Asymptotic behaviour of some anisotropic problems

Abstract

The goal of this paper is to explore the asymptotic behaviour of anisotropic problems governed by operators of the pseudo p-Laplacian type when the size of the domain goes to infinity in different directions.

Keywords

Anisotropic operators nonlinear elliptic operators asymptotic behaviour cylinder like domains

1. Basic notation

When Ω is a bounded open set of $R^{n}$ , we denote by $W_{0}^{1, r} (Ω)$ , $r > 1$ , the usual Sobolev space constructed on $L^{r} (Ω)$ , of functions vanishing on the boundary of Ω. That is to say we set $\begin{matrix} (1.1) & W^{1, r} (Ω) = {v \in L^{r} (Ω) | \partial_{x_{i}} v \in L^{r} (Ω) \forall i = 1, \dots, n} . \end{matrix}$ We equip this space with the norm $\begin{matrix} (1.2) & ‖ v ‖_{1, r, Ω} = {(\int_{Ω} | v |^{r} + \sum_{i = 1}^{n} | \partial_{x_{i}} v |^{r} d x)}^{\frac{1}{r}} \end{matrix}$ and we set $\begin{matrix} (1.3) & W_{0}^{1, r} (Ω) = \overline{D (Ω)} = the closure of D (Ω) in W^{1, r} (Ω) . \end{matrix}$ ( $D (Ω)$ denotes the space of $C^{\infty}$ -functions with compact support in Ω). It is well known that $W_{0}^{1, r} (Ω)$ is a reflexive Banach space which can be equipped with the equivalent norm $\begin{matrix} (1.4) & ‖ \nabla v ‖_{r, Ω} = {(\int_{Ω} {| \nabla v (x) |}^{r} d x)}^{\frac{1}{r}} . \end{matrix}$ (∇ denotes the usual gradient an $| |$ the euclidean norm, i.e. $| \nabla v (x) | = {(\sum_{1}^{n} {(\partial_{x_{i}} v)}^{2})}^{\frac{1}{2}}$ , $| |_{r, Ω}$ denotes the $L^{r}$ -norm on Ω). The dual of $W_{0}^{1, r} (Ω)$ is denoted by $W^{- 1, r^{'}} (Ω)$ , $r^{'} = \frac{r}{r - 1}$ and consists in the distributions of the form $\begin{matrix} (1.5) & f = f_{0} - \sum_{i = 1}^{n} \partial_{x_{i}} f_{i}, f_{i} \in L^{r^{'}} (Ω) . \end{matrix}$ We use the notation $\begin{matrix} (1.6) & ⟨ f, v ⟩ = \int_{Ω} f_{0} v + \sum_{i = 1}^{n} f_{i} \partial_{x_{i}} v d x . \end{matrix}$ The paper is organised as follows. In the next section we address the case of a simple problem set on a rectangle with one side going to infinity. We consider all the possible values of $(p, q)$ , $p, q > 1$ for the pseudo $(p, q)$ -operator at hand allowing to present a variety of techniques. Some of them are issued of previous works. We refer the reader to [3,5,10,11,15], for details. The Section 3 relies on the experience acquired on the simple model investigated in Section 1 to extend some results to more complex situations. The operators at hand are Euler equations of some anisotropic functionals of calculus of variations introduced for other reasons in [17], see also [18]. For basic notions on Sobolev spaces we refer to [1,12–14,16].

2. A model problem

We denote by $Ω_{ℓ}$ the open subset of $R^{2}$ defined as $\begin{matrix} (2.1) & Ω_{ℓ} = (- ℓ, ℓ) \times (- 1, 1) . \end{matrix}$ We will set $ω = (- 1, 1)$ and $\partial Ω_{ℓ}$ will denote the boundary of $Ω_{ℓ}$ , see the Fig. 1 below.

Fig. 1.

The domain $Ω_{ℓ}$ .

If $p, q > 1$ are two positive numbers we would like to consider $u_{ℓ}$ solution to $\begin{matrix} (2.2) & \{\begin{array}{l} - \partial_{x_{1}} (| \partial_{x_{1}} u_{ℓ} |^{p - 2} \partial_{x_{1}} u_{ℓ}) - \partial_{x_{2}} (| \partial_{x_{2}} u_{ℓ} |^{q - 2} \partial_{x_{2}} u_{ℓ}) = f in Ω_{ℓ}, \\ u_{ℓ} = 0 on \partial Ω_{ℓ} . \end{array} \end{matrix}$ More precisely we are interested to the asymptotic behaviour of $u_{ℓ}$ when $ℓ \to + \infty$ . f is a function or distribution depending only on $x_{2}$ . A natural candidate for the limit of the problem is $u_{\infty}$ solution to $\begin{matrix} (2.3) & \{\begin{array}{l} - \partial_{x_{2}} (| \partial_{x_{2}} u_{\infty} |^{q - 2} \partial_{x_{2}} u_{\infty}) = f in ω, \\ u_{\infty} = 0 on \partial ω, \end{array} \end{matrix}$ where $\partial ω = {- 1, 1}$ is the boundary of ω. First let us recast these problems under their natural weak form.

We can first introduce the weak formulation of (2.3). If $f \in W^{- 1, q^{'}} (ω)$ is given by $\begin{matrix} (2.4) & f = f (x_{2}) = f_{0} (x_{2}) - \partial_{x_{2}} f_{1} (x_{2}), \end{matrix}$ where $f_{0}, f_{1} \in L^{q^{'}} (ω)$ then, the weak formulation to (2.3) corresponding to f reads $\begin{matrix} (2.5) & \{\begin{array}{l} u_{\infty} \in W_{0}^{1, q} (ω), \\ \int_{ω} | \partial_{x_{2}} u_{\infty} |^{q - 2} \partial_{x_{2}} u_{\infty} \partial_{x_{2}} v d x_{2} = ⟨ f, v ⟩ = \int_{ω} f_{0} v + f_{1} \partial_{x_{2}} v d x_{2} \forall v \in W_{0}^{1, q} (ω) . \end{array} \end{matrix}$ To arrive to a weak formulation for (2.2) one introduces $\begin{matrix} (2.6) & W^{1, p, q} (Ω_{ℓ}) = {v \in L^{p} (Ω_{ℓ}) \cap L^{q} (Ω_{ℓ}) | \partial_{x_{1}} v \in L^{p} (Ω_{ℓ}), \partial_{x_{2}} v \in L^{q} (Ω_{ℓ})} . \end{matrix}$ It is a reflexive Banach space when equipped with the norm $\begin{matrix} (2.7) & ‖ v ‖_{1, p, q, Ω_{ℓ}} = | v |_{p, Ω_{ℓ}} + | v |_{q, Ω_{ℓ}} + | \partial_{x_{1}} v |_{p, Ω_{ℓ}} + | \partial_{x_{2}} v |_{q, Ω_{ℓ}} . \end{matrix}$ Then we define $\begin{matrix} (2.8) & W_{0}^{1, p, q} (Ω_{ℓ}) = \overline{D (Ω_{ℓ})} = the closure of D (Ω_{ℓ}) in W^{1, p, q} (Ω_{ℓ}) . \end{matrix}$ If f is defined by (2.4) if follows easily that there exists a unique $u_{ℓ}$ weak solution to (2.2) i.e. satisfying $\begin{matrix} (2.9) & \{\begin{array}{l} u_{ℓ} \in W_{0}^{1, p, q} (Ω_{ℓ}), \\ \int_{Ω_{ℓ}} | \partial_{x_{1}} u_{ℓ} |^{p - 2} \partial_{x_{1}} u_{ℓ} \partial_{x_{1}} v + | \partial_{x_{2}} u_{ℓ} |^{q - 2} \partial_{x_{2}} u_{ℓ} \partial_{x_{2}} v d x_{1} d x_{2} \\ = ⟨ f, v ⟩ = \int_{Ω_{ℓ}} f_{0} v + f_{1} \partial_{x_{2}} v d x_{1} d x_{2} \forall v \in W_{0}^{1, p, q} (Ω_{ℓ}) . \end{array} \end{matrix}$ We are interested in showing that $u_{ℓ} \to u_{\infty}$ when $ℓ \to \infty$ , but also to investigate at what speed. We will now denote $d x_{1} d x_{2}$ by $d x$ .

The operators defined by (2.2), (2.3) are strictly monotone, hemicontinuous, coercive from $W_{0}^{1, p, q} (Ω_{ℓ})$ , $W_{0}^{1, q} (ω)$ into their duals. Existence and uniqueness of a solution for (2.9), (2.5) follows from classical arguments (see [4,12,16]).

Let us first prove the following lemma.

Lemma 2.1.

Suppose that f is given by ( 2.4 ). If $u_{ℓ}$ is the solution to ( 2.9 ) there exists a constant C independent of ℓ such that $\begin{matrix} (2.10) & \int_{Ω_{ℓ}} | \partial_{x_{1}} u_{ℓ} |^{p} + | \partial_{x_{2}} u_{ℓ} |^{q} d x ⩽ C ℓ . \end{matrix}$

Proof.

Taking $v = u_{ℓ}$ in (2.9) we get $\begin{matrix} (2.11) & \begin{aligned} \int_{Ω_{ℓ}} | \partial_{x_{1}} u_{ℓ} |^{p} + | \partial_{x_{2}} u_{ℓ} |^{q} d x & = ⟨ f, u_{ℓ} ⟩ = \int_{Ω_{ℓ}} f_{0} u_{ℓ} + f_{1} \partial_{x_{2}} u_{ℓ} d x \\ ⩽ | f_{0} |_{q^{'}, Ω_{ℓ}} | u_{ℓ} |_{q, Ω_{ℓ}} + | f_{1} |_{q^{'}, Ω_{ℓ}} | \partial_{x_{2}} u_{ℓ} |_{q, Ω_{ℓ}} \\ ⩽ (C | f_{0} |_{q^{'}, Ω_{ℓ}} + | f_{1} |_{q^{'}, Ω_{ℓ}}) | \partial_{x_{2}} u_{ℓ} |_{q, Ω_{ℓ}} \end{aligned} \end{matrix}$ this by the Hölder and the Poincaré inequality. Let us recall regarding this last point an argument that we will use several times later on. If $u \in W_{0}^{1, p, q} (Ω_{ℓ})$ , let $u_{n} \in D (Ω_{ℓ})$ such that $u_{n} \to u$ in $W_{0}^{1, p, q} (Ω_{ℓ})$ . By the Poincaré inequality on ω one has for some constant C independent of ℓ $\begin{matrix} \int_{ω} {| u_{n} (x_{1}, x_{2}) |}^{q} d x_{2} ⩽ C^{q} \int_{ω} {| \partial_{x_{2}} u_{n} (x_{1}, x_{2}) |}^{q} d x_{2}, a.e. x_{1} \in (- ℓ, ℓ) . \end{matrix}$ Integrating in $x_{1}$ we deduce $\begin{matrix} | u_{n} |_{q, Ω_{ℓ}} ⩽ C | \partial_{x_{2}} u_{n} |_{q, Ω_{ℓ}} \end{matrix}$ and passing to the limit in n the same inequality holds for u or $u_{ℓ}$ . Then let us notice that for $i = 0, 1$ one has $\begin{matrix} | f_{i} |_{q^{'}, Ω_{ℓ}} = {(\int_{- ℓ}^{ℓ} \int_{ω} {| f_{i} (x_{2}) |}^{q^{'}} d x_{2} d x_{1})}^{\frac{1}{q^{'}}} = {(2 ℓ)}^{\frac{1}{q^{'}}} | f_{i} |_{q^{'}, ω} . \end{matrix}$ Thus from (2.11) we derive for some constant $C = C (q, f)$ $\begin{matrix} | \partial_{x_{2}} u_{ℓ} |_{q, Ω_{ℓ}}^{q} ⩽ C ℓ^{\frac{1}{q^{'}}} | \partial_{x_{2}} u_{ℓ} |_{q, Ω_{ℓ}} \end{matrix}$ Since $q^{'} = \frac{q}{q - 1}$ this is equivalent for some new constant to $\begin{matrix} | \partial_{x_{2}} u_{ℓ} |_{q, Ω_{ℓ}} ⩽ C ℓ^{\frac{1}{q}} . \end{matrix}$ Going back to (2.11), the result follows. □

Somehow one can ignore f thanks to the following remark.

Lemma 2.2.

If $u_{ℓ}$ is the solution to ( 2.9 ) and $u_{\infty}$ solution to ( 2.5 ) one has $\begin{matrix} (2.12) & \begin{aligned} \int_{Ω_{ℓ}} | \partial_{x_{1}} u_{ℓ} |^{p - 2} \partial_{x_{1}} u_{ℓ} \partial_{x_{1}} v + {| \partial_{x_{2}} u_{ℓ} |^{q - 2} \partial_{x_{2}} u_{ℓ} - | \partial_{x_{2}} u_{\infty} |^{q - 2} \partial_{x_{2}} u_{\infty}} \partial_{x_{2}} v d x = 0 \\ \forall v \in W_{0}^{1, p, q} (Ω_{ℓ}) . \end{aligned} \end{matrix}$

Proof.

First by (2.9) if $v \in W_{0}^{1, p, q} (Ω_{ℓ})$ one has $\begin{matrix} (2.13) & \int_{Ω_{ℓ}} | \partial_{x_{1}} u_{ℓ} |^{p - 2} \partial_{x_{1}} u_{ℓ} \partial_{x_{1}} v + | \partial_{x_{2}} u_{ℓ} |^{q - 2} \partial_{x_{2}} u_{ℓ} \partial_{x_{2}} v = \int_{Ω_{ℓ}} f_{0} v + f_{1} \partial_{x_{2}} v d x \end{matrix}$ If $v \in W_{0}^{1, p, q} (Ω_{ℓ})$ one has for almost every $x_{1}$ $\begin{matrix} v (x_{1}, \cdot) \in W_{0}^{1, q} (ω) . \end{matrix}$ Thus by (2.5) $\begin{matrix} \int_{ω} | \partial_{x_{2}} u_{\infty} |^{q - 2} \partial_{x_{2}} u_{\infty} \partial_{x_{2}} v (x_{1}, x_{2}) d x_{2} = \int_{ω} f_{0} v + f_{1} \partial_{x_{2}} v d x_{2} . \end{matrix}$ Integrating in $x_{1}$ it comes $\begin{matrix} (2.14) & \int_{Ω_{ℓ}} | \partial_{x_{2}} u_{\infty} |^{q - 2} \partial_{x_{2}} u_{\infty} \partial_{x_{2}} v d x = \int_{Ω_{ℓ}} f_{0} v + f_{1} \partial_{x_{2}} v d x . \end{matrix}$ Subtracting from (2.13), (2.12) follows. □

Let us recall the following result (see [4,6]) which garanties also the strict monotonicity of the operators at hand.

Lemma 2.3.

For any $q > 1$ there exist positive constants $c_{q}$ , $C_{q}$ such that $\begin{array}{c} (2.15) & | | ξ |^{q - 2} ξ - | η |^{q - 2} η | ⩽ C_{q} | ξ - η | {(| ξ | + | η |)}^{q - 2} \forall ξ, η \in R^{n}, \\ (2.16) & (| ξ |^{q - 2} ξ - | η |^{q - 2} η) \cdot (ξ - η) ⩾ c_{q} | ξ - η |^{2} {(| ξ | + | η |)}^{q - 2} \forall ξ, η \in R^{n} . \end{array}$

Then one has:

Lemma 2.4.

Let $u_{ℓ} = u_{ℓ} (f)$ be the solution to ( 2.9 ) and $u_{\infty} = u_{\infty} (f)$ be the solution to ( 2.5 ). Suppose that $f_{1} ⩾ f_{2}$ , $f ⩾ 0$ then one has $\begin{matrix} (2.17) & u_{ℓ} (f_{2}) ⩽ u_{ℓ} (f_{1}), 0 ⩽ u_{ℓ} (f) ⩽ u_{\infty} (f) . \end{matrix}$ (If f is not a function, $f ⩾ 0$ means $⟨ f, v ⟩ ⩾ 0 \forall v \in W_{0}^{1, q} (ω)$ , $v ⩾ 0$ ).

Proof.

We use the notation $u_{i} = u_{ℓ} (f_{i})$ . From (2.9) by subtraction we get $\begin{aligned} \int_{Ω_{ℓ}} {| \partial_{x_{1}} u_{2} |^{p - 2} \partial_{x_{1}} u_{2} - | \partial_{x_{1}} u_{1} |^{p - 2} \partial_{x_{1}} u_{1}} \partial_{x_{1}} v \\ + {| \partial_{x_{2}} u_{2} |^{q - 2} \partial_{x_{2}} u_{2} - | \partial_{x_{2}} u_{1} |^{q - 2} \partial_{x_{1}} u_{1}} \partial_{x_{2}} v = ⟨ f_{2} - f_{1}, v ⟩ . \end{aligned}$ Taking $v = {(u_{2} - u_{1})}^{+}$ one deduces easily using the lemma 2.3 that ${(u_{2} - u_{1})}^{+} = 0$ i.e. $u_{1} ⩾ u_{2}$ (see below (2.18) for a similar argument).

If $f ⩾ 0$ taking $f_{1} = f$ , $f_{2} = 0$ one gets $0 ⩽ u_{ℓ} (f)$ .

Regarding $u_{\infty}$ , taking $v = u_{\infty}^{-}$ in (2.5) one gets for $f ⩾ 0$ $\begin{matrix} \int_{Ω_{ℓ}} | \partial_{x_{1}} u_{\infty} |^{p - 2} \partial_{x_{1}} u_{\infty} \partial_{x_{1}} u_{\infty}^{-} + | \partial_{x_{2}} u_{\infty} |^{q - 2} \partial_{x_{2}} u_{\infty} \partial_{x_{2}} u_{\infty}^{-} = \int_{Ω_{ℓ}} f_{0} u_{\infty}^{-} + f_{1} \partial_{x_{2}} u_{\infty}^{-} d x ⩾ 0 . \end{matrix}$ This reads also $\begin{matrix} \int_{Ω_{ℓ}} {| \partial_{x_{1}} u_{\infty}^{-} |}^{p - 2} \partial_{x_{1}} u_{\infty}^{-} \partial_{x_{1}} u_{\infty}^{-} + {| \partial_{x_{2}} u_{\infty}^{-} |}^{q - 2} \partial_{x_{2}} u_{\infty}^{-} \partial_{x_{2}} u_{\infty}^{-} ⩽ 0 . \end{matrix}$ Thus $u_{\infty}^{-} = 0$ and $u_{\infty} (f) ⩾ 0$ . Then taking $v = {(u_{ℓ} - u_{\infty})}^{+}$ in (2.12) one gets $\begin{matrix} (2.18) & \begin{aligned} \int_{Ω_{ℓ}} | \partial_{x_{1}} u_{ℓ} |^{p - 2} \partial_{x_{1}} u_{ℓ} \partial_{x_{1}} {(u_{ℓ} - u_{\infty})}^{+} \\ + {| \partial_{x_{2}} u_{ℓ} |^{q - 2} \partial_{x_{2}} u_{ℓ} - | \partial_{x_{2}} u_{\infty} |^{q - 2} \partial_{x_{2}} u_{\infty}} \partial_{x_{2}} {(u_{ℓ} - u_{\infty})}^{+} d x = 0 \end{aligned} \end{matrix}$ i.e. $\begin{aligned} \int_{{u_{ℓ} - u_{\infty} > 0}} | \partial_{x_{1}} u_{ℓ} |^{p - 2} \partial_{x_{1}} u_{ℓ} \partial_{x_{1}} u_{ℓ} \\ + {| \partial_{x_{2}} u_{ℓ} |^{q - 2} \partial_{x_{2}} u_{ℓ} - | \partial_{x_{2}} u_{\infty} |^{q - 2} \partial_{x_{2}} u_{\infty}} \partial_{x_{2}} (u_{ℓ} - u_{\infty}) d x = 0 . \end{aligned}$ This implies that the set ${u_{ℓ} - u_{\infty} > 0}$ is of measure 0 since $\nabla (u_{ℓ} - u_{\infty}) = 0$ on this set i.e. $\nabla {(u_{ℓ} - u_{\infty})}^{+} = 0$ (see [14]). This completes the proof of the Lemma. □

Let us now show:

Lemma 2.5.

If $u_{ℓ}$ is the solution to ( 2.9 ) and $u_{\infty}$ solution to ( 2.5 ) one has for every smooth function $φ = φ (x_{1})$ vanishing at ${- ℓ . ℓ}$ $\begin{matrix} (2.19) & \begin{aligned} \int_{Ω_{ℓ}} {| \partial_{x_{1}} u_{ℓ} |^{p} + (| \partial_{x_{2}} u_{ℓ} |^{q - 2} \partial_{x_{2}} u_{ℓ} - | \partial_{x_{2}} u_{\infty} |^{q - 2} \partial_{x_{2}} u_{\infty}) \partial_{x_{2}} (u_{ℓ} - u_{\infty})} φ d x \\ ⩽ \int_{Ω_{ℓ}} | \partial_{x_{1}} u_{ℓ} |^{p - 1} | \partial_{x_{1}} φ | | u_{ℓ} - u_{\infty} | d x . \end{aligned} \end{matrix}$

Proof.

Taking $v = (u_{ℓ} - u_{\infty}) φ$ in (2.12) one gets $\begin{matrix} (2.20) & \begin{aligned} \int_{Ω_{ℓ}} {| \partial_{x_{1}} u_{ℓ} |^{p} + (| \partial_{x_{2}} u_{ℓ} |^{q - 2} \partial_{x_{2}} u_{ℓ} - | \partial_{x_{2}} u_{\infty} |^{q - 2} \partial_{x_{2}} u_{\infty}) \partial_{x_{2}} (u_{ℓ} - u_{\infty})} φ d x \\ = - \int_{Ω_{ℓ}} | \partial_{x_{1}} u_{ℓ} |^{p - 2} \partial_{x_{1}} u_{ℓ} \partial_{x_{1}} φ (u_{ℓ} - u_{\infty}) d x . \end{aligned} \end{matrix}$ (Recall that $u_{\infty}$ is independent of $x_{1}$ ). Then (2.19) follows easily. □

Denote by $ρ = ρ (x_{1})$ a smooth function such that $\begin{matrix} (2.21) & 0 ⩽ ρ ⩽ 1, ρ = 1 on (- \frac{1}{2}, \frac{1}{2}), ρ = 0 near {- 1, 1}, | \partial_{x_{1}} ρ | ⩽ C \end{matrix}$ and set $\begin{matrix} φ = ρ^{α} = ρ^{α} (\frac{x_{1}}{ℓ}), \end{matrix}$ where $α > 0$ . We can now prove:

Lemma 2.6.

Let $f = f_{0} \in L^{q^{'}} (ω)$ and $u_{ℓ}$ , $u_{\infty}$ be the solutions to ( 2.9 ), ( 2.5 ). Then it holds for some constant C independent of ℓ $\begin{matrix} (2.22) & I = \int_{Ω_{ℓ}} {| \partial_{x_{1}} u_{ℓ} |^{p} + (| \partial_{x_{2}} u_{ℓ} |^{q - 2} \partial_{x_{2}} u_{ℓ} - | \partial_{x_{2}} u_{\infty} |^{q - 2} \partial_{x_{2}} u_{\infty}) \partial_{x_{2}} (u_{ℓ} - u_{\infty})} ρ^{α} d x ⩽ \frac{C}{ℓ^{p - 1}} . \end{matrix}$

Proof.

From (2.19) one derives $\begin{matrix} (2.23) & \begin{aligned} \int_{Ω_{ℓ}} {| \partial_{x_{1}} u_{ℓ} |^{p} + (| \partial_{x_{2}} u_{ℓ} |^{q - 2} \partial_{x_{2}} u_{ℓ} - | \partial_{x_{2}} u_{\infty} |^{q - 2} \partial_{x_{2}} u_{\infty}) \partial_{x_{2}} (u_{ℓ} - u_{\infty})} ρ^{α} d x \\ ⩽ \frac{α C}{ℓ} \int_{Ω_{ℓ}} | \partial_{x_{1}} u_{ℓ} |^{p - 1} | u_{ℓ} - u_{\infty} | ρ^{α - 1} d x . \end{aligned} \end{matrix}$ Noting that $ρ^{α - 1} = ρ^{\frac{α}{p^{'}}} ρ^{\frac{α}{p} - 1}$ and using Hölder’s inequality it comes $\begin{matrix} (2.24) & \begin{aligned} I = & \int_{Ω_{ℓ}} {| \partial_{x_{1}} u_{ℓ} |^{p} + (| \partial_{x_{2}} u_{ℓ} |^{q - 2} \partial_{x_{2}} u_{ℓ} - | \partial_{x_{2}} u_{\infty} |^{q - 2} \partial_{x_{2}} u_{\infty}) \partial_{x_{2}} (u_{ℓ} - u_{\infty})} ρ^{α} d x \\ ⩽ \frac{α C}{ℓ} {(\int_{Ω_{ℓ}} | \partial_{x_{1}} u_{ℓ} |^{p} ρ^{α} d x)}^{\frac{1}{p^{'}}} {(\int_{Ω_{ℓ}} | u_{ℓ} - u_{\infty} |^{p} ρ^{α - p} d x)}^{\frac{1}{p}} . \end{aligned} \end{matrix}$ Thus it follows that $\begin{matrix} (2.25) & I ⩽ {(\frac{α C}{ℓ})}^{p} \int_{Ω_{ℓ}} | u_{ℓ} - u_{\infty} |^{p} ρ^{α - p} d x ⩽ {(\frac{α C}{ℓ})}^{p} \int_{Ω_{ℓ}} | u_{ℓ} - u_{\infty} |^{p} d x, \end{matrix}$ provided we chose $α > p$ . From the lemma 2.4 one has $\begin{matrix} u_{ℓ} (f) ⩽ u_{ℓ} (f^{+}) ⩽ u_{\infty} (f^{+}), u_{\infty} (- f^{-}) ⩽ u_{ℓ} (- f^{-}) ⩽ u_{ℓ} (f), \end{matrix}$ (notice that $u_{ℓ} (- f) = - u_{ℓ} (f)$ ). Then one derives $\begin{matrix} | u_{ℓ} - u_{\infty} | ⩽ | u_{ℓ} | + | u_{\infty} | ⩽ max {u_{\infty} (f^{+}), u_{\infty} (f^{-})} + | u_{\infty} (f) | . \end{matrix}$ Since this last function is independent of $x_{1}$ one derives from (2.25) $\begin{matrix} I = \int_{Ω_{ℓ}} {| \partial_{x_{1}} u_{ℓ} |^{p} + (| \partial_{x_{2}} u_{ℓ} |^{q - 2} \partial_{x_{2}} u_{ℓ} - | \partial_{x_{2}} u_{\infty} |^{q - 2} \partial_{x_{2}} u_{\infty}) \partial_{x_{2}} (u_{ℓ} - u_{\infty})} ρ^{α} d x ⩽ \frac{C}{ℓ^{p - 1}} \end{matrix}$ for some new constant C. This is (2.22). This completes the proof of the lemma. □

We give now a very simple proof of the convergence of $u_{ℓ}$ toward $u_{\infty}$ which is valid for every p and q.

Theorem 2.1.

Let $f = f_{0} \in L^{q^{'}} (ω)$ and $u_{ℓ}$ , $u_{\infty}$ be the solutions to ( 2.9 ), ( 2.5 ). Then for any $ℓ_{0}$ it holds when $ℓ \to + \infty$ $\begin{matrix} (2.26) & \partial_{x_{1}} u_{ℓ} \to 0 in L^{p} (Ω_{ℓ_{0}}), \partial_{x_{2}} u_{ℓ} \to \partial_{x_{2}} u_{\infty} in L^{q} (Ω_{ℓ_{0}}) . \end{matrix}$

Proof.

The first part of (2.26) follows immediately from (2.22) if one chooses $\frac{ℓ}{2} > ℓ_{0}$ . For the second part let us consider a smooth function $ρ = ρ (x_{1})$ such that for $ℓ_{0} < ℓ - 1$ fixed $\begin{matrix} 0 ⩽ ρ ⩽ 1, ρ = 1 on (- ℓ_{0}, ℓ_{0}), ρ has compact support in (- ℓ_{0} - 1, ℓ_{0} + 1), | \partial_{x_{1}} ρ | ⩽ C . \end{matrix}$ Since $ρ^{α} u_{ℓ} \in W_{0}^{1, p, q} (Ω_{ℓ})$ one gets from (2.9) $\begin{aligned} \int_{Ω_{ℓ_{0} + 1}} {| \partial_{x_{1}} u_{ℓ} |^{p} + | \partial_{x_{2}} u_{ℓ} |^{q}} ρ^{α} d x & = - \int_{Ω_{ℓ_{0} + 1}} | \partial_{x_{1}} u_{ℓ} |^{p - 2} \partial_{x_{1}} u_{ℓ} \partial_{x_{1}} ρ^{α} u_{ℓ} + \int_{Ω_{ℓ_{0} + 1}} f_{0} u_{ℓ} ρ^{α} d x \\ ⩽ α C \int_{Ω_{ℓ_{0} + 1}} | \partial_{x_{1}} u_{ℓ} |^{p - 1} | u_{ℓ} | ρ^{α - 1} + \int_{Ω_{ℓ_{0} + 1}} | f_{0} | | u_{ℓ} | d x . \end{aligned}$ Noticing that $ρ^{α - 1} = ρ^{\frac{α}{p^{'}}} ρ^{\frac{α}{p} - 1}$ and using the Young inequality $a b ⩽ \frac{1}{p^{'}} a^{p^{'}} + \frac{1}{p} b^{p}$ , $a, b > 0$ we get for some new constant C $\begin{aligned} \int_{Ω_{ℓ_{0} + 1}} {| \partial_{x_{1}} u_{ℓ} |^{p} + | \partial_{x_{2}} u_{ℓ} |^{q}} ρ^{α} d x \\ ⩽ \frac{1}{p^{'}} \int_{Ω_{ℓ_{0} + 1}} | \partial_{x_{1}} u_{ℓ} |^{p} ρ^{α} d x + C \int_{Ω_{ℓ_{0} + 1}} | u_{ℓ} |^{p} ρ^{α - p} d x + \int_{Ω_{ℓ_{0} + 1}} | f_{0} | | u_{ℓ} | d x . \end{aligned}$ Using the inequality $\begin{matrix} | u_{ℓ} | ⩽ max {u_{\infty} (f_{0}^{+}), u_{\infty} (f_{0}^{-})} \end{matrix}$ which is due to $\begin{matrix} u_{ℓ} (f_{0}) ⩽ u_{ℓ} (f_{0}^{+}) ⩽ u_{\infty} (f_{0}^{+}), u_{ℓ} (- f) = - u_{ℓ} (f), \end{matrix}$ we derive easily taking $α > p$ that $\begin{matrix} \int_{Ω_{ℓ_{0}}} | \partial_{x_{2}} u_{ℓ} |^{q} d x ⩽ C (ℓ_{0}) \end{matrix}$ where $C (ℓ_{0})$ is independent of ℓ. Thus up to a subsequence there exists $v_{\infty} \in L^{q} (Ω_{ℓ_{0}})$ , $w_{\infty} \in L^{q^{'}} (Ω_{ℓ_{0}})$ such that $\begin{matrix} \partial_{x_{2}} u_{ℓ} ⇀ v_{\infty} \in L^{q} (Ω_{ℓ_{0}}), | \partial_{x_{2}} u_{ℓ} |^{q - 2} \partial_{x_{2}} u_{ℓ} ⇀ w_{\infty} \in L^{q^{'}} (Ω_{ℓ_{0}}) . \end{matrix}$ From (2.22) one derives that up to a subsequence $\begin{matrix} {| \partial_{x_{2}} u_{ℓ} |^{q - 2} \partial_{x_{2}} u_{ℓ} - | \partial_{x_{2}} u_{\infty} |^{q - 2} \partial_{x_{2}} u_{\infty}} \partial_{x_{2}} (u_{ℓ} - u_{\infty}) \to 0 a.e in Ω_{ℓ_{0}} . \end{matrix}$ Thus, up to a subsequence, $\partial_{x_{2}} u_{ℓ} \to \partial_{x_{2}} u_{\infty}$ a.e on $Ω_{ℓ_{0}}$ . To see this point one notices that by (2.16) one has $\begin{matrix} \partial_{x_{2}} (u_{ℓ} - u_{\infty}) \partial_{x_{2}} (u_{ℓ} - u_{\infty}) {| \partial_{x_{2}} u_{ℓ} | + | \partial_{x_{2}} u_{\infty} |}^{q - 2} \to 0 a.e in Ω_{ℓ_{0}} . \end{matrix}$ If $\partial_{x_{2}} (u_{ℓ} - u_{\infty}) ↛ 0$ then $\begin{matrix} \partial_{x_{2}} (u_{ℓ} - u_{\infty}) {| \partial_{x_{2}} u_{ℓ} | + | \partial_{x_{2}} u_{\infty} |}^{q - 2} \to 0 a.e in Ω_{ℓ_{0}} \end{matrix}$ and by (2.15) it follows that $\begin{matrix} {| \partial_{x_{2}} u_{ℓ} |^{q - 2} \partial_{x_{2}} u_{ℓ} - | \partial_{x_{2}} u_{\infty} |^{q - 2} \partial_{x_{2}} u_{\infty}} \to 0 a.e in Ω_{ℓ_{0}} \end{matrix}$ and again due to the strict monotonicity of the function $| x |^{q - 2} x$ one has $\partial_{x_{2}} u_{ℓ} \to \partial_{x_{2}} u_{\infty}$ a.e on $Ω_{ℓ_{0}}$ . From this it follows (see for instance [19] lemma 8.3) for a proof that $\begin{matrix} \partial_{x_{2}} u_{ℓ} ⇀ \partial_{x_{2}} u_{\infty} \in L^{q} (Ω_{ℓ_{0}}), | \partial_{x_{2}} u_{ℓ} |^{q - 2} \partial_{x_{2}} u_{ℓ} ⇀ | \partial_{x_{2}} u_{\infty} |^{q - 2} \partial_{x_{2}} u_{\infty} \in L^{q^{'}} (Ω_{ℓ_{0}}) . \end{matrix}$ Now from (2.22) one has $\begin{matrix} \int_{Ω_{ℓ_{0}}} {| \partial_{x_{2}} u_{ℓ} |^{q - 2} \partial_{x_{2}} u_{ℓ} - | \partial_{x_{2}} u_{\infty} |^{q - 2} \partial_{x_{2}} u_{\infty}} \partial_{x_{2}} (u_{ℓ} - u_{\infty}) \to 0, \end{matrix}$ that is $\begin{matrix} \int_{Ω_{ℓ_{0}}} | \partial_{x_{2}} u_{ℓ} |^{q} - | \partial_{x_{2}} u_{ℓ} |^{q - 2} \partial_{x_{2}} u_{ℓ} \partial_{x_{2}} u_{\infty} - | \partial_{x_{2}} u_{\infty} |^{q - 2} \partial_{x_{2}} u_{\infty} \partial_{x_{2}} u_{ℓ} + | \partial_{x_{2}} u_{\infty} |^{q} d x \to 0 . \end{matrix}$ It follows that $\begin{matrix} \int_{Ω_{ℓ_{0}}} | \partial_{x_{2}} u_{ℓ} |^{q} d x \to \int_{Ω_{ℓ_{0}}} | \partial_{x_{2}} u_{\infty} |^{q} d x \end{matrix}$ and the result, i.e. the strong convergence, follows. □

One can estimate the convergence rate in some situations. Indeed one has:

Theorem 2.2.

Let $f = f_{0} \in L^{q^{'}} (ω)$ and $u_{ℓ}$ , $u_{\infty}$ be the solutions to ( 2.9 ), ( 2.5 ). Then it holds for some constant C independent of ℓ $\begin{matrix} (2.27) & \begin{aligned} \int_{Ω_{\frac{ℓ}{2}}} {| \partial_{x_{1}} (u_{ℓ} - u_{\infty}) |}^{p} + {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q} d x & = \int_{Ω_{\frac{ℓ}{2}}} | \partial_{x_{1}} u_{ℓ} |^{p} + {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q} d x \\ ⩽ \frac{C}{ℓ^{\frac{p}{2} q \land 2 - 1}} \end{aligned} \end{matrix}$ where $q \land 2$ denotes the minimum of q and 2.

Proof.

It follows from the lemmas 2.3 and 2.6 that $\begin{matrix} \tilde{I} = \int_{Ω_{ℓ}} {| \partial_{x_{1}} u_{ℓ} |^{p} + {(| \partial_{x_{2}} u_{ℓ} | + | \partial_{x_{2}} u_{\infty} |)}^{q - 2} \partial_{x_{2}} {(u_{ℓ} - u_{\infty})}^{2}} ρ^{α} d x ⩽ \frac{C}{ℓ^{p - 1}} . \end{matrix}$ If $q ⩾ 2$ , since $| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) | ⩽ | \partial_{x_{2}} u_{ℓ} | + | \partial_{x_{2}} u_{\infty} |$ , one derives immediately (2.27), i.e. $\begin{matrix} \int_{Ω_{\frac{ℓ}{2}}} {| \partial_{x_{1}} (u_{ℓ} - u_{\infty}) |}^{p} + {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q} d x = \int_{Ω_{\frac{ℓ}{2}}} | \partial_{x_{1}} u_{ℓ} |^{p} + {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q} d x ⩽ \frac{C}{ℓ^{p - 1}} . \end{matrix}$ If $q < 2$ one has thanks to Hölder’s inequality $\begin{matrix} (2.28) & \begin{aligned} \int_{Ω_{ℓ}} {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q} ρ^{α - q} d x \\ = \int_{Ω_{ℓ}} {(| \partial_{x_{2}} u_{ℓ} | + | \partial_{x_{2}} u_{\infty} |)}^{(q - 2) \frac{q}{2}} {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q} ρ^{α - q} {(| \partial_{x_{2}} u_{ℓ} | + | \partial_{x_{2}} u_{\infty} |)}^{(2 - q) \frac{q}{2}} d x \\ ⩽ {(\int_{Ω_{ℓ}} {(| \partial_{x_{2}} u_{ℓ} | + | \partial_{x_{2}} u_{\infty} |)}^{(q - 2)} {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{2} ρ^{(α - q) \frac{2}{q}} d x)}^{\frac{q}{2}} \\ {(\int_{Ω_{ℓ}} {(| \partial_{x_{2}} u_{ℓ} | + | \partial_{x_{2}} u_{\infty} |)}^{q} d x)}^{1 - \frac{q}{2}} . \end{aligned} \end{matrix}$ Chosing $(α - q) \frac{2}{q} > α$ and taking into account the lemmas 2.1, 2.3 we get for different constants independent of ℓ $\begin{matrix} (2.29) & \begin{aligned} \int_{Ω_{ℓ}} {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q} ρ^{α - q} d x \\ ⩽ {(\int_{Ω_{ℓ}} {(| \partial_{x_{2}} u_{ℓ} | + | \partial_{x_{2}} u_{\infty} |)}^{(q - 2)} {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{2} ρ^{α} d x)}^{\frac{q}{2}} \\ {(\int_{Ω_{ℓ}} {(| \partial_{x_{2}} u_{ℓ} | + | \partial_{x_{2}} u_{\infty} |)}^{q} d x)}^{1 - \frac{q}{2}} \\ ⩽ C {\tilde{I}}^{\frac{q}{2}} {(\int_{Ω_{ℓ}} {(| \partial_{x_{2}} u_{ℓ} | + | \partial_{x_{2}} u_{\infty} |)}^{q} d x)}^{1 - \frac{q}{2}} \\ ⩽ C {\tilde{I}}^{\frac{q}{2}} {(\int_{Ω_{ℓ}} | \partial_{x_{2}} u_{ℓ} |^{q} + | \partial_{x_{2}} u_{\infty} |^{q} d x)}^{1 - \frac{q}{2}} ⩽ C {\tilde{I}}^{\frac{q}{2}} ℓ^{1 - \frac{q}{2}} ⩽ \frac{C}{ℓ^{(p - 1) \frac{q}{2}}} ℓ^{1 - \frac{q}{2}} . \end{aligned} \end{matrix}$ (Note that ${| a | + | b |}^{q} ⩽ 2^{q - 1} {| a |^{q} + | b |^{q}}$ ). Choosing also $α > q$ we are ending up with $\begin{matrix} \int_{Ω_{\frac{ℓ}{2}}} {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q} d x ⩽ \frac{C}{ℓ^{\frac{p}{2} q - 1}} . \end{matrix}$ Combining this with (2.22) we arrive also to (2.27). □

In the case where $p < q$ one can consider a general f and not only assume that it is in $L^{q^{'}} (ω_{2})$ . Indeed one has first:

Lemma 2.7.

Suppose that $p < q$ . If $ρ^{α}$ is defined by ( 2.21 ) and if α is chosen such that $α \frac{q}{p} - q > α$ it holds for some constant C $\begin{matrix} (2.30) & \begin{aligned} I & = \int_{Ω_{ℓ}} {| \partial_{x_{1}} u_{ℓ} |^{p} + (| \partial_{x_{2}} u_{ℓ} |^{q - 2} \partial_{x_{2}} u_{ℓ} - | \partial_{x_{2}} u_{\infty} |^{q - 2} \partial_{x_{2}} u_{\infty}) \partial_{x_{2}} (u_{ℓ} - u_{\infty})} ρ^{α} d x \\ ⩽ \frac{C}{ℓ^{p}} {(\int_{Ω_{ℓ}} {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q} ρ^{α} d x)}^{\frac{p}{q}} ℓ^{1 - \frac{p}{q}} . \end{aligned} \end{matrix}$

Proof.

Since (2.19) is valid for a general f one derives as in (2.23), (2.24) $\begin{matrix} (2.31) & \begin{aligned} I & = \int_{Ω_{ℓ}} {| \partial_{x_{1}} u_{ℓ} |^{p} + (| \partial_{x_{2}} u_{ℓ} |^{q - 2} \partial_{x_{2}} u_{ℓ} - | \partial_{x_{2}} u_{\infty} |^{q - 2} \partial_{x_{2}} u_{\infty}) \partial_{x_{2}} (u_{ℓ} - u_{\infty})} ρ^{α} d x \\ ⩽ \frac{α C}{ℓ} {(\int_{Ω_{ℓ}} | \partial_{x_{1}} u_{ℓ} |^{p} ρ^{α} d x)}^{\frac{1}{p^{'}}} {(\int_{Ω_{ℓ}} | u_{ℓ} - u_{\infty} |^{p} ρ^{α - p} d x)}^{\frac{1}{p}} \\ ⩽ \frac{α C}{ℓ} I^{\frac{1}{p^{'}}} {(\int_{Ω_{ℓ}} | u_{ℓ} - u_{\infty} |^{p} ρ^{α - p} d x)}^{\frac{1}{p}} . \end{aligned} \end{matrix}$ From this inequality it follows since $α \frac{q}{p} - q > α$ for various constant C $\begin{matrix} (2.32) & \begin{aligned} I & ⩽ {(\frac{α C}{ℓ})}^{p} \int_{Ω_{ℓ}} | u_{ℓ} - u_{\infty} |^{p} ρ^{α - p} d x \\ ⩽ {(\frac{α C}{ℓ})}^{p} {(\int_{Ω_{ℓ}} | u_{ℓ} - u_{\infty} |^{p \frac{q}{p}} ρ^{(α - p) \frac{q}{p}} d x)}^{\frac{p}{q}} {(\int_{Ω_{ℓ}} 1 d x)}^{1 - \frac{p}{q}} \\ ⩽ \frac{C}{ℓ^{p}} {(\int_{Ω_{ℓ}} | u_{ℓ} - u_{\infty} |^{q} ρ^{α} d x)}^{\frac{p}{q}} {(\int_{Ω_{ℓ}} 1 d x)}^{1 - \frac{p}{q}} \\ ⩽ \frac{C}{ℓ^{p}} {(\int_{Ω_{ℓ}} {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q} ρ^{α} d x)}^{\frac{p}{q}} ℓ^{1 - \frac{p}{q}} . \end{aligned} \end{matrix}$ (In the last inequality we used the Poincaré inequality on ω). This completes the proof of the lemma. □

Then we have:

Theorem 2.3.

Suppose that $p < q$ . One has $\begin{matrix} (2.33) & \int_{Ω_{\frac{ℓ}{2}}} | \partial_{x_{1}} u_{ℓ} |^{p} + {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q} d x ⩽ \frac{C}{ℓ^{\frac{p q}{q - p} - 1}} \end{matrix}$

Proof.

If $q ⩾ 2$ one has by (2.16) $\begin{matrix} (2.34) & (| ξ |^{q - 2} ξ - | η |^{q - 2} η) \cdot (ξ - η) ⩾ c_{q} | ξ - η |^{2} {(| ξ | + | η |)}^{q - 2} ⩾ c_{q} | ξ - η |^{q} \forall ξ, η \in R^{n} . \end{matrix}$ Thus from (2.30) one deduces for some constant C $\begin{aligned} J & = \int_{Ω_{ℓ}} {| \partial_{x_{1}} u_{ℓ} |^{p} + {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q}} ρ^{α} d x \\ ⩽ \frac{C}{ℓ^{p}} {(\int_{Ω_{ℓ}} {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q} ρ^{α} d x)}^{\frac{p}{q}} ℓ^{1 - \frac{p}{q}} . \end{aligned}$ From this it follows that $\begin{matrix} J ⩽ \frac{C}{ℓ^{p}} J^{\frac{p}{q}} ℓ^{1 - \frac{p}{q}} \Leftrightarrow J ⩽ \frac{C}{ℓ^{\frac{p q}{q - p} - 1}}, \end{matrix}$ and (2.33) follows by definition of ρ. In the case when $p < q < 2$ , noting $S = | \partial_{x_{2}} (u_{ℓ}) | + | \partial_{x_{2}} (u_{\infty}) |$ one derives from (2.16), (2.30) for some constant $\begin{matrix} (2.35) & \begin{aligned} \tilde{I} & = \int_{Ω_{ℓ}} {| \partial_{x_{1}} u_{ℓ} |^{p} + S^{q - 2} \partial_{x_{2}} {(u_{ℓ} - u_{\infty})}^{2}} ρ^{α} d x \\ ⩽ \frac{C}{ℓ^{p}} {(\int_{Ω_{ℓ}} {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q} ρ^{α} d x)}^{\frac{p}{q}} ℓ^{1 - \frac{p}{q}} . \end{aligned} \end{matrix}$ Thanks to Hölder’s inequality one has as in (2.28) $\begin{aligned} \int_{Ω_{ℓ}} {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q} ρ^{α} d x & = \int_{Ω_{ℓ}} S^{\frac{q}{2} (q - 2)} {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q} ρ^{α \frac{q}{2}} S^{\frac{q}{2} (2 - q)} ρ^{α (1 - \frac{q}{2})} d x \\ ⩽ {(\int_{Ω_{ℓ}} S^{q - 2} {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{2} ρ^{α} d x)}^{\frac{q}{2}} {(\int_{Ω_{ℓ}} S^{q} d x)}^{1 - \frac{q}{2}} \\ ⩽ {\tilde{I}}^{\frac{q}{2}} {(\int_{Ω_{ℓ}} {| \partial_{x_{2}} (u_{ℓ}) | + | \partial_{x_{2}} (u_{\infty}) |}^{q} d x)}^{1 - \frac{q}{2}} . \end{aligned}$ It follows from (2.10) that $\begin{matrix} \int_{Ω_{ℓ}} {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q} ρ^{α} d x ⩽ C {\tilde{I}}^{\frac{q}{2}} {(\int_{Ω_{ℓ}} {| \partial_{x_{2}} (u_{ℓ}) |}^{q} + {| \partial_{x_{2}} (u_{\infty}) |}^{q} d x)}^{1 - \frac{q}{2}} ⩽ C {\tilde{I}}^{\frac{q}{2}} ℓ^{1 - \frac{q}{2}} . \end{matrix}$ Going back to (2.35) we obtain $\begin{matrix} \tilde{I} ⩽ \frac{C}{ℓ^{p}} {({\tilde{I}}^{\frac{q}{2}} ℓ^{1 - \frac{q}{2}})}^{\frac{p}{q}} ℓ^{1 - \frac{p}{q}} = \frac{C}{ℓ^{p}} {\tilde{I}}^{\frac{p}{2}} ℓ^{1 - \frac{p}{2}} . \end{matrix}$ Hence $\begin{matrix} \tilde{I} ⩽ \frac{C}{ℓ^{\frac{2 p}{2 - p} - 1}}, \end{matrix}$ and $\begin{matrix} \int_{Ω_{ℓ}} {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q} ρ^{α} d x ⩽ C \frac{1}{ℓ^{\frac{p q}{2 - p} - 1}} . \end{matrix}$ The inequality (2.32) follows from these two estimates. □

In the case $p ⩾ q ⩾ 2$ one can show that $u_{ℓ} \to u_{\infty}$ exponentially quickly (see [6,15] and also this issue in the next section). Indeed one has:

Theorem 2.4.

Suppose that $p ⩾ q ⩾ 2$ , $f \in L^{1} (ω)$ . It holds for some positive constants C, α $\begin{matrix} (2.36) & \int_{Ω_{\frac{ℓ}{2}}} | \partial_{x_{1}} u_{ℓ} |^{p} + {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q} d x ⩽ C e^{- α ℓ} . \end{matrix}$

Proof.

Since $f \in L^{1} (ω)$ one has $\begin{matrix} - \partial_{x_{2}} (| \partial_{x_{2}} u_{\infty} |^{q - 2} \partial_{x_{2}} u_{\infty}) = f \Leftrightarrow | \partial_{x_{2}} u_{\infty} |^{q - 2} \partial_{x_{2}} u_{\infty} = - \int_{0}^{x_{2}} f (ξ) d ξ + C . \end{matrix}$ This implies that $u_{\infty}$ is a $C^{1}$ -function which is bounded as $u_{ℓ}$ is (see the lemmas 2.4 and 2.6). Let us set $A = | \partial_{x_{1}} u_{ℓ} |^{p} + | \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |^{q}$ . For $σ > 0$ consider as in [2] $\begin{matrix} φ = (e^{- σ | x_{1} |} - e^{- σ ℓ}) \end{matrix}$ in (2.19). Taking into account the lemma 2.3 and the fact that $\partial_{x_{1}} φ = \pm σ e^{- σ | x_{1} |}$ we get $\begin{matrix} (2.37) & \begin{aligned} \int_{Ω_{ℓ}} {| \partial_{x_{1}} u_{ℓ} |^{p} + {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q}} φ d x \\ ⩽ C \int_{Ω_{ℓ}} {| \partial_{x_{1}} u_{ℓ} |^{p} + (| \partial_{x_{2}} u_{ℓ} |^{q - 2} \partial_{x_{2}} u_{ℓ} - | \partial_{x_{2}} u_{\infty} |^{q - 2} \partial_{x_{2}} u_{\infty}) \partial_{x_{2}} (u_{ℓ} - u_{\infty})} φ d x \\ ⩽ σ C \int_{Ω_{ℓ}} e^{- σ | x_{1} |} | \partial_{x_{1}} u_{ℓ} |^{p - 1} | u_{ℓ} - u_{\infty} | d x . \end{aligned} \end{matrix}$ Using the Young inequality in this last integral i.e. $| a | | b | ⩽ \frac{1}{p^{'}} | a |^{p^{'}} + \frac{1}{p} | b |^{p}$ we get for some new constants $\begin{matrix} (2.38) & \begin{aligned} \int_{Ω_{ℓ}} A (e^{- σ | x_{1} |} - e^{- σ ℓ}) d x \\ ⩽ σ C \int_{Ω_{ℓ}} e^{- σ | x_{1} |} {| \partial_{x_{1}} u_{ℓ} |^{p} + | u_{ℓ} - u_{\infty} |^{p}} d x . \\ ⩽ σ C \int_{Ω_{ℓ}} e^{- σ | x_{1} |} {| \partial_{x_{1}} u_{ℓ} |^{p} + | u_{ℓ} - u_{\infty} |^{q}} d x . \\ ⩽ σ C \int_{Ω_{ℓ}} e^{- σ | x_{1} |} {| \partial_{x_{1}} u_{ℓ} |^{p} + {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q}} d x = σ C \int_{Ω_{ℓ}} A e^{- σ | x_{1} |} d x . \end{aligned} \end{matrix}$ (In the above, we used the fact that $u_{ℓ}$ and $u_{\infty}$ are uniformly bounded independently of ℓ and the Poincaré inequality on the section ω). Choosing $σ C = \frac{1}{2}$ it comes $\begin{matrix} (2.39) & \frac{1}{2} \int_{Ω_{ℓ}} A e^{- σ | x_{1} |} d x ⩽ e^{- σ ℓ} \int_{Ω_{ℓ}} A d x \end{matrix}$ that is to say $\begin{matrix} (2.40) & e^{- σ \frac{ℓ}{2}} \int_{Ω_{\frac{ℓ}{2}}} A d x ⩽ 2 e^{- σ ℓ} \int_{Ω_{ℓ}} A d x . \end{matrix}$ It follows from the lemma 2.1 that $\begin{matrix} (2.41) & \int_{Ω_{\frac{ℓ}{2}}} A d x = \int_{Ω_{\frac{ℓ}{2}}} | \partial_{x_{1}} u_{ℓ} |^{p} + {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q} d x ⩽ 2 e^{- σ \frac{ℓ}{2}} \int_{Ω_{ℓ}} A d x ⩽ C ℓ e^{- σ \frac{ℓ}{2}} . \end{matrix}$ The result follows by choosing $α < \frac{σ}{2}$ . □

The last case to address is when $p ⩾ q$ , $q < 2$ . In this case one can prove:

Theorem 2.5.

Suppose that $p ⩾ q$ , $q < 2$ , $f \in L^{1} (ω)$ . It holds for some positive constants C $\begin{matrix} (2.42) & \int_{Ω_{\frac{ℓ}{2}}} | \partial_{x_{1}} u_{ℓ} |^{p} + {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q} d x ⩽ \frac{C}{ℓ^{\frac{p q}{2 - q} - 1}} . \end{matrix}$

Proof.

Choosing ρ as in (2.21) one has – see (2.16), (2.23) $\begin{matrix} (2.43) & \begin{aligned} \tilde{I} & = \int_{Ω_{ℓ}} | \partial_{x_{1}} u_{ℓ} |^{p} + S^{q - 2} {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{2} ρ^{α} d x \\ ⩽ \frac{C}{ℓ} \int_{Ω_{ℓ}} | \partial_{x_{1}} u_{ℓ} |^{p - 1} | u_{ℓ} - u_{\infty} | ρ^{α - 1} d x \\ ⩽ \frac{C}{ℓ} {(\int_{Ω_{ℓ}} | \partial_{x_{1}} u_{ℓ} |^{p} ρ^{α} d x)}^{\frac{1}{p^{'}}} {(\int_{Ω_{ℓ}} | u_{ℓ} - u_{\infty} |^{p} ρ^{α - p} d x)}^{\frac{1}{p}} \\ ⩽ \frac{C}{ℓ} {\tilde{I}}^{\frac{1}{p^{'}}} {(\int_{Ω_{ℓ}} | u_{ℓ} - u_{\infty} |^{q} ρ^{α - p} d x)}^{\frac{1}{p}} ⩽ \frac{C}{ℓ} {\tilde{I}}^{\frac{1}{p^{'}}} {(\int_{Ω_{ℓ}} {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q} ρ^{α - p} d x)}^{\frac{1}{p}} . \end{aligned} \end{matrix}$ Recall that $S = | \partial_{x_{2}} (u_{ℓ}) | + | \partial_{x_{2}} (u_{\infty}) |$ . In the two last inequalities we used the fact that $u_{ℓ}$ and $u_{\infty}$ are uniformly bounded and the Poincaré inequality. Arguing as before we have $\begin{matrix} (2.44) & \begin{aligned} \int_{Ω_{ℓ}} {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q} ρ^{α - p} d x & = \int_{Ω_{ℓ}} S^{(q - 2) \frac{q}{2}} {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q} ρ^{α \frac{q}{2}} S^{(2 - q) \frac{q}{2}} ρ^{α (1 - \frac{q}{2}) - p} d x \\ ⩽ {(\int_{Ω_{ℓ}} S^{(q - 2)} {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{2} ρ^{α})}^{\frac{q}{2}} {(\int_{Ω_{ℓ}} S^{q} d x)}^{1 - \frac{q}{2}} \\ ⩽ C {\tilde{I}}^{\frac{q}{2}} ℓ^{1 - \frac{q}{2}} \end{aligned} \end{matrix}$ provided $α (1 - \frac{q}{2}) - p > 0$ . Thus from (2.43) we derive $\begin{matrix} (2.45) & \tilde{I} ⩽ \frac{C}{ℓ} {\tilde{I}}^{\frac{1}{p^{'}}} {({\tilde{I}}^{\frac{q}{2}} ℓ^{1 - \frac{q}{2}})}^{\frac{1}{p}} \Leftrightarrow {\tilde{I}}^{\frac{1}{p} - \frac{q}{2 p}} ⩽ \frac{C}{ℓ} ℓ^{\frac{1}{p} - \frac{q}{2 p}} \Leftrightarrow \tilde{I} ⩽ \frac{C}{ℓ^{\frac{2 p}{2 - q} - 1}} . \end{matrix}$ Going back to (2.44) one has if $α > p$ $\begin{matrix} \int_{Ω_{\frac{ℓ}{2}}} {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q} d x ⩽ \int_{Ω_{ℓ}} {| \partial_{x_{2}} (u_{ℓ} - u_{\infty}) |}^{q} ρ^{α - p} d x ⩽ \frac{C}{{(ℓ^{\frac{2 p}{2 - q} - 1})}^{\frac{q}{2}}} ℓ^{1 - \frac{q}{2}} = \frac{C}{ℓ^{\frac{p q}{2 - q} - 1}} . \end{matrix}$ Combining this with (2.45) leads easily to (2.42) since $p q < 2 p$ . This completes the proof. □

3. Some generalisations

Let us denote by $ω_{1}$ a bounded convex domain of $R^{m}$ containing 0 and by $ω_{2}$ a bounded domain in $R^{n - m}$ , $m ⩾ 1$ . Let us set for $ℓ > 0$ $\begin{matrix} Ω_{ℓ} = ℓ ω_{1} \times ω_{2} . \end{matrix}$ We will denote the points in $Ω_{ℓ}$ by $\begin{matrix} x = (X_{1}, X_{2}) \end{matrix}$ where $X_{1} = x_{1}, \dots, x_{m}$ , $X_{2} = x_{m + 1}, \dots, x_{n}$ . If $p_{i}, i = 1, \dots, m$ , $q_{j}, j = m + 1, \dots, n$ are numbers larger than 1 set $\begin{matrix} \vec{p} = (p_{1}, \dots, p_{m}), \vec{q} = (q_{m + 1}, \dots, q_{n}) . \end{matrix}$ Then we define $\begin{matrix} (3.1) & \begin{aligned} W^{1, \vec{p}, \vec{q}} (Ω_{ℓ}) = {v \in L^{p_{i}} (Ω_{ℓ}) \cap L^{q_{j}} (Ω_{ℓ}) | \partial_{x_{i}} v \in L^{p_{i}} (Ω_{ℓ}), \partial_{x_{j}} v \in L^{q_{j}} (Ω_{ℓ}), \forall i, j}, \\ W^{1, \vec{q}} (ω_{2}) = {v \in L^{q_{j}} (ω_{2}) | \partial_{x_{j}} v \in L^{q_{j}} (ω_{2}), \forall j} . \end{aligned} \end{matrix}$ In the definition above the indices i are running from 1 to m and the indices j from $m + 1$ to n. Clearly $W^{1, \vec{p}, \vec{q}} (Ω_{ℓ})$ , $W^{1, \vec{q}} (ω_{2})$ are reflexive Banach spaces when equipped with the norms $\begin{array}{c} (3.2) & \begin{aligned} ‖ v ‖_{1, \vec{p}, \vec{q}} = \sum_{i = 1}^{m} (| v |_{p_{i}, Ω_{ℓ}} + | \partial_{x_{i}} v |_{p_{i}, Ω_{ℓ}}) + \sum_{j = m + 1}^{n} (| v |_{q_{j}, Ω_{ℓ}} + | \partial_{x_{j}} v |_{q_{j}, Ω_{ℓ}}) \\ ‖ v ‖_{1, \vec{q}} = \sum_{j = m + 1}^{n} (| v |_{q_{j}, Ω_{ℓ}} + | \partial_{x_{j}} v |_{q_{j}, Ω_{ℓ}}) . \end{aligned} \end{array}$ One denotes by $W_{0}^{1, \vec{p}, \vec{q}} (Ω_{ℓ})$ (respectively $W_{0}^{1, \vec{q}} (ω_{2})$ ) the closure of $D (Ω_{ℓ})$ (respectively $D (ω_{2})$ ) in these spaces and by $u_{ℓ}$ the solution to $\begin{matrix} (3.3) & \{\begin{array}{l} u_{ℓ} \in W_{0}^{1, \vec{p}, \vec{q}} (Ω_{ℓ}), \\ \int_{Ω_{ℓ}} | \partial_{x_{i}} u_{ℓ} |^{p_{i} - 2} \partial_{x_{i}} u_{ℓ} \partial_{x_{i}} v + | \partial_{x_{j}} u_{ℓ} |^{q_{j} - 2} \partial_{x_{j}} u_{ℓ} \partial_{x_{j}} v d x = ⟨ f, v ⟩ \forall v \in W_{0}^{1, \vec{p}, \vec{q}} (Ω_{ℓ}) . \end{array} \end{matrix}$ In the formula above we make the summation convention, i.e. we are summing in i and j. $f = f (X_{2})$ is a continuous linear form on $W_{0}^{1, \vec{q}} (ω_{2})$ defined as $\begin{matrix} (3.4) & \begin{aligned} f = f_{0} - \sum_{j = m + 1}^{n} \partial_{x_{j}} f_{j}, f_{0} \in \cap_{j} L^{q_{j}^{'}} (ω_{2}), f_{j} \in L^{q_{j}^{'}} (ω_{2}), \\ ⟨ f, v ⟩ = \sum_{ω_{2}} f_{0} v + \sum_{j = m + 1}^{n} f_{j} \partial_{x_{j}} v d x . \end{aligned} \end{matrix}$ We would like to sketch some behaviour of $u_{ℓ}$ when $ℓ \to \infty$ , in particular to show that $u_{ℓ} \to u_{\infty}$ where $u_{\infty}$ is the solution to $\begin{matrix} (3.5) & \{\begin{array}{l} u_{\infty} \in W_{0}^{1, \vec{q}} (ω_{2}), \\ \int_{ω_{2}} | \partial_{x_{j}} u_{\infty} |^{q_{j} - 2} \partial_{x_{j}} u_{\infty} \partial_{x_{j}} v d X_{2} = ⟨ f, v ⟩ \forall v \in W_{0}^{1, \vec{q}} (ω_{2}) . \end{array} \end{matrix}$ Note that by the same arguments as in Section 2 the problems (3.3), (3.5) admit a unique solution. The analogue of lemma 2.1 is the following.

Lemma 3.1.

Let $u_{ℓ}$ be the solution of ( 3.3 ) for f given by ( 3.4 ). There exists a constant C independent of ℓ such that $\begin{matrix} (3.6) & \int_{Ω_{ℓ}} \sum_{i = 1}^{m} | \partial_{x_{i}} u_{ℓ} |^{p_{i}} + \sum_{j = m + 1}^{n} | \partial_{x_{j}} u_{ℓ} |^{q_{j}} d x ⩽ C ℓ^{m} . \end{matrix}$

Proof.

Let $q = max (q_{j}) = q_{j_{0}}$ for some $j_{0}$ . Taking $v = u_{ℓ}$ in (3.3) we get with the summation convention in i, j $\begin{matrix} (3.7) & \begin{aligned} \int_{Ω_{ℓ}} | \partial_{x_{i}} u_{ℓ} |^{p_{i}} + | \partial_{x_{j}} u_{ℓ} |^{q_{j}} d x \\ = \int_{Ω_{ℓ}} {f_{0} u_{ℓ} + f_{j_{0}} \partial_{x_{j_{0}}} u_{ℓ} + \sum_{j \neq j_{0}} f_{j} \partial_{x_{j}} u_{ℓ}} d x \\ ⩽ \int_{Ω_{ℓ}} {| f_{0} | | u_{ℓ} | + | f_{j_{0}} | | \partial_{x_{j_{0}}} u_{ℓ} | + \sum_{j \neq j_{0}} | f_{j} | | \partial_{x_{j}} u_{ℓ} |} d x \\ ⩽ | f_{0} |_{q^{'}, Ω_{ℓ}} | u_{ℓ} |_{q, Ω_{ℓ}} + | f_{j_{0}} |_{q_{j_{0}}^{'}, Ω_{ℓ}} | \partial_{x_{j_{0}}} u_{ℓ} |_{q_{j_{0}}, Ω_{ℓ}} + \sum_{j \neq j_{0}} | f_{j} |_{q_{j}^{'}, Ω_{ℓ}} | \partial_{x_{j}} u_{ℓ} |_{q_{j}, Ω_{ℓ}} . \end{aligned} \end{matrix}$ Using the Poincaré inequality $\begin{matrix} | u_{ℓ} |_{q, Ω_{ℓ}} = | u_{ℓ} |_{q_{j_{0}}, Ω_{ℓ}} ⩽ C | \partial_{x_{j_{0}}} u_{ℓ} |_{q_{j_{0}}, Ω_{ℓ}} \end{matrix}$ we derive $\begin{matrix} \int_{Ω_{ℓ}} | \partial_{x_{i}} u_{ℓ} |^{p_{i}} + | \partial_{x_{j}} u_{ℓ} |^{q_{j}} d x ⩽ {C | f_{0} |_{q_{j_{0}}^{'}, Ω_{ℓ}} + | f_{j_{0}} |_{q_{j_{0}}^{'}, Ω_{ℓ}}} | \partial_{x_{j_{0}}} u_{ℓ} |_{q_{j_{0}}, Ω_{ℓ}} + \sum_{j \neq j_{0}} | f_{j} |_{q_{j}^{'}, Ω_{ℓ}} | \partial_{x_{j}} u_{ℓ} |_{q_{j}, Ω_{ℓ}} . \end{matrix}$ Note now that for $f \in L^{q^{'}} (ω_{2})$ one has for some constant C independent of ℓ $\begin{matrix} (3.8) & | f |_{q^{'}, Ω_{ℓ}} = {(\int_{ℓ ω_{1}} \int_{ω_{2}} {| f (X_{2}) |}^{q^{'}} d X_{2} d X_{1})}^{\frac{1}{q^{'}}} ⩽ C ℓ^{\frac{m}{q^{'}}} . \end{matrix}$ Thus we get $\begin{matrix} (3.9) & \begin{aligned} \int_{Ω_{ℓ}} | \partial_{x_{i}} u_{ℓ} |^{p_{i}} + | \partial_{x_{j}} u_{ℓ} |^{q_{j}} d x & ⩽ C \sum_{j} ℓ^{\frac{m}{q_{j}^{'}}} | \partial_{x_{j}} u_{ℓ} |_{q_{j}, Ω_{ℓ}} \\ ⩽ ϵ \sum_{j} | \partial_{x_{j}} u_{ℓ} |_{q_{j}, Ω_{ℓ}}^{q_{j}} + C_{ϵ} ℓ^{m}, \end{aligned} \end{matrix}$ using the Young inequality $| a b | ⩽ ϵ | a |^{q} + C_{ϵ} | b |^{q^{'}}$ . The result follows by choosing $ϵ = \frac{1}{2}$ . □

With the same proofs we have the analogues of Lemmas 2.2 and 2.4 namely with the summation convention $\begin{matrix} (3.10) & \begin{aligned} \int_{Ω_{ℓ}} | \partial_{x_{i}} u_{ℓ} |^{p_{i} - 2} \partial_{x_{i}} u_{ℓ} \partial_{x_{i}} v + {| \partial_{x_{j}} u_{ℓ} |^{q_{j} - 2} \partial_{x_{j}} u_{ℓ} - | \partial_{x_{j}} u_{\infty} |^{q_{j} - 2} \partial_{x_{j}} u_{\infty}} \partial_{x_{j}} v d x = 0 \\ \forall v \in W_{0}^{1, \vec{p}, \vec{q}} (Ω_{ℓ}) . \end{aligned} \end{matrix}$ Similarly if $u_{ℓ} = u_{ℓ} (f)$ is the solution to (3.3) and $u_{\infty} = u_{\infty} (f)$ the solution to (3.5) and if $f_{1} ⩾ f_{2}$ , $f ⩾ 0$ then one has $\begin{matrix} (3.11) & u_{ℓ} (f_{2}) ⩽ u_{ℓ} (f_{1}), 0 ⩽ u_{ℓ} (f) ⩽ u_{\infty} (f) . \end{matrix}$

Remark 1.

Note that (3.10) allows a perhaps simpler proof of (3.6) where, however, the dependence in f is lost. Indeed taking $v = u_{ℓ}$ in (3.10) we get with the summation convention in i and j $\begin{aligned} \int_{Ω_{ℓ}} | \partial_{x_{i}} u_{ℓ} |^{p_{i}} + | \partial_{x_{j}} u_{ℓ} |^{q_{j}} d x & = \int_{Ω_{ℓ}} | \partial_{x_{j}} u_{\infty} |^{q_{j} - 2} \partial_{x_{j}} u_{\infty} \partial_{x_{j}} u_{ℓ} d x \\ ⩽ \int_{Ω_{ℓ}} | \partial_{x_{j}} u_{\infty} |^{q_{j} - 1} | \partial_{x_{j}} u_{ℓ} | d x . \end{aligned}$ Using the Young inequality $| a b | ⩽ \frac{1}{q_{j}^{'}} | a |^{q_{j}^{'}} + \frac{1}{q_{j}} | b |^{q_{j}}$ it comes $\begin{aligned} \int_{Ω_{ℓ}} | \partial_{x_{i}} u_{ℓ} |^{p_{i}} + | \partial_{x_{j}} u_{ℓ} |^{q_{j}} d x & ⩽ \int_{Ω_{ℓ}} \frac{1}{q_{j}^{'}} | \partial_{x_{j}} u_{\infty} |^{q_{j}} + \frac{1}{q_{j}} | \partial_{x_{j}} u_{ℓ} |^{q_{j}} d x \\ ⩽ \int_{Ω_{ℓ}} \frac{1}{{min}_{j} q_{j}} | \partial_{x_{j}} u_{ℓ} |^{q_{j}} + \frac{1}{{min}_{j} q_{j}^{'}} | \partial_{x_{j}} u_{\infty} |^{q_{j}} d x \end{aligned}$ and thus for some constant C since ${min}_{j} q_{j} > 1$ $\begin{matrix} \int_{Ω_{ℓ}} | \partial_{x_{i}} u_{ℓ} |^{p_{i}} + | \partial_{x_{j}} u_{ℓ} |^{q_{j}} d x ⩽ C \int_{Ω_{ℓ}} | \partial_{x_{j}} u_{\infty} |^{q_{j}} d x ⩽ C ℓ^{m} . \end{matrix}$

Then we can turn to the generalisation of lemma 2.6. Denote by $ρ = ρ (X_{1})$ a smooth function such that $\begin{matrix} (3.12) & 0 ⩽ ρ ⩽ 1, ρ = 1 on \frac{1}{2} ω_{1}, ρ = 0 near \partial ω_{1}, | \nabla_{X_{1}} ρ | ⩽ C, \end{matrix}$ where $\nabla_{X_{1}} ρ$ denotes the gradient of ρ in $X_{1}$ , i.e. $\nabla_{X_{1}} ρ = (\partial_{x_{1}} ρ, \dots, \partial_{x_{m}} ρ)$ .

We can show:

Lemma 3.2.

Let $f = f_{0} \in L^{q^{'}} (ω_{2})$ , $q = max q_{j}$ and $u_{ℓ}$ , $u_{\infty}$ be the solutions to ( 3.3 ), ( 3.5 ). Then it holds for some constant C independent of ℓ $\begin{matrix} (3.13) & \begin{aligned} I & = \int_{Ω_{ℓ}} {| \partial_{x_{i}} u_{ℓ} |^{p_{i}} + (| \partial_{x_{j}} u_{ℓ} |^{q_{j} - 2} \partial_{x_{j}} u_{ℓ} - | \partial_{x_{j}} u_{\infty} |^{q_{j} - 2} \partial_{x_{j}} u_{\infty}) \partial_{x_{j}} (u_{ℓ} - u_{\infty})} ρ^{α} d x \\ ⩽ \frac{C}{ℓ^{p - m}} . \end{aligned} \end{matrix}$ ( $p = {min}_{i} (p_{i})$ denotes the smallest $p_{i}$ )

Proof.

From (3.10) taking $v = ρ^{α} (\frac{X_{1}}{ℓ}) (u_{ℓ} - u_{\infty})$ one derives easily with the summation convention in i and j $\begin{matrix} (3.14) & \begin{aligned} \int_{Ω_{ℓ}} {| \partial_{x_{i}} u_{ℓ} |^{p_{i}} + (| \partial_{x_{j}} u_{ℓ} |^{q_{j} - 2} \partial_{x_{j}} u_{ℓ} - | \partial_{x_{j}} u_{\infty} |^{q_{j} - 2} \partial_{x_{j}} u_{\infty}) \partial_{x_{j}} (u_{ℓ} - u_{\infty})} ρ^{α} d x \\ ⩽ \frac{α C}{ℓ} \int_{Ω_{ℓ}} | \partial_{x_{i}} u_{ℓ} |^{p_{i} - 1} | u_{ℓ} - u_{\infty} | ρ^{α - 1} d x . \end{aligned} \end{matrix}$ Noting that $ρ^{α - 1} = ρ^{\frac{α}{p_{i}^{'}}} ρ^{\frac{α}{p_{i}} - 1}$ and using Young’s inequality it comes $\begin{matrix} (3.15) & \begin{aligned} I & = \int_{Ω_{ℓ}} {| \partial_{x_{i}} u_{ℓ} |^{p_{i}} + (| \partial_{x_{j}} u_{ℓ} |^{q_{j} - 2} \partial_{x_{j}} u_{ℓ} - | \partial_{x_{j}} u_{\infty} |^{q_{j} - 2} \partial_{x_{j}} u_{\infty}) \partial_{x_{j}} (u_{ℓ} - u_{\infty})} ρ^{α} d x \\ ⩽ \frac{1}{p_{i}^{'}} \int_{Ω_{ℓ}} | \partial_{x_{i}} u_{ℓ} |^{p_{i}} ρ^{α} d x + \frac{C}{ℓ^{p_{i}}} \int_{Ω_{ℓ}} | u_{ℓ} - u_{\infty} |^{p_{i}} ρ^{α - p_{i}} d x . \end{aligned} \end{matrix}$ Recalling our summation in i and the fact that $p_{i}^{'} > 1$ it follows that for some constant C $\begin{matrix} (3.16) & I ⩽ C \sum_{i} \frac{1}{ℓ^{p_{i}}} \int_{Ω_{ℓ}} | u_{ℓ} - u_{\infty} |^{p_{i}} ρ^{α - p_{i}} d x ⩽ C \sum_{i} \frac{1}{ℓ^{p_{i}}} \int_{Ω_{ℓ}} | u_{ℓ} - u_{\infty} |^{p_{i}} d x, \end{matrix}$ provided we chose α large enough. Note that at this point we did not use the assumption $f = f_{0} \in L^{q^{'}} (ω_{2})$ , $q = max q_{j}$ . Arguing now like in Lemma 2.6 one can bound $| u_{ℓ} - u_{\infty} |$ by something depending only on $X_{2}$ to get $\begin{matrix} (3.17) & I ⩽ C \sum_{i} \frac{1}{ℓ^{p_{i} - m}} . \end{matrix}$ This completes the proof of the lemma. □

The convergence of $u_{ℓ}$ toward $u_{\infty}$ is insured for general $p_{i}$ , $q_{j}$ by the following result.

Theorem 3.1.

Let $f = f_{0} \in L^{q^{'}} (ω_{2})$ , $q = max q_{j}$ . Let $u_{ℓ}$ , $u_{\infty}$ be the solutions to ( 3.3 ), ( 3.5 ) respectively. If $p_{i} > m \forall i$ one has for every $ℓ_{0} > 0$ when $ℓ \to + \infty$ $\begin{matrix} (3.18) & \partial_{x_{i}} u_{ℓ} \to 0 in L^{p_{i}} (Ω_{ℓ_{0}}), \partial_{x_{j}} u_{ℓ} \to \partial_{x_{j}} u_{\infty} in L^{q_{j}} (Ω_{ℓ_{0}}) . \end{matrix}$

Proof.

The first part of (3.18) follows directly from (3.13). For the second part let us consider a smooth function ρ such that for $ℓ_{0} < ℓ - 1$ $\begin{matrix} 0 ⩽ ρ ⩽ 1, ρ = 1 on ℓ_{0} ω_{1}, ρ has a support in (ℓ_{0} + 1) ω_{1}, | \nabla_{X_{1}} ρ | ⩽ C . \end{matrix}$ Then $ρ^{α} u_{ℓ}$ is a test function for (3.3) and one has $\begin{aligned} \int_{Ω_{ℓ_{0} + 1}} {| \partial_{x_{i}} u_{ℓ} |^{p_{i}} + | \partial_{x_{j}} u_{ℓ} |^{q_{j}}} ρ^{α} d x \\ = - α \int_{Ω_{ℓ_{0} + 1}} | \partial_{x_{i}} u_{ℓ} |^{p_{i} - 2} \partial_{x_{i}} u_{ℓ} \partial_{x_{i}} ρ u_{ℓ} ρ^{α - 1} d x + \int_{Ω_{ℓ_{0} + 1}} f_{0} u_{ℓ} d x \\ ⩽ α C \int_{Ω_{ℓ_{0} + 1}} | \partial_{x_{i}} u_{ℓ} |^{p_{i} - 1} | u_{ℓ} | ρ^{α - 1} d x + \int_{Ω_{ℓ_{0} + 1}} | f_{0} | | u_{ℓ} | d x . \end{aligned}$ Using the fact that $ρ^{α - 1} = ρ^{\frac{1}{p_{i}^{'}}} ρ^{\frac{1}{p_{i}} - 1}$ we get by Young’s inequality for some constant C $\begin{aligned} \int_{Ω_{ℓ_{0} + 1}} {| \partial_{x_{i}} u_{ℓ} |^{p_{i}} + | \partial_{x_{j}} u_{ℓ} |^{q_{j}}} ρ^{α} d x \\ ⩽ \frac{1}{p_{i}^{'}} \int_{Ω_{ℓ_{0} + 1}} | \partial_{x_{i}} u_{ℓ} |^{p_{i}} ρ^{α} d x + C \int_{Ω_{ℓ_{0} + 1}} | u_{ℓ} |^{p_{i}} ρ^{α - p_{i}} d x + \int_{Ω_{ℓ_{0} + 1}} | f_{0} | | u_{ℓ} | d x . \end{aligned}$ We assume here that $α > p_{i}$ . We know that by (3.11) $\begin{matrix} | u_{ℓ} | ⩽ max {u_{\infty} (f_{0}^{+}), u_{\infty} (f_{0}^{-})} \end{matrix}$ and since this bound is independent of ℓ we get $\begin{matrix} | \partial_{x_{j}} u_{ℓ} |_{q_{j}, Ω_{ℓ_{0}}} ⩽ C (ℓ_{0}) . \end{matrix}$ The rest of the proof follows as in Theorem 2.1 since by (3.13) we have for every j $\begin{matrix} \int_{Ω_{ℓ_{0}}} (| \partial_{x_{j}} u_{ℓ} |^{q_{j} - 2} \partial_{x_{j}} u_{ℓ} - | \partial_{x_{j}} u_{\infty} |^{q_{j} - 2} \partial_{x_{j}} u_{\infty}) \partial_{x_{j}} (u_{ℓ} - u_{\infty}) d x . \to 0 . \end{matrix}$ This completes the proof of the theorem. □

We have then an analogue to theorem 2.3 for a general f.

Theorem 3.2.

Let $u_{ℓ}$ , $u_{\infty}$ be the solutions to ( 3.3 ), ( 3.5 ). Suppose that $\begin{matrix} (3.19) & \forall i = 1, \dots, m, \exists j_{i} \in {m + 1, \dots, n} such that p_{i} < q_{j_{i}} . \end{matrix}$ Then there exists a constant C such that $\begin{matrix} (3.20) & \begin{aligned} I & = \int_{Ω_{ℓ}} {| \partial_{x_{i}} u_{ℓ} |^{p_{i}} + (| \partial_{x_{j}} u_{ℓ} |^{q_{j} - 2} \partial_{x_{j}} u_{ℓ} - | \partial_{x_{j}} u_{\infty} |^{q_{j} - 2} \partial_{x_{j}} u_{\infty}) \partial_{x_{j}} (u_{ℓ} - u_{\infty})} ρ^{α} d x \\ ⩽ C \sum_{q_{j_{i}} ⩾ 2} \frac{1}{ℓ^{\frac{p_{i} q_{j_{i}}}{q_{j_{i}} - p_{i}} - m}} + C \sum_{q_{j_{i}} < 2} \frac{1}{ℓ^{\frac{2 p_{i}}{2 - p_{i}} - m}} . \end{aligned} \end{matrix}$ $ρ^{α} = ρ^{α} (\frac{X_{1}}{ℓ})$ is as in Lemma 3.2 .

Proof.

Going back to (3.16) one has if $(α - p_{i}) \frac{q_{j_{i}}}{p_{i}} > α \forall i$ $\begin{aligned} I & ⩽ C \sum_{i} \frac{1}{ℓ^{p_{i}}} \int_{Ω_{ℓ}} | u_{ℓ} - u_{\infty} |^{p_{i}} ρ^{α - p_{i}} d x \\ ⩽ C \sum_{i} \frac{1}{ℓ^{p_{i}}} {(\int_{Ω_{ℓ}} | u_{ℓ} - u_{\infty} |^{q_{j_{i}}} ρ^{(α - p_{i}) \frac{q_{j_{i}}}{p_{i}}} d x)}^{\frac{p_{i}}{q_{j_{i}}}} {(\int_{Ω_{ℓ}} 1 d x)}^{1 - \frac{p_{i}}{q_{j_{i}}}} \\ ⩽ C \sum_{i} \frac{1}{ℓ^{p_{i}}} {(\int_{Ω_{ℓ}} | u_{ℓ} - u_{\infty} |^{q_{j_{i}}} ρ^{α} d x)}^{\frac{p_{i}}{q_{j_{i}}}} ℓ^{m (1 - \frac{p_{i}}{q_{j_{i}}})} . \end{aligned}$ Using the Poincaré inequality we get $\begin{matrix} (3.21) & I ⩽ C \sum_{i} \frac{1}{ℓ^{p_{i}}} {(\int_{Ω_{ℓ}} {| \partial_{x_{j_{i}}} (u_{ℓ} - u_{\infty}) |}^{q_{j_{i}}} ρ^{α} d x)}^{\frac{p_{i}}{q_{j_{i}}}} ℓ^{m (1 - \frac{p_{i}}{q_{j_{i}}})} . \end{matrix}$ If $q_{j_{i}} ⩾ 2$ one has $\begin{matrix} (3.22) & {| \partial_{x_{j_{i}}} u_{ℓ} |^{q_{j_{i}} - 2} \partial_{x_{j_{i}}} u_{ℓ} - | \partial_{x_{j_{i}}} u_{\infty} |^{q_{j_{i}} - 2} \partial_{x_{j_{i}}} u_{\infty}} \partial_{x_{j_{i}}} (u_{ℓ} - u_{\infty}) ⩾ C_{q_{j_{i}}} | \partial_{x_{j_{i}}} (u_{ℓ} - u_{\infty} |^{q_{j_{i}}} \end{matrix}$ and thus for some constant $\begin{matrix} (3.23) & \int_{Ω_{ℓ}} {| \partial_{x_{j_{i}}} (u_{ℓ} - u_{\infty}) |}^{q_{j_{i}}} ρ^{α} d x ⩽ C I . \end{matrix}$ If $p_{i} < q_{j_{i}} < 2$ one has (see the theorem 2.3) $\begin{matrix} (3.24) & \begin{aligned} \int_{Ω_{ℓ}} {| \partial_{x_{j_{i}}} (u_{ℓ} - u_{\infty}) |}^{q_{j_{i}}} ρ^{α} d x \\ ⩽ C I^{\frac{q_{j_{i}}}{2}} {(\int_{Ω_{ℓ}} | \partial_{x_{j_{i}}} u_{ℓ} |^{q_{j_{i}}} + | \partial_{x_{j_{i}}} u_{\infty} |^{q_{j_{i}}} d x)}^{1 - \frac{q_{j_{i}}}{2}} \\ ⩽ C I^{\frac{q_{j_{i}}}{2}} ℓ^{m (1 - \frac{q_{j_{i}}}{2})} . \end{aligned} \end{matrix}$ Thus from (3.21) we derive replacing $p_{i}$ by p and $q_{j_{i}}$ by q $\begin{matrix} (3.25) & \begin{aligned} I & ⩽ C \sum_{i} \frac{1}{ℓ^{p}} I^{\frac{p}{q}} ℓ^{m (1 - \frac{p}{q})} + C \sum_{i} \frac{1}{ℓ^{p}} I^{\frac{q}{2} \frac{p}{q}} ℓ^{m (1 - \frac{q}{2}) \frac{p}{q}} ℓ^{m (1 - \frac{p}{q})} \\ ⩽ C \sum_{i} \frac{1}{ℓ^{p}} I^{\frac{p}{q}} ℓ^{m (1 - \frac{p}{q})} + C \sum_{i} \frac{1}{ℓ^{p}} I^{\frac{p}{2}} ℓ^{m (1 - \frac{p}{2})} . \end{aligned} \end{matrix}$ The first sum is for i such that $q_{j_{i}} ⩾ 2$ , the second one for the i’s such that $q_{j_{i}} < 2$ . Using the Young inequality with ϵ we get $\begin{matrix} (3.26) & I ⩽ ϵ I + C_{ϵ} \sum_{i} {(\frac{ℓ^{m (1 - \frac{p}{q})}}{ℓ^{p}})}^{\frac{1}{1 - \frac{p}{q}}} + ϵ I + C_{ϵ} \sum_{i} {(\frac{ℓ^{m (1 - \frac{p}{2})}}{ℓ^{p}})}^{\frac{1}{1 - \frac{p}{2}}} . \end{matrix}$ Choosing ϵ small enough we get $\begin{matrix} (3.27) & I ⩽ C_{ϵ} \sum_{i} \frac{1}{ℓ^{\frac{p q}{q - p} - m}} + C_{ϵ} \sum_{i} \frac{1}{ℓ^{\frac{2 p}{2 - p} - m}} . \end{matrix}$ Coming back to our notation in $p_{i}$ , $q_{j_{i}}$ (3.19) follows. This completes the proof of he theorem. □

Theorem 3.3.

We suppose that $p_{i} ⩾ 2$ , $\forall i = 1, \dots, m$ . In addition we assume that $\begin{matrix} (3.28) & \forall i = 1, \dots, m, \exists j_{i} \in {m + 1, \dots, n} such that p_{i} = q_{j_{i}} . \end{matrix}$ Then there exists constants C, α independent of ℓ such that $\begin{matrix} (3.29) & \begin{aligned} \int_{Ω_{\frac{ℓ}{2}}} \sum_{i = 1}^{m} {| \partial_{x_{i}} (u_{ℓ} - u_{\infty}) |}^{p_{i}} + {| \partial_{x_{j_{i}}} (u_{ℓ} - u_{\infty}) |}^{p_{i}} \\ + \sum_{j \neq j_{i}} {| \partial_{x_{j}} u_{ℓ} |^{q_{j} - 2} \partial_{x_{j}} u_{ℓ} - | \partial_{x_{j}} u_{\infty} |^{q_{j} - 2} \partial_{x_{j}} u_{\infty}} \partial_{x_{j}} (u_{ℓ} - u_{\infty}) d x ⩽ C e^{- α ℓ} . \end{aligned} \end{matrix}$ Note that when $j \neq j_{i}$ , $q_{j}$ is only assumed to be greater than 1 and that the $q_{j_{i}}$ are not necessarily distinct as the $p_{i}$ .

Proof.

For $ℓ_{1} ⩽ ℓ - 1$ we denote by $ρ_{ℓ_{1}} = ρ_{ℓ_{1}} (X_{1})$ a smooth function satisfying $\begin{matrix} (3.30) & \begin{aligned} 0 ⩽ ρ_{ℓ_{1}} ⩽ 1, ρ_{ℓ_{1}} = 1 on ℓ_{1} ω_{1}, \\ ρ_{ℓ_{1}} = 0 outside (ℓ_{1} + 1) ω_{1}, | \nabla_{X_{1}} ρ_{ℓ_{1}} | ⩽ C \end{aligned} \end{matrix}$ where C is some positive constant. Taking $v = ρ_{ℓ_{1}} (u_{ℓ} - u_{\infty})$ as test function in (3.10) we get $\begin{matrix} (3.31) & \begin{aligned} \int_{Ω_{ℓ_{1} + 1}} {\sum_{i = 1}^{m} | \partial_{x_{i}} u_{ℓ} |^{p_{i}} \\ + \sum_{j = m + 1}^{n} {| \partial_{x_{j}} u_{ℓ} |^{q_{j} - 2} \partial_{x_{j}} u_{ℓ} - | \partial_{x_{j}} u_{\infty} |^{q_{j} - 2} \partial_{x_{j}} u_{\infty}} \partial_{x_{j}} (u_{ℓ} - u_{\infty})} ρ_{ℓ_{1}} d x \\ = - \int_{Ω_{ℓ_{1} + 1} ∖ Ω_{ℓ_{1}}} \sum_{i = 1}^{m} | \partial_{x_{i}} u_{ℓ} |^{p_{i} - 2} \partial_{x_{i}} u_{ℓ} \partial_{x_{i}} ρ_{ℓ_{1}} (u_{ℓ} - u_{\infty}) d x \\ ⩽ C \sum_{i = 1}^{m} {(\int_{D_{ℓ_{1}}} | \partial_{x_{i}} u_{ℓ} |^{p_{i}} d x)}^{\frac{1}{p_{i}^{'}}} {(\int_{D_{ℓ_{1}}} | u_{ℓ} - u_{\infty} |^{p_{i}} d x)}^{\frac{1}{p_{i}}} \end{aligned} \end{matrix}$ where we have set $D_{ℓ_{1}} = Ω_{ℓ_{1} + 1} ∖ Ω_{ℓ_{1}}$ . Let us define A as $\begin{matrix} (3.32) & \begin{aligned} A & = \sum_{i = 1}^{m} {| \partial_{x_{i}} (u_{ℓ} - u_{\infty}) |}^{p_{i}} + {| \partial_{x_{j_{i}}} (u_{ℓ} - u_{\infty}) |}^{q_{j_{i}}} \\ + \sum_{j \neq j_{i}} {| \partial_{x_{j}} u_{ℓ} |^{q_{j} - 2} \partial_{x_{j}} u_{ℓ} - | \partial_{x_{j}} u_{\infty} |^{q_{j} - 2} \partial_{x_{j}} u_{\infty}} \partial_{x_{j}} (u_{ℓ} - u_{\infty}) . \end{aligned} \end{matrix}$ Using the lemma 2.3, (3.32) and the Poincaré inequality on the section of the domain one deduces from (3.31) $\begin{matrix} (3.33) & \begin{aligned} \int_{Ω_{ℓ_{1}}} A d x & ⩽ \int_{Ω_{ℓ_{1} + 1}} A ρ_{ℓ_{1}} d x \\ ⩽ C \sum_{i = 1}^{m} {(\int_{D_{ℓ_{1}}} | \partial_{x_{i}} u_{ℓ} |^{p_{i}} d x)}^{\frac{1}{p_{i}^{'}}} {(\int_{D_{ℓ_{1}}} {| \partial_{x_{j_{i}}} (u_{ℓ} - u_{\infty}) |}^{q_{j_{i}}} d x)}^{\frac{1}{p_{i}}} \\ ⩽ C \sum_{i = 1}^{m} {(\int_{D_{ℓ_{1}}} A d x)}^{\frac{1}{p_{i}^{'}}} {(\int_{D_{ℓ_{1}}} A d x)}^{\frac{1}{p_{i}}} = C \int_{D_{ℓ_{1}}} A d x . \end{aligned} \end{matrix}$ It follows that $\begin{matrix} (3.34) & \int_{Ω_{ℓ_{1}}} A d x ⩽ \frac{C}{C + 1} \int_{Ω_{ℓ_{1} + 1}} A \end{matrix}$ Denote by $[\frac{ℓ}{2}]$ the integer part of $\frac{ℓ}{2}$ . Setting $a = \frac{C}{C + 1}$ and iterating this formula $[\frac{ℓ}{2}]$ times starting from $\frac{ℓ}{2}$ we obtain easily taking into account the inequality $\frac{ℓ}{2} - 1 < [\frac{ℓ}{2}] ⩽ \frac{ℓ}{2}$ $\begin{matrix} (3.35) & \int_{Ω_{\frac{ℓ}{2}}} A d x ⩽ a^{[\frac{ℓ}{2}]} \int_{Ω_{\frac{ℓ}{2} + [\frac{ℓ}{2}]}} A d x ⩽ a^{\frac{ℓ}{2} - 1} \int_{Ω_{ℓ}} A d x . \end{matrix}$ To evaluate this last integral one relies on the lemma 3.1. Indeed using the lemma 2.3 one has $\begin{matrix} A ⩽ \sum_{i = 1}^{m} {(| \partial_{x_{i}} u | + | \partial_{x_{i}} u_{\infty} |)}^{p_{i}} + (| \partial_{x_{j_{i}}} u_{ℓ} | + | \partial_{x_{j_{i}}} u_{\infty}) |^{p_{i}} + \sum_{j \neq j_{i}} C_{q_{j}} (| \partial_{x_{j}} u_{ℓ} | + | \partial_{x_{j}} u_{\infty}) |^{q_{j}} \end{matrix}$ Using again the formula ${(| a | + | b |)}^{q} ⩽ 2^{q - 1} (| a |^{q} + | b |^{q})$ one derives for some constant $\begin{matrix} A ⩽ C {\sum_{i = 1}^{m} | \partial_{x_{i}} u |^{p_{i}} + | \partial_{x_{i}} u_{\infty} |^{p_{i}} + \sum_{j = m + 1}^{n} | \partial_{x_{j}} u_{ℓ} |^{q_{j}} + | \partial_{x_{j}} u_{\infty} |^{q_{j}}} \end{matrix}$ Since $u_{\infty}$ is independent of $X_{1}$ it follows from (3.6) that $\begin{matrix} \int_{Ω_{ℓ}} A d x ⩽ C ℓ^{m} \end{matrix}$ and from (3.35) one derives $\begin{matrix} (3.36) & \int_{Ω_{\frac{ℓ}{2}}} A d x ⩽ C e^{- ℓ \frac{1}{2} ln \frac{1}{a}} ℓ^{m} . \end{matrix}$ This leads to (3.29) provided we chose $α < \frac{1}{2} ln \frac{1}{a}$ . □

In the case where $f \in L^{\infty} (ω_{2})$ one can show the following.

Theorem 3.4.

We suppose that $f \in L^{\infty} (ω_{2})$ and $p_{i} ⩾ 2$ , $\forall i = 1, \dots, m$ . In addition we assume that $\begin{matrix} (3.37) & \forall i = 1, \dots, m, \exists j_{i} \in {m + 1, \dots, n} such that p_{i} ⩾ q_{j_{i}} ⩾ 2 . \end{matrix}$ Then there exists constants C, α independent of ℓ such that $\begin{matrix} (3.38) & \begin{aligned} \int_{Ω_{\frac{ℓ}{2}}} \sum_{i = 1}^{m} {| \partial_{x_{i}} (u_{ℓ} - u_{\infty}) |}^{p_{i}} + {| \partial_{x_{j_{i}}} (u_{ℓ} - u_{\infty}) |}^{p_{i}} \\ + \sum_{j \neq j_{i}} {| \partial_{x_{j}} u_{ℓ} |^{q_{j} - 2} \partial_{x_{j}} u_{ℓ} - | \partial_{x_{j}} u_{\infty} |^{q_{j} - 2} \partial_{x_{j}} u_{\infty}} \partial_{x_{j}} (u_{ℓ} - u_{\infty}) d x ⩽ C e^{- α ℓ} . \end{aligned} \end{matrix}$

Proof.

As in Theorem 3.3 we derive (3.31) that is $\begin{matrix} (3.39) & \begin{aligned} \int_{Ω_{ℓ_{1} + 1}} {\sum_{i = 1}^{m} | \partial_{x_{i}} u_{ℓ} |^{p_{i}} \\ + \sum_{j = m + 1}^{n} {| \partial_{x_{j}} u_{ℓ} |^{q_{j} - 2} \partial_{x_{j}} u_{ℓ} - | \partial_{x_{j}} u_{\infty} |^{q_{j} - 2} \partial_{x_{j}} u_{\infty}} \partial_{x_{j}} (u_{ℓ} - u_{\infty})} ρ_{ℓ_{1}} d x \\ ⩽ C \sum_{i = 1}^{m} {(\int_{D_{ℓ_{1}}} | \partial_{x_{i}} u_{ℓ} |^{p_{i}} d x)}^{\frac{1}{p_{i}^{'}}} {(\int_{D_{ℓ_{1}}} | u_{ℓ} - u_{\infty} |^{p_{i}} d x)}^{\frac{1}{p_{i}}}, \end{aligned} \end{matrix}$ where $D_{ℓ_{1}} = Ω_{ℓ_{1} + 1} ∖ Ω_{ℓ_{1}}$ . One claims that for some constant C independent of ℓ one has $\begin{matrix} (3.40) & | u_{ℓ} - u_{\infty} | ⩽ C . \end{matrix}$ Then, we derive that $\begin{matrix} {(\int_{D_{ℓ_{1}}} | u_{ℓ} - u_{\infty} |^{p_{i}} d x)}^{\frac{1}{p_{i}}} ⩽ C {(\int_{D_{ℓ_{1}}} | u_{ℓ} - u_{\infty} |^{q_{j_{1}}} d x)}^{\frac{1}{p_{i}}} . \end{matrix}$ Recalling the notation (3.32), (3.33) follows easily and the rest of the proof as well.

To prove (3.40), suppose that $ω_{2}$ is contained in the strip $\begin{matrix} {X_{2} | | x_{n} | ⩽ s} \end{matrix}$ for some positive s (recall that $ω_{2}$ is supposed to be bounded in $R^{n - m}$ ). Then set $\begin{matrix} β = 1 + \frac{1}{q_{n} - 1}, g = s^{β} - | x_{n} |^{β} . \end{matrix}$ One finds easily since $(β - 1) (q_{n} - 1) = 1$ that $\begin{matrix} \partial_{x_{n}} g = - β | x_{n} |^{β - 2} x_{n}, | \partial_{x_{n}} g |^{q_{n} - 2} \partial_{x_{n}} g = - β^{q_{n} - 1} x_{n}, - \partial_{x_{n}} (| \partial_{x_{n}} g |^{q_{n} - 2} \partial_{x_{n}} g) = β^{q_{n} - 1} . \end{matrix}$ If $| f |_{\infty}$ denotes the $L^{\infty}$ -norm of f setting $\begin{matrix} h = {(| f |_{\infty})}^{\frac{1}{q_{n} - 1}} \frac{g}{β} \end{matrix}$ one has (see (3.3)) $\begin{aligned} - \sum_{i = 1}^{m} \partial_{x_{i}} (| \partial_{x_{i}} u_{ℓ} |^{p_{i} - 2} \partial_{x_{i}} u_{ℓ}) - \sum_{j = m + 1}^{n} \partial_{x_{j}} (| \partial_{x_{j}} u_{ℓ} |^{q_{j} - 2} \partial_{x_{j}} u_{ℓ}) \\ = f ⩽ | f |_{\infty} \\ = - \partial_{x_{n}} (| \partial_{x_{n}} h |^{q_{n} - 2} \partial_{x_{n}} h) = - \sum_{i = 1}^{m} \partial_{x_{i}} (| \partial_{x_{i}} h |^{p_{i} - 2} \partial_{x_{i}} h) - \sum_{j = m + 1}^{n} \partial_{x_{j}} (| \partial_{x_{j}} h |^{q_{j} - 2} \partial_{x_{j}} h) . \end{aligned}$ Using in the weak formulation $v = {(u_{ℓ} - h)}^{+}$ one deduces easily that $\begin{matrix} u_{ℓ} (f) ⩽ h ⩽ {(| f |_{\infty})}^{\frac{1}{q_{n} - 1}} \frac{s^{β}}{β} . \end{matrix}$ Since $- u_{ℓ} (f) = u_{ℓ} (- f)$ , (3.40) follows easily. This completes the proof of the theorem. □

Remark 2.

One could try to mix assumptions of the type of Theorem 3.2 and 3.4 however it will make the result regarding the speed of convergence messy, the convergence being insured by the theorem 3.1 for the $p_{i}$ ’s large enough. In the case of theorems 3.3, 3.4 one can take advantage of the exponential speed of convergence to get existence results in unbounded domains in the spirit of [7,9].

Remark 3.

The operators that we have considered here are the sum of p-Laplacians in one dimension. One can consider also operators sums of p-Laplacians in larger dimensions. For instance, with the notation of this section, if $u_{ℓ}$ is the weak solution to $\begin{matrix} \{\begin{array}{l} - \nabla_{X_{1}} \cdot (| \nabla_{X_{1}} u_{ℓ} |^{p - 2} \nabla_{X_{1}} u_{ℓ}) - \nabla_{X_{2}} \cdot (| \nabla_{X_{2}} u_{ℓ} |^{q - 2} \nabla_{X_{2}} u_{ℓ}) = f in Ω_{ℓ} \\ u_{ℓ} = 0 on \partial Ω_{ℓ}, \end{array} \end{matrix}$ one can show if $p = q ⩾ 2$ , using the technique of theorem 3.3, that $u_{ℓ}$ converges exponentially quickly toward the solution to. $\begin{matrix} \{\begin{array}{l} - \nabla_{X_{2}} \cdot (| \nabla_{X_{2}} u_{\infty} |^{q - 2} \nabla_{X_{2}} u_{\infty}) = f in ω_{2} \\ u_{\infty} = 0 on \partial ω_{2} . \end{array} \end{matrix}$ ( $\nabla_{X_{i}} \cdot$ denotes the divergence in $R^{m}$ or $R^{n - m}$ ). Similarly one can consider operators sums of p-Laplacians of different dimensions i.e. problems of the type $\begin{matrix} \{\begin{array}{l} - \sum_{i} \nabla_{Y_{i}} \cdot (| \nabla_{Y_{i}} u_{ℓ} |^{p_{i} - 2} \nabla_{Y_{i}} u_{ℓ}) = f in Ω_{ℓ} \\ u_{ℓ} = 0 on \partial Ω_{ℓ}, \end{array} \end{matrix}$ where $Y_{i}$ denotes some subset of the coordinates, and develop results similar to the ones of this note. The case of the sum of n-dimensional p-Laplacians was considered in [8].

References

Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext., Springer, New York, 2011.

Bruyère, Comportement asymptotique de problèmes posés dans des cylindres. Problèmes d’unicité pour les systèmes de Boussinesq, PhD thesis, Université de Rouen, 2007.

Chipot, ℓ Goes to Plus Infinity, Birkhäuser Advanced Text, 2002.

Chipot, Elliptic Equations: An Introductory Course, Birkhäuser Advanced Texts, Basel, 2009, 2nd edn. Birkhäuser Advanced Texts, Basler Lehrbücher, 2024.

Chipot, ℓ goes to plus infinity: An update, J. KSIAM 18(2) (2014), 107–127.

Chipot, Asymptotic Issues for Some Partial Differential Equations, Imperial College Press, 2016, 2nd edn, 2024, World Scientific.

Chipot, On some elliptic problems in unbounded domains, Chinese Ann. Math. Ser. B 39(3) (2018), 597–606. doi:10.1007/s11401-018-0083-7.

Chipot, Asymptotic behaviour of operators sum of p-Laplacians, Discrete and Continuous Dynamical Systems – Series S. doi:10.3934/dcdss.2023117.

Chipot and

Mardare, Asymptotic behaviour of the Stokes problem in cylinders becoming unbounded in one direction, J. Math. Pures Appl. 90 (2008), 133–159. doi:10.1016/j.matpur.2008.04.002.

10.

Chipot and

Xie, On the asymptotic behaviour of the p-Laplace equation in cylinders becoming unbounded, in: Proceedings of the International Conference: Nonlinear PDE’s and Their Applications,

Kenmochi,

Ôtani and

Zheng, eds, Gakkotosho, 2004, pp. 16–27.

11.

Chipot and

Xie, Some issues on the p-Laplace equation in cylindrical domains, Proceedings of the Steklov Institue of Mathematics 261 (2008), 287–294. doi:10.1134/S0081543808020235.

12.

P.G.

Ciarlet, Linear and Nonlinear Functional Analysis with Applications, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013.

13.

L.C.

Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, 1998.

14.

Gilbarg and

N.S.

Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer, Berlin, Heidelberg, New-York, 2001. Reprint of the 1998 edition.

15.

Jana, Anisotropic p-Laplace equations on long cylindrical domain, Opuscula Math. 44(2) (2024), 249–265, https://urldefense.com/v3/https://doi.org/10.7494/OpMath.2024.44.2.249 . doi:10.7494/OpMath.2024.44.2.249.

16.

Kinderlehrer and

Stampacchia, An Introduction to Variational Inequalities and Their Applications, Classic Appl. Math., Vol. 31, SIAM, Philadelphia, 2000.

17.

Marcellini, Regularity and existence of solutions of elliptic equations with

(p, q)

-growth conditions, J. Differ. Equ. 90 (1991), 1–30. doi:10.1016/0022-0396(91)90158-6.

18.

Marcellini, Anisotropic and p, q-nonlinear partial differential equations, Rendiconti Lincei, Scienze Fisiche e Naturali 31 (2020), 295–301, https://urldefense.com/v3/https://doi.org/10.1007/s12210-020-00885-y . doi:10.1007/s12210-020-00885-y.

19.

J.C.

Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Text in Applied Mathematics, Cambridge University Press, 2001.