^{2} {| ρ \nabla_{X_{1}} R_{ε}^{N} |}_{2, Ω}^{2} + {| ρ \nabla_{X_{2}} R_{ε}^{N} |}_{2, Ω}^{2} \\ ⩽ \frac{2 ε^{2} \sqrt{C_{ω_{2}}}}{λ} | A_{11} |_{\infty, ω_{1}^{″} \times ω_{2}} | \nabla_{X_{1}} ρ |_{\infty, ω_{1}^{″}} {| \nabla_{X_{1}} R_{ε}^{N - 1} |}_{2, ω_{1}^{″} \times ω_{2}} {| ρ \nabla_{X_{2}} R_{ε}^{N} |}_{2, Ω} \\ + \frac{2 ε^{N + 1} \sqrt{C_{ω_{2}}}}{λ} | A_{11} |_{\infty, ω_{1}^{″} \times ω_{2}} | \nabla_{X_{1}} ρ |_{\infty, ω_{1}^{″}} | \nabla_{X_{1}} u_{N - 1} |_{2, Ω} {| ρ \nabla_{X_{2}} R_{ε}^{N} |}_{2, Ω} \\ + \frac{2 ε \sqrt{C_{ω_{2}}}}{λ} | A_{12} |_{\infty, ω_{1}^{″} \times ω_{2}} | \nabla_{X_{1}} ρ |_{\infty, ω_{1}^{″}} {| \nabla_{X_{2}} R_{ε}^{N - 1} |}_{2, ω_{1}^{″} \times ω_{2}} {| ρ \nabla_{X_{2}} R_{ε}^{N} |}_{2, Ω} \\ + \frac{ε^{N + 1}}{λ} | A_{11} |_{\infty, ω_{1}^{″} \times ω_{2}} | \nabla_{X_{1}} u_{N} |_{2, Ω} ε {| ρ \nabla_{X_{1}} R_{ε}^{N} |}_{2, Ω} + \frac{ε^{N + 1}}{λ} | A_{21} |_{\infty, ω_{1}^{″} \times ω_{2}} | \nabla_{X_{1}} u_{N} |_{2, Ω} {| ρ \nabla_{X_{2}} R_{ε}^{N} |}_{2, Ω} \\ - \frac{ε^{N + 1}}{λ} \int_{Ω} ρ^{2} A_{11} \nabla_{X_{1}} u_{N - 1} \cdot \nabla_{X_{1}} R_{ε}^{N} d x - \frac{ε^{N + 1}}{λ} \int_{Ω} ρ^{2} A_{12} \nabla_{X_{2}} u_{N} \cdot \nabla_{X_{1}} R_{ε}^{N} d x, \end{array}$ where $C_{ω_{2}}$ is the Poincaré constant. Then, using Young’s inequality and the convergences (3.8) on $ω_{1}^{″}$ for $N - 1$ , we derive $\begin{array}{l} ε^{2} {| ρ \nabla_{X_{1}} R_{ε}^{N} |}_{2, Ω}^{2} + {| ρ \nabla_{X_{2}} R_{ε}^{N} |}_{2, Ω}^{2} \\ ⩽ C ε^{2 N + 2} - \frac{2 ε^{N + 1}}{λ} \int_{Ω} ρ^{2} A_{11} \nabla_{X_{1}} u_{N - 1} \cdot \nabla_{X_{1}} R_{ε}^{N} d x \\ (3.11) & - \frac{2 ε^{N + 1}}{λ} \int_{Ω} ρ^{2} A_{12} \nabla_{X_{2}} u_{N} \cdot \nabla_{X_{1}} R_{ε}^{N} d x . \end{array}$ Now we treat the last two integrals of the above inequality separately. For the first one we have $\begin{array}{rcl} \int_{Ω} ρ^{2} A_{11} \nabla_{X_{1}} u_{N - 1} \cdot \nabla_{X_{1}} R_{ε}^{N} d x & = & \sum_{i, j = 1}^{p} \int_{Ω} ρ^{2} a_{i j} \partial_{x_{j}} u_{N - 1} \partial_{x_{i}} R_{ε}^{N} d x \\ = & - \sum_{i, j = 1}^{p} \int_{Ω} (\partial_{x_{i}} a_{i j} ρ + 2 a_{i j} \partial_{x_{i}} ρ) ρ R_{ε}^{N} \partial_{x_{j}} u_{N - 1} d x \\ - \sum_{i, j = 1}^{p} \int_{Ω} a_{i j} ρ^{2} R_{ε}^{N} \partial_{x_{i}} \partial_{x_{j}} u_{N - 1} d x . \end{array}$

Of course the last integral is well defined since, by Remark 2, $u_{N - 1} \in H^{2} (ω_{1}; H_{0}^{1} (ω_{2}))$ . Also, the last integral in (3.11) can be rewritten as $\begin{array}{rcl} \int_{Ω} ρ^{2} A_{12} \nabla_{X_{2}} u_{N} \cdot \nabla_{X_{1}} R_{ε}^{N} d x & = & \sum_{i = 1}^{p} \sum_{j = p + 1}^{n} \int_{Ω} ρ^{2} a_{i j} \partial_{x_{j}} u_{N} \partial_{x_{i}} R_{ε}^{N} d x \\ = & - \sum_{i = 1}^{p} \sum_{j = p + 1}^{n} \int_{Ω} (\partial_{x_{i}} a_{i j} ρ + 2 a_{i j} \partial_{x_{i}} ρ) ρ R_{ε}^{N} \partial_{x_{j}} u_{N} d x \\ - \sum_{i = 1}^{p} \sum_{j = p + 1}^{n} \int_{Ω} a_{i j} ρ^{2} R_{ε}^{N} \partial_{x_{i}} \partial_{x_{j}} u_{N} d x . \end{array}$ Here also, thanks to Remark 2, we have $u_{N} \in H^{1} (ω_{1}; H_{0}^{1} (ω_{2}))$ . Now if we use the above new forms of the last integrals in (3.11) and arguing as we did above, we get $\begin{matrix} (3.12) & ε^{2} {| ρ \nabla_{X_{1}} R_{ε}^{N} |}_{2, Ω}^{2} + {| ρ \nabla_{X_{2}} R_{ε}^{N} |}_{2, Ω}^{2} ⩽ C ε^{2 N + 2} . \end{matrix}$ Then, according to the boundedness of $\frac{1}{ε^{N}} | \nabla_{X_{1}} R_{ε}^{N} |_{2, ω_{1}^{'} \times ω_{2}}$ from the above estimate (3.12), we can extract a weakly converging subsequence from $\frac{1}{ε^{N}} \nabla_{X_{1}} R_{ε}^{N}$ in $L^{2} (ω_{1}^{'} \times ω_{2})$ . It follows then that the whole sequence converges weakly to zero, i.e. $\begin{matrix} \frac{1}{ε^{N}} \nabla_{X_{1}} [u_{ε} - (u_{0} + ε u_{1} + \dots + ε^{N} u_{N})] ⇀ 0 weakly in L^{2} (ω_{1}^{'} \times ω_{2}) . \end{matrix}$ Of course, this is thanks to the density of $D (ω_{1}^{'} \times ω_{2})$ in $L^{2} (ω_{1}^{'} \times ω_{2})$ and the continuity of the derivation in $D^{'} (ω_{1}^{'} \times ω_{2})$ which allows to use the convergence of the sequence $\frac{1}{ε^{N}} R_{ε}^{N}$ in $L^{2} (ω_{1}^{'} \times ω_{2})$ to zero. More details about the argument can be found in [8]. This completes the proof of the theorem. □
Remark 3.
If we look for (3.8) to be satisfied on some given $ω_{1}^{'} \subset \subset ω_{1}$ we can replace $ω_{1}$ in (3.5) and (3.6) by any $ω_{1}^{″}$ such that $ω_{1}^{'} \subset \subset ω_{1}^{″} \subset \subset ω_{1}$ . Nevertheless we keep the hypotheses (3.5), (3.6) as they are since they also serve for the existence of the coefficients $u_{N}$ for arbitrary $ω_{1}^{'} \subset \subset ω_{1}$ .
Remark 4.
We can easily see from (3.1) that (3.2) can be weakened as follows $\begin{matrix} A_{11}^{'}, A_{12}^{'} \in L^{\infty} (Ω), D_{X_{1}}^{2} u_{N - 2}, D_{X_{1}}^{1} u_{N - 1}, D_{X_{1}}^{1} \nabla_{X_{2}} u_{N - 1} \in L^{2} (Ω), \end{matrix}$ where $A_{11}^{'} = {(\partial_{x_{i}} a_{i j})}_{i, j = 1, \dots, p}$ and $A_{12}^{'} = {(\partial_{x_{i}} a_{i j})}_{i = 1, \dots, p, j = p + 1, \dots, n}$ . This also allows to weaken (3.5) and replace it by $\begin{matrix} (3.13) & A_{12}^{'}, D_{X_{1}}^{1} A_{22} \in L^{\infty} (Ω), D_{X_{1}}^{1} f \in L^{2} (Ω) \end{matrix}$ to guarantee the existence of $u_{1}$ in $V_{0} (Ω)$ , and for $u_{N}$ ( $N > 1$ ) we may assume $\begin{matrix} (3.14) & D_{X_{1}}^{N - 2} A_{11}^{'}, D_{X_{1}}^{N - 1} A_{12}^{'}, D_{X_{1}}^{N - 1} A_{21}, D_{X_{1}}^{N} A_{22} \in L^{\infty} (Ω), D_{X_{1}}^{N} f \in L^{2} (Ω) . \end{matrix}$

In fact, even if the particular case $N = 0$ is shown in [5, Theorem 3], it is possible to improve the results of such theorem if we take into account Theorem 2 for $N = 1$ . Indeed, let us show it for any N. Suppose that (3.8) holds for $N + 1$ , then in particular we have $\begin{matrix} {| \nabla_{X_{1}} R_{ε}^{N + 1} |}_{2, ω_{1}^{'} \times ω_{2}} = O (ε^{N + 1}) . \end{matrix}$ This implies $\begin{array}{rcl} {| \nabla_{X_{1}} R_{ε}^{N} |}_{2, ω_{1}^{'} \times ω_{2}} & = & {| \nabla_{X_{1}} R_{ε}^{N + 1} + ε^{N + 1} \nabla_{X_{1}} u_{N + 1} |}_{2, ω_{1}^{'} \times ω_{2}} \\ ⩽ & {| \nabla_{X_{1}} R_{ε}^{N + 1} |}_{2, ω_{1}^{'} \times ω_{2}} + ε^{N + 1} | \nabla_{X_{1}} u_{N + 1} |_{2, ω_{1}^{'} \times ω_{2}} ⩽ C ε^{N + 1} . \end{array}$

Then we end up with Corollary 2.
Under the assumptions ( 1.1 ), ( 1.2 ) and ( 3.13 ) for $N = 0$ or ( 3.14 ) (satisfied for $N + 2$ ) if $N > 0$ , we have $\begin{matrix} u_{ε} - (u_{0} + ε u_{1} + \dots + ε^{N} u_{N}) = O (ε^{N + 1}) in H^{1} (ω_{1}^{'} \times ω_{2}), \forall ω_{1}^{'} \subset \subset ω_{1} . \end{matrix}$

4. Sharper convergence and estimate results

4.1. General view

An expected natural improvement of the convergence rate (3.8) may be expressed, for any $ω_{1}^{'} \subset \subset ω_{1}$ , as $\begin{matrix} (4.1) & \frac{1}{ε^{N + 1}} R_{ε}^{N} ⇀ 0 in L^{2} (ω_{1}^{'} \times ω_{2}) . \end{matrix}$ In fact this means, by (3.9), that $\begin{matrix} (4.2) & \nabla_{X_{1}} \cdot (A_{11} \nabla_{X_{1}} u_{N - 1}) + \nabla_{X_{1}} \cdot (A_{12} \nabla_{X_{2}} u_{N}) + \nabla_{X_{2}} \cdot (A_{21} \nabla_{X_{1}} u_{N}) = 0 in Ω, \end{matrix}$ where $u_{- 1} = 0$ . Indeed, in addition to the weak convergence (4.1) we also have $\begin{matrix} (4.3) & \frac{1}{ε^{N + 1}} \nabla_{X_{2}} R_{ε}^{N} ⇀ 0 in L^{2} (ω_{1}^{'} \times ω_{2}), \end{matrix}$ thanks to the boundedness of $\frac{1}{ε^{N + 1}} | \nabla_{X_{2}} R_{ε}^{N} |_{2, ω_{1}^{'} \times ω_{2}}$ (see (3.8)). Now rewriting the identity (3.9), for $v \in D (ω_{1}^{'} \times ω_{2})$ , as $\begin{array}{l} \frac{1}{ε^{N - 1}} \int_{Ω} A_{11} \nabla_{X_{1}} R_{ε}^{N} \cdot \nabla_{X_{1}} v d x + \frac{1}{ε^{N}} \int_{Ω} A_{12} \nabla_{X_{2}} R_{ε}^{N} \cdot \nabla_{X_{1}} v d x \\ + \frac{1}{ε^{N}} \int_{Ω} A_{21} \nabla_{X_{1}} R_{ε}^{N} \cdot \nabla_{X_{2}} v d x \\ + \frac{1}{ε^{N + 1}} \int_{Ω} A_{22} \nabla_{X_{2}} R_{ε}^{N} \cdot \nabla_{X_{2}} v d x \\ = - ε \int_{Ω} A_{11} \nabla_{X_{1}} u_{N} \cdot \nabla_{X_{1}} v d x - \int_{Ω} A_{11} \nabla_{X_{1}} u_{N - 1} \cdot \nabla_{X_{1}} v d x \\ - \int_{Ω} A_{12} \nabla_{X_{2}} u_{N} \cdot \nabla_{X_{1}} v d x - \int_{Ω} A_{21} \nabla_{X_{1}} u_{N} \cdot \nabla_{X_{2}} v d x . \end{array}$ Using (3.8) and (4.3), then passing to the limit in each term of the above identity we end up with (4.2). By consequence, we deduce, from (3.1) and (4.2), that $u_{N + 1} = 0$ iff we have (4.2). This means that the existence of $u_{N + 1}$ solution to (3.1) is ensured by the above identity (4.2) in the distributional sense and we do not need to pass by Corollary 1 to ensure this existence.

Now, if we expect a little bit more as $\begin{matrix} \frac{1}{ε^{N + 2}} \nabla_{X_{2}} R_{ε}^{N} ⇀ 0 in L^{2} (ω_{1}^{'} \times ω_{2}), \end{matrix}$ we likewise deduce $\begin{matrix} u_{N + 1} = 0, u_{N + 2} = 0 . \end{matrix}$ As above, this holds iff we have (4.2) and $\begin{matrix} (4.4) & \nabla_{X_{1}} \cdot (A_{11} \nabla_{X_{1}} u_{N}) = 0 in D^{'} (Ω) . \end{matrix}$ Here also, $u_{N + 2}$ exists and is null since the right-hand side of (3.1) is vanishing.

Nevertheless the last expectations are not little, we can easily see from (3.1), that $\begin{matrix} u_{k} = 0, \forall k > N . \end{matrix}$ Actually, it follows that $R_{ε}^{N}$ goes to zero in $H^{1} (ω_{1}^{'} \times ω_{2})$ faster than any power of ε, i.e. for any positive constant τ $\begin{matrix} R_{ε}^{N} = O (ε^{τ}) in H^{1} (ω_{1}^{'} \times ω_{2}) . \end{matrix}$ Of course, this lets us to think about a sharper rate of convergence as it is summarized in the following theorem.

Theorem 3.
Given the assumptions of Theorem 2 and consider the following assertions, for any $ω_{1}^{'} \subset \subset ω_{1}$ ,
$u_{N + 1} = 0$ ,

the condition ( 4.2 ) holds,

$\frac{1}{ε^{N + 1}} [u_{ε} - (u_{0} + ε u_{1} + \dots + ε^{N} u_{N})] ⇀ 0 in L^{2} (ω_{1}^{'} \times ω_{2}), as ε ⟶ 0$ ,

$u_{ε} - (u_{0} + ε u_{1} + \dots + ε^{N} u_{N}) = o (ε^{N + 1}) in V (ω_{1}^{'} \times ω_{2}), as ε ⟶ 0$ ,

$u_{N + 1} = 0, u_{N + 2} = 0$ ,

the conditions ( 4.2 ) and ( 4.4 ) hold,

$u_{k} = 0, \forall k > N$ ,

$\frac{1}{ε^{N + 2}} \nabla_{X_{2}} [u_{ε} - (u_{0} + ε u_{1} + \dots + ε^{N} u_{N})] ⇀ 0 in L^{2} (ω_{1}^{'} \times ω_{2}), as ε ⟶ 0$ ,

$u_{ε} - (u_{0} + ε u_{1} + \dots + ε^{N} u_{N}) = O (exp (\frac{- η}{ε})) in H^{1} (ω_{1}^{'} \times ω_{2}), as ε ⟶ 0$ for some $η > 0$ .

Then we have $\begin{matrix} (i) \Leftrightarrow (ii) \Leftrightarrow (iii) \Leftrightarrow (iv) \Leftarrow (v) \Leftrightarrow (vi) \Leftrightarrow (vii) \Leftrightarrow (viii) \Leftrightarrow (ix) . \end{matrix}$
Proof.
We can easily see, from what is done above, that $\begin{matrix} (i) \Leftrightarrow (ii) \Leftarrow (iii) \Leftarrow (iv) . \end{matrix}$ Then let us show, for example, how (ii) ensures the strong convergence (iv) which also guarantees the equivalence between the first four assertions. Going back to (3.9) and taking into account (4.2), we get for $v \in H_{0}^{1} (Ω)$ $\begin{array}{l} ε^{2} \int_{Ω} A_{11} \nabla_{X_{1}} R_{ε}^{N} \cdot \nabla_{X_{1}} v d x + ε \int_{Ω} A_{12} \nabla_{X_{2}} R_{ε}^{N} \cdot \nabla_{X_{1}} v d x \\ + ε \int_{Ω} A_{21} \nabla_{X_{1}} R_{ε}^{N} \cdot \nabla_{X_{2}} v d x + \int_{Ω} A_{22} \nabla_{X_{2}} R_{ε}^{N} \cdot \nabla_{X_{2}} v d x \\ = - ε^{N + 2} \int_{Ω} A_{11} \nabla_{X_{1}} u_{N} \cdot \nabla_{X_{1}} v d x . \end{array}$ As above, we take as a test function $v = R_{ε}^{N} (\cdot) ρ^{2} (X_{1}) \in H_{0}^{1} (Ω)$ , we obtain $\begin{array}{l} \frac{1}{ε^{2 N + 2}} \int_{Ω} ρ^{2} A_{ε} \nabla R_{ε}^{N} \cdot \nabla R_{ε}^{N} d x \\ = - \frac{2}{ε^{2 N}} \int_{Ω} A_{11} \nabla_{X_{1}} R_{ε}^{N} \cdot \nabla_{X_{1}} ρ ρ R_{ε}^{N} d x - \frac{2}{ε^{2 N + 1}} \int_{Ω} A_{12} \nabla_{X_{2}} R_{ε}^{N} \cdot \nabla_{X_{1}} ρ ρ R_{ε}^{N} d x \\ (4.5) & - \frac{1}{ε^{N}} \int_{Ω} ρ^{2} A_{11} \nabla_{X_{1}} u_{N} \cdot \nabla_{X_{1}} R_{ε}^{N} d x - \frac{2}{ε^{N}} \int_{Ω} A_{11} \nabla_{X_{1}} u_{N} \cdot \nabla_{X_{1}} ρ ρ R_{ε}^{N} d x . \end{array}$ Applying the ellipticity assumption and taking into account the convergences appearing in (3.8), we get $\begin{matrix} (4.6) & {| \nabla_{X_{1}} R_{ε}^{N} |}_{2, ω_{1}^{'} \times ω_{2}} = o (ε^{N}) and {| \nabla_{X_{2}} R_{ε}^{N} |}_{2, ω_{1}^{'} \times ω_{2}} = o (ε^{N + 1}) . \end{matrix}$ Note that, in the first two integrals of the right hand side of (4.5), we use (3.8) to pass to the limit in a scalar product of weakly and strongly converging sequences. Then applying the Poincaré inequality in $X_{2}$ -direction in the last identity we end up with (iv). Similarly, we can also see that $\begin{matrix} (v) \Leftrightarrow (vi) \Leftrightarrow (vii) \Leftarrow (viii) \Leftarrow (ix) . \end{matrix}$ Now, we have just to prove for example (v)⇒(ix) to ensure the equivalence between the last five assertions and by consequence to end the proof. We proceed as in [10] or [13]. Without loss of generality, we assume that $dist (ω_{1}^{'}, ω_{1} ∖ ω_{1}^{″}) > 1$ where $ω_{1}^{'} \subset \subset ω_{1}^{″} \subset \subset ω_{1}$ . For ε small enough we can always construct a sequence ${(D_{k})}_{0 ⩽ k ⩽ m + 1}$ with $m = [\frac{1}{ε}]$ , of strictly increasing sets such that $\begin{array}{l} ω_{1}^{'} = : D_{0} \subset \subset D_{1} \subset \subset \dots \subset \subset D_{m} \subset \subset D_{m + 1} : = ω_{1}^{″}, \\ dist (D_{k}, ω_{1} ∖ D_{k + 1}) ⩾ ε, k = 0, \dots, m . \end{array}$ Let ${(ρ_{k})}_{0 ⩽ k ⩽ m}$ be a family of functions depending only on $X_{1}$ such that $supp ρ_{k} \subset \overline{D_{k + 1}}$ , $ρ_{k} = 1$ on $D_{k}$ , $0 ⩽ ρ_{k} ⩽ 1$ and $| \nabla_{X_{1}} ρ_{k} |_{\infty, R^{p}} ⩽ \frac{C}{ε}$ for some constant $C$ independent of ε. Next, rewriting (3.10) for $N + 2$ , replacing ρ by $ρ_{k}$ and taking into account the fact that $u_{N + 1} = u_{N + 2} = 0$ , we get $\begin{array}{rcl} \int_{Ω} ρ_{k}^{2} A_{ε} \nabla R_{ε}^{N} \cdot \nabla R_{ε}^{N} d x & = & - 2 ε^{2} \int_{Ω} A_{11} \nabla_{X_{1}} R_{ε}^{N} \cdot \nabla_{X_{1}} ρ_{k} ρ_{k} R_{ε}^{N} d x \\ - 2 ε \int_{Ω} A_{12} \nabla_{X_{2}} R_{ε}^{N} \cdot \nabla_{X_{1}} ρ_{k} ρ_{k} R_{ε}^{N} d x . \end{array}$ Applying the ellipticity assumption, the Young and the Poincaré inequalities yield $\begin{array}{rcl} {| \nabla_{ε} R_{ε}^{N} |}_{2, D_{k} \times ω_{2}}^{2} & ⩽ & \frac{(| A_{11} |_{\infty, Ω}^{2} + | A_{12} |_{\infty, Ω}^{2}) C^{2} C_{ω_{2}}}{λ^{2}} {| \nabla_{ε} R_{ε}^{N} |}_{2, (D_{k + 1} ∖ D_{k}) \times ω_{2}}^{2} \\ = & C {| \nabla_{ε} R_{ε}^{N} |}_{2, D_{k + 1} \times ω_{2}}^{2} - C {| \nabla_{ε} R_{ε}^{N} |}_{2, D_{k} \times ω_{2}}^{2}, \end{array}$ where $\nabla_{ε} \cdot = (\binom{ε \nabla_{X_{1}} \cdot}{\nabla_{X_{2}} \cdot})$ . We used the identity $\nabla_{X_{1}} ρ_{k} = 0$ on $D_{k}$ and $ε | \nabla_{X_{1}} ρ_{k} |_{\infty, R^{p}} ⩽ C$ . Thus $\begin{matrix} {| \nabla_{ε} R_{ε}^{N} |}_{2, D_{k} \times ω_{2}}^{2} ⩽ r {| \nabla_{ε} R_{ε}^{N} |}_{2, D_{k + 1} \times ω_{2}}^{2}, \end{matrix}$ where $r = \frac{C}{1 + C} < 1$ , $C = \frac{(| A_{11} |_{\infty, Ω}^{2} + | A_{12} |_{\infty, Ω}^{2}) C^{2} C_{ω_{2}}}{λ^{2}}$ . Iterating this inequality for $k = 0, \dots, m$ , we derive $\begin{matrix} {| \nabla_{ε} R_{ε}^{N} |}_{2, ω_{1}^{'} \times ω_{2}}^{2} ⩽ r^{[\frac{1}{ε}] + 1} {| \nabla_{ε} R_{ε}^{N} |}_{2, ω_{1}^{″} \times ω_{2}}^{2}, \end{matrix}$ i.e. $\begin{matrix} {| \nabla_{ε} R_{ε}^{N} |}_{2, ω_{1}^{'} \times ω_{2}}^{2} ⩽ C e^{\frac{ln r}{ε}} . \end{matrix}$ This completes the proof by setting $η = - ln r$ . □
Remark 5.
It is clear from (4.6) that the assertion (iii) also implies a convergence faster than $ε^{N}$ ensured in a standard Sobolev space, as $ε ⟶ 0$ , i.e. $\begin{matrix} u_{ε} - (u_{0} + ε u_{1} + \dots + ε^{N} u_{N}) = o (ε^{N}) in H^{1} (ω_{1}^{'} \times ω_{2}) for any ω_{1}^{'} \subset \subset ω_{1} . \end{matrix}$
Remark 6.
Keeping the same results of the above theorem, the weak convergence (viii) can be weakened as follows $\begin{matrix} (4.7) & \frac{1}{ε^{N + 2}} R_{ε}^{N} ⇀ 0 in L^{2} (ω_{1}^{'} \times ω_{2}) for any ω_{1}^{'} \subset \subset ω_{1}, \end{matrix}$ if we assume more smoothness as $\begin{matrix} {(A_{21}^{T})}^{'} = {(\partial_{x_{j}} a_{i j})}_{j = 1, \dots, p, i = p + 1, \dots, n}, {(A_{22}^{T})}^{'} = {(\partial_{x_{j}} a_{i j})}_{j, i = p + 1, \dots, n} \in L^{\infty} (Ω) . \end{matrix}$ Indeed, thanks to the Poincaré inequality, we deduce, from (viii), that the sequence $\frac{1}{ε^{N + 2}} R_{ε}^{N}$ is bounded in $L^{2} (ω_{1}^{'} \times ω_{2})$ . Then for $φ \in D (ω_{1}^{'} \times ω_{2})$ extended by zero on $ω_{1}^{'} \times R^{n - p}$ , we have $\begin{array}{rcl} \int_{Ω} \frac{1}{ε^{N + 2}} R_{ε}^{N} φ d x & = & \int_{Ω} \frac{1}{ε^{N + 2}} R_{ε}^{N} \partial_{x_{i}} \int_{- \infty}^{x_{i}} φ (\dots, s, \dots) d s d x \\ = & - \int_{Ω} \frac{1}{ε^{N + 2}} \partial_{x_{i}} R_{ε}^{N} \int_{- \infty}^{x_{i}} φ (\dots, s, \dots) d s d x \to 0, \end{array}$ where $i = p + 1, \dots, n$ . This means that (viii) implies (4.7). It is now enough to show, for instance, that (4.7) implies (4.2) and (4.4). Of course, (4.2) is clearly guaranteed since (4.7) implies (iii) . Now to achieve the second integral identity, we rewrite (3.9) as $\begin{array}{l} ε^{- N} \int_{Ω} A_{11} \nabla_{X_{1}} R_{ε}^{N} \cdot \nabla_{X_{1}} v d x + ε^{- N - 1} \int_{Ω} A_{12} \nabla_{X_{2}} R_{ε}^{N} \cdot \nabla_{X_{1}} v d x \\ - ε \int_{Ω} ε^{- N - 2} R_{ε}^{N} \nabla_{X_{1}} \cdot (A_{21}^{T} \nabla_{X_{2}} v) d x - \int_{Ω} ε^{- N - 2} R_{ε}^{N} \nabla_{X_{2}} \cdot (A_{22}^{T} \nabla_{X_{2}} v) d x \\ = - \int_{Ω} A_{11} \nabla_{X_{1}} u_{N} \cdot \nabla_{X_{1}} v d x, \end{array}$ where “ $\cdot^{T}$ ” denotes the transpose of the matrix. Of course, (4.2) is taken into account and the integrals are well defined since ${(A_{21}^{T})}^{'}, {(A_{22}^{T})}^{'} \in L^{\infty} (Ω)$ . Finally, we take into account (3.8) and (iv) (which is implied by (4.2)) in the first line and (4.7) in the second line to pass to the limit in the above identity and end up with (4.4).
Remark 7.
Assuming (i), it hints at a sharper convergence than what is mentioned in (iv) since we have $R_{ε}^{N} = R_{ε}^{N + 1}$ . This allows to think about $\begin{matrix} (4.8) & u_{ε} - (u_{0} + ε u_{1} + \dots + ε^{N} u_{N}) = O (ε^{N + 2}) in V (ω_{1}^{'} \times ω_{2}), \end{matrix}$ for any $ω_{1}^{'} \subset \subset ω_{1}$ . Here also the smoothness of the data may improve this as it is expected and done above many times. However, if we assume in addition that (3.5) is satisfied for $N + 2$ , the above theorem remains true if we replace (iv) by (4.8). Always, thanks to (4.8), we also have $\begin{matrix} \frac{1}{ε^{N + 1}} \nabla_{X_{1}} [u_{ε} - (u_{0} + ε u_{1} + \dots + ε^{N} u_{N})] ⇀ 0 weakly in L^{2} (ω_{1}^{'} \times ω_{2}), \end{matrix}$ for any $ω_{1}^{'} \subset \subset ω_{1}$ , and moreover if (3.5) is satisfied for $N + 3$ , it holds $\begin{matrix} u_{ε} - (u_{0} + ε u_{1} + \dots + ε^{N} u_{N}) = O (ε^{N + 2}) in H^{1} (ω_{1}^{'} \times ω_{2}) \end{matrix}$ for any $ω_{1}^{'} \subset \subset ω_{1}$ .

In [5], an exponential rate of convergence is shown under the condition (4.2) satisfied for $N = 0$ and the independence of $u_{0}$ of $X_{1}$ which implies (4.4) for $N = 0$ . They were sufficient conditions, however, we are here faced with sufficient and necessary conditions. In the following we formulate the above analysis in terms of data.
4.2. Diagonal block matrices

We here assume that $\begin{matrix} (4.9) & A_{12} = A_{21}^{T} = 0 . \end{matrix}$ It is clear that (4.2) holds for $N = 0$ which implies that $u_{1} = 0$ . Then the condition (4.2) written for $N = 2$ is reduced to $\begin{matrix} \nabla_{X_{1}} \cdot (A_{12} \nabla_{X_{2}} u_{2}) + \nabla_{X_{2}} \cdot (A_{21} \nabla_{X_{1}} u_{2}) = 0 in Ω, \end{matrix}$ which is also held thanks to (4.9) and by consequence $u_{3} = 0$ . Following the same argument we end up with $\begin{matrix} u_{2 k + 1} = 0, \forall k \in N, \end{matrix}$ and consequently the even order coefficients $u_{2 k}$ , $k \in N ∖ {0}$ are solutions to the system $\begin{matrix} \{\begin{matrix} - \nabla_{X_{2}} \cdot (A_{22} \nabla_{X_{2}} u_{2 k} (X_{1}, \cdot)) = \nabla_{X_{1}} \cdot (A_{11} \nabla_{X_{1}} u_{2 k - 2} (X_{1}, \cdot)) & in ω_{2}, \\ u_{2 k} (X_{1}, \cdot) \in H_{0}^{1} (ω_{2}), \end{matrix} \end{matrix}$ with assumptions (3.14) for $N = 2 k (k > 0)$ reduced to $\begin{matrix} (4.10) & D_{X_{1}}^{N - 2} A_{11}^{'}, D_{X_{1}}^{N} A_{22} \in L^{\infty} (Ω), D_{X_{1}}^{N} f \in L^{2} (Ω) . \end{matrix}$ Then, under the assumptions (1.2), (4.9), (4.10) for $N = 2 k + 1 (k > 0)$ and thanks to Theorem 3, we have for any $ω_{1}^{'} \subset \subset ω_{1}$ , as $ε \to 0$ , $\begin{array}{l} u_{ε} - (u_{0} + ε^{2} u_{2} + \dots + ε^{2 k} u_{2 k}) = o (ε^{2 k + 1}) in V (ω_{1}^{'} \times ω_{2}), \\ \nabla_{X_{1}} [u_{ε} - (u_{0} + ε^{2} u_{2} + \dots + ε^{2 k} u_{2 k})] = o (ε^{2 k}) in L^{2} (ω_{1}^{'} \times ω_{2}) . \end{array}$ Moreover if (4.10) is assumed to be held for $N = 2 k + 2 (k > 0)$ (respect. for $N = 2 k + 3$ ), then in view of Remark 7, we have for any $ω_{1}^{'} \subset \subset ω_{1}$ , $\begin{array}{l} u_{ε} - (u_{0} + ε^{2} u_{2} + \dots + ε^{2 k} u_{2 k}) = O (ε^{2 k + 2}) in V (ω_{1}^{'} \times ω_{2}), \\ \frac{1}{ε^{2 k + 1}} \nabla_{X_{1}} [u_{ε} - (u_{0} + ε^{2} u_{2} + \dots + ε^{2 k} u_{2 k})] ⇀ 0 in L^{2} (ω_{1}^{'} \times ω_{2}), \end{array}$ respect $\begin{matrix} u_{ε} - (u_{0} + ε^{2} u_{2} + \dots + ε^{2 k} u_{2 k}) = O (ε^{2 k + 2}) in H^{1} (ω_{1}^{'} \times ω_{2}) . \end{matrix}$

Remark 8.
Since we have $\begin{matrix} \nabla_{X_{1}} \cdot (A_{12} \nabla_{X_{2}} v) + \nabla_{X_{2}} \cdot (A_{21} \nabla_{X_{1}} v) = 0, \forall v \in D (Ω), \end{matrix}$ if the matrix A satisfies $\begin{matrix} A_{12} = - A_{21}^{T}, A_{12}^{'} = 0, \end{matrix}$ the matrices A and $(\begin{matrix} A_{11} & 0 \\ 0 & A_{22} \end{matrix})$ define the same problem (1.4) and by consequence the subsection result maintains for the above case.

4.3. Polynomial data

In the present subsection we consider a polynomial structure of the data as follows. We suppose that the matrix A still satisfied the hypotheses (1.1), (1.2) and moreover $\begin{matrix} (4.11) & A = (\begin{matrix} A_{11} (X_{1}, X_{2}) & A_{12} (X_{2}) \\ A_{21} (X_{2}) & A_{22} (X_{2}) \end{matrix}), \end{matrix}$ where $A_{11}$ is a first order polynomial in $X_{1}$ and f is a polynomial of degree $k \in N ∖ {0}$ in $X_{1}$ , i.e. $\begin{matrix} (4.12) & f (X_{1}, X_{2}) = \sum_{| α | ⩽ k} X_{1}^{α} f_{α} (X_{2}), f_{α} \in L^{2} (ω_{2}), \end{matrix}$ where $X_{1}^{α} : = x_{1}^{α_{1}} \dots x_{p}^{α_{p}}$ , for some multi-index $α \in N^{p}$ . Let $u_{0}^{α}$ be the unique solution to the following elliptic problem, $\begin{matrix} \{\begin{matrix} - \nabla_{X_{2}} \cdot (A_{22} \nabla_{X_{2}} u_{0}^{α}) = f_{α} & in ω_{2}, \\ u_{0}^{α} = 0 & on \partial ω_{2} . \end{matrix} \end{matrix}$ Thanks to the linearity of the problems, the solution of problem (2.1) can be expressed as $\begin{matrix} (4.13) & u_{0} (X_{1}, X_{2}) = \sum_{| α | ⩽ k} X_{1}^{α} u_{0}^{α} (X_{2}), \end{matrix}$ which is also a polynomial of degree k in $X_{1}$ . This means that if the right-hand side of the problem (2.1) is a polynomial of degree k in $X_{1}$ the solution is also a polynomial of the same degree in $X_{1}$ . Going back now to problem (2.6), for $u_{0}$ defined in (4.13), we deduce that $u_{1}$ is a polynomial of degree at most $k - 1$ in $X_{1}$ . Next regarding (2.7) rewritten for $N = 2$ , $u_{2}$ is a polynomial of degree at most $k - 1$ also, i.e. we cannot guarantee the reduction of the polynomial degree in this step. Applying again the same argument we can see that $u_{3}$ and $u_{4}$ are polynomials of degree at most $k - 2$ in $X_{1}$ . For a general statement we can show that $u_{2 m - 1}$ and $u_{2 m}$ are polynomials of degree at most $k - m$ in $X_{1}$ for $m = 1, \dots, k$ . This leads to end with two coefficients $u_{2 k - 1}$ and $u_{2 k}$ of (2.4) that are independent of $X_{1}$ . Consequently it follows that $u_{2 k + 1} = u_{2 k + 2} = 0$ and thanks to Theorem 3 we end up with an exponential rate of convergence. To summarize we have

Corollary 3.
Under the assumptions ( 1.1 ), ( 1.2 ), ( 4.11 ) and ( 4.12 ), there exists a constant $η > 0$ independent of $ε > 0$ such that $\begin{matrix} u_{ε} - (u_{0} + ε u_{1} + \dots + ε^{2 k} u_{2 k}) = O (exp (\frac{- η}{ε})) in H^{1} (ω_{1}^{'} \times ω_{2}), \end{matrix}$ for any $ω_{1}^{'} \subset \subset ω_{1}$ .
Remark 9.
If the matrix A is independent of $X_{1}$ the above development will be reduced as $\begin{matrix} u_{ε} - (u_{0} + ε u_{1} + \dots + ε^{k} u_{k}) = O (exp (\frac{- η}{ε})) in H^{1} (ω_{1}^{'} \times ω_{2}) . \end{matrix}$
Remark 10.
In the case where the function f is independent of $X_{1}$ and the matrix A satisfies (1.1), (1.2) and $\begin{matrix} A = (\begin{matrix} A_{11} (X_{1}, X_{2}) & A_{12} (X_{1}, X_{2}) \\ A_{21} (X_{1}, X_{2}) & A_{22} (X_{2}) \end{matrix}) \end{matrix}$ we can get the following exponential rate of convergence, when $ε ⟶ 0$ , $\begin{matrix} u_{ε} - u_{0} = O (exp (\frac{- η}{ε})) in H^{1} (ω_{1}^{'} \times ω_{2}), \end{matrix}$ for some constant $η > 0$ independent of $ε > 0$ iff the limit $u_{0}$ , which is independent of $X_{1}$ , satisfies $\begin{matrix} \nabla_{X_{1}} \cdot (A_{12} \nabla_{X_{2}} u_{0}) = 0 in Ω . \end{matrix}$ The last hypothesis holds for example if $A_{12}^{'} = 0$ .

4.4. Special example

In the last two cases as well as in many works in the bibliography, some symmetries properties are always the unique concrete examples for which the significant improvement (as the exponential rate of convergence) takes place. The following example shows that the exponential rate of convergence can be maintained for a different class of problems. We take, $\begin{array}{l} Ω = & ω_{1} \times ω_{2} = (0, 1) \times (0, π), X_{1} = x_{1}, X_{2} = x_{2}, \\ A = & (\begin{matrix} \frac{m e^{- x_{1}} cos \frac{x_{2}}{4}}{cos (\frac{1}{2} e^{x_{1}} sin x_{2})} & \frac{x_{1} e^{- x_{1}}}{cos (\frac{1}{2} e^{x_{1}} sin x_{2})} \\ \frac{- e^{- x_{1}}}{cos (\frac{1}{2} e^{x_{1}} sin x_{2})} & 2 m (1 + x_{1} + x_{2}) \end{matrix}), \\ A_{11} = & \frac{m e^{- x_{1}} cos \frac{x_{2}}{4}}{cos (\frac{1}{2} e^{x_{1}} sin x_{2})}, A_{12} = \frac{x_{1} e^{- x_{1}}}{cos (\frac{1}{2} e^{x_{1}} sin x_{2})}, A_{21} = \frac{- e^{- x_{1}}}{cos (\frac{1}{2} e^{x_{1}} sin x_{2})}, \\ A_{22} = & 2 m (1 + x_{1} + x_{2}) . \\ f = & m e^{x_{1}} {[(1 + x_{1} + x_{2}) sin x_{2} - cos x_{2}] cos (\frac{1}{2} e^{x_{1}} sin x_{2}) \\ + \frac{1}{2} e^{x_{1}} (1 + x_{1} + x_{2}) {cos}^{2} x_{2} sin (\frac{1}{2} e^{x_{1}} sin x_{2})} . \end{array}$ For a sufficient large positive integer m, the matrix A satisfies the ellipticity assumption (1.2) and of course there exists a unique $u_{ε} \in H_{0}^{1} (Ω)$ solution to (1.4). The limit solution to (2.1) can be found explicitly, i.e. $\begin{matrix} u_{0} = sin (\frac{1}{2} e^{x_{1}} sin x_{2}) \in H_{0}^{1} (ω_{2}), for a.e. x_{1} \in (0, 1) . \end{matrix}$ For this choice the identities (4.2) and (4.4) are fulfilled for $N = 0$ , and thanks to Theorem 3 we end up with the exponential rate of convergence $\begin{matrix} u_{ε} - u_{0} = O (exp (\frac{- η}{ε})) in H^{1} (ω_{1}^{'} \times ω_{2}) \end{matrix}$ for some $η > 0$ .

Footnotes

Acknowledgements

The second author has been supported by Qassim university. He is very grateful to this institution and its deanship of scientific research.

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