On the Leray problem and the Poiseuille flow

Abstract

The goal of this note is to present a simple proof of existence and uniqueness of the solution of the Leray problem for high viscosities and homogeneous or non homogeneous boundary conditions. Furthermore we address the issue of uniqueness of the Poiseuille flow in a pipe.

Keywords

Nonlinear elliptic equations Navier Stokes stationary problem Poiseuille flow

1. Introduction and notation

We will denote by Ω a smooth open set of $R^{n}$ , $n ⩾ 2$ . If V denotes a Banach space we will denote by $V$ the space of n copies of it and by $\hat{V}$ the space of solenoidal vectors fields belonging to it. For instance, if for $r ⩾ 1$ , $L^{r} (Ω)$ denotes the usual $L^{r}$ -space on Ω we will denote by $L^{r} (Ω)$ the space defined as $\begin{array}{l} L^{r} (Ω) = {v = (v_{1}, \dots, v_{n}) | v_{i} \in L^{r} (Ω) \forall i = 1, \dots, n} . \end{array}$ We denote by ${\hat{L}}^{r} (Ω)$ the space defined as $\begin{array}{l} {\hat{L}}^{r} (Ω) = {v \in L^{r} (Ω) | div v = 0 in Ω}, \end{array}$ ( $div v = \sum_{i = 1}^{n} \partial_{x_{i}} v_{i}$ , the derivative being taken in the distributional sense). Note that ${\hat{L}}^{r} (Ω)$ is a closed subspace of the Banach space $L^{r} (Ω)$ that we will suppose equipped with the usual $L^{r} (Ω)$ -norm defined as $\begin{array}{l} | v |_{r} = {(\int_{Ω} {| v (x) |}^{r} d x)}^{\frac{1}{r}}, \end{array}$ with, in the integral, $| \cdot |$ denoting the usual euclidean norm in $R^{n}$ . Similarly we will denote by $H^{1} (Ω)$ , $H_{0}^{1} (Ω)$ the spaces $\begin{array}{l} H^{1} (Ω) = {v = (v_{1}, \dots, v_{n}) | v_{i} \in H^{1} (Ω) \forall i = 1, \dots, n}, \\ H_{0}^{1} (Ω) = {v = (v_{1}, \dots, v_{n}) | v_{i} \in H_{0}^{1} (Ω) \forall i = 1, \dots, n} \end{array}$ and $\begin{array}{l} {\hat{H}}^{1} (Ω) = {v \in H^{1} (Ω) | div v = 0 in Ω}, \\ {\hat{H}}_{0}^{1} (Ω) = {v \in H_{0}^{1} (Ω) | div v = 0 in Ω}, \end{array}$ where $H^{1} (Ω)$ , $H_{0}^{1} (Ω)$ are the usual Sobolev space built on Ω (Cf. [5,6,9–11]). These spaces will be equipped with their usual norms and for $H_{0}^{1} (Ω)$ we will use $\begin{array}{l} | | \nabla v | |_{2} = {(\int_{Ω} {| \nabla v (x) |}^{2} d x)}^{\frac{1}{2}}, \end{array}$ where $| \nabla v |$ denotes the euclidean norm of the Jacobian matrix $\nabla v$ given by $\begin{array}{l} | \nabla v |^{2} = \sum_{i, k} {(\partial_{x_{i}} v_{k})}^{2} . \end{array}$ We will sometimes omit the sum using the usual summation convention. It is well known that the spaces defined above are reflexive as Hilbert spaces. We will denote the dual of $H_{0}^{1} (Ω)$ by $H^{- 1} (Ω)$ . Moreover assuming Ω smooth enough (see [10]) one has for $n = 2$ or 3 the Sobolev embedding $\begin{array}{l} (1.1) & H^{1} (Ω) \subset L^{4} (Ω) \end{array}$ and for some constant $C_{S}$ $\begin{array}{l} (1.2) & C_{S} | v |_{4} ⩽ | | \nabla v | |_{2} \forall v \in H_{0}^{1} (Ω) . \end{array}$ For $f \in H^{- 1} (Ω)$ , $g \in {\hat{H}}^{1} (Ω)$ , $ν > 0$ we would like to find a couple $(u, p)$ solution to $\begin{array}{l} \{\begin{matrix} u - g \in {\hat{H}}_{0}^{1} (Ω), \\ - ν Δ u + (u \cdot \nabla) u + \nabla p = f in Ω . \end{matrix} \end{array}$ Note that if $u \in {\hat{H}}^{1} (Ω)$ , by the divergence theorem, one has $\begin{array}{l} \int_{\partial Ω} u \cdot n d σ = 0 . \end{array}$ (n denotes the outward unit normal to $\partial Ω$ , “·” the canonical euclidean scalar product). Thus the flux of u through the boundary of $\partial Ω$ has to vanish and u cannot fit any boundary condition. This is why we chose $g \in {\hat{H}}^{1} (Ω)$ . The boundary of Ω can have here several inner parts. In a weak form, eliminating the pressure, one wants to find u such that $\begin{array}{l} (1.3) & \{\begin{matrix} u - g \in {\hat{H}}_{0}^{1} (Ω), \\ ν \int_{Ω} \nabla u \cdot \nabla w d x + \int_{Ω} (u \cdot \nabla) u \cdot w d x = ⟨ f, w ⟩ & \forall v \in {\hat{H}}_{0}^{1} (Ω) . \end{matrix} \end{array}$ We will denote by t the trilinear form appearing above and defined as $\begin{array}{l} t (u, w, v) = \int_{Ω} (u \cdot \nabla) w \cdot v d x = \int_{Ω} u_{k} \partial_{x_{k}} w_{i} v_{i} d x \end{array}$ the summation convention being used in the integral above. The problem has attracted a lot of attention since the seminal paper of J. Leray [17]. Many of them are based on the so called Leray inequality or are using some special properties of the domain like its symmetry or else some particular fluxes (see [1,7–9,12–14,16,18–20]). Our technique for large viscosities is very elementary and is simply based on the Lax–Milgram theorem together with some compactness argument allowing us to use the Schauder fixed point theorem. Usually these kinds of problems are more relevant in the physical cases of $n = 2$ or 3 (see [1,15,17,21]) and we will restrict ourselves to these two cases except perhaps at the end of this note. Regarding the trilinear form t one will notice using the Cauchy–Schwarz inequality repeatedly that $\begin{array}{l} | t (u, v, w) | & = | \int_{Ω} (u \cdot \nabla) v \cdot w d x | ⩽ \int_{Ω} | \partial_{x_{k}} v_{i} | | u_{k} w_{i} | d x \\ ⩽ \int_{Ω} | \nabla v | | u | | w | d x ⩽ {\int_{Ω} | \nabla v |^{2} d x}^{\frac{1}{2}} {\int_{Ω} | u |^{2} | w |^{2} d x}^{\frac{1}{2}} \\ (1.4) & ⩽ | u |_{4} | | \nabla v | |_{2} | w |_{4} \forall u, v, w \in H^{1} (Ω) . \end{array}$ In the last section of the paper we will address the issue of uniqueness of the so called Poiseuille flow.

2. A result of existence

The main result of this section is the following:

Theorem 2.1.
Suppose that we are under the assumptions above i.e. suppose that $f \in H^{- 1} (Ω)$ , $g \in {\hat{H}}^{1} (Ω)$ then if $\begin{array}{l} (2.1) & ν . C_{S} > | g |_{4} \end{array}$ there exists a solution to ( 1.3 ).
Proof.
For $v \in {\hat{L}}^{4} (Ω)$ let us define a as $\begin{array}{l} a (u, w) = ν \int_{Ω} \nabla u \cdot \nabla w d x - t (v + g, w, u) . \end{array}$ For convenience we drop the dependence in v in our notation. a is for any such v a continuous bilinear form on ${\hat{H}}_{0}^{1} (Ω)$ (see (1.2), (1.4)) which in addition is coercive since $\begin{array}{l} a (u, u) = ν \int_{Ω} \nabla u \cdot \nabla u d x = ν | | \nabla u | |_{2}^{2} \forall u \in H_{0}^{1} (Ω) . \end{array}$ This follows from $\begin{array}{l} t (v + g, u, u) = \int_{Ω} (v_{k} + g_{k}) \partial_{x_{k}} u_{i} u_{i} d x = \int_{Ω} (v_{k} + g_{k}) \partial_{x_{k}} {\frac{1}{2} | u |^{2}} d x = 0 . \end{array}$ (The equality is clear for a smooth u with compact support and extends by continuity to $H_{0}^{1} (Ω)$ ). Then, since the right hand side of the equation below is a continuous linear form on $H_{0}^{1} (Ω)$ , by the Lax–Milgram theorem, there exists a unique $U = S (v)$ such that $\begin{array}{l} (2.2) & \{\begin{matrix} U \in {\hat{H}}_{0}^{1} (Ω), \\ ν \int_{Ω} \nabla U \cdot \nabla w d x - t (v + g, w, U) \\ = - ν \int_{Ω} \nabla g \cdot \nabla w d x + t (v + g, w, g) + ⟨ f, w ⟩ \forall w \in {\hat{H}}_{0}^{1} (Ω) . \end{matrix} \end{array}$ If S has a fixed point u it will be solution to $\begin{array}{l} (2.3) & \{\begin{matrix} u \in {\hat{H}}_{0}^{1} (Ω), \\ ν \int_{Ω} \nabla (u + g) \cdot \nabla w d x - t (u + g, w, u + g) = ⟨ f, w ⟩ \forall w \in {\hat{H}}_{0}^{1} (Ω) . \end{matrix} \end{array}$ Since $\begin{array}{l} - t (u + g, w, u + g) & = - \int_{Ω} (u_{k} + g_{k}) \partial_{x_{k}} w_{i} (u_{i} + g_{i}) d x \\ = - \int_{Ω} \partial_{x_{k}} {(u_{k} + g_{k}) w_{i} (u_{i} + g_{i})} d x + \int_{Ω} {(u_{k} + g_{k}) w_{i} \partial_{x_{k}} (u_{i} + g_{i})} d x \\ = t (u + g, u + g, w) \end{array}$ $u + g$ will be solution to (1.3). Taking $w = U$ in (2.2) one gets easily, $| f |_{}$ denoting the strong dual norm of f, $\begin{array}{l} ν | | \nabla U | |_{2}^{2} ⩽ ν | | \nabla g | |_{2} | | \nabla U | |_{2} + | g |_{4}^{2} | | \nabla U | |_{2} + | f |_{} | | \nabla U | |_{2} + | v |_{4} | g |_{4} | | \nabla U | |_{2} \end{array}$ which leads to $\begin{array}{l} ν | | \nabla U | |_{2} ⩽ ν | | \nabla g | |_{2} + | g |_{4}^{2} + | f |_{} + | v |_{4} | g |_{4} . \end{array}$ Using (1.2) it comes $\begin{array}{l} (2.4) & ν . C_{S} | U |_{4} ⩽ ν | | \nabla g | |_{2} + | g |_{4}^{2} + | f |_{} + | v |_{4} | g |_{4} . \end{array}$ We suppose now that (2.1) holds that is to say that $ν . C_{S} > | g |_{4}$ . Note that in the case where $g = 0$ we have no restriction on ν. Consider the ball $\begin{array}{l} (2.5) & B = {v \in {\hat{L}}^{4} (Ω) : | v |_{4} ⩽ \frac{K}{(ν . C_{S} - | g |_{4})}} \end{array}$ where $\begin{array}{l} (2.6) & K = ν | | \nabla g | |_{2} + | g |_{4}^{2} + | f |_{*} . \end{array}$ For $v \in B$ the corresponding $U = s (v)$ satisfies $\begin{array}{l} ν . C_{S} | U |_{4} ⩽ K + | g |_{4} \frac{K}{(ν . C_{S} - | g |_{4})} \end{array}$ which is also $\begin{array}{l} | U |_{4} ⩽ \frac{K}{(ν . C_{S} - | g |_{4})} . \end{array}$ Thus S maps B into B and $S (B)$ is relatively compact in B, since $| | \nabla U | |_{2}$ is bounded independently of $v \in B$ . By the Schauder fixed point theorem S will have a fixed point provided it is continuous. Let us set $\begin{array}{l} U_{i} = S (v_{i}), i = 1, 2 . \end{array}$ From (2.2) written for $i = 1, 2$ we deduce easily $\begin{array}{l} ν \int_{Ω} \nabla (U_{1} - U_{2}) \cdot \nabla w d x - t (v_{1} + g, w, U_{1} + g) + t (v_{2} + g, w, U_{2} + g) = 0 \forall w \in {\hat{H}}_{0}^{1} (Ω) . \end{array}$ Taking $w = U_{1} - U_{2}$ it comes $\begin{array}{l} ν | | \nabla (U_{1} - U_{2}) | |_{2}^{2} & = t (v_{1} + g, U_{1} - U_{2}, U_{1} + g) - t (v_{2} + g, U_{1} - U_{2}, U_{2} + g) \\ = t (v_{1} + g, U_{1} - U_{2}, U_{1} - U_{2} + U_{2} + g) - t (v_{2} + g, U_{1} - U_{2}, U_{2} + g) \\ = t (v_{1} - v_{2}, U_{1} - U_{2}, U_{2} + g) \\ ⩽ | v_{1} - v_{2} |_{4} | | \nabla (U_{1} - U_{2}) | |_{2} | U_{2} + g |_{4} \end{array}$ and thus $\begin{array}{l} ν | | \nabla (U_{1} - U_{2}) | |_{2} ⩽ | v_{1} - v_{2} |_{4} | U_{2} + g |_{4} . \end{array}$ Using (2.2) one deduces $\begin{array}{l} (2.7) & ν . C_{S} | U_{1} - U_{2}) |_{4} ⩽ | v_{1} - v_{2} |_{4} | U_{2} + g |_{4} . \end{array}$ The continuity of S from B into itself follows. This completes the proof of the theorem. □
Remark 1.
Note that for a fixed ν, if one can find an extension $\tilde{g}$ of g on the boundary such that $\begin{array}{l} | \tilde{g} |_{4} < ν . C_{S}, \tilde{g} - g \in {\hat{H}}_{0}^{1} (Ω), \end{array}$ then the existence of a solution holds true.

3. Uniqueness issue

It is well known that in the case where $g = 0$ uniqueness might fail (Cf. [9]) unless some conditions on f or ν are set. In the non homogeneous case the same uniqueness holds provide ν is large enough. One has

Theorem 3.1.
Under the assumption ( 2.1 ) the problem ( 1.3 ) has a unique solution when $ν = ν (g, f)$ is large enough. More precisely when $\begin{array}{l} (3.1) & \frac{ν | | \nabla g | |_{2} + ν . C_{S} | g |_{4} + | f |_{}}{ν . C_{S} (ν . C_{S} - | g |_{4})} < 1 . \end{array}$
Proof.
$u_{1}, u_{2}$ are two solutions of (1.3), if and only if $U_{i} = u_{i} - g$ , $i = 1, 2$ , are solutions to (2.3). Then by (2.7) it comes $\begin{array}{l} (3.2) & ν . C_{S} | U_{1} - U_{2}) |_{4} ⩽ | U_{1} - U_{2} |_{4} | U_{2} + g |_{4} . \end{array}$ If $ν . C_{S} > | g |_{4}$ one has (see (2.4)) $\begin{array}{l} (ν . C_{S} - | g |_{4}) | U_{2} |_{4} ⩽ ν | | \nabla g | |_{2} + | g |_{4}^{2} + | f |_{} \end{array}$ i.e. $\begin{array}{l} | U_{2} |_{4} ⩽ \frac{ν | | \nabla g | |_{2} + | g |_{4}^{2} + | f |_{}}{(ν . C_{S} - | g |_{4})} . \end{array}$ Thus one has $\begin{array}{l} | U_{2} + g |_{4} & ⩽ \frac{ν | | \nabla g | |_{2} + | g |_{4}^{2} + | f |_{}}{(ν . C_{S} - | g |_{4})} + | g |_{4} \\ ⩽ \frac{ν | | \nabla g | |_{2} + | g |_{4}^{2} + | f |_{} + ν . C_{S} | g |_{4} - | g |_{4}^{2}}{(ν . C_{S} - | g |_{4})} \\ ⩽ \frac{ν | | \nabla g | |_{2} + ν . C_{S} | g |_{4} + | f |_{}}{(ν . C_{S} - | g |_{4})} . \end{array}$ Then, by (3.2), uniqueness holds when $\begin{array}{l} \frac{ν | | \nabla g | |_{2} + ν . C_{S} | g |_{4} + | f |_{}}{ν . C_{S} (ν . C_{S} - | g |_{4})} < 1 . \end{array}$ This completes the proof of the theorem. □
Remark 2.
Note that the uniqueness holds for any solution to (1.3) and not only for the one such that $u - g \in B$ . However, any solution to (1.3) is such that $U = u - g$ is a fixed point for S and by (2.4), as a fixed point for S, is automatically in B. In the case where $g = 0$ , as already mentioned, (3.1) reduces to the classical condition $\frac{| f |_{}}{ν^{2} C_{S}^{2}} < 1$ .

4. Uniqueness results for the Poiseuille flow

Let us denote by Ω the unbounded domain defined as $\begin{array}{l} Ω = R \times ω \end{array}$ where ω is a bounded domain of $R^{n - 1}$ . We will denote by $x = (x_{1}, x^{'})$ the points in Ω with an obvious notation for $x^{'}$ and for $ℓ > 0$ we will define $Ω_{ℓ}$ as $\begin{array}{l} Ω_{ℓ} = (- ℓ, ℓ) \times ω . \end{array}$ If $u \in {\hat{H}}^{1} (Ω_{ℓ})$ for every $ℓ > 0$ , and if for every $ℓ > 0$ , $u = 0$ on $(- ℓ, ℓ) \times \partial ω$ by the divergence formula it is easy to see that $\begin{array}{l} \int_{ω} u_{1} (x_{1}, x^{'}) d x^{'} is constant for a.e. x_{1} \end{array}$ i.e. the flux of u is constant through the chanel Ω and if $u \in {\hat{H}}_{0}^{1} (Ω_{ℓ})$ then this flux vanishes since it vanishes at $(\pm ℓ, x^{'})$ i.e. we have $\begin{array}{l} (4.1) & \int_{ω} u_{1} (x_{1}, x^{'}) d x^{'} = 0 a.e. x_{1} \in (- ℓ, ℓ), \end{array}$ (see for instance [3,4]). Let us denote by h the weak solution to $\begin{array}{l} (4.2) & \{\begin{matrix} h \in H_{0}^{1} (ω), \\ - Δ h = 1 in ω . \end{matrix} \end{array}$ It is clear that h is a smooth bounded function. For $F \in R$ one sets $\begin{array}{l} (4.3) & {\tilde{u}}_{1} = F . \frac{h}{\int_{ω} h d x^{'}}, \tilde{u} = ({\tilde{u}}_{1}, 0), \end{array}$ where 0 stands for the 0 in $R^{n - 1}$ . We will denote by $H_{\infty}$ the $L^{\infty} (ω)$ -norm of $\frac{h}{\int_{ω} h d x^{'}}$ that is we set $\begin{array}{l} (4.4) & H_{\infty} = \underset{ω}{esssup} | \frac{h}{\int_{ω} h d x^{'}} | . \end{array}$ Moreover we recall the Poincaré inequality. For some constant $C_{ω}$ we have $\begin{array}{l} C_{ω}^{2} \int_{ω} u^{2} d x^{'} ⩽ \int_{ω} {| \nabla^{'} u |}^{2} d x^{'} \forall u \in H_{0}^{1} (ω) . \end{array}$ The inequality extend trivialement component by component and one has $\begin{array}{l} (4.5) & C_{ω}^{2} \int_{ω} | u |^{2} d x^{'} ⩽ \int_{ω} {| \nabla^{'} u |}^{2} d x^{'} \forall u \in H_{0}^{1} (ω) . \end{array}$ ( $\nabla^{'}$ is the gradient in $R^{n - 1}$ , we denote by $| |_{r, A}$ the usual $L^{r} (A)$ -norm). Combining (4.1)–(4.3) it is easy to see that $\tilde{u}$ satisfies for every $ℓ > 0$ : $\begin{matrix} (4.6) & \{\begin{matrix} \tilde{u} \in {\hat{H}}^{1} (Ω_{ℓ}), \tilde{u} = 0 on (- ℓ, ℓ) \times \partial ω, \\ \int_{ω} {\tilde{u}}_{1} (x_{1}, x^{'}) d x^{'} = F, a.e. x_{1} \in (- ℓ, ℓ), \\ ν \int_{Ω_{ℓ}} \nabla \tilde{u} \cdot \nabla w d x = ν \int_{Ω_{ℓ}} \nabla \tilde{u} \cdot \nabla w d x + t (\tilde{u}, \tilde{u}, w) = 0 \forall w \in {\hat{H}}_{0}^{1} (Ω_{ℓ}) . \end{matrix} \end{matrix}$ Indeed it is clear that $div \tilde{u} = 0$ in Ω and thus $\begin{array}{l} ν \int_{Ω_{ℓ}} \nabla \tilde{u} \cdot \nabla w d x & = ν \int_{Ω_{ℓ}} \nabla \tilde{u_{1}} \cdot \nabla w_{1} d x = ν \int_{Ω_{ℓ}} - Δ \tilde{u_{1}} w_{1} d x \\ = C \int_{Ω_{ℓ}} w_{1} d x = 0 \forall w \in {\hat{H}}_{0}^{1} (Ω_{ℓ}), \end{array}$ by (4.1), C being some constant. Moreover, $\begin{array}{l} t (\tilde{u}, \tilde{u}, w) = \int_{Ω_{ℓ}} {\tilde{u}}_{k} \partial_{x_{k}} {\tilde{u}}_{i} w_{i} d x = \int_{Ω_{ℓ}} {\tilde{u}}_{1} \partial_{x_{1}} {\tilde{u}}_{i} w_{i} d x = 0 \forall w \in {\hat{H}}_{0}^{1} (Ω_{ℓ}) \end{array}$ and $\tilde{u}$ is solution to (4.6). $\tilde{u}$ is known as the Poiseuille flow solution to (4.6). An important issue is to know if it is the only solution of the equations in (4.6). It is well known for the Laplace operator that the problem $\begin{array}{l} - Δ u = 0 in Ω = R \times ω, u = 0 on R \times \partial ω \end{array}$ does not admit 0 as unique solution. The solution of this problem is unique only if one limits its growth (we refer to [2] Chap. 6 for some simple argument regarding this issue). Thus in order to have uniqueness for the solution to (4.6) it is not surprising to have to impose also some limitation to the solution. We have the following two results:

Theorem 4.1.

There exists a positive constant $σ_{0}$ such that if u satisfies for every $ℓ > 0$ $\begin{matrix} (4.7) & \{\begin{matrix} u \in {\hat{H}}^{1} (Ω_{ℓ}), u = 0 on (- ℓ, ℓ) \times \partial ω, \\ \int_{ω} u_{1} (x_{1}, x^{'}) d x^{'} = F, a.e. x_{1} \in (- ℓ, ℓ), \\ ν \int_{Ω_{ℓ}} \nabla u \cdot \nabla w d x = 0 \forall w \in {\hat{H}}_{0}^{1} (Ω_{ℓ}), \\ | | \nabla u | |_{2, Ω_{ℓ}} ⩽ e^{σ ℓ} for σ < σ_{0}, \end{matrix} \end{matrix}$ then one has $u = \tilde{u}$ .

This problem correspond to the so called Stokes problem. For the complete Navier–Stokes problem we have to put more restriction on u and chose ν large or F small. More precisely we have:

Theorem 4.2.

Set $D_{ℓ} = Ω_{ℓ + 1} ∖ Ω_{ℓ}$ . Assume that $\begin{array}{l} (4.8) & ν > F H_{\infty} C_{ω}, | u |_{4, D_{ℓ}} ⩽ K \forall ℓ > 0, \end{array}$ where K is a constant independent of ℓ. Then there exists a positive constant $σ_{0}$ such that if u satisfies for every $ℓ > 0$ $\begin{matrix} (4.9) & \{\begin{matrix} u \in {\hat{H}}^{1} (Ω_{ℓ}), u = 0 on (- ℓ, ℓ) \times \partial ω, \\ \int_{ω} u_{1} (x_{1}, x^{'}) d x^{'} = F, a.e. x_{1} \in (- ℓ, ℓ), \\ ν \int_{Ω_{ℓ}} \nabla u \cdot \nabla w d x + t (u, u, w) = 0 \forall w \in {\hat{H}}_{0}^{1} (Ω_{ℓ}), \\ | | \nabla u | |_{2, Ω_{ℓ}} ⩽ e^{σ ℓ} for σ < σ_{0}, \end{matrix} \end{matrix}$ then one has $u = \tilde{u}$ .

Proof of Theorem 4.1.

Let u be solution to (4.7). For a fixed ℓ consider the function $ρ = ρ (x_{1})$ defined by $\begin{array}{l} (4.10) & \{\begin{matrix} ρ = 1 & on [- ℓ, ℓ], \\ ρ = x + ℓ + 1 & on [- ℓ - 1, - ℓ], \\ ρ = - x + ℓ + 1 & on [ℓ, ℓ + 1], \\ ρ = 0 & outside [- ℓ - 1, ℓ + 1] . \end{matrix} \end{array}$ One has $\begin{array}{l} div (ρ (u - \tilde{u})) = \partial_{x_{1}} ρ . (u_{1} - {\tilde{u}}_{1}) = \pm (u_{1} - {\tilde{u}}_{1}) on D_{ℓ} . \end{array}$ Since $\int_{ω} (u_{1} - {\tilde{u}}_{1}) d x^{'} = 0$ one can find $β \in H_{0}^{1} (D_{ℓ})$ such that $\begin{array}{l} (4.11) & \{\begin{matrix} div β = - \partial_{x_{1}} ρ . (u_{1} - {\tilde{u}}_{1}) in D_{ℓ}, \\ | | \nabla β | |_{2, D_{ℓ}} ⩽ C | u_{1} - {\tilde{u}}_{1} |_{2, D_{ℓ}}, \end{matrix} \end{array}$ for some constant C independent of ℓ (we refer to [4,9] for a similar argument, note that we solve the above problem on each of the connected components of $D_{ℓ}$ ). Then $ρ . (u - \tilde{u}) + β \in {\hat{H}}_{0}^{1} (Ω_{ℓ + 1})$ and we have $\begin{array}{l} ν \int_{Ω_{ℓ + 1}} \nabla (u - \tilde{u}) \cdot \nabla {ρ . (u - \tilde{u}) + β} d x = 0 . \end{array}$ This leads to $\begin{array}{l} ν \int_{Ω_{ℓ + 1}} {| \nabla (u - \tilde{u}) |}^{2} ρ d x \\ = - ν \int_{D_{ℓ}} \partial_{x_{1}} (u_{i} - {\tilde{u}}_{i}) \partial_{x_{1}} ρ (u_{i} - {\tilde{u}}_{i}) d x - ν \int_{D_{ℓ}} \nabla (u - \tilde{u}) \cdot \nabla β d x \\ ⩽ ν \int_{D_{ℓ}} | \nabla (u - \tilde{u}) | | u - \tilde{u} | d x + ν \int_{D_{ℓ}} | \nabla (u - \tilde{u}) | | \nabla β | d x \\ (4.12) & ⩽ ν C \int_{D_{ℓ}} {| \nabla (u - \tilde{u}) |}^{2} d x . \end{array}$ by applying Cauchy–Schwarz inequality and the Poincaré inequality (4.5) on the section of $Ω_{ℓ + 1}$ . C is some constant independent of ℓ. Indeed integrating (4.5) in $x_{1}$ one gets easily $\begin{array}{l} C_{ω}^{2} \int_{D_{ℓ}} | u |^{2} d x ⩽ \int_{D_{ℓ}} | \nabla u |^{2} d x, \end{array}$ since for a.e. $x_{1}$ , $u (x_{1}, \cdot) \in H_{0}^{1} (ω)$ (see [3]).

Since $ρ = 1$ on $Ω_{ℓ}$ this leads easily to $\begin{array}{l} \int_{Ω_{ℓ}} {| \nabla (u - \tilde{u}) |}^{2} d x ⩽ \frac{C}{C + 1} \int_{Ω_{ℓ + 1}} {| \nabla (u - \tilde{u}) |}^{2} d x \end{array}$ for every ℓ. Starting from $\frac{ℓ}{2}$ and iterating this inequality $[\frac{ℓ}{2}]$ times with $[\frac{ℓ}{2}]$ denoting the integer part of $\frac{ℓ}{2}$ we get $\begin{array}{l} \int_{Ω_{\frac{ℓ}{2}}} {| \nabla (u - \tilde{u}) |}^{2} d x & ⩽ {(\frac{C}{C + 1})}^{[\frac{ℓ}{2}]} \int_{Ω_{\frac{ℓ}{2} + [\frac{ℓ}{2}]}} {| \nabla (u - \tilde{u}) |}^{2} d x \\ ⩽ {(\frac{C}{C + 1})}^{\frac{ℓ}{2} - 1} \int_{Ω_{ℓ}} {| \nabla (u - \tilde{u}) |}^{2} d x, \end{array}$ since $\frac{ℓ}{2} - 1 ⩽ [\frac{ℓ}{2}] ⩽ \frac{ℓ}{2}$ . Set $\begin{array}{l} σ_{0} = - \frac{1}{4} ln \frac{C}{C + 1} . \end{array}$ Then one has $\begin{array}{l} \int_{Ω_{\frac{ℓ}{2}}} {| \nabla (u - \tilde{u}) |}^{2} d x ⩽ 2 \frac{C + 1}{C} e^{- 2 σ_{0} ℓ} e^{2 σ ℓ} \to 0 \end{array}$ when $ℓ \to \infty$ . This shows that $u = \tilde{u}$ and completes the proof of the theorem. □

Proof of Theorem 4.2.

Let u be solution to (4.9). Using as in the previous proof the function $ρ . (u - \tilde{u}) + β \in {\hat{H}}_{0}^{1} (Ω_{ℓ + 1})$ one gets $\begin{array}{l} ν \int_{Ω_{ℓ + 1}} \nabla (u - \tilde{u}) \cdot \nabla {ρ . (u - \tilde{u}) + β} d x + t (u, u, {ρ . (u - \tilde{u}) + β}) = 0 . \end{array}$ Expanding the first integral above like in (4.12) we get $\begin{array}{l} (4.13) & ν \int_{Ω_{ℓ + 1}} {| \nabla (u - \tilde{u}) |}^{2} ρ d x ⩽ ν C \int_{D_{ℓ}} {| \nabla (u - \tilde{u}) |}^{2} d x - t (u, u, {ρ . (u - \tilde{u}) + β}) . \end{array}$ One has $\begin{array}{l} (4.14) & t (u, u, {ρ . (u - \tilde{u}) + β}) = t (u, u - \tilde{u}, {ρ . (u - \tilde{u}) + β}) + t (u, \tilde{u}, {ρ . (u - \tilde{u}) + β}) . \end{array}$ Let us first estimate the first term of the right hand side above. One has $\begin{array}{l} t (u, u - \tilde{u}, {ρ . (u - \tilde{u}) + β}) & = \int_{Ω_{ℓ + 1}} u_{k} \partial_{x_{k}} (u_{i} - {\tilde{u}}_{i}) {ρ (u_{i} - {\tilde{u}}_{i}) + β_{i}} d x \\ = \int_{Ω_{ℓ + 1}} u_{k} ρ \partial_{x_{k}} \frac{| u - \tilde{u} |^{2}}{2} d x + \int_{D_{ℓ}} u_{k} \partial_{x_{k}} (u_{i} - {\tilde{u}}_{i}) β_{i} d x \\ = - \int_{D_{ℓ}} u_{1} \partial_{x_{1}} ρ \frac{| u - \tilde{u} |^{2}}{2} d x - \int_{D_{ℓ}} u_{k} (u_{i} - {\tilde{u}}_{i}) \partial_{x_{k}} β_{i} d x . \end{array}$ Thus we derive easily for some constant C independant of ℓ $\begin{array}{l} | t (u, u - \tilde{u}, {ρ . (u - \tilde{u}) + β}) | \\ ⩽ \frac{1}{2} | u |_{4, D_{ℓ}} | u - \tilde{u} |_{4, D_{ℓ}} | u - \tilde{u} |_{2, D_{ℓ}} + | u |_{4, D_{ℓ}} | u - \tilde{u} |_{4, D_{ℓ}} | | \nabla β | |_{2, D_{ℓ}} \\ ⩽ C | u - \tilde{u} |_{4, D_{ℓ}} | u - \tilde{u} |_{2, D_{ℓ}} \end{array}$ we used here (1.4), (4.8), (4.10), (4.11). Next using the poincaré inequality on the section of the domain and the embedding of $H^{1}$ into $L^{4}$ one get easily for a constant $C_{S}^{'}$ independent of ℓ $\begin{array}{l} (4.15) & C_{ω} | u - \tilde{u} |_{2, D_{ℓ}} ⩽ | | \nabla (u - \tilde{u}) | |_{2, D_{ℓ}}, C_{S}^{'} | u - \tilde{u} |_{4, D_{ℓ}} ⩽ | | \nabla (u - \tilde{u}) | |_{2, D_{ℓ}} . \end{array}$ Thus for some constant C independent of ℓ we get $\begin{array}{l} (4.16) & | t (u, u - \tilde{u}, {ρ . (u - \tilde{u}) + β}) | ⩽ C | | \nabla (u - \tilde{u}) | |_{2, D_{ℓ}}^{2} . \end{array}$

For the second term of (4.14) we get $\begin{array}{l} t (u, \tilde{u}, {ρ . (u - \tilde{u}) + β}) = & t (u - \tilde{u}, \tilde{u}, {ρ . (u - \tilde{u}) + β}) \\ = & \int_{Ω_{ℓ + 1}} (u_{k} - {\tilde{u}}_{k}) \partial_{x_{k}} {\tilde{u}}_{i} {ρ (u_{i} - {\tilde{u}}_{i}) + β_{i}} d x \\ = & \int_{Ω_{ℓ + 1}} (u_{k} - {\tilde{u}}_{k}) \partial_{x_{k}} {\tilde{u}}_{1} {ρ (u_{1} - {\tilde{u}}_{1}) + β_{1}} d x \\ = & - \int_{Ω_{ℓ + 1}} (u_{k} - {\tilde{u}}_{k}) {\tilde{u}}_{1} \partial_{x_{k}} {ρ (u_{1} - {\tilde{u}}_{1}) + β_{1}} d x \\ = & - \int_{Ω_{ℓ + 1}} (u_{k} - {\tilde{u}}_{k}) {\tilde{u}}_{1} ρ \partial_{x_{k}} (u_{1} - {\tilde{u}}_{1}) d x \\ (4.17) & - \int_{D_{ℓ}} (u_{k} - {\tilde{u}}_{k}) {\tilde{u}}_{1} {(\partial_{x_{k}} ρ) (u_{1} - {\tilde{u}}_{1}) + \partial_{x_{k}} β_{1}} d x . \end{array}$ Thus we derive $\begin{array}{l} | t (u, \tilde{u}, {ρ . (u - \tilde{u}) + β}) | \\ ⩽ F H_{\infty} \int_{Ω_{ℓ + 1}} | u - \tilde{u} | ρ | \nabla (u - \tilde{u}) | d x \\ (4.18) & + | u - \tilde{u} |_{4, D_{ℓ}} | \tilde{u} |_{4, D_{ℓ}} {| u - \tilde{u} |_{2, D_{ℓ}} + | | \nabla β | |_{2, D_{ℓ}}} . \end{array}$ Thus by (4.8), (4.11) and by the Cauchy–Schwarz inequality we get for some constant C $\begin{array}{l} | t (u, \tilde{u}, {ρ . (u - \tilde{u}) + β}) | \\ ⩽ F H_{\infty} \int_{(- (ℓ + 1), ℓ + 1)} ρ {(\int_{ω} | u - \tilde{u} |^{2} d x^{'})}^{\frac{1}{2}} {(\int_{ω} {| \nabla (u - \tilde{u}) |}^{2} d x^{'})}^{\frac{1}{2}} d x_{1} \\ + C | u - \tilde{u} |_{4, D_{ℓ}} | u - \tilde{u} |_{2, D_{ℓ}} . \end{array}$ Applying the Poincaré inequality on the section ω and (4.15) it comes $\begin{array}{l} | t (u, \tilde{u}, {ρ . (u - \tilde{u}) + β}) | \\ (4.19) & ⩽ F H_{\infty} C_{ω} \int_{Ω_{ℓ + 1}} ρ {| \nabla (u - \tilde{u}) |}^{2} d x + C | | \nabla (u - \tilde{u}) | |_{2, D_{ℓ}}^{2} . \end{array}$ Collecting (4.13), (4.16), (4.19) we arrive to $\begin{array}{l} (ν - F H_{\infty} C_{ω}) \int_{Ω_{ℓ + 1}} {| \nabla (u - \tilde{u}) |}^{2} ρ d x ⩽ C | | \nabla (u - \tilde{u}) | |_{2, D_{ℓ}}^{2}, \end{array}$ It follows that $\begin{array}{l} (4.20) & (ν - F H_{\infty} C_{ω}) \int_{Ω_{ℓ}} {| \nabla (u - \tilde{u}) |}^{2} d x ⩽ C | | \nabla (u - \tilde{u}) | |_{2, D_{ℓ}}^{2} . \end{array}$ Then one can conclude as after (4.12) in the previous theorem. This completes the proof of the theorem. □

The Theorem 4.1 is true for any n. Due to the use of the Sobolev embedding in the second inequality of (4.15) Theorem 4.2 is only true for $n ⩽ 4$ . However, for $n ⩾ 5$ with some more constraints and limitation of oscillations one can prove:

Theorem 4.3.

Suppose $n ⩾ 5$ . Let u satisfying for every $ℓ > 0$ $\begin{matrix} (4.21) & \{\begin{matrix} u \in {\hat{H}}^{1} (Ω_{ℓ}), u = 0 on (- ℓ, ℓ) \times \partial ω, \\ \int_{ω} u_{1} (x_{1}, x^{'}) d x^{'} = F, a.e. x_{1} \in (- ℓ, ℓ), \\ ν \int_{Ω_{ℓ}} \nabla u \cdot \nabla w d x + t (u, u, w) = 0 \forall w \in {\hat{H}}_{0}^{1} (Ω_{ℓ}) . \end{matrix} \end{matrix}$ If for some constant independent of ℓ $\begin{array}{l} (4.22) & | | \nabla u | |_{2, D_{ℓ}}, | u |_{4, D_{ℓ}} ⩽ K \end{array}$ then for $\begin{array}{l} (4.23) & ν > F H_{\infty} C_{ω} \end{array}$ one has $u = \tilde{u}$ .

Proof of Theorem 4.3.

(4.13) remains unchanged. Using (4.22), (4.16) becomes for some constant C independent of ℓ $\begin{array}{l} (4.24) & | t (u, u - \tilde{u}, {ρ . (u - \tilde{u}) + β}) | ⩽ C | | \nabla (u - \tilde{u}) | |_{2, D_{ℓ}} . \end{array}$ Similarly (4.19) becomes $\begin{array}{l} (4.25) & | t (u, \tilde{u}, {ρ . (u - \tilde{u}) + β}) | ⩽ C | | \nabla (u - \tilde{u}) | |_{2, D_{ℓ}} . \end{array}$ Collecting (4.13), (4.14), (4.24), (4.25) we arrive to $\begin{array}{l} (4.26) & (ν - F H_{\infty} C_{ω}) \int_{Ω_{ℓ}} {| \nabla (u - \tilde{u}) |}^{2} d x ⩽ C | | \nabla (u - \tilde{u}) | |_{2, D_{ℓ}} . \end{array}$ Thus, by (4.22), $\int_{Ω_{ℓ}} | \nabla (u - \tilde{u}) |^{2} d x$ is bounded independently of ℓ and since it is nondecreasing in ℓ it has a limit when $ℓ \to \infty$ . It follows that $\begin{array}{l} | | \nabla (u - \tilde{u}) | |_{2, D_{ℓ}}^{2} = \int_{Ω_{ℓ + 1}} {| \nabla (u - \tilde{u}) |}^{2} d x - \int_{Ω_{ℓ}} {| \nabla (u - \tilde{u}) |}^{2} d x \to 0 \end{array}$ when $ℓ \to \infty$ and by (4.26) that $u = \tilde{u}$ . This completes the proof of the theorem. □

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