On pressure-driven Hele–Shaw flow of power-law fluids

Abstract

We analyze the asymptotic behavior of a non-Newtonian Stokes system, posed in a Hele–Shaw cell, i.e. a thin three-dimensional domain which is confined between two curved surfaces and contains a cylindrical obstacle. The fluid is assumed to be of power-law type defined by the exponent $1<p<\mathrm{\infty }$. By letting the thickness of the domain tend to zero we obtain a generalized form of the Poiseuille law, i.e. the limit velocity is a nonlinear function of the limit pressure gradient. The flow is assumed to be driven by an external pressure which is applied as a normal stress along the lateral part of the boundary. On the remaining part of the boundary we impose a no-slip condition. The two-dimensional limit problem for the pressure is a generalized form of the $p^{\mathrm{\prime }}$-Laplace equation, $1/p+1/p^{\mathrm{\prime }}=1$, with a coefficient called ‘flow factor’, which depends on the geometry as well as the power-law exponent. The boundary conditions are preserved in the limit as a Dirichlet condition for the pressure on the lateral boundary and as a Neumann condition for the pressure on the solid obstacle.

Keywords:

• Stress boundary condition
• Hele–Shaw cell
• power-law fluid
• p-Laplace equation
• thin film flow

2010 Mathematics Subject Classifications:

• 76A05
• 76D27
• 76A20

1. Introduction

In this paper we consider the stationary flow of an incompressible non-Newtonian fluid in a thin domain ${\mathrm{\Omega }}^{ϵ}$ in ${\mathbb{R}}^3$, where ε is a positive small parameter related to the thickness of the domain. More precisely, we consider a Hele–Shaw cell, i.e. the domain is confined between two curved surfaces in close proximity and contains a cylindrical obstacle. The fluid is modeled as a power-law fluid, i.e. the viscosity depends on the rate of strain via a power-law. The boundary $\mathrm{\partial }{\mathrm{\Omega }}^{ϵ}$ consists of two disjoint part ${\mathrm{\Gamma }}_D^{ϵ}$ (Dirichlet boundary) and ${\mathrm{\Gamma }}_N^{ϵ}$ (Neumann boundary). The Dirichlet part corresponds to the upper and lower surfaces which are separated by a non-uniform thickness, as well as the lateral surface of the obstacle. The Neumann part, which corresponds to the lateral boundary of the cell, is regarded as a penetrable inlet/outlet zone where an external pressure gradient is applied as a surface force along ${\mathrm{\Gamma }}_N^{ϵ}$. Moreover, the flow is assumed to be governed by the Stokes equation, therefore inertial effects are neglected. It is also assumed that the flow is isothermal and that gravity can be neglected.

The main aim of the present work is to derive a lower-dimensional model for the flow by studying the asymptotic behavior of the system described above, as $ϵ\to 0$. This dimensional reduction will be achieved by using an adaptation of a technique called two-scale convergence for thin domains which was developed by Marušić and Marušić-Paloka [1]. The present results generalize [2] where the Newtonian case was considered without any obstacle in the geometry. Moreover, we generalize the classical Poiseuille law, see e.g. [3, p. 222], for Newtonian flow. In particular the Newtonian Hele–Shaw flow is suitable to visualize streamlines around an obstacle [4], as well as, connecting the velocity to the gradient of the pressure by linear dependence. The main result of the present paper is a nonlinear Poiseuille law, in which the limit velocity and the limit pressure gradient follows a power-law [5, Sec. 7.3.1]. A novelty is that we consider pressure-driven flow that can be found in many realistic applications, see e.g. [6–9].

Non-Newtonian flow in Hele–Shaw like domains appears in many industrial applications, e.g. polymer processing, transportation of oil in pipelines, hydrology, as well as in food processing. In particular, we focus on some works where analysis and mathematical modeling of Poiseuille flow of power-law fluids are considered. Aronsson and Janfalk [10] derived the equations of motion and proved some exact solutions and a representation theorem. A more rigorous study was performed by Mikelić and Tapiéro [11] who derived a nonlinear Poiseuille law as a limit case of the Navier–Stokes equations when the thickness of the cell tends to zero. However, to avoid technical difficulties the authors chose to employ a no-slip condition on the whole boundary. The typical situation in a Hele–Shaw cell is that some part of the boundary should be penetrable, thus allowing fluid particles to enter and leave the domain. At such boundaries it is natural to prescribe the mass flux or impose a boundary condition for the pressure, i.e. by prescribing the momentum flux. Note also that, most examples in Hele–Shaw's original paper showed domains where a solid obstacle was placed in the interior of the domain, which is another characteristic of Hele–Shaw flow. Thus, the present work complements [11] in the sense that a mixed boundary condition is employed, both surfaces may be curved and the domain may contain an interior obstacle. Related models that take thermal effects into account can be found in [12,13]. Non-Newtonian Hele–shaw flow of other type than power-law relation has been studied in [14,15].

Poiseuille's law also plays an important role in lubrication theory, formulated in the Newtonian case by Reynolds [16], see also [17, Ch. 22]. Recall that the fundamental problem in lubrication theory is to describe fluid flow in the gap between adjacent surfaces which are in relative motion to each other and in general there are no obstacles in the domain. Bayada and Chambat [18], gave the first rigorous mathematical derivation of Reynolds equation, considering the Stokes problem in a thin domain and assuming that the velocity field satisfies a Dirichlet condition on the whole boundary. The authors proved that the limit pressure satisfies the classical Reynolds equation with a Neumann condition. In contrast, our limit equation has a Dirichlet condition for the pressure. The connection between Hele–Shaw theory and lubrication theory has been further explored in [19]. A lubrication problem with power-law fluids was considered in [20], under the assumption that the lower surface is flat and moving whereas the upper surface is rough and stationary. Flow of power-law fluids through a thin porous medium, i.e. periodic array of vertical cylinders confined between two parallel plates with no-slip boundary condition on the whole boundary was studied in [21].

The paper is organized as follows: In Section 2 we give a precise description of the Hele–shaw cell and present some notation and preliminary results. In Section 3 we set the problem and formulate our main result. To prove the main result we need several a priori estimates and some results related to two-scale convergence for thin domains. These results are proved in Section 4– 5. Finally, the main result is proved in Section 5–6. In addition, several technical results are proved in Appendices 1–4.

2. Preliminaries, notation and some technical results

2.1. Euclidean structure

Let ${\mathbb{R}}^{m\times n}$ denote the set of real $m\times n$ matrices $X=(x_{ij}{)}_{i,j=1}^{m,n}$ equipped with the Euclidean scalar product $X:Y=\mathrm{t}\mathrm{r}\left(X^{T}Y\right)=\sum _{j=1}^n\sum _{i=1}^m{x}_{ij}{y}_{ij},|X|=(X:X{)}^{1/2},$where $X^T=(x_{ji}{)}_{i,j=1}^{n,m}$ in ${\mathbb{R}}^{m\times n}$ denotes the transpose of X, the symmetrical and skew-symmetric parts of X are defined respectively $e\left(X\right)=\frac{1}{2}(X+X^T),\omega \left(X\right)=\frac{1}{2}(X-X^T).$The identity element in ${\mathbb{R}}^{n\times n}$ is denoted as $I=({\delta }_{ij}{)}_{i,j=1}^{n,n}$. We identify ${\mathbb{R}}^n$ with ${\mathbb{R}}^{n\times 1}$ (column vectors) and denote the scalar product in ${\mathbb{R}}^n$ as $x\cdot y=x^{T}y=\sum _{i=1}^n{x}_i{y}_i,|x|=(x\cdot x{)}^{1/2}.$

2.2. Geometry of the domain

Let ω denote a perforated Lipschitz domain in ${\mathbb{R}}^2,$ say $\omega =(-L,L)\times (-L,L)-D,$where L>0 and D is the closed disc $D=\left\{\right(x_1,x_2):x_1^2+x_2^2\le R^2\}$of radius 0<R<L. The boundary of ω is decomposed into two disjoint parts $\begin{array}{rl}{\gamma }_D& =\mathrm{\partial }D\\ {\gamma }_N& =\mathrm{\partial }\omega -{\gamma }_D.\end{array}$Thus, both ${\gamma }_D$ and ${\gamma }_N$ have positive measure (arc length). Note that the main result of this paper is valid for more general domains ω that share the essential properties of the definition given above.

The (unscaled) fluid domain is defined as $\mathrm{\Omega }=\left\{\right(x_1,x_2,x_3)\in {\mathbb{R}}^3:(x_1,x_2)\in \omega ,h^{-}(x_1,x_2)<x_3<h^{+}(x_1,x_2\left)\right\},$where $h^{+},h^{-}$ are Lipschitz functions defined on the closure of ω, i.e. $h^{+},h^{-}\in C^{0,1}\left(\overline{\omega }\right)$ and satisfy the conditions (1) $\begin{array}{rl}& -\frac{1}{2}\le h^{-}(x_1,x_2)\le -\frac{1}{4}\mathrm{a}\mathrm{n}\mathrm{d}\frac{1}{4}\le h^{+}(x_1,x_2)\le \frac{1}{2}\\ & h\left(x^{\mathrm{\prime }}\right)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{h}^{+}(x_1,x_2)-h^{-}(x_1,x_2)\mathrm{a}\mathrm{n}\mathrm{d}m(x_1,x_2)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{h^{+}(x_1,x_2)+h^{-}(x_1,x_2)}{2},\end{array}$(1) where $h(x_1,x_2)$ with $\frac{1}{2}\le h(x_1,x_2)\le 1$ is the thickness of the fluid domain and $m(x_1,x_2)$ is the center surface of the fluid domain.

A point in the domain Ω will be denoted as $(x^{\mathrm{\prime }},y),$ where $x^{\mathrm{\prime }}=(x_1,x_2)$ and $y=x_3.$ Similarly, the components of a vector function $u:\mathrm{\Omega }\to {\mathbb{R}}^3$ are denoted as $u=(u^{\mathrm{\prime }},u_3)$, where $u^{\mathrm{\prime }}=(u_1,u_2)$.

The boundary of Ω is divided into two disjoint parts $\begin{array}{rl}{\mathrm{\Gamma }}_D& =\mathrm{\partial }\mathrm{\Omega }\cap \left\{\right(x^{\mathrm{\prime }},y):y=h^{+}(x^{\mathrm{\prime }})\text{ or }y=h^{-}(x^{\mathrm{\prime }})\text{ or }{x}^{\mathrm{\prime }}\in \mathrm{\partial }D\}\\ {\mathrm{\Gamma }}_N& =\mathrm{\partial }\mathrm{\Omega }-{\mathrm{\Gamma }}_D,\end{array}$where ${\mathrm{\Gamma }}_N$ is referred to as the ‘lateral’ surface.

The thin domain ${\mathrm{\Omega }}^{ϵ}$ is defined by anisotropic scaling of the unscaled domain Ω of unit thickness, using the parameter $0<ϵ\le 1$, i.e. ${\mathrm{\Omega }}^{ϵ}=\left\{\right(x^{\mathrm{\prime }},x_3)\in {\mathbb{R}}^3:x^{\mathrm{\prime }}\in \omega ,ϵh^{-}(x^{\mathrm{\prime }})<x_3<ϵh^{+}(x^{\mathrm{\prime }}\left)\right\}.$In other words ${\mathrm{\Omega }}^{ϵ}$ is a Hele–Shaw cell of thickness $ϵh\left(x^{\mathrm{\prime }}\right)$. Also, the boundary of ${\mathrm{\Omega }}^{ϵ}$ is divided into two disjoint parts $\begin{array}{rl}{\mathrm{\Gamma }}_D^{ϵ}& =\mathrm{\partial }{\mathrm{\Omega }}^{ϵ}\cap \left\{\right(x^{\mathrm{\prime }},x_3):x_3=ϵh^{+}(x^{\mathrm{\prime }})\text{ or}{x}_3=ϵh^{-}(x^{\mathrm{\prime }})\text{ or}{x}^{\mathrm{\prime }}\in \mathrm{\partial }D\}\\ {\mathrm{\Gamma }}_N^{ϵ}& =\mathrm{\partial }{\mathrm{\Omega }}^{ϵ}-{\mathrm{\Gamma }}_D^{ϵ}.\end{array}$

2.3. Function spaces

We shall work mainly with the following function spaces $\begin{array}{rl}{W}^p\left({\mathrm{\Omega }}^{ϵ}\right)& =\{v\in W^{1,p}({\mathrm{\Omega }}^{ϵ};{\mathbb{R}}^3):v=0\mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_D^{ϵ}\}\\ V^p\left({\mathrm{\Omega }}^{ϵ}\right)& =\{v\in W^p({\mathrm{\Omega }}^{ϵ}):\mathrm{d}\mathrm{i}\mathrm{v}v=0\text{ in }{\mathrm{\Omega }}^{ϵ},v=0\mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_D^{ϵ}\},\end{array}$where $1<p<\mathrm{\infty }$ with conjugate exponent $p^{\mathrm{\prime }}=p/(p-1)$ and the divergence operator $\mathrm{d}\mathrm{i}\mathrm{v}:W^p\left({\mathrm{\Omega }}^{ϵ}\right)\to L^p\left({\mathrm{\Omega }}^{ϵ}\right)$ is defined by (2) $\mathrm{d}\mathrm{i}\mathrm{v}v=\sum _{i=1}^3\frac{\mathrm{\partial }{v}_i}{\mathrm{\partial }{x}_i}.$(2) $V^p\left({\mathrm{\Omega }}^{ϵ}\right)\subset W^p\left({\mathrm{\Omega }}^{ϵ}\right)$ are closed subspace of $W^{1,p}({\mathrm{\Omega }}^{ϵ};{\mathbb{R}}^3)$. Since ${\mathrm{\Gamma }}_D^{ϵ}$ has positive surface measure we can equip $W^p\left({\mathrm{\Omega }}^{ϵ}\right)$ with the norm $\parallel v{\parallel }_{W^p\left({\mathrm{\Omega }}^{ϵ}\right)}=\parallel e\left(\mathrm{\nabla }v\right){\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}={\left({\int }_{{\mathrm{\Omega }}^{ϵ}}{∣e\left(\mathrm{\nabla }v\right)∣}^p\mathrm{d}x\right)}^{1/p}$which is equivalent to the usual $W^{1,p}$-norm by Korn's inequality [22]. Moreover, $W^p\left({\mathrm{\Omega }}^{ϵ}\right)/V^p\left({\mathrm{\Omega }}^{ϵ}\right)$ denotes the quotient space of $W^p\left({\mathrm{\Omega }}^{ϵ}\right)$ by $V^p\left({\mathrm{\Omega }}^{ϵ}\right)$ defined as the set of all equivalence classes $\left\{u\right\}=u+V^p\left({\mathrm{\Omega }}^{ϵ}\right),$equipped with the norm $\parallel \left\{v\right\}{\parallel }_{W^p\left({\mathrm{\Omega }}^{ϵ}\right)/V^p\left({\mathrm{\Omega }}^{ϵ}\right)}=\underset{u\in V^p\left({\mathrm{\Omega }}^{ϵ}\right)}{inf}\parallel v-u{\parallel }_{W^p\left({\mathrm{\Omega }}^{ϵ}\right)}.$For more details concerning the characterization of this space, see e.g. [23,24].

We introduce the operator $a:{\mathbb{R}}^{3\times 3}\to {\mathbb{R}}^{3\times 3}$ defined by (3) $a\left(X\right)=2{\mu }_0|e\left(X\right){|}^{p-2}e\left(X\right)(X\in {\mathbb{R}}^{3\times 3}),$(3) where $e\left(X\right)$ is the symmetrical part of X. Then a has the following properties

1. a is continuous $(1<p<\mathrm{\infty })$, with $a\left(0\right)=0$;

2. a is $(p-1)$-homogeneous, i.e. (4) $a\left(\lambda X\right)={\lambda }^{p-1}a\left(X\right),(\lambda >0);$(4)

3. a is monotone, i.e. (5) $\left(a\right(X)-a(Y\left)\right):(X-Y)\ge 0,$(5) holds for all $X,Y\in {\mathbb{R}}^{3\times 3}.$

For more details, see e.g. [25].

In the following subsections, we give some variants of Korn's inequality, Bogovski and de Rham's operators for our thin domain.

2.4. Korn's inequality

These estimates show how the constants depend on the parameter in the thin domain ${\mathrm{\Omega }}^{ϵ}$.

Theorem 2.1

Korn's inequality

There exist constants $K_1$ and $K_2$ depending only on Ω and ${\mathrm{\Gamma }}_D$ such that (6) $\begin{array}{r}||v|{|}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\le ϵK_1||e\left(\mathrm{\nabla }v\right)|{|}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\end{array}$(6) (7) $\begin{array}{r}||\mathrm{\nabla }v|{|}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\le K_2||e\left(\mathrm{\nabla }v\right)|{|}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)},\end{array}$(7) for all $v\in W^p\left({\mathrm{\Omega }}^{ϵ}\right)$. These constants are such that $\underset{0<ϵ\le 1}{sup}{ϵ}^{-1}{K}_1^{ϵ}\le K_1,\underset{0<ϵ\le 1}{sup}{K}_2^{ϵ}\le K_2,$where $K_1^{ϵ}$ and $K_2^{ϵ}$ denote the best constants in the inequalities (6(6) $\begin{array}{r}||v|{|}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\le ϵK_1||e\left(\mathrm{\nabla }v\right)|{|}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\end{array}$(6) ) and (7(7) $\begin{array}{r}||\mathrm{\nabla }v|{|}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\le K_2||e\left(\mathrm{\nabla }v\right)|{|}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)},\end{array}$(7) ).

Proof.

See Appendix 1.

2.5. The Bogovski operator

The divergence operator is onto, see [24, Theorem 5.4] where the case p = 2 was considered. The generalization to $1<p<\mathrm{\infty }$ is straight-forward since the vector field $\hat{v}$ in [24, Lemma 5.1] is of class $C^{0,1}.$ Related results can also be found in [26–28]. However, the divergence operator is not one-to-one since $\mathrm{N}\mathrm{u}\mathrm{l}\mathrm{l}\mathrm{d}\mathrm{i}\mathrm{v}=V^p\left({\mathrm{\Omega }}^{ϵ}\right)$ is always nontrivial. By collapsing the nullspace to zero we can construct an invertible operator $A:W^p\left({\mathrm{\Omega }}^{ϵ}\right)/V^p\left({\mathrm{\Omega }}^{ϵ}\right)\to L^p\left({\mathrm{\Omega }}^{ϵ}\right)$defined by $A\left\{u\right\}=\mathrm{d}\mathrm{i}\mathrm{v}u$, for all $u\in W^p\left({\mathrm{\Omega }}^{ϵ}\right)$, which is one-to-one, onto and therefore A is an isomorphism. From Jensen's inequality it follows that $||A\left\{u\right\}|{|}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}=||\mathrm{d}\mathrm{i}\mathrm{v}u|{|}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\le \sqrt[p^{\mathrm{\prime }}]{3}||\mathrm{\nabla }u|{|}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)},$for all $u\in \left\{u\right\}$, hence A is continuous. The Open Mapping theorem asserts that the inverse $B:L^p\left({\mathrm{\Omega }}^{ϵ}\right)\to W^p\left({\mathrm{\Omega }}^{ϵ}\right)/V^p\left({\mathrm{\Omega }}^{ϵ}\right)$of A is continuous. The operator B is called the Bogovski operator.

2.6. De Rham's operator

Since B is a continuous isomorphism, we can identify ${\left(W^p\left({\mathrm{\Omega }}^{ϵ}\right)/V^p\left({\mathrm{\Omega }}^{ϵ}\right)\right)}^{\mathrm{\prime }}\simeq V^p({\mathrm{\Omega }}^{ϵ}{)}^{\perp },$where $V^p({\mathrm{\Omega }}^{ϵ}{)}^{\perp }$ is the annihilator of $V\left({\mathrm{\Omega }}^{ϵ}\right)$ defined as $V^p({\mathrm{\Omega }}^{ϵ}{)}^{\perp }=\{F\in W^p({\mathrm{\Omega }}^{ϵ}{)}^{\mathrm{\prime }}:〈F,\phi {〉}_{W^p({\mathrm{\Omega }}^{ϵ}{)}^{\mathrm{\prime }},W^p({\mathrm{\Omega }}^{ϵ})}=0,\mathrm{\forall }\phi \in V^p({\mathrm{\Omega }}^{ϵ}\left)\right\},$and by the Riesz Representation theorem, we have $\left(L^p\right({\mathrm{\Omega }}^{ϵ}){)}^{\mathrm{\prime }}\simeq L^{p^{\mathrm{\prime }}}({\mathrm{\Omega }}^{ϵ})$. The dual operator $B^{\mathrm{\prime }}:V^p({\mathrm{\Omega }}^{ϵ}{)}^{\perp }\to L^{p^{\mathrm{\prime }}}({\mathrm{\Omega }}^{ϵ}),$of B is usually called De Rham's operator or the ‘pressure operator’. Note that $B^{\mathrm{\prime }}$ is also continuous with $||B^{\mathrm{\prime }}||=||B||$. To emphasize that B depends on the domain ${\mathrm{\Omega }}^{ϵ}$, we write $B=B_{{\mathrm{\Omega }}^{ϵ}}$. Thus, to obtain a $L^{p^{\mathrm{\prime }}}$-bound for the pressure we need to investigate how the operator norm $||B_{{\mathrm{\Omega }}^{ϵ}}^{\mathrm{\prime }}||=||B_{{\mathrm{\Omega }}^{ϵ}}||\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\underset{\begin{array}{c}h\in L^p\left({\mathrm{\Omega }}^{ϵ}\right)\\ \parallel h\parallel =1\end{array}}{sup}\underset{\begin{array}{c}v\in W^p\left({\mathrm{\Omega }}^{ϵ}\right)\\ \mathrm{d}\mathrm{i}\mathrm{v}v=h\end{array}}{inf}{\left({\int }_{{\mathrm{\Omega }}^{ϵ}}|e\left(\mathrm{\nabla }v\right){|}^p\mathrm{d}x\right)}^{1/p}$depends on ε.

Theorem 2.2

Let $B_{{\mathrm{\Omega }}^{ϵ}}:L^p\left({\mathrm{\Omega }}^{ϵ}\right)\to W^p\left({\mathrm{\Omega }}^{ϵ}\right)/V^p\left({\mathrm{\Omega }}^{ϵ}\right),$denote the Bogovski operator. Then there exist positive constants $C_1$ and $C_2$ depending only on Ω and ${\mathrm{\Gamma }}_D$ such that (8) $C_1\le \underset{0<ϵ\le 1}{inf}ϵ\parallel B_{{\mathrm{\Omega }}^{ϵ}}\parallel \le \underset{0<ϵ\le 1}{sup}ϵ\parallel B_{{\mathrm{\Omega }}^{ϵ}}\parallel \le C_2.$(8)

Proof.

See Appendix 2.

Remark 2.3

We would like to point out that the lower bound in the Theorem 2.2 for the norm of the Bogovski operator is not needed to prove the main result, it is included only to show that the norm $\parallel B_{{\mathrm{\Omega }}^{ϵ}}\parallel \to \mathrm{\infty }$ as $ϵ\to 0$ and that an upper bound of the form $\parallel B_{{\mathrm{\Omega }}^{ϵ}}\parallel \le {ϵ}^{-1}{C}_2$ is the best possible estimate that can be obtained.

Theorem 2.4

De Rham's Theorem

Let $F\in W^p({\mathrm{\Omega }}^{ϵ}{)}^{\mathrm{\prime }}$ be such that $〈F,\phi {〉}_{W^p({\mathrm{\Omega }}^{ϵ}{)}^{\mathrm{\prime }},W^p({\mathrm{\Omega }}^{ϵ})}=0,\text{for all}\phi \in V^p\left({\mathrm{\Omega }}^{ϵ}\right).$Then, there exists a unique $q\in L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)$, such that $〈F,\phi {〉}_{W^p({\mathrm{\Omega }}^{ϵ}{)}^{\mathrm{\prime }},W^p({\mathrm{\Omega }}^{ϵ})}={\int }_{{\mathrm{\Omega }}^{ϵ}}q\mathrm{d}\mathrm{i}\mathrm{v}\phi \mathrm{d}x,\text{for all}\phi \in W^p\left({\mathrm{\Omega }}^{ϵ}\right).$

Proof.

Since $F\in W^p({\mathrm{\Omega }}^{ϵ}{)}^{\mathrm{\prime }}$ satisfying $〈F,\phi 〉=0$ for all $\phi \in V^p\left({\mathrm{\Omega }}^{ϵ}\right)$ this implies that $F\in V^p({\mathrm{\Omega }}^{ϵ}{)}^{\perp }$, therefore, we can choose q in $L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)$ such that $q=B^{\mathrm{\prime }}F$. Furthermore, by duality we have $〈B^{\mathrm{\prime }}F,\varphi {〉}_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right),L^p\left({\mathrm{\Omega }}^{ϵ}\right)}=〈F,B\varphi {〉}_{W^p({\mathrm{\Omega }}^{ϵ}{)}^{\mathrm{\prime }},W^p({\mathrm{\Omega }}^{ϵ})}.$Then, we define $\left\{\phi \right\}=B\varphi \in W^p\left({\mathrm{\Omega }}^{ϵ}\right)/V^p\left({\mathrm{\Omega }}^{ϵ}\right)$, which implies that $\varphi =A\left\{\phi \right\}=\mathrm{d}\mathrm{i}\mathrm{v}\phi$, consequently, $〈q,\mathrm{d}\mathrm{i}\mathrm{v}\phi {〉}_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right),L^p\left({\mathrm{\Omega }}^{ϵ}\right)}=〈F,\phi {〉}_{W^p({\mathrm{\Omega }}^{ϵ}{)}^{\mathrm{\prime }},W^p({\mathrm{\Omega }}^{ϵ})},$i.e. $〈F,\phi {〉}_{W^p({\mathrm{\Omega }}^{ϵ}{)}^{\mathrm{\prime }},W^p({\mathrm{\Omega }}^{ϵ})}={\int }_{{\mathrm{\Omega }}^{ϵ}}q\mathrm{d}\mathrm{i}\mathrm{v}\phi \mathrm{d}x,$as desired.

3. Setting the problem and the main result

3.1. Power-law fluid

Let us suppose that $u^{ϵ}$ is the velocity field, $q^{ϵ}$ is the pressure and $\mathrm{\nabla }{u}^{ϵ}=(\mathrm{\partial }{u}_i^{ϵ}/\mathrm{\partial }{x}_j{)}_{i,j=1}^{3,3}$ is the gradient of velocity. The rate of strain tensor $e=(e_{ij}{)}_{i,j=1}^{3,3}$ is defined by $e_{ij}=\frac{1}{2}(\frac{\mathrm{\partial }{u}_i^{ϵ}}{\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }{u}_j^{ϵ}}{\mathrm{\partial }{x}_i}).$The stress tensor ${\sigma }^{ϵ}=({\sigma }_{ij}^{ϵ}{)}_{i,j=1}^{3,3}$ is defined by the following non-Newtonian constitutive law (9) ${\sigma }_{ij}^{ϵ}=-q^{ϵ}{\delta }_{ij}+2\mu \left(∣e∣\right)e_{ij}.$(9) When the function μ (dynamic viscosity) is constant, we say that the fluid is Newtonian. In the present paper we study the flow of a power-law fluid, where the viscosity $\mu :[0,\mathrm{\infty })\to \mathbb{R}$ is a nonlinear function. More precisely, we assume that (10) $\mu \left(t\right)={\mu }_0{t}^{p-2}(t>0),$(10) which defines the rheology of the fluid, where $1<p<\mathrm{\infty }$ and ${\mu }_0>0$ are constant parameters called power-law index and the consistency respectively. For more details see [29, pp. 16–22] or [30, p. 55]. The parameter p is dimensionless while ${\mu }_0$ has units which depend on the value of p. When p = 2 we recover the case of a Newtonian fluid studied in [2].

3.2. Boundary value problem

Let us consider stationary pressure-driven isothermal fluid flow in the Hele–Shaw cell ${\mathrm{\Omega }}^{ϵ}$. We assume that inertial and body forces can be neglected. Moreover, we assume that the fluid is incompressible and obeys the non-Newtonian constitutive relation (9(9) ${\sigma }_{ij}^{ϵ}=-q^{ϵ}{\delta }_{ij}+2\mu \left(∣e∣\right)e_{ij}.$(9) ). We model the flow by the following boundary value problem in ${\mathrm{\Omega }}^{ϵ}$ (11) $\left\{\begin{array}{ll}\mathrm{d}\mathrm{i}\mathrm{v}{\sigma }^{ϵ}=0& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ \mathrm{d}\mathrm{i}\mathrm{v}{u}^{ϵ}=0& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ u^{ϵ}=0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_D^{ϵ}\\ {\sigma }^{ϵ}\hat{n}=-p_b\hat{n}& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_N^{ϵ},\end{array}\right.$(11) where $u^{ϵ}$ and $q^{ϵ}$ are the unknowns, $\hat{n}$ is the outward unit normal of $\mathrm{\partial }{\mathrm{\Omega }}^{ϵ}$ and $p_b$, the external pressure, is a given function. We assume that $p_b$ is defined on the whole domain, more precisely $p_b\in W^{1,p^{\mathrm{\prime }}}\left(\omega \right)$, i.e. $p_b$ depends only on the variable $x^{\mathrm{\prime }}$.

3.3. Existence and uniqueness

In order to prove existence and uniqueness of solutions, we consider the problem (11(11) $\left\{\begin{array}{ll}\mathrm{d}\mathrm{i}\mathrm{v}{\sigma }^{ϵ}=0& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ \mathrm{d}\mathrm{i}\mathrm{v}{u}^{ϵ}=0& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ u^{ϵ}=0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_D^{ϵ}\\ {\sigma }^{ϵ}\hat{n}=-p_b\hat{n}& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_N^{ϵ},\end{array}\right.$(11) ) in the general form (12) $\left\{\begin{array}{ll}\mathrm{d}\mathrm{i}\mathrm{v}{\sigma }^{ϵ}=f& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ \mathrm{d}\mathrm{i}\mathrm{v}{u}^{ϵ}=0& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ u^{ϵ}=0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_D^{ϵ}\\ {\sigma }^{ϵ}\hat{n}=g& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_N^{ϵ},\end{array}\right.$(12) where $f\in L^{p^{\mathrm{\prime }}}({\mathrm{\Omega }}^{ϵ};{\mathbb{R}}^3)$ and $g\in L^{p^{\mathrm{\prime }}}({\mathrm{\Gamma }}_N^{ϵ};{\mathbb{R}}^3)$ are given vector functions, which are interpreted as body force and surface force respectively.

We use the standard methods of the calculus of variations to interpret the problem (12(12) $\left\{\begin{array}{ll}\mathrm{d}\mathrm{i}\mathrm{v}{\sigma }^{ϵ}=f& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ \mathrm{d}\mathrm{i}\mathrm{v}{u}^{ϵ}=0& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ u^{ϵ}=0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_D^{ϵ}\\ {\sigma }^{ϵ}\hat{n}=g& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_N^{ϵ},\end{array}\right.$(12) ) as a problem of minimization.

Lemma 3.1

Let J be the functional defined as $J\left(v\right)={\int }_{{\mathrm{\Omega }}^{ϵ}}\left(\frac{2{\mu }_0}{p}{∣e\left(\mathrm{\nabla }v\right)∣}^p-f\cdot v\right)\mathrm{d}x-{\int }_{{\mathrm{\Gamma }}_N^{ϵ}}g\cdot v\mathrm{d}S.$Then, there exists a unique minimizer $u^{ϵ}$ in $V^p\left({\mathrm{\Omega }}^{ϵ}\right)$, i.e. $J\left(u^{ϵ}\right)\le J\left(v\right)$ for all v in the admissible class $V^p\left({\mathrm{\Omega }}^{ϵ}\right)$.

Proof.

See Appendix 3.

3.4. Weak formulation

Take a test function $v\in W^p\left({\mathrm{\Omega }}^{ϵ}\right)$ and apply the Divergence theorem to ${\sigma }^{ϵ}\hat{n}\cdot v$, we have $\begin{array}{rl}{\int }_{\mathrm{\partial }{\mathrm{\Omega }}^{ϵ}}{\sigma }^{ϵ}\hat{n}\cdot v\mathrm{d}S& ={\int }_{\mathrm{\partial }{\mathrm{\Omega }}^{ϵ}}\sum _{i,j=1}^n{\sigma }_{ij}^{ϵ}{n}_j{v}_i\mathrm{d}S=\sum _{i=1}^n{\int }_{{\mathrm{\Omega }}^{ϵ}}\sum _{j=1}^n\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}\left({\sigma }_{ij}^{ϵ}{v}_i\right)\mathrm{d}x\\ & =\sum _{i,j=1}^n{\int }_{{\mathrm{\Omega }}^{ϵ}}\left(v_i\frac{\mathrm{\partial }{\sigma }_{ij}^{ϵ}}{\mathrm{\partial }{x}_j}+{\sigma }_{ij}^{ϵ}\frac{\mathrm{\partial }{v}_i}{\mathrm{\partial }{x}_j}\right)\mathrm{d}x={\int }_{{\mathrm{\Omega }}^{ϵ}}\left(v\cdot \mathrm{d}\mathrm{i}\mathrm{v}{\sigma }^{ϵ}+{\sigma }^{ϵ}:\mathrm{\nabla }v\right)\mathrm{d}x.\end{array}$Since v = 0 on ${\mathrm{\Gamma }}_D^{ϵ}$ and ${\sigma }^{ϵ}\hat{n}=g$, from the definition of the stress tensor ${\sigma }^{ϵ}$ we see that, $\begin{array}{rl}{\int }_{{\mathrm{\Gamma }}_N^{ϵ}}g\cdot v\mathrm{d}S+{\int }_{{\mathrm{\Omega }}^{ϵ}}f\cdot v\mathrm{d}x& ={\int }_{{\mathrm{\Omega }}^{ϵ}}(-q^{ϵ}I+2\mu (|e\left(\mathrm{\nabla }{u}^{ϵ}\right)|\left)e\right(\mathrm{\nabla }{u}^{ϵ}\left)\right):\mathrm{\nabla }v\mathrm{d}x\\ & =-{\int }_{{\mathrm{\Omega }}^{ϵ}}{q}^{ϵ}\mathrm{d}\mathrm{i}\mathrm{v}v\mathrm{d}x+{\int }_{{\mathrm{\Omega }}^{ϵ}}2{\mu }_0{∣e\left(\mathrm{\nabla }{u}^{ϵ}\right)∣}^{p-2}e\left(\mathrm{\nabla }{u}^{ϵ}\right):\mathrm{\nabla }v\mathrm{d}x.\end{array}$Thus, the weak formulation of problem (12(12) $\left\{\begin{array}{ll}\mathrm{d}\mathrm{i}\mathrm{v}{\sigma }^{ϵ}=f& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ \mathrm{d}\mathrm{i}\mathrm{v}{u}^{ϵ}=0& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ u^{ϵ}=0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_D^{ϵ}\\ {\sigma }^{ϵ}\hat{n}=g& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_N^{ϵ},\end{array}\right.$(12) ) is (13) ${\int }_{{\mathrm{\Omega }}^{ϵ}}{q}^{ϵ}\mathrm{d}\mathrm{i}\mathrm{v}v\mathrm{d}x={\int }_{{\mathrm{\Omega }}^{ϵ}}\left(2{\mu }_0{∣e\left(\mathrm{\nabla }{u}^{ϵ}\right)∣}^{p-2}e\right(\mathrm{\nabla }{u}^{ϵ}):\mathrm{\nabla }v-f\cdot v)\mathrm{d}x-{\int }_{{\mathrm{\Gamma }}_N^{ϵ}}g\cdot v\mathrm{d}S,$(13) for all v in $W^p\left({\mathrm{\Omega }}^{ϵ}\right).$

Theorem 3.2

Given $f\in L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)$ and $g\in L^{p^{\mathrm{\prime }}}\left({\mathrm{\Gamma }}_N^{ϵ}\right)$, there exists a unique weak solution $(u^{ϵ},q^{ϵ})\in V^p\left({\mathrm{\Omega }}^{ϵ}\right)\times L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)$ of the problem (12(12) $\left\{\begin{array}{ll}\mathrm{d}\mathrm{i}\mathrm{v}{\sigma }^{ϵ}=f& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ \mathrm{d}\mathrm{i}\mathrm{v}{u}^{ϵ}=0& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ u^{ϵ}=0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_D^{ϵ}\\ {\sigma }^{ϵ}\hat{n}=g& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_N^{ϵ},\end{array}\right.$(12) ) satisfying the weak formulation (13(13) ${\int }_{{\mathrm{\Omega }}^{ϵ}}{q}^{ϵ}\mathrm{d}\mathrm{i}\mathrm{v}v\mathrm{d}x={\int }_{{\mathrm{\Omega }}^{ϵ}}\left(2{\mu }_0{∣e\left(\mathrm{\nabla }{u}^{ϵ}\right)∣}^{p-2}e\right(\mathrm{\nabla }{u}^{ϵ}):\mathrm{\nabla }v-f\cdot v)\mathrm{d}x-{\int }_{{\mathrm{\Gamma }}_N^{ϵ}}g\cdot v\mathrm{d}S,$(13) ).

Proof.

By Lemma 3.1, there exists a unique minimizer $u^{ϵ}\in V^p\left({\mathrm{\Omega }}^{ϵ}\right)$ which satisfies the Euler–Lagrange equation, (14) ${\int }_{{\mathrm{\Omega }}^{ϵ}}2{\mu }_0{∣e\left(\mathrm{\nabla }{u}^{ϵ}\right)∣}^{p-2}e\left(\mathrm{\nabla }{u}^{ϵ}\right):\mathrm{\nabla }v\mathrm{d}x={\int }_{{\mathrm{\Omega }}^{ϵ}}f\cdot v\mathrm{d}x+{\int }_{{\mathrm{\Gamma }}_N^{ϵ}}g\cdot v\mathrm{d}S,$(14) for all $v\in V^p\left({\mathrm{\Omega }}^{ϵ}\right)$. Define $F^{ϵ}\in W^p({\mathrm{\Omega }}^{ϵ}{)}^{\mathrm{\prime }}$ by $〈F^{ϵ},v〉\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{\int }_{{\mathrm{\Omega }}^{ϵ}}\left(2{\mu }_0{∣e\left(\mathrm{\nabla }{u}^{ϵ}\right)∣}^{p-2}e\right(\mathrm{\nabla }{u}^{ϵ}):\mathrm{\nabla }v-f\cdot v)\mathrm{d}x-{\int }_{{\mathrm{\Gamma }}_N^{ϵ}}g\cdot v\mathrm{d}S.$In this notation (14(14) ${\int }_{{\mathrm{\Omega }}^{ϵ}}2{\mu }_0{∣e\left(\mathrm{\nabla }{u}^{ϵ}\right)∣}^{p-2}e\left(\mathrm{\nabla }{u}^{ϵ}\right):\mathrm{\nabla }v\mathrm{d}x={\int }_{{\mathrm{\Omega }}^{ϵ}}f\cdot v\mathrm{d}x+{\int }_{{\mathrm{\Gamma }}_N^{ϵ}}g\cdot v\mathrm{d}S,$(14) ) becomes $〈F^{ϵ},v〉=0\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}v\in V^p\left({\mathrm{\Omega }}^{ϵ}\right).$From De Rham's theorem we deduce the existence and uniqueness of a pressure function $q^{ϵ}\in L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right),$ such that $〈F^{ϵ},v〉={\int }_{{\mathrm{\Omega }}^{ϵ}}{q}^{ϵ}\mathrm{d}\mathrm{i}\mathrm{v}v\mathrm{d}x,\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}v\in W^p\left({\mathrm{\Omega }}^{ϵ}\right).$This is the weak formulation (13(13) ${\int }_{{\mathrm{\Omega }}^{ϵ}}{q}^{ϵ}\mathrm{d}\mathrm{i}\mathrm{v}v\mathrm{d}x={\int }_{{\mathrm{\Omega }}^{ϵ}}\left(2{\mu }_0{∣e\left(\mathrm{\nabla }{u}^{ϵ}\right)∣}^{p-2}e\right(\mathrm{\nabla }{u}^{ϵ}):\mathrm{\nabla }v-f\cdot v)\mathrm{d}x-{\int }_{{\mathrm{\Gamma }}_N^{ϵ}}g\cdot v\mathrm{d}S,$(13) ) and therefore $(u^{ϵ},q^{ϵ})\in V^p\left({\mathrm{\Omega }}^{ϵ}\right)\times L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)$ is the unique weak solution of (12(12) $\left\{\begin{array}{ll}\mathrm{d}\mathrm{i}\mathrm{v}{\sigma }^{ϵ}=f& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ \mathrm{d}\mathrm{i}\mathrm{v}{u}^{ϵ}=0& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ u^{ϵ}=0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_D^{ϵ}\\ {\sigma }^{ϵ}\hat{n}=g& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_N^{ϵ},\end{array}\right.$(12) ).

3.5. Main result

Taking f = 0 and $g=-p_b\hat{n}$ in the weak formulation (13(13) ${\int }_{{\mathrm{\Omega }}^{ϵ}}{q}^{ϵ}\mathrm{d}\mathrm{i}\mathrm{v}v\mathrm{d}x={\int }_{{\mathrm{\Omega }}^{ϵ}}\left(2{\mu }_0{∣e\left(\mathrm{\nabla }{u}^{ϵ}\right)∣}^{p-2}e\right(\mathrm{\nabla }{u}^{ϵ}):\mathrm{\nabla }v-f\cdot v)\mathrm{d}x-{\int }_{{\mathrm{\Gamma }}_N^{ϵ}}g\cdot v\mathrm{d}S,$(13) ) we have that the weak solution $u^{ϵ}\in V^p\left({\mathrm{\Omega }}^{ϵ}\right)$ of (11(11) $\left\{\begin{array}{ll}\mathrm{d}\mathrm{i}\mathrm{v}{\sigma }^{ϵ}=0& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ \mathrm{d}\mathrm{i}\mathrm{v}{u}^{ϵ}=0& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ u^{ϵ}=0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_D^{ϵ}\\ {\sigma }^{ϵ}\hat{n}=-p_b\hat{n}& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_N^{ϵ},\end{array}\right.$(11) ) satisfies (15) ${\int }_{{\mathrm{\Omega }}^{ϵ}}{q}^{ϵ}\mathrm{d}\mathrm{i}\mathrm{v}v\mathrm{d}x={\int }_{{\mathrm{\Omega }}^{ϵ}}2{\mu }_0{∣e\left(\mathrm{\nabla }{u}^{ϵ}\right)∣}^{p-2}e\left(\mathrm{\nabla }{u}^{ϵ}\right):\mathrm{\nabla }v\mathrm{d}x+{\int }_{\mathrm{\partial }{\mathrm{\Omega }}^{ϵ}}{p}_{b}v\cdot \hat{n}\mathrm{d}S,$(15) for all $v\in W^p\left({\mathrm{\Omega }}^{ϵ}\right).$ Applying the Divergence theorem to the surface integral and using that v = 0 on ${\mathrm{\Gamma }}_D^{ϵ}$, yields (16) ${\int }_{{\mathrm{\Omega }}^{ϵ}}{\pi }^{ϵ}\mathrm{d}\mathrm{i}\mathrm{v}v\mathrm{d}x={\int }_{{\mathrm{\Omega }}^{ϵ}}\left(2{\mu }_0{∣e\left(\mathrm{\nabla }{u}^{ϵ}\right)∣}^{p-2}e\right(\mathrm{\nabla }{u}^{ϵ}):\mathrm{\nabla }v+\mathrm{\nabla }{p}_b\cdot v)\mathrm{d}x,$(16) for all $v\in W^p\left({\mathrm{\Omega }}^{ϵ}\right),$ where ${\pi }^{ϵ}$ denotes the normalized pressure defined by (17) ${\pi }^{ϵ}=q^{ϵ}-p_b.$(17) Recall that, by assumption, $p_b$ depends only of $x^{\mathrm{\prime }}\in \omega$. This implies that the third component in $\mathrm{\nabla }{p}_b$ is zero, i.e. $\mathrm{\nabla }{p}_b=(\mathrm{\partial }{p}_b/\mathrm{\partial }{x}_1,\mathrm{\partial }{p}_b/\mathrm{\partial }{x}_2,0).$

To give a precise asymptotic description of the system (11(11) $\left\{\begin{array}{ll}\mathrm{d}\mathrm{i}\mathrm{v}{\sigma }^{ϵ}=0& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ \mathrm{d}\mathrm{i}\mathrm{v}{u}^{ϵ}=0& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ u^{ϵ}=0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_D^{ϵ}\\ {\sigma }^{ϵ}\hat{n}=-p_b\hat{n}& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_N^{ϵ},\end{array}\right.$(11) ), we need the definition of two-scale convergence for thin domains introduced by Marušić and Marušić-Paloka in [1].

Definition 3.3

We say that a sequence $u^{ϵ}$, where $ϵ>0$, in $L^p\left({\mathrm{\Omega }}^{ϵ}\right)$ two-scale converges to u in $L^p\left(\mathrm{\Omega }\right)$ provided that (18) $\underset{ϵ\to 0}{lim}\frac{1}{∣{\mathrm{\Omega }}^{ϵ}∣}{\int }_{{\mathrm{\Omega }}^{ϵ}}{u}^{ϵ}\left(x\right)v\left(x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3\right)\mathrm{d}x=\frac{1}{∣\mathrm{\Omega }∣}{\int }_{\mathrm{\Omega }}u(x^{\mathrm{\prime }},y)v(x^{\mathrm{\prime }},y)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y,$(18) for all v in $L^{p^{\mathrm{\prime }}}\left(\mathrm{\Omega }\right)$ and we write $u^{ϵ}\stackrel{2}{\rightharpoonup }u.$ Moreover, we say that $u^{ϵ}$ two-scale converges strongly to u if (19) $\underset{ϵ\to 0}{lim}\frac{1}{∣{\mathrm{\Omega }}^{ϵ}∣}{\int }_{{\mathrm{\Omega }}^{ϵ}}{∣u^{ϵ}\left(x\right)-u(x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3)∣}^p\mathrm{d}x=0$(19) and we write $u^{ϵ}\stackrel{2}{\to }u$ (strongly).

To characterize the limit velocity, we introduce the so called permeability function and the flow factor for a thin domain, both depending on the rheology of the fluid and the geometry of the domain.

Definition 3.4

The solution ψ of the boundary value problem (20) $\left\{\begin{array}{ll}-{\mathrm{\Delta }}_{p,y}\psi =1& \mathrm{i}\mathrm{n}\mathrm{\Omega }\\ \psi =0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_D^{\pm },\end{array}\right.$(20) where ${\mathrm{\Gamma }}_D^{\pm }=\left\{(x^{\mathrm{\prime }},y)\in {\mathrm{\Gamma }}_D:y=h^{\pm }\left(x^{\mathrm{\prime }}\right)\right\},$is called the permeability function of Ω. That is, for each $x^{\mathrm{\prime }}\in \omega$, $\psi (x^{\mathrm{\prime }},\cdot )$ is the solution of $\left\{\begin{array}{ll}-{\mathrm{\Delta }}_{p,y}\psi (x^{\mathrm{\prime }},\cdot )=1& \mathrm{i}\mathrm{n}\left(h^{-}\left(x^{\mathrm{\prime }}\right),h^{+}\left(x^{\mathrm{\prime }}\right)\right)\\ \psi (x^{\mathrm{\prime }},\cdot )=0& \mathrm{o}\mathrm{n}y=h^{\pm }\left(x^{\mathrm{\prime }}\right),\end{array}\right.$where ${\mathrm{\Delta }}_{p,y}(\cdot )=\frac{\mathrm{\partial }}{\mathrm{\partial }y}\left(|\frac{\mathrm{\partial }}{\mathrm{\partial }y}\right(\cdot \left){|}^{p-2}\frac{\mathrm{\partial }}{\mathrm{\partial }y}\right(\cdot \left)\right)$ is the p-laplacian in the variable y.

Definition 3.5

The flow factor of Ω is defined as (21) $\varrho \left(x^{\mathrm{\prime }}\right)={\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}\psi (x^{\mathrm{\prime }},y)\mathrm{d}y,x^{\mathrm{\prime }}\in \omega ,$(21) where ψ is the permeability function, i.e. the solution of boundary value problem (20(20) $\left\{\begin{array}{ll}-{\mathrm{\Delta }}_{p,y}\psi =1& \mathrm{i}\mathrm{n}\mathrm{\Omega }\\ \psi =0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_D^{\pm },\end{array}\right.$(20) ).

Lemma 3.6

For our geometry the permeability function ψ and the flow factor ϱ for Ω, are given by (22) $\psi (x^{\mathrm{\prime }},y)=\frac{1}{p^{\mathrm{\prime }}}\left({\left|\frac{h\left(x^{\mathrm{\prime }}\right)}{2}\right|}^{p^{\mathrm{\prime }}}-|m\left(x^{\mathrm{\prime }}\right)-y{|}^{p^{\mathrm{\prime }}}\right),$(22) for all $y\in \left[h^{-}\right(x^{\mathrm{\prime }}),h^{+}(x^{\mathrm{\prime }}\left)\right]$ and (23) $\varrho \left(x^{\mathrm{\prime }}\right)=\frac{2^{-p^{\mathrm{\prime }}}}{p^{\mathrm{\prime }}+1}{h}^{p^{\mathrm{\prime }}+1}\left(x^{\mathrm{\prime }}\right),$(23) for all $x^{\mathrm{\prime }}\in \omega$ where $h\left(x^{\mathrm{\prime }}\right)$ and $m\left(x^{\mathrm{\prime }}\right)$ are defined in (1(1) $\begin{array}{rl}& -\frac{1}{2}\le h^{-}(x_1,x_2)\le -\frac{1}{4}\mathrm{a}\mathrm{n}\mathrm{d}\frac{1}{4}\le h^{+}(x_1,x_2)\le \frac{1}{2}\\ & h\left(x^{\mathrm{\prime }}\right)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{h}^{+}(x_1,x_2)-h^{-}(x_1,x_2)\mathrm{a}\mathrm{n}\mathrm{d}m(x_1,x_2)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{h^{+}(x_1,x_2)+h^{-}(x_1,x_2)}{2},\end{array}$(1) ).

Proof.

The proof of this lemma will be given in Section 6.

Remark 3.7

The smoothness of ψ depends on the smoothness of $\mathrm{\partial }\mathrm{\Omega }$. From (22(22) $\psi (x^{\mathrm{\prime }},y)=\frac{1}{p^{\mathrm{\prime }}}\left({\left|\frac{h\left(x^{\mathrm{\prime }}\right)}{2}\right|}^{p^{\mathrm{\prime }}}-|m\left(x^{\mathrm{\prime }}\right)-y{|}^{p^{\mathrm{\prime }}}\right),$(22) ) we see that ψ is of class $C^{0,1}\left(\overline{\mathrm{\Omega }}\right)$, since h is a Lipschitz function on $\overline{\omega }$. From (23(23) $\varrho \left(x^{\mathrm{\prime }}\right)=\frac{2^{-p^{\mathrm{\prime }}}}{p^{\mathrm{\prime }}+1}{h}^{p^{\mathrm{\prime }}+1}\left(x^{\mathrm{\prime }}\right),$(23) ) and the fact that h is bounded from below by the positive constant by (1(1) $\begin{array}{rl}& -\frac{1}{2}\le h^{-}(x_1,x_2)\le -\frac{1}{4}\mathrm{a}\mathrm{n}\mathrm{d}\frac{1}{4}\le h^{+}(x_1,x_2)\le \frac{1}{2}\\ & h\left(x^{\mathrm{\prime }}\right)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{h}^{+}(x_1,x_2)-h^{-}(x_1,x_2)\mathrm{a}\mathrm{n}\mathrm{d}m(x_1,x_2)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{h^{+}(x_1,x_2)+h^{-}(x_1,x_2)}{2},\end{array}$(1) ), we deduce also that the flow factor ϱ and $1/\varrho$ belong to $W^{1,\mathrm{\infty }}\left(\omega \right)$.

Before we formulate the main result, let us introduce the following definitions that will be used throughout this paper: Let ϕ be a scalar function and $u=(u_1,u_2,u_3)$ be a vector function, then we define ${\mathrm{\nabla }}_{x^{\mathrm{\prime }}}$, ${\mathrm{\nabla }}_y$, ${\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}$ and ${\mathrm{d}\mathrm{i}\mathrm{v}}_y$ as ${\mathrm{\nabla }}_{x^{\mathrm{\prime }}}\varphi =\left(\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }{x}_1},\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }{x}_2},0\right),{\mathrm{\nabla }}_y\varphi =\left(0,0,\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }y}\right)$and ${\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}u=\frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }{x}_1}+\frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }{x}_2},{\mathrm{d}\mathrm{i}\mathrm{v}}_{y}u=\frac{\mathrm{\partial }{u}_3}{\mathrm{\partial }y}.$

Theorem 3.8

Main result

For each $0<ϵ\le 1$ the boundary value problem (11(11) $\left\{\begin{array}{ll}\mathrm{d}\mathrm{i}\mathrm{v}{\sigma }^{ϵ}=0& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ \mathrm{d}\mathrm{i}\mathrm{v}{u}^{ϵ}=0& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ u^{ϵ}=0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_D^{ϵ}\\ {\sigma }^{ϵ}\hat{n}=-p_b\hat{n}& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_N^{ϵ},\end{array}\right.$(11) ) has a unique solution $(u^{ϵ},q^{ϵ})\in V^p\left({\mathrm{\Omega }}^{ϵ}\right)\times L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)$ such that (24) $\begin{array}{r}\begin{array}{rl}& \frac{1}{|{\mathrm{\Omega }}^{ϵ}{|}^{1/p}}\left(\parallel {ϵ}^{-p^{\mathrm{\prime }}}{u}^{ϵ}{\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}+\parallel {ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{u}^{ϵ}{\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\right)\le C_1\parallel \mathrm{\nabla }{p}_b{\parallel }_{L^{p^{\mathrm{\prime }}}\left(\mathrm{\Omega }\right)}^{p^{\mathrm{\prime }}-1}\\ & \frac{1}{|{\mathrm{\Omega }}^{ϵ}{|}^{1/p^{\mathrm{\prime }}}}\left(\parallel q^{ϵ}{\parallel }_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}+||{ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right)|{|}_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}\right)\le C_2\parallel \mathrm{\nabla }{p}_b{\parallel }_{L^{p^{\mathrm{\prime }}}\left(\mathrm{\Omega }\right)},\end{array}\end{array}$(24) where the constants $C_1$ and $C_2$ depends only on Ω and ${\mathrm{\Gamma }}_D$. Let ψ be the permeability function defined by (22(22) $\psi (x^{\mathrm{\prime }},y)=\frac{1}{p^{\mathrm{\prime }}}\left({\left|\frac{h\left(x^{\mathrm{\prime }}\right)}{2}\right|}^{p^{\mathrm{\prime }}}-|m\left(x^{\mathrm{\prime }}\right)-y{|}^{p^{\mathrm{\prime }}}\right),$(22) ). Then, the following two-scale convergence holds, ${ϵ}^{-p^{\mathrm{\prime }}}{u}^{ϵ}\stackrel{2}{\rightharpoonup }u\mathrm{a}\mathrm{n}\mathrm{d}{q}^{ϵ}\stackrel{2}{\rightharpoonup }q,$where $u\in L^p(\mathrm{\Omega };{\mathbb{R}}^3)$ is the limit velocity defined by $u(x^{\mathrm{\prime }},y)=-\psi (x^{\mathrm{\prime }},y){\left|\frac{1}{{\mu }_0}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}q\left(x^{\mathrm{\prime }}\right)\right|}^{p^{\mathrm{\prime }}-2}\frac{1}{{\mu }_0}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}q\left(x^{\mathrm{\prime }}\right),$and $q\in W^{1,p^{\mathrm{\prime }}}\left(\omega \right)$ is the unique solution of the boundary value problem (25) $\left\{\begin{array}{ll}{\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}\left(\varrho |{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}q{|}^{p^{\mathrm{\prime }}-2}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}q\right)=0& \mathrm{i}\mathrm{n}\omega \\ q=p_b& \mathrm{o}\mathrm{n}{\gamma }_N\\ \varrho |{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}q{|}^{p^{\mathrm{\prime }}-2}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}q\cdot \hat{n}=0& \mathrm{o}\mathrm{n}{\gamma }_D,\end{array}\right.$(25) where ϱ is the flow factor defined by (23(23) $\varrho \left(x^{\mathrm{\prime }}\right)=\frac{2^{-p^{\mathrm{\prime }}}}{p^{\mathrm{\prime }}+1}{h}^{p^{\mathrm{\prime }}+1}\left(x^{\mathrm{\prime }}\right),$(23) ) and $\hat{n}$ is the outward unit normal to the surface of the obstacle.

We observe that the Dirichlet condition on ${\mathrm{\Gamma }}_D^{ϵ}$ for the velocity field in the original problem becomes a Neumann condition on ${\gamma }_D$ for the limit pressure and the Neumann condition on ${\mathrm{\Gamma }}_N^{ϵ}$ for the stress tensor in the original problem becomes a Dirichlet condition on ${\gamma }_N$ for the limit pressure.

Furthermore, by contrast, if one imposes a non-homogeneous Dirichlet condition for the velocity field on ${\mathrm{\Gamma }}_N^{ϵ}$ one ends up with a Neumann condition on ${\gamma }_N$ (this is shown in [18, equation (5.7)]). A major difference between the two boundary conditions is that the limit pressure is uniquely determined by the Dirichlet condition, whereas it is only determined up to an additive constant in the case of a Neumann condition. Moreover, under the Neumann condition the obstacle becomes a streamline of the y-averaged flow (see Lemma 5.4 (iv)). The limit velocity u is uniquely determined in both formulations and the equation $\frac{\mathrm{\partial }q}{\mathrm{\partial }y}=0\mathrm{i}\mathrm{n}\mathrm{\Omega },$tells us that pressure variation in the thickness of Ω can be neglected. In the Newtonian case, p = 2, the flow factor is given by $\varrho \left(x^{\mathrm{\prime }}\right)=h^3\left(x^{\mathrm{\prime }}\right)/12$ and therefore, the pressure equation (25(25) $\left\{\begin{array}{ll}{\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}\left(\varrho |{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}q{|}^{p^{\mathrm{\prime }}-2}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}q\right)=0& \mathrm{i}\mathrm{n}\omega \\ q=p_b& \mathrm{o}\mathrm{n}{\gamma }_N\\ \varrho |{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}q{|}^{p^{\mathrm{\prime }}-2}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}q\cdot \hat{n}=0& \mathrm{o}\mathrm{n}{\gamma }_D,\end{array}\right.$(25) ) becomes the classical coefficient in the Reynolds equation formulated in [16]. This shows that our results can be extended to the context of lubrication theory, see e.g. [19]. In addition, for future work, it would be interesting to allow the upper and lower surfaces to be in contact, i.e. $h^{+}\left(x^{\mathrm{\prime }}\right)=h^{-}\left(x^{\mathrm{\prime }}\right)$ in some points $x^{\mathrm{\prime }}$ of ω, thus creating more obstacles in the domain. Then, the problem (12(12) $\left\{\begin{array}{ll}\mathrm{d}\mathrm{i}\mathrm{v}{\sigma }^{ϵ}=f& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ \mathrm{d}\mathrm{i}\mathrm{v}{u}^{ϵ}=0& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ u^{ϵ}=0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_D^{ϵ}\\ {\sigma }^{ϵ}\hat{n}=g& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_N^{ϵ},\end{array}\right.$(12) ) becomes a singular Hele–Shaw flow problem and therefore, the limit problem (25(25) $\left\{\begin{array}{ll}{\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}\left(\varrho |{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}q{|}^{p^{\mathrm{\prime }}-2}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}q\right)=0& \mathrm{i}\mathrm{n}\omega \\ q=p_b& \mathrm{o}\mathrm{n}{\gamma }_N\\ \varrho |{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}q{|}^{p^{\mathrm{\prime }}-2}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}q\cdot \hat{n}=0& \mathrm{o}\mathrm{n}{\gamma }_D,\end{array}\right.$(25) ) will be degenerate. For some results in the Newtonian case and $h^{+}=h^{-}$ on ${\gamma }_D$ see [1, Sec. 3.3.].

4. Estimates

To prove the main result (Theorem 3.8) we need uniform a priori estimates for the velocity, the monotone operator and the pressure with respect to the parameter ε.

Theorem 4.1

Velocity estimates

For the velocity field $u^{ϵ}$ the following estimates hold (26) $\begin{array}{rl}{∣∣{ϵ}^{-p^{\mathrm{\prime }}}{u}^{ϵ}∣∣}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}& \le K_1{\left(\frac{K_1}{2{\mu }_0}{∣∣\mathrm{\nabla }{p}_b∣∣}_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}\right)}^{p^{\mathrm{\prime }}-1}\\ {∣∣{ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{u}^{ϵ}∣∣}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}& \le K_2{\left(\frac{K_1}{2{\mu }_0}{∣∣\mathrm{\nabla }{p}_b∣∣}_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}\right)}^{p^{\mathrm{\prime }}-1},\end{array}$(26) where $K_1$ and $K_2$ are the constants in Theorem 2.1.

Proof.

Taking $v=u^{ϵ}$ in (16(16) ${\int }_{{\mathrm{\Omega }}^{ϵ}}{\pi }^{ϵ}\mathrm{d}\mathrm{i}\mathrm{v}v\mathrm{d}x={\int }_{{\mathrm{\Omega }}^{ϵ}}\left(2{\mu }_0{∣e\left(\mathrm{\nabla }{u}^{ϵ}\right)∣}^{p-2}e\right(\mathrm{\nabla }{u}^{ϵ}):\mathrm{\nabla }v+\mathrm{\nabla }{p}_b\cdot v)\mathrm{d}x,$(16) ) and using the fact that $\mathrm{d}\mathrm{i}\mathrm{v}{u}^{ϵ}=0$ we have, (27) ${\int }_{{\mathrm{\Omega }}^{ϵ}}2{\mu }_0|e\left(\mathrm{\nabla }{u}^{ϵ}\right){|}^p\mathrm{d}x={\int }_{{\mathrm{\Omega }}^{ϵ}}-\mathrm{\nabla }{p}_b\cdot u^{ϵ}\mathrm{d}x.$(27) Applying the Hölder inequality in (27(27) ${\int }_{{\mathrm{\Omega }}^{ϵ}}2{\mu }_0|e\left(\mathrm{\nabla }{u}^{ϵ}\right){|}^p\mathrm{d}x={\int }_{{\mathrm{\Omega }}^{ϵ}}-\mathrm{\nabla }{p}_b\cdot u^{ϵ}\mathrm{d}x.$(27) ) and the Korn inequality (6(6) $\begin{array}{r}||v|{|}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\le ϵK_1||e\left(\mathrm{\nabla }v\right)|{|}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\end{array}$(6) ) we find ${\int }_{{\mathrm{\Omega }}^{ϵ}}2{\mu }_0|e\left(\mathrm{\nabla }{u}^{ϵ}\right){|}^p\mathrm{d}x\le \parallel u^{ϵ}{\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\parallel \mathrm{\nabla }{p}_b{\parallel }_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}\le ϵK_1\parallel e\left(\mathrm{\nabla }{u}^{ϵ}\right){\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\parallel \mathrm{\nabla }{p}_b{\parallel }_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}.$Thus, (28) $\parallel e\left(\mathrm{\nabla }{u}^{ϵ}\right){\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\le {\left(\frac{ϵK_1}{2{\mu }_0}\parallel \mathrm{\nabla }{p}_b{\parallel }_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}\right)}^{p^{\mathrm{\prime }}-1}.$(28) This together with the first and second inequalities in the Korn inequality again in the left hand side, we obtain $\begin{array}{rl}\parallel u^{ϵ}{\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}& \le ϵK_1{\left(\frac{ϵK_1}{2{\mu }_0}\parallel \mathrm{\nabla }{p}_b{\parallel }_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}\right)}^{p^{\mathrm{\prime }}-1}\\ \parallel \mathrm{\nabla }{u}^{ϵ}{\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}& \le K_2{\left(\frac{ϵK_1}{2{\mu }_0}\parallel \mathrm{\nabla }{p}_b{\parallel }_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}\right)}^{p^{\mathrm{\prime }}-1}.\text{■}\end{array}$

Theorem 4.2

Monotone operator estimate

The monotone operator (3(3) $a\left(X\right)=2{\mu }_0|e\left(X\right){|}^{p-2}e\left(X\right)(X\in {\mathbb{R}}^{3\times 3}),$(3) ) satisfies the following estimate (29) $||{ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right)|{|}_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}\le K_1||\mathrm{\nabla }{p}_b|{|}_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)},$(29) where $K_1$ is the constants in Theorem 2.1.

Proof.

Follows from the estimate obtained in the proof of Theorem 4.1.

Theorem 4.3

Pressure estimate

The normalized pressure ${\pi }^{ϵ}$ satisfies the estimate (30) $\parallel {\pi }^{ϵ}{\parallel }_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}\le C_2{K}_1(K_2+1)\parallel \mathrm{\nabla }{p}_b{\parallel }_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)},$(30) where $K_1,K_2$ and $C_2$ are the constants defined in Theorems 2.1 and 2.2 respectively.

Proof.

Let $F^{ϵ}\in W^p({\mathrm{\Omega }}^{ϵ}{)}^{\mathrm{\prime }}$ be defined by $〈F^{ϵ},v〉={\int }_{{\mathrm{\Omega }}^{ϵ}}\left(2{\mu }_0{∣e\left(\mathrm{\nabla }{u}^{ϵ}\right)∣}^{p-2}e\right(\mathrm{\nabla }{u}^{ϵ}):\mathrm{\nabla }v+\mathrm{\nabla }{p}_b\cdot v)\mathrm{d}x.$From (16(16) ${\int }_{{\mathrm{\Omega }}^{ϵ}}{\pi }^{ϵ}\mathrm{d}\mathrm{i}\mathrm{v}v\mathrm{d}x={\int }_{{\mathrm{\Omega }}^{ϵ}}\left(2{\mu }_0{∣e\left(\mathrm{\nabla }{u}^{ϵ}\right)∣}^{p-2}e\right(\mathrm{\nabla }{u}^{ϵ}):\mathrm{\nabla }v+\mathrm{\nabla }{p}_b\cdot v)\mathrm{d}x,$(16) ) we see that $F^{ϵ}$ belongs to $\left(V^p\right(\mathrm{\Omega }){)}^{\perp }$, and by the Hölder inequality and the Korn inequalities (6(6) $\begin{array}{r}||v|{|}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\le ϵK_1||e\left(\mathrm{\nabla }v\right)|{|}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\end{array}$(6) ) and (7(7) $\begin{array}{r}||\mathrm{\nabla }v|{|}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\le K_2||e\left(\mathrm{\nabla }v\right)|{|}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)},\end{array}$(7) ) we obtain, $\begin{array}{rl}|〈F^{ϵ},v〉|& \le {\int }_{{\mathrm{\Omega }}^{ϵ}}\left(2{\mu }_0|e\right(\mathrm{\nabla }{u}^{ϵ}){|}^{p-1}|\mathrm{\nabla }v|+|\mathrm{\nabla }{p}_b|\cdot |v|)\mathrm{d}x\\ & \le 2{\mu }_0{K}_2\parallel e\left(\mathrm{\nabla }{u}^{ϵ}\right){\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}^{p-1}\parallel e\left(\mathrm{\nabla }v\right){\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}+ϵK_1\parallel \mathrm{\nabla }{p}_b{\parallel }_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}\parallel e\left(\mathrm{\nabla }v\right){\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}.\end{array}$This together with inequality (28(28) $\parallel e\left(\mathrm{\nabla }{u}^{ϵ}\right){\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\le {\left(\frac{ϵK_1}{2{\mu }_0}\parallel \mathrm{\nabla }{p}_b{\parallel }_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}\right)}^{p^{\mathrm{\prime }}-1}.$(28) ) it follows $|〈F^{ϵ},v〉|\le ϵK_1(K_2+1)\parallel \mathrm{\nabla }{p}_b{\parallel }_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}\parallel v{\parallel }_{W^p\left({\mathrm{\Omega }}^{ϵ}\right)},$whence, we deduce that $\parallel F^{ϵ}{\parallel }_{W^p({\mathrm{\Omega }}^{ϵ}{)}^{\mathrm{\prime }}}\le ϵK_1(K_2+1)\parallel \mathrm{\nabla }{p}_b{\parallel }_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}.$Let ${\pi }^{ϵ}$ in $L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)$ be defined by the pressure operator, i.e. ${\pi }^{ϵ}=B_{{\mathrm{\Omega }}^{ϵ}}^{\mathrm{\prime }}{F}^{ϵ}⟺{\int }_{{\mathrm{\Omega }}^{ϵ}}{\pi }^{ϵ}\mathrm{d}\mathrm{i}\mathrm{v}v\mathrm{d}x=〈F^{ϵ},v〉\mathrm{\forall }v\in W^p\left({\mathrm{\Omega }}^{ϵ}\right).$This means that the unknown pressure function in (11(11) $\left\{\begin{array}{ll}\mathrm{d}\mathrm{i}\mathrm{v}{\sigma }^{ϵ}=0& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ \mathrm{d}\mathrm{i}\mathrm{v}{u}^{ϵ}=0& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ u^{ϵ}=0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_D^{ϵ}\\ {\sigma }^{ϵ}\hat{n}=-p_b\hat{n}& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_N^{ϵ},\end{array}\right.$(11) ) can be recovered via (17(17) ${\pi }^{ϵ}=q^{ϵ}-p_b.$(17) ), i.e. $q^{ϵ}={\pi }^{ϵ}+p_b$. Note that $q^{ϵ}$ depends only on the boundary values of $p_b$ on ${\mathrm{\Gamma }}_N^{ϵ}$. From (8(8) $C_1\le \underset{0<ϵ\le 1}{inf}ϵ\parallel B_{{\mathrm{\Omega }}^{ϵ}}\parallel \le \underset{0<ϵ\le 1}{sup}ϵ\parallel B_{{\mathrm{\Omega }}^{ϵ}}\parallel \le C_2.$(8) ) in Theorem 2.2 we infer, $\begin{array}{rl}\parallel {\pi }^{ϵ}{\parallel }_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}& \le \parallel B_{{\mathrm{\Omega }}^{ϵ}}^{\mathrm{\prime }}\parallel \parallel F^{ϵ}{\parallel }_{W^p({\mathrm{\Omega }}^{ϵ}{)}^{\mathrm{\prime }}}\\ & \le ϵK_1(K_2+1)\parallel B_{{\mathrm{\Omega }}^{ϵ}}^{\mathrm{\prime }}\parallel \parallel \mathrm{\nabla }{p}_b{\parallel }_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}\le C_2{K}_1(K_2+1)\parallel \mathrm{\nabla }{p}_b{\parallel }_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}.\text{■}\end{array}$

5. Two-scale convergence results

Here we prove some compactness results for two-scale convergence

5.1. Compactness results

Since $p_b\in W^{1,p^{\mathrm{\prime }}}\left(\omega \right)$ we have $\frac{1}{|{\mathrm{\Omega }}^{ϵ}{|}^{1/p^{\mathrm{\prime }}}}\parallel \mathrm{\nabla }{p}_b{\parallel }_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}=\frac{1}{|\mathrm{\Omega }{|}^{1/p^{\mathrm{\prime }}}}\parallel \mathrm{\nabla }{p}_b{\parallel }_{L^{p^{\mathrm{\prime }}}\left(\mathrm{\Omega }\right)}.$So, raising both sides to power $p^{\mathrm{\prime }}-1$ and using the Theorems 4.1, 4.2 and 4.3 yield $\parallel {ϵ}^{-p^{\mathrm{\prime }}}{u}^{ϵ}{\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}+\parallel {ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{u}^{ϵ}{\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\le {\left(\frac{K_1}{2{\mu }_0}\right)}^{p^{\mathrm{\prime }}-1}\left(K_1+K_2\right)\parallel \mathrm{\nabla }{p}_b{\parallel }_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}^{p^{\mathrm{\prime }}-1}$and $\parallel {\pi }^{ϵ}{\parallel }_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}+||{ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right)|{|}_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}\le K_1\left[C_2\right(K_2+1)+1]\parallel \mathrm{\nabla }{p}_b{\parallel }_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}.$Thus, we can choose some constants C and $\tilde{C}$, which are independent of ε as $C={\left(\frac{K_1}{2{\mu }_0}\right)}^{p^{\mathrm{\prime }}-1}(K_1+K_2)\frac{1}{|\mathrm{\Omega }{|}^{1/p}}\parallel \mathrm{\nabla }{p}_b{\parallel }_{L^{p^{\mathrm{\prime }}}\left(\mathrm{\Omega }\right)}^{p^{\mathrm{\prime }}-1}\mathrm{a}\mathrm{n}\mathrm{d}\tilde{C}=K_1\left[C_2\right(K_2+1)+1]\frac{1}{|\mathrm{\Omega }{|}^{1/p}}\parallel \mathrm{\nabla }{p}_b{\parallel }_{L^{p^{\mathrm{\prime }}}\left(\mathrm{\Omega }\right)}$such that (31) $\begin{array}{rl}& \frac{1}{|{\mathrm{\Omega }}^{ϵ}{|}^{1/p}}\left(\parallel {ϵ}^{-p^{\mathrm{\prime }}}{u}^{ϵ}{\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}+\parallel {ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{u}^{ϵ}{\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\right)\le C,\\ & \frac{1}{|{\mathrm{\Omega }}^{ϵ}{|}^{1/p^{\mathrm{\prime }}}}\left(\parallel {\pi }^{ϵ}{\parallel }_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}+||{ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right)|{|}_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}\right)\le \tilde{C}.\end{array}$(31)

Lemma 5.1

For any sequence $(u^{ϵ},{\pi }^{ϵ})$ with $0<ϵ\le 1$ in $W^p\left({\mathrm{\Omega }}^{ϵ}\right)\times L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)$ satisfying the bounds (31(31) $\begin{array}{rl}& \frac{1}{|{\mathrm{\Omega }}^{ϵ}{|}^{1/p}}\left(\parallel {ϵ}^{-p^{\mathrm{\prime }}}{u}^{ϵ}{\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}+\parallel {ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{u}^{ϵ}{\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\right)\le C,\\ & \frac{1}{|{\mathrm{\Omega }}^{ϵ}{|}^{1/p^{\mathrm{\prime }}}}\left(\parallel {\pi }^{ϵ}{\parallel }_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}+||{ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right)|{|}_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}\right)\le \tilde{C}.\end{array}$(31) ), there exist a $u\in L^p(\mathrm{\Omega };{\mathbb{R}}^3)$ with $\mathrm{\partial }u/\mathrm{\partial }y\in L^p(\mathrm{\Omega },{\mathbb{R}}^{3\times 1})$, and a ${\pi }^0\in L^{p^{\mathrm{\prime }}}\left(\mathrm{\Omega }\right)$ such that, up to a subsequence

1. ${ϵ}^{-p^{\mathrm{\prime }}}{u}^{ϵ}\stackrel{2}{\rightharpoonup }u=(u^{\mathrm{\prime }},u_3)$,

2. ${ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{u}^{ϵ}\stackrel{2}{\rightharpoonup }D\left(u\right)=\left(\begin{array}{cc}0& {\displaystyle \frac{\mathrm{\partial }{u}^{\mathrm{\prime }}}{\mathrm{\partial }y}}\\ 0& {\displaystyle \frac{\mathrm{\partial }{u}_3}{\mathrm{\partial }y}}\end{array}\right),$

3. ${ϵ}^{1-p^{\mathrm{\prime }}}e\left(\mathrm{\nabla }{u}^{ϵ}\right)\stackrel{2}{\rightharpoonup }e\left(D\right(u\left)\right)={\displaystyle \frac{1}{2}}\left(\begin{array}{cc}0& {\displaystyle \frac{\mathrm{\partial }{u}^{\mathrm{\prime }}}{\mathrm{\partial }y}}\\ {\left({\displaystyle \frac{\mathrm{\partial }{u}^{\mathrm{\prime }}}{\mathrm{\partial }y}}\right)}^T& 2{\displaystyle \frac{\mathrm{\partial }{u}_3}{\mathrm{\partial }y}}\end{array}\right),$

4. ${\pi }^{ϵ}\stackrel{2}{\rightharpoonup }{\pi }^0.$

Proof.

For the sake of completeness we provide the proof which follows the same ideas as in [1, Theorem 1]. Since (31(31) $\begin{array}{rl}& \frac{1}{|{\mathrm{\Omega }}^{ϵ}{|}^{1/p}}\left(\parallel {ϵ}^{-p^{\mathrm{\prime }}}{u}^{ϵ}{\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}+\parallel {ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{u}^{ϵ}{\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\right)\le C,\\ & \frac{1}{|{\mathrm{\Omega }}^{ϵ}{|}^{1/p^{\mathrm{\prime }}}}\left(\parallel {\pi }^{ϵ}{\parallel }_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}+||{ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right)|{|}_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}\right)\le \tilde{C}.\end{array}$(31) ) holds, there exist a subsequence which is still denoted by $u^{ϵ}$ and $u\in L^p(\mathrm{\Omega };{\mathbb{R}}^3)$ such that $\underset{ϵ\to 0}{lim}\frac{1}{∣{\mathrm{\Omega }}^{ϵ}∣}{\int }_{{\mathrm{\Omega }}^{ϵ}}{ϵ}^{-p^{\mathrm{\prime }}}{u}^{ϵ}\left(x\right)\cdot v\left(x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3\right)\mathrm{d}x=\frac{1}{∣\mathrm{\Omega }∣}{\int }_{\mathrm{\Omega }}u(x^{\mathrm{\prime }},y)\cdot v(x^{\mathrm{\prime }},y)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y,$for any $v\in L^{p^{\mathrm{\prime }}}(\mathrm{\Omega };{\mathbb{R}}^3)$. Similarly, there exists $d\in L^p(\mathrm{\Omega };{\mathbb{R}}^{3\times 3})$ such that, up to a subsequence $\underset{ϵ\to 0}{lim}\frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}{ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{u}^{ϵ}\left(x\right):\mathrm{\Phi }(x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3)\mathrm{d}x=\frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}d(x^{\mathrm{\prime }},y):\mathrm{\Phi }(x^{\mathrm{\prime }},y)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y,$for any $\mathrm{\Phi }\in L^{p^{\mathrm{\prime }}}(\mathrm{\Omega };{\mathbb{R}}^{3\times 3})$.

Since $u^{ϵ}$ vanishes on ${\mathrm{\Gamma }}_D^{ϵ}$ we have (32) $\begin{array}{rl}0& ={\int }_{\mathrm{\partial }{\mathrm{\Omega }}^{ϵ}}{u}^{ϵ}\left(x\right)\mathrm{\Phi }(x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3)\cdot \hat{n}\mathrm{d}S={\int }_{{\mathrm{\Omega }}^{ϵ}}\text{div }\left({\mathrm{\Phi }}^T{u}^{ϵ}\right)\mathrm{d}x\\ & ={\int }_{{\mathrm{\Omega }}^{ϵ}}\left(\mathrm{\nabla }{u}^{ϵ}\left(x\right):\mathrm{\Phi }(x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3)+u^{ϵ}\left(x\right)\cdot \left[{\text{div }}_{x^{\mathrm{\prime }}}\mathrm{\Phi }+{ϵ}^{-1}{\text{div }}_y\mathrm{\Phi }\right](x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3)\right)\mathrm{d}x,\end{array}$(32) for all $\mathrm{\Phi }\in W^{1,p^{\mathrm{\prime }}}(\mathrm{\Omega };{\mathbb{R}}^{3\times 3})$ such that $\mathrm{\Phi }\cdot \hat{n}=0$ on ${\mathrm{\Gamma }}_N$. Multiplying (32(32) $\begin{array}{rl}0& ={\int }_{\mathrm{\partial }{\mathrm{\Omega }}^{ϵ}}{u}^{ϵ}\left(x\right)\mathrm{\Phi }(x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3)\cdot \hat{n}\mathrm{d}S={\int }_{{\mathrm{\Omega }}^{ϵ}}\text{div }\left({\mathrm{\Phi }}^T{u}^{ϵ}\right)\mathrm{d}x\\ & ={\int }_{{\mathrm{\Omega }}^{ϵ}}\left(\mathrm{\nabla }{u}^{ϵ}\left(x\right):\mathrm{\Phi }(x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3)+u^{ϵ}\left(x\right)\cdot \left[{\text{div }}_{x^{\mathrm{\prime }}}\mathrm{\Phi }+{ϵ}^{-1}{\text{div }}_y\mathrm{\Phi }\right](x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3)\right)\mathrm{d}x,\end{array}$(32) ) by ${ϵ}^{1-p^{\mathrm{\prime }}}/|{\mathrm{\Omega }}^{ϵ}|$ and passing to the limit as $ϵ\to 0$ we see that (33) ${\int }_{\mathrm{\Omega }}\left(d(x^{\mathrm{\prime }},y):\mathrm{\Phi }(x^{\mathrm{\prime }},y)+u(x^{\mathrm{\prime }},y)\cdot {\text{div }}_y\mathrm{\Phi }(x^{\mathrm{\prime }},y)\mathrm{d}{x}^{\mathrm{\prime }}\right)\mathrm{d}y=0.$(33) Let us characterize the subsequence limit d, by considering it block by block, i.e. $d=\left(\begin{array}{cc}{d}_{11}& d_{12}\\ d_{21}& d_{22}\end{array}\right)=\left(\begin{array}{cc}2\times 2& 2\times 1\\ 1\times 2& 1\times 1\end{array}\right).$Similarly the blocks of the matrix Φ are denoted as $\mathrm{\Phi }=\left(\begin{array}{cc}{\mathrm{\Phi }}_{11}& {\mathrm{\Phi }}_{12}\\ {\mathrm{\Phi }}_{21}& {\mathrm{\Phi }}_{22}\end{array}\right)=\left(\begin{array}{cc}2\times 2& 2\times 1\\ 1\times 2& 1\times 1\end{array}\right).$Choosing ${\mathrm{\Phi }}_{12}=0$ and ${\mathrm{\Phi }}_{22}=0$ in (33(33) ${\int }_{\mathrm{\Omega }}\left(d(x^{\mathrm{\prime }},y):\mathrm{\Phi }(x^{\mathrm{\prime }},y)+u(x^{\mathrm{\prime }},y)\cdot {\text{div }}_y\mathrm{\Phi }(x^{\mathrm{\prime }},y)\mathrm{d}{x}^{\mathrm{\prime }}\right)\mathrm{d}y=0.$(33) ) gives ${\int }_{\mathrm{\Omega }}(d_{11}:{\mathrm{\Phi }}_{11}+d_{21}:{\mathrm{\Phi }}_{21})\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y=0\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{\Phi }\in C_c^{\mathrm{\infty }}(\mathrm{\Omega };{\mathbb{R}}^{3\times 3}).$We conclude that $d_{11}=0$ and $d_{21}=0$. Next, choosing ${\mathrm{\Phi }}_{11}=0$ and ${\mathrm{\Phi }}_{21}=0$ in (33(33) ${\int }_{\mathrm{\Omega }}\left(d(x^{\mathrm{\prime }},y):\mathrm{\Phi }(x^{\mathrm{\prime }},y)+u(x^{\mathrm{\prime }},y)\cdot {\text{div }}_y\mathrm{\Phi }(x^{\mathrm{\prime }},y)\mathrm{d}{x}^{\mathrm{\prime }}\right)\mathrm{d}y=0.$(33) ) gives ${\int }_{\mathrm{\Omega }}\left(d_{12}\cdot {\mathrm{\Phi }}_{12}+d_{22}{\mathrm{\Phi }}_{22}+u^{\mathrm{\prime }}\cdot \frac{\mathrm{\partial }{\mathrm{\Phi }}_{12}}{\mathrm{\partial }y}+u_3\frac{\mathrm{\partial }{\mathrm{\Phi }}_{22}}{\mathrm{\partial }y}\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y=0\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{\Phi }\in C_c^{\mathrm{\infty }}(\mathrm{\Omega };{\mathbb{R}}^{3\times 3}).$By definition of weak derivative we deduce that $d_{12}=\mathrm{\partial }{u}^{\mathrm{\prime }}/\mathrm{\partial }y$ and $d_{22}=\mathrm{\partial }{u}_3/\mathrm{\partial }y$. Thus, $d=\left(\begin{array}{ll}0& \frac{\mathrm{\partial }{u}^{\mathrm{\prime }}}{\mathrm{\partial }y}\\ 0& \frac{\mathrm{\partial }{u}_3}{\mathrm{\partial }y}\end{array}\right)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}D\left(u\right).$Hence the trace of u on ${\mathrm{\Gamma }}_D$ is well defined by [1, Lemma 4 (i)]. Applying the Divergence theorem to (33(33) ${\int }_{\mathrm{\Omega }}\left(d(x^{\mathrm{\prime }},y):\mathrm{\Phi }(x^{\mathrm{\prime }},y)+u(x^{\mathrm{\prime }},y)\cdot {\text{div }}_y\mathrm{\Phi }(x^{\mathrm{\prime }},y)\mathrm{d}{x}^{\mathrm{\prime }}\right)\mathrm{d}y=0.$(33) ) we obtain $0={\int }_{{\mathrm{\Gamma }}_D}u(x^{\mathrm{\prime }},y)\mathrm{\Phi }(x^{\mathrm{\prime }},y)\cdot \hat{n}\mathrm{d}S,$for all $\mathrm{\Phi }\in W^{1,p}(\mathrm{\Omega };{\mathbb{R}}^{3\times 3})$ such that $\mathrm{\Phi }\cdot \hat{n}=0$ on ${\mathrm{\Gamma }}_N$. We conclude that u satisfies the Dirichlet condition u = 0 on ${\mathrm{\Gamma }}_D$. The convergence of the symmetrical part of gradient u follows by linearity. The two-scale convergence for the normalized pressure follows directly from the definition.

5.2. Two-scale convergence and the monotone operator

We introduce the monotone operator (3(3) $a\left(X\right)=2{\mu }_0|e\left(X\right){|}^{p-2}e\left(X\right)(X\in {\mathbb{R}}^{3\times 3}),$(3) ) in the weak formulation (16(16) ${\int }_{{\mathrm{\Omega }}^{ϵ}}{\pi }^{ϵ}\mathrm{d}\mathrm{i}\mathrm{v}v\mathrm{d}x={\int }_{{\mathrm{\Omega }}^{ϵ}}\left(2{\mu }_0{∣e\left(\mathrm{\nabla }{u}^{ϵ}\right)∣}^{p-2}e\right(\mathrm{\nabla }{u}^{ϵ}):\mathrm{\nabla }v+\mathrm{\nabla }{p}_b\cdot v)\mathrm{d}x,$(16) ), i.e. (34) ${\int }_{{\mathrm{\Omega }}^{ϵ}}(-{\pi }^{ϵ}\mathrm{d}\mathrm{i}\mathrm{v}v+a(\mathrm{\nabla }{u}^{ϵ}):\mathrm{\nabla }v+\mathrm{\nabla }{p}_b\cdot v)\mathrm{d}x=0,\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}v\in W^p\left(\mathrm{\Omega }\right).$(34)

Lemma 5.2

Given $v\in W^p\left(\mathrm{\Omega }\right)$, let $v^{ϵ}\in W^p\left({\mathrm{\Omega }}^{ϵ}\right)$ be defined as $v^{ϵ}\left(x\right)={ϵ}^{p^{\mathrm{\prime }}}v(x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3).$ Then, we have the strong two-scale convergences

1. ${ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{v}^{ϵ}\stackrel{2}{\to }D\left(v\right)$,

2. $a\left({ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{v}^{ϵ}\right)\stackrel{2}{\to }a\left(D\right(v\left)\right)$.

In particular, $v^{ϵ}$ is an admissible test function in the two-scale sense.

Proof.

The gradient of $v^{ϵ}\in W^p\left({\mathrm{\Omega }}^{ϵ}\right)$ is defined as $\mathrm{\nabla }{v}^{ϵ}\left(x\right)=\left(\begin{array}{cc}{ϵ}^{p^{\mathrm{\prime }}}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{v}^{\mathrm{\prime }}& {ϵ}^{p^{\mathrm{\prime }}-1}\frac{\mathrm{\partial }{v}^{\mathrm{\prime }}}{\mathrm{\partial }y}\\ {ϵ}^{p^{\mathrm{\prime }}}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{v}_3& {ϵ}^{p^{\mathrm{\prime }}-1}\frac{\mathrm{\partial }{v}_3}{\mathrm{\partial }y}\end{array}\right)(x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3),$for all $v\in W^p\left({\mathrm{\Omega }}^{ϵ}\right)$. Therefore, the weak two-scale convergence $\left(i\right)$ follows from Lemma 5.1 (ii). The weak two-scale convergence $\left(ii\right)$ is deduced from the homogeneity property of a defined in (4(4) $a\left(\lambda X\right)={\lambda }^{p-1}a\left(X\right),(\lambda >0);$(4) ). Thus, it is enough to prove (35) $\underset{ϵ\to 0}{lim}\frac{1}{\mid {\mathrm{\Omega }}^{ϵ}{\mid }^{\frac{1}{p^{\mathrm{\prime }}}}}\parallel a\left({ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{v}^{ϵ}\right){\parallel }_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}=\frac{1}{\mid \mathrm{\Omega }{\mid }^{\frac{1}{p^{\mathrm{\prime }}}}}\parallel a\left(D\right(v\left)\right){\parallel }_{L^{p^{\mathrm{\prime }}}\left(\mathrm{\Omega }\right)}.$(35) Then, by the Change of Variable theorem, and noting that $|{\mathrm{\Omega }}^{ϵ}|=ϵ|\mathrm{\Omega }|$, we obtain $\frac{1}{|{\mathrm{\Omega }}^{ϵ}{|}^{1/p^{\mathrm{\prime }}}}{\left({\int }_{{\mathrm{\Omega }}^{ϵ}}|a\left(\mathrm{\nabla }{v}^{ϵ}\right){|}^{p^{\mathrm{\prime }}}\mathrm{d}x\right)}^{1/p^{\mathrm{\prime }}}=\frac{1}{|\mathrm{\Omega }{|}^{1/p^{\mathrm{\prime }}}}{\left({\int }_{\mathrm{\Omega }}{\left|a\left(\begin{array}{cc}{ϵ}^{p^{\mathrm{\prime }}}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{v}^{\mathrm{\prime }}& {ϵ}^{p^{\mathrm{\prime }}-1}\frac{\mathrm{\partial }{v}^{\mathrm{\prime }}}{\mathrm{\partial }y}\\ {ϵ}^{p^{\mathrm{\prime }}}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{v}_3& {ϵ}^{p^{\mathrm{\prime }}-1}\frac{\mathrm{\partial }{v}_3}{\mathrm{\partial }y}\end{array}\right)\right|}^{p^{\mathrm{\prime }}}\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y\right)}^{1/p^{\mathrm{\prime }}}.$Multiplying by ${ϵ}^{-1}={ϵ}^{-(p-1)(p^{\mathrm{\prime }}-1)}$ and homogeneity property, we get $\frac{1}{|{\mathrm{\Omega }}^{ϵ}{|}^{1/p^{\mathrm{\prime }}}}{\left({\int }_{{\mathrm{\Omega }}^{ϵ}}|a\left({ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{v}^{ϵ}\right){|}^{p^{\mathrm{\prime }}}dx\right)}^{1/p^{\mathrm{\prime }}}=\frac{1}{|\mathrm{\Omega }{|}^{1/p^{\mathrm{\prime }}}}{\left({\int }_{\mathrm{\Omega }}{\left|a\left(\begin{array}{cc}ϵ{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{v}^{\mathrm{\prime }}& \frac{\mathrm{\partial }{v}^{\mathrm{\prime }}}{\mathrm{\partial }y}\\ ϵ{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{v}_3& \frac{\mathrm{\partial }{v}_3}{\mathrm{\partial }y}\end{array}\right)\right|}^{p^{\mathrm{\prime }}}\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y\right)}^{1/p^{\mathrm{\prime }}}.$Passing to the limit as $ϵ\to 0$, using the Dominated Convergence theorem and the continuity of a, we obtain (35(35) $\underset{ϵ\to 0}{lim}\frac{1}{\mid {\mathrm{\Omega }}^{ϵ}{\mid }^{\frac{1}{p^{\mathrm{\prime }}}}}\parallel a\left({ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{v}^{ϵ}\right){\parallel }_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}=\frac{1}{\mid \mathrm{\Omega }{\mid }^{\frac{1}{p^{\mathrm{\prime }}}}}\parallel a\left(D\right(v\left)\right){\parallel }_{L^{p^{\mathrm{\prime }}}\left(\mathrm{\Omega }\right)}.$(35) ).

Lemma 5.3

The sequence $a\left(\mathrm{\nabla }{u}^{ϵ}\right),\left(0<ϵ\le 1\right)$ in $L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)$ satisfies the bound (31(31) $\begin{array}{rl}& \frac{1}{|{\mathrm{\Omega }}^{ϵ}{|}^{1/p}}\left(\parallel {ϵ}^{-p^{\mathrm{\prime }}}{u}^{ϵ}{\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}+\parallel {ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{u}^{ϵ}{\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\right)\le C,\\ & \frac{1}{|{\mathrm{\Omega }}^{ϵ}{|}^{1/p^{\mathrm{\prime }}}}\left(\parallel {\pi }^{ϵ}{\parallel }_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}+||{ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right)|{|}_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}\right)\le \tilde{C}.\end{array}$(31) ). There exists $\zeta \in L^{p^{\mathrm{\prime }}}(\mathrm{\Omega };{\mathbb{R}}^{3\times 3})$ such that up to a subsequence ${ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right)\stackrel{2}{\rightharpoonup }\zeta .$Moreover, (36) $\underset{ϵ\to 0}{lim}inf\frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}{ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right):{ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{u}^{ϵ}\mathrm{d}x\ge \frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}\zeta :D\left(u\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y$(36) and if (36(36) $\underset{ϵ\to 0}{lim}inf\frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}{ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right):{ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{u}^{ϵ}\mathrm{d}x\ge \frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}\zeta :D\left(u\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y$(36) ) holds as an equality, then (37) ${\int }_{\mathrm{\Omega }}\zeta :D\left(v\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y={\int }_{\mathrm{\Omega }}a\left(D\right(u\left)\right):D\left(v\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y,$(37) for all test functions v in $W^p\left(\mathrm{\Omega }\right)$.

Proof.

In view of the monotonicity condition (5(5) $\left(a\right(X)-a(Y\left)\right):(X-Y)\ge 0,$(5) ), we have (38) ${\int }_{{\mathrm{\Omega }}^{ϵ}}\left(a\right(\mathrm{\nabla }{u}^{ϵ})-a(\mathrm{\nabla }{v}^{ϵ}\left)\right):(\mathrm{\nabla }{u}^{ϵ}-\mathrm{\nabla }{v}^{ϵ})\mathrm{d}x\ge 0,$(38) where $v^{ϵ}$ is a test function of the same form as in Lemma 5.2. This inequality yields ${\int }_{{\mathrm{\Omega }}^{ϵ}}a\left(\mathrm{\nabla }{u}^{ϵ}\right):\mathrm{\nabla }{u}^{ϵ}\mathrm{d}x\ge {\int }_{{\mathrm{\Omega }}^{ϵ}}\left(a\right(\mathrm{\nabla }{u}^{ϵ}):\mathrm{\nabla }{v}^{ϵ}+a(\mathrm{\nabla }{v}^{ϵ}):(\mathrm{\nabla }{u}^{ϵ}-\mathrm{\nabla }{v}^{ϵ}\left)\right)\mathrm{d}x.$Multiplying this inequality by the factor $\frac{{ϵ}^{-p^{\mathrm{\prime }}}}{|{\mathrm{\Omega }}^{ϵ}|}=\frac{{ϵ}^{-1}}{|{\mathrm{\Omega }}^{ϵ}{|}^{1/p^{\mathrm{\prime }}}}\cdot \frac{{ϵ}^{1-p^{\mathrm{\prime }}}}{|{\mathrm{\Omega }}^{ϵ}{|}^{1/p}},$ we find (39) $\begin{array}{rl}& \frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}{ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right):{ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{u}^{ϵ}\mathrm{d}x\ge \frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}{ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right):{ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{v}^{ϵ}\mathrm{d}x\\ & +\frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}{ϵ}^{-1}a\left(\mathrm{\nabla }{v}^{ϵ}\right):{ϵ}^{1-p^{\mathrm{\prime }}}(\mathrm{\nabla }{u}^{ϵ}-\mathrm{\nabla }{v}^{ϵ})\mathrm{d}x.\end{array}$(39) Let us analyze the integrals on the right hand side in (39(39) $\begin{array}{rl}& \frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}{ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right):{ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{u}^{ϵ}\mathrm{d}x\ge \frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}{ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right):{ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{v}^{ϵ}\mathrm{d}x\\ & +\frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}{ϵ}^{-1}a\left(\mathrm{\nabla }{v}^{ϵ}\right):{ϵ}^{1-p^{\mathrm{\prime }}}(\mathrm{\nabla }{u}^{ϵ}-\mathrm{\nabla }{v}^{ϵ})\mathrm{d}x.\end{array}$(39) ) in terms of limits as $ϵ\to 0$. The limit of the first integral exists by hypothesis and is given by (40) $\underset{ϵ\to 0}{lim}\frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}{ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right):{ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{v}^{ϵ}\mathrm{d}x=\frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}\zeta :D\left(v\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y.$(40) The second integral is convergent by Lemma 5.1 (ii) and Lemma 5.2, as the integrand is a product of strongly two-scale convergent and two-scale convergent sequences and [1, Lemma 2]. (41) $\underset{ϵ\to 0}{lim}\frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}{ϵ}^{-1}a\left(\mathrm{\nabla }{v}^{ϵ}\right):{ϵ}^{1-p^{\mathrm{\prime }}}(\mathrm{\nabla }{u}^{ϵ}-\mathrm{\nabla }{v}^{ϵ})\mathrm{d}x=\frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}a\left(D\right(v\left)\right):D(u-v)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y.$(41) Thus, we can estimate the inferior limit of the integral on the left hand side in (39(39) $\begin{array}{rl}& \frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}{ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right):{ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{u}^{ϵ}\mathrm{d}x\ge \frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}{ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right):{ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{v}^{ϵ}\mathrm{d}x\\ & +\frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}{ϵ}^{-1}a\left(\mathrm{\nabla }{v}^{ϵ}\right):{ϵ}^{1-p^{\mathrm{\prime }}}(\mathrm{\nabla }{u}^{ϵ}-\mathrm{\nabla }{v}^{ϵ})\mathrm{d}x.\end{array}$(39) ), by means of limits (40(40) $\underset{ϵ\to 0}{lim}\frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}{ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right):{ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{v}^{ϵ}\mathrm{d}x=\frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}\zeta :D\left(v\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y.$(40) ) and (41(41) $\underset{ϵ\to 0}{lim}\frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}{ϵ}^{-1}a\left(\mathrm{\nabla }{v}^{ϵ}\right):{ϵ}^{1-p^{\mathrm{\prime }}}(\mathrm{\nabla }{u}^{ϵ}-\mathrm{\nabla }{v}^{ϵ})\mathrm{d}x=\frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}a\left(D\right(v\left)\right):D(u-v)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y.$(41) ), as $ϵ\to 0$, obtaining (42) $\begin{array}{rl}& \underset{ϵ\to 0}{lim}inf\frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}{ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right):{ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{u}^{ϵ}\mathrm{d}x\\ & \ge \frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}(\zeta :D(v)+a(D\left(v\right)):D(u-v\left)\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y,\end{array}$(42) and the inequality (36(36) $\underset{ϵ\to 0}{lim}inf\frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}{ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right):{ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{u}^{ϵ}\mathrm{d}x\ge \frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}\zeta :D\left(u\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y$(36) ) is established by taking u = v.

Assuming that equality holds in (36(36) $\underset{ϵ\to 0}{lim}inf\frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}{ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right):{ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{u}^{ϵ}\mathrm{d}x\ge \frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}\zeta :D\left(u\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y$(36) ), we see from (42(42) $\begin{array}{rl}& \underset{ϵ\to 0}{lim}inf\frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}{ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right):{ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{u}^{ϵ}\mathrm{d}x\\ & \ge \frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}(\zeta :D(v)+a(D\left(v\right)):D(u-v\left)\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y,\end{array}$(42) ) that $\frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}\zeta :D\left(u\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y\ge \frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}(\zeta :D(v)+a(D\left(v\right)):D(u-v\left)\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y.$After some rearranging, it follows that (43) ${\int }_{\mathrm{\Omega }}(\zeta -D(v\left)\right):\left(D\right(u)-D(v\left)\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y\ge 0.$(43) Fix $w\in W^p\left(\mathrm{\Omega }\right)$ and set $v\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}u-tw$ with $(t>0)$ in (43(43) ${\int }_{\mathrm{\Omega }}(\zeta -D(v\left)\right):\left(D\right(u)-D(v\left)\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y\ge 0.$(43) ), we obtain $t{\int }_{\mathrm{\Omega }}(\zeta -a(D\left(u\right)-tD\left(w\right)\left)\right):D\left(w\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y\ge 0.$Dividing by t, using the continuity of a, and letting $t\to 0$ give ${\int }_{\mathrm{\Omega }}(\zeta -a(D\left(u\right)\left)\right):D\left(w\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y\ge 0.$Finally, replacing w by $-w$, we deduce that ${\int }_{\mathrm{\Omega }}\zeta :D\left(w\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y={\int }_{\mathrm{\Omega }}a\left(D\right(u\left)\right):D\left(w\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y,$for all $w\in W^p\left(\mathrm{\Omega }\right)$.

5.3. Conservation of volume

In this subsection we investigate the conservation of the incompressibility of the flow when we let $ϵ\to 0$. To this end it is convenient to introduce the following space of functions.

Definition 5.4

Let $W_y^p\left(\mathrm{\Omega }\right)$ be defined as the closure of $W^p\left(\mathrm{\Omega }\right)$ in the norm $\parallel v{\parallel }_{W_y^p\left(\mathrm{\Omega }\right)}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\parallel v{\parallel }_{L^p\left(\mathrm{\Omega }\right)}+{∥\frac{\mathrm{\partial }v}{\mathrm{\partial }y}∥}_{L^p\left(\mathrm{\Omega }\right)}.$

Lemma 5.5

Let $u^{ϵ},\left(0<ϵ\le 1\right)$ be a sequence as in Lemma 5.1 with the additional condition that $\text{div }{u}^{ϵ}=0$ in ${\mathrm{\Omega }}^{ϵ}$. Then any subsequential two-scale limit $u=(u^{\mathrm{\prime }},u_3)$ satisfies (44) $u_3=0in\mathrm{\Omega }$(44) and (45) $\left\{\begin{array}{ll}{\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}\left({\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}{u}^{\mathrm{\prime }}(x^{\mathrm{\prime }},y)\mathrm{d}y\right)=0& \mathrm{i}\mathrm{n}\omega \\ \left({\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}{u}^{\mathrm{\prime }}(x^{\mathrm{\prime }},y)\mathrm{d}y\right)\cdot \hat{n}=0& \mathrm{o}\mathrm{n}{\gamma }_D.\end{array}\right.$(45)

Proof.

As in [1, Proposition 4], the divergence free condition for $u^{ϵ}$ can be stated as ${\int }_{\mathrm{\partial }{\mathrm{\Omega }}^{ϵ}}{u}^{ϵ}v\cdot \hat{n}\mathrm{d}S={\int }_{{\mathrm{\Omega }}^{ϵ}}{u}^{ϵ}\cdot \mathrm{\nabla }v\mathrm{d}x,$for all scalar test functions $v\in W^{1,p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)$. It follows that (46) $0={\int }_{{\mathrm{\Omega }}^{ϵ}}\left(\left(u^{ϵ}{)}^{\mathrm{\prime }}\right(x)\cdot {\mathrm{\nabla }}_{x^{\mathrm{\prime }}}v(x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3)+{ϵ}^{-1}{u}_3^{ϵ}(x\left)\frac{\mathrm{\partial }v}{\mathrm{\partial }y}\right(x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3)\right)\mathrm{d}x,$(46) for all $v\in W^{1,p^{\mathrm{\prime }}}\left(\mathrm{\Omega }\right)$ such that v = 0 on ${\mathrm{\Gamma }}_N$. Multiplying (46(46) $0={\int }_{{\mathrm{\Omega }}^{ϵ}}\left(\left(u^{ϵ}{)}^{\mathrm{\prime }}\right(x)\cdot {\mathrm{\nabla }}_{x^{\mathrm{\prime }}}v(x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3)+{ϵ}^{-1}{u}_3^{ϵ}(x\left)\frac{\mathrm{\partial }v}{\mathrm{\partial }y}\right(x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3)\right)\mathrm{d}x,$(46) ) by ${ϵ}^{1-p^{\mathrm{\prime }}}/|{\mathrm{\Omega }}^{ϵ}|$ and passing to the limit, as $ϵ\to 0$ yield $\frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}{u}_3(x^{\mathrm{\prime }},y)\frac{\mathrm{\partial }v}{\mathrm{\partial }y}(x^{\mathrm{\prime }},y)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y=0,$hence $\mathrm{\partial }{u}_3/\mathrm{\partial }y=0$. From this and the no-slip boundary conditions on ${\mathrm{\Gamma }}_D^{\pm }$ we deduce (44(44) $u_3=0in\mathrm{\Omega }$(44) ).

By choosing v in (46(46) $0={\int }_{{\mathrm{\Omega }}^{ϵ}}\left(\left(u^{ϵ}{)}^{\mathrm{\prime }}\right(x)\cdot {\mathrm{\nabla }}_{x^{\mathrm{\prime }}}v(x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3)+{ϵ}^{-1}{u}_3^{ϵ}(x\left)\frac{\mathrm{\partial }v}{\mathrm{\partial }y}\right(x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3)\right)\mathrm{d}x,$(46) ) which does not depend on y, i.e. $v=v\left(x^{\mathrm{\prime }}\right)$, such that v = 0 on ${\gamma }_N$ and passing to the limit, as $ϵ\to 0$, we obtain $0=\frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}{u}^{\mathrm{\prime }}(x^{\mathrm{\prime }},y)\cdot {\mathrm{\nabla }}_{x^{\mathrm{\prime }}}v\left(x^{\mathrm{\prime }}\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y=\frac{1}{|\mathrm{\Omega }|}{\int }_{\omega }\left({\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}{u}^{\mathrm{\prime }}(x^{\mathrm{\prime }},y)\mathrm{d}y\right)\cdot {\mathrm{\nabla }}_{x^{\mathrm{\prime }}}v\left(x^{\mathrm{\prime }}\right)\mathrm{d}{x}^{\mathrm{\prime }},$for all $v\in W^{1,p^{\mathrm{\prime }}}\left(\omega \right)$ such that v = 0 on ${\gamma }_N$. This implies that the line integral on $\mathrm{\partial }\omega$ depends only on ${\gamma }_D$, whence by the Divergence theorem, we have ${\int }_{\omega }{\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}\left({\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}{u}^{\mathrm{\prime }}(x^{\mathrm{\prime }},y)\mathrm{d}y\right)v\left(x^{\mathrm{\prime }}\right)\mathrm{d}{x}^{\mathrm{\prime }}={\int }_{{\gamma }_D}\left({\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}{u}^{\mathrm{\prime }}(x^{\mathrm{\prime }},y)\mathrm{d}y\right)v\left(x^{\mathrm{\prime }}\right)\cdot \hat{n}\mathrm{d}S.$Since $u^{\mathrm{\prime }}\in L^p(\mathrm{\Omega };{\mathbb{R}}^2)$ we see that ${\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}\left({\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}{u}^{\mathrm{\prime }}(x^{\mathrm{\prime }},y)\mathrm{d}y\right)\in L^p\left(\omega \right).$It follows from [22, Proposition 2] that the trace in the dual sense is the restriction of $\left({\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}{u}^{\mathrm{\prime }}\right(x^{\mathrm{\prime }},y\left)\mathrm{d}y\right)\cdot \hat{n}$ on ${\gamma }_D$, which is well defined as, (47) $\left({\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}{u}^{\mathrm{\prime }}(x^{\mathrm{\prime }},y)\mathrm{d}y\right)\cdot \hat{n}\in W^{-1/p^{\mathrm{\prime }},p^{\mathrm{\prime }}}({\gamma }_D;{\mathbb{R}}^2).$(47) Thus, (45(45) $\left\{\begin{array}{ll}{\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}\left({\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}{u}^{\mathrm{\prime }}(x^{\mathrm{\prime }},y)\mathrm{d}y\right)=0& \mathrm{i}\mathrm{n}\omega \\ \left({\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}{u}^{\mathrm{\prime }}(x^{\mathrm{\prime }},y)\mathrm{d}y\right)\cdot \hat{n}=0& \mathrm{o}\mathrm{n}{\gamma }_D.\end{array}\right.$(45) ) is proved.

Lemma 5.6

All functions $v\in W_y^p\left(\mathrm{\Omega }\right)$ satisfy

1. $\mathrm{\partial }v/\mathrm{\partial }y\in L^p(\mathrm{\Omega };{\mathbb{R}}^{3\times 1})$,

2. v = 0 on ${\mathrm{\Gamma }}_D$,

3. ${\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}\left({\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}v(x^{\mathrm{\prime }},y)\mathrm{d}y\right)\in L^p\left(\omega \right)$,

4. $\left({\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}v(x^{\mathrm{\prime }},y)\mathrm{d}y\right)\cdot \hat{n}=0$ on ${\gamma }_D.$

Proof.

The proofs of $\left(i\right)$ –$\left(ii\right)$ follow by the same arguments used in the proof of Lemma 5.1 and (iii)–(iv) can be deduced by following the ideas used in the proof of Lemma 5.5.

5.4. Momentum equation

We end this Section, by proving the main result of this paper. Indeed, we will derive a lower-dimensional model for the flow from the Stokes Equation (11(11) $\left\{\begin{array}{ll}\mathrm{d}\mathrm{i}\mathrm{v}{\sigma }^{ϵ}=0& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ \mathrm{d}\mathrm{i}\mathrm{v}{u}^{ϵ}=0& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ u^{ϵ}=0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_D^{ϵ}\\ {\sigma }^{ϵ}\hat{n}=-p_b\hat{n}& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_N^{ϵ},\end{array}\right.$(11) ).

Lemma 5.7

The thin film equation

Let $u=(u^{\mathrm{\prime }},u_3)$ and ${\pi }^0$ be as in Lemma 5.1 and let ζ denote a subsequential limit of ${ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right)$ as in Lemma 5.3. Then the first component $u^{\mathrm{\prime }}$ of u and ${\pi }^0$ satisfy (48) $\left\{\begin{array}{ll}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}(-{\pi }^0+p_b)+{\mu }_0{\mathrm{\Delta }}_{p,y}{u}^{\mathrm{\prime }}=0& \mathrm{i}\mathrm{n}\mathrm{\Omega }\\ -\frac{\mathrm{\partial }{\pi }^0}{\mathrm{\partial }y}=0& \mathrm{i}\mathrm{n}\mathrm{\Omega }\\ u^{\mathrm{\prime }}=0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_D\\ \left(\begin{array}{ll}-{\pi }^0& {\mu }_0{\left|\frac{\mathrm{\partial }{u}^{\mathrm{\prime }}}{\mathrm{\partial }y}\right|}^{p-2}\frac{\mathrm{\partial }{u}^{\mathrm{\prime }}}{\mathrm{\partial }y}\\ 0& -{\pi }^0\end{array}\right)\hat{n}=0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_N.\end{array}\right.$(48)

Proof.

By choosing test functions in (34(34) ${\int }_{{\mathrm{\Omega }}^{ϵ}}(-{\pi }^{ϵ}\mathrm{d}\mathrm{i}\mathrm{v}v+a(\mathrm{\nabla }{u}^{ϵ}):\mathrm{\nabla }v+\mathrm{\nabla }{p}_b\cdot v)\mathrm{d}x=0,\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}v\in W^p\left(\mathrm{\Omega }\right).$(34) ), of the form $v^{ϵ}\left(x\right)=\left(v^{\mathrm{\prime }}\right(x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3),ϵv_3(x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3\left)\right),$we have $\begin{array}{rl}& \frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}\left\{\phantom{\left(\begin{array}{c}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{v}^{\mathrm{\prime }}{ϵ}^{-1}\frac{\mathrm{\partial }{v}^{\mathrm{\prime }}}{\mathrm{\partial }y}\\ {\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{v}_3\frac{\mathrm{\partial }{v}_3}{\mathrm{\partial }y}\end{array}\right)}-{\pi }^{ϵ}\left(x\right)\left[{\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}v+\frac{\mathrm{\partial }{v}_3}{\mathrm{\partial }y}\right](x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3)\right.\\ & +\left.{ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right)\left(x\right):ϵ\left(\begin{array}{c}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{v}^{\mathrm{\prime }}{ϵ}^{-1}\frac{\mathrm{\partial }{v}^{\mathrm{\prime }}}{\mathrm{\partial }y}\\ {\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{v}_3\frac{\mathrm{\partial }{v}_3}{\mathrm{\partial }y}\end{array}\right)(x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3)+{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{p}_b\left(x\right)\cdot v^{ϵ}\left(x\right)\right\}\mathrm{d}x=0.\end{array}$Passing to the limit as $ϵ\to 0$, we obtain (49) $\frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}\left\{-{\pi }^0\left[{\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}v+\frac{\mathrm{\partial }{v}_3}{\mathrm{\partial }y}\right]+\zeta :\left(\begin{array}{cc}0& \frac{\mathrm{\partial }{v}^{\mathrm{\prime }}}{\mathrm{\partial }y}\\ 0& 0\end{array}\right)+{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{p}_b\cdot v^{\mathrm{\prime }}\mathrm{d}{x}^{\mathrm{\prime }}\right\}\mathrm{d}y=0,$(49) for all $(v^{\mathrm{\prime }},v_3)\in W^p\left(\mathrm{\Omega }\right)$. Choosing $v^{\mathrm{\prime }}=0$, we see that $0={\int }_{\mathrm{\Omega }}-{\pi }^0\frac{\mathrm{\partial }{v}_3}{\mathrm{\partial }y}\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}{v}_3\in W^p\left(\mathrm{\Omega }\right),$or in other words, $\mathrm{\partial }{\pi }^0/\mathrm{\partial }y=0\mathrm{i}\mathrm{n}\mathrm{\Omega }.$ This shows that ${\pi }^0$ does not depend of variable y, i.e. ${\pi }^0(x^{\mathrm{\prime }},y)={\pi }^0\left(x^{\mathrm{\prime }}\right)$.

Furthermore, multiplying (34(34) ${\int }_{{\mathrm{\Omega }}^{ϵ}}(-{\pi }^{ϵ}\mathrm{d}\mathrm{i}\mathrm{v}v+a(\mathrm{\nabla }{u}^{ϵ}):\mathrm{\nabla }v+\mathrm{\nabla }{p}_b\cdot v)\mathrm{d}x=0,\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}v\in W^p\left(\mathrm{\Omega }\right).$(34) ) with ${ϵ}^{-p^{\mathrm{\prime }}}/|{\mathrm{\Omega }}^{ϵ}|$, choosing $v^{ϵ}=u^{ϵ}$ and using that $\mathrm{d}\mathrm{i}\mathrm{v}{u}^{ϵ}=0$ in ${\mathrm{\Omega }}^{ϵ}$, we can assert that (50) $\begin{array}{rl}& \underset{ϵ\to 0}{lim}\frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}{ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right):{ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{u}^{ϵ}\mathrm{d}x\\ & =-\underset{ϵ\to 0}{lim}\frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}\mathrm{\nabla }{p}_b\cdot {ϵ}^{-p^{\mathrm{\prime }}}{u}^{ϵ}\mathrm{d}x=-\frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{p}_b\cdot u^{\mathrm{\prime }}\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y.\end{array}$(50) Using ${\pi }^0(x^{\mathrm{\prime }},y)={\pi }^0\left(x^{\mathrm{\prime }}\right)$, we can rewrite the first term in (49(49) $\frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}\left\{-{\pi }^0\left[{\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}v+\frac{\mathrm{\partial }{v}_3}{\mathrm{\partial }y}\right]+\zeta :\left(\begin{array}{cc}0& \frac{\mathrm{\partial }{v}^{\mathrm{\prime }}}{\mathrm{\partial }y}\\ 0& 0\end{array}\right)+{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{p}_b\cdot v^{\mathrm{\prime }}\mathrm{d}{x}^{\mathrm{\prime }}\right\}\mathrm{d}y=0,$(49) ) as ${\int }_{\mathrm{\Omega }}{\pi }^0\left[{\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}v+\frac{\mathrm{\partial }{v}_3}{\mathrm{\partial }y}\right]\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y={\int }_{\omega }{\pi }^0\left(x^{\mathrm{\prime }}\right){\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}\left({\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}v\mathrm{d}y\right)\mathrm{d}{x}^{\mathrm{\prime }}$we deduce (51) ${\int }_{\omega }-{\pi }^0{\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}\left({\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}v\mathrm{d}y\right)\mathrm{d}{x}^{\mathrm{\prime }}+{\int }_{\mathrm{\Omega }}\left\{\zeta :\left(\begin{array}{ll}0& \frac{\mathrm{\partial }{v}^{\mathrm{\prime }}}{\mathrm{\partial }y}\\ 0& 0\end{array}\right)+{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{p}_b\cdot v^{\mathrm{\prime }}\right\}\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y=0,$(51) which holds for all $v=(v^{\mathrm{\prime }},v_3)\in W_y^p\left(\mathrm{\Omega }\right)$ (see Definition 5.4), since $W^p\left(\mathrm{\Omega }\right)$ is dense in $W_y^p\left(\mathrm{\Omega }\right)$. Then taking v = u in (51(51) ${\int }_{\omega }-{\pi }^0{\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}\left({\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}v\mathrm{d}y\right)\mathrm{d}{x}^{\mathrm{\prime }}+{\int }_{\mathrm{\Omega }}\left\{\zeta :\left(\begin{array}{ll}0& \frac{\mathrm{\partial }{v}^{\mathrm{\prime }}}{\mathrm{\partial }y}\\ 0& 0\end{array}\right)+{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{p}_b\cdot v^{\mathrm{\prime }}\right\}\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y=0,$(51) ) and using Lemma 5.5, i.e. (44(44) $u_3=0in\mathrm{\Omega }$(44) ) and (45(45) $\left\{\begin{array}{ll}{\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}\left({\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}{u}^{\mathrm{\prime }}(x^{\mathrm{\prime }},y)\mathrm{d}y\right)=0& \mathrm{i}\mathrm{n}\omega \\ \left({\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}{u}^{\mathrm{\prime }}(x^{\mathrm{\prime }},y)\mathrm{d}y\right)\cdot \hat{n}=0& \mathrm{o}\mathrm{n}{\gamma }_D.\end{array}\right.$(45) ), we have (52) ${\int }_{\mathrm{\Omega }}\zeta :D\left(u\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y=-{\int }_{\mathrm{\Omega }}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{p}_b\cdot u^{\mathrm{\prime }}\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y.$(52) From (50(50) $\begin{array}{rl}& \underset{ϵ\to 0}{lim}\frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}{ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right):{ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{u}^{ϵ}\mathrm{d}x\\ & =-\underset{ϵ\to 0}{lim}\frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}\mathrm{\nabla }{p}_b\cdot {ϵ}^{-p^{\mathrm{\prime }}}{u}^{ϵ}\mathrm{d}x=-\frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{p}_b\cdot u^{\mathrm{\prime }}\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y.\end{array}$(50) ) and (52(52) ${\int }_{\mathrm{\Omega }}\zeta :D\left(u\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y=-{\int }_{\mathrm{\Omega }}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{p}_b\cdot u^{\mathrm{\prime }}\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y.$(52) ) we deduce that equality holds in (36(36) $\underset{ϵ\to 0}{lim}inf\frac{1}{|{\mathrm{\Omega }}^{ϵ}|}{\int }_{{\mathrm{\Omega }}^{ϵ}}{ϵ}^{-1}a\left(\mathrm{\nabla }{u}^{ϵ}\right):{ϵ}^{1-p^{\mathrm{\prime }}}\mathrm{\nabla }{u}^{ϵ}\mathrm{d}x\ge \frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}\zeta :D\left(u\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y$(36) ). Thus (37(37) ${\int }_{\mathrm{\Omega }}\zeta :D\left(v\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y={\int }_{\mathrm{\Omega }}a\left(D\right(u\left)\right):D\left(v\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y,$(37) ) implies that (49(49) $\frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}\left\{-{\pi }^0\left[{\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}v+\frac{\mathrm{\partial }{v}_3}{\mathrm{\partial }y}\right]+\zeta :\left(\begin{array}{cc}0& \frac{\mathrm{\partial }{v}^{\mathrm{\prime }}}{\mathrm{\partial }y}\\ 0& 0\end{array}\right)+{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{p}_b\cdot v^{\mathrm{\prime }}\mathrm{d}{x}^{\mathrm{\prime }}\right\}\mathrm{d}y=0,$(49) ) can be written as (53) $\begin{array}{rl}& {\int }_{\omega }-{\pi }^0{\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}\left({\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}v\mathrm{d}y\right)\mathrm{d}{x}^{\mathrm{\prime }}\\ & +{\int }_{\mathrm{\Omega }}\left\{a\left(D\right(u\left)\right):\left(\begin{array}{cc}0& \frac{\mathrm{\partial }{v}^{\mathrm{\prime }}}{\mathrm{\partial }y}\\ 0& 0\end{array}\right)+{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{p}_b\cdot v^{\mathrm{\prime }}\right\}\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y=0.\end{array}$(53) From the definition of the operator a, i.e. equality (3(3) $a\left(X\right)=2{\mu }_0|e\left(X\right){|}^{p-2}e\left(X\right)(X\in {\mathbb{R}}^{3\times 3}),$(3) ) and (44(44) $u_3=0in\mathrm{\Omega }$(44) ), we obtain $a\left(D\right(u\left)\right)=\frac{{\mu }_0}{2^{p-2}}{\left|\left(\begin{array}{cc}0& \frac{\mathrm{\partial }{u}^{\mathrm{\prime }}}{\mathrm{\partial }y}\\ {\left(\frac{\mathrm{\partial }{u}^{\mathrm{\prime }}}{\mathrm{\partial }y}\right)}^T& 0\end{array}\right)\right|}^{p-2}\left(\begin{array}{cc}0& \frac{\mathrm{\partial }{u}^{\mathrm{\prime }}}{\mathrm{\partial }y}\\ {\left(\frac{\mathrm{\partial }{u}^{\mathrm{\prime }}}{\mathrm{\partial }y}\right)}^T& 0\end{array}\right).$Inserting this into (53(53) $\begin{array}{rl}& {\int }_{\omega }-{\pi }^0{\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}\left({\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}v\mathrm{d}y\right)\mathrm{d}{x}^{\mathrm{\prime }}\\ & +{\int }_{\mathrm{\Omega }}\left\{a\left(D\right(u\left)\right):\left(\begin{array}{cc}0& \frac{\mathrm{\partial }{v}^{\mathrm{\prime }}}{\mathrm{\partial }y}\\ 0& 0\end{array}\right)+{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{p}_b\cdot v^{\mathrm{\prime }}\right\}\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y=0.\end{array}$(53) ) yields (54) $\begin{array}{rl}& {\int }_{\omega }\left(-{\pi }^0{\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}\left({\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}v\mathrm{d}y\right)\right.\\ & +\left.{\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}\left\{{\mu }_0{\left|\frac{\mathrm{\partial }{u}^{\mathrm{\prime }}}{\mathrm{\partial }y}\right|}^{p-2}\frac{\mathrm{\partial }{u}^{\mathrm{\prime }}}{\mathrm{\partial }y}\cdot \frac{\mathrm{\partial }{v}^{\mathrm{\prime }}}{\mathrm{\partial }y}+{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{p}_b\cdot v^{\mathrm{\prime }}\right\}\mathrm{d}y\right)\mathrm{d}{x}^{\mathrm{\prime }}=0,\end{array}$(54) for all $(v^{\mathrm{\prime }},0)\in W_y^p\left(\mathrm{\Omega }\right)$. This is the weak formulation of the boundary value problem (48(48) $\left\{\begin{array}{ll}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}(-{\pi }^0+p_b)+{\mu }_0{\mathrm{\Delta }}_{p,y}{u}^{\mathrm{\prime }}=0& \mathrm{i}\mathrm{n}\mathrm{\Omega }\\ -\frac{\mathrm{\partial }{\pi }^0}{\mathrm{\partial }y}=0& \mathrm{i}\mathrm{n}\mathrm{\Omega }\\ u^{\mathrm{\prime }}=0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_D\\ \left(\begin{array}{ll}-{\pi }^0& {\mu }_0{\left|\frac{\mathrm{\partial }{u}^{\mathrm{\prime }}}{\mathrm{\partial }y}\right|}^{p-2}\frac{\mathrm{\partial }{u}^{\mathrm{\prime }}}{\mathrm{\partial }y}\\ 0& -{\pi }^0\end{array}\right)\hat{n}=0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_N.\end{array}\right.$(48) ).

6. Regularity of pressure and pressure equation

In this section, we will prove some properties of solutions $(u,q)$ to the lower-dimensional problem. Indeed, we will prove a regularity result for the pressure $q={\pi }^0+p_b$ and a nonlinear relation between the velocity u and the gradient of q. However, to be able to do this we first present the proof of Lemma 3.6.

Proof

Proof of Lemma 3.6

We note that the boundary value problem (20(20) $\left\{\begin{array}{ll}-{\mathrm{\Delta }}_{p,y}\psi =1& \mathrm{i}\mathrm{n}\mathrm{\Omega }\\ \psi =0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_D^{\pm },\end{array}\right.$(20) ) is formally an ordinary differential equation in variable y, with the parameter $x^{\mathrm{\prime }}\in \omega$, i.e. $-\frac{\mathrm{\partial }}{\mathrm{\partial }y}\left({\left|\frac{\mathrm{\partial }\psi }{\mathrm{\partial }y}(x^{\mathrm{\prime }},y)\right|}^{p-2}\frac{\mathrm{\partial }\psi }{\mathrm{\partial }y}(x^{\mathrm{\prime }},y)\right)=1.$By formal integration, we see that there exist a scalar function $C\left(x^{\mathrm{\prime }}\right)$ such that ${\left|\frac{\mathrm{\partial }\psi }{\mathrm{\partial }y}(x^{\mathrm{\prime }},y)\right|}^{p-2}\frac{\mathrm{\partial }\psi }{\mathrm{\partial }y}(x^{\mathrm{\prime }},y)=C\left(x^{\mathrm{\prime }}\right)-y.$From the observation, $\left.\begin{array}{l}{\left|\frac{\mathrm{\partial }\psi }{\mathrm{\partial }y}(x^{\mathrm{\prime }},y)\right|}^{p-1}=|C\left(x^{\mathrm{\prime }}\right)-y|\\ \left|\frac{\mathrm{\partial }\psi }{\mathrm{\partial }y}(x^{\mathrm{\prime }},y)\right|=|C\left(x^{\mathrm{\prime }}\right)-y{|}^{p^{\mathrm{\prime }}-1}\end{array}\right\}\Rightarrow {\left|\frac{\mathrm{\partial }\psi }{\mathrm{\partial }y}\right|}^{p-2}=|C\left(x^{\mathrm{\prime }}\right)-y{|}^{2-p^{\mathrm{\prime }}},$it follows that $\frac{\mathrm{\partial }\psi }{\mathrm{\partial }y}(x^{\mathrm{\prime }},y)=|C\left(x^{\mathrm{\prime }}\right)-y{|}^{p^{\mathrm{\prime }}-2}\left(C\right(x^{\mathrm{\prime }})-y).$Whence by integration once again the last equality becomes (55) $\begin{array}{rl}\psi (x^{\mathrm{\prime }},y)& ={\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^y|C\left(x^{\mathrm{\prime }}\right)-s{|}^{p^{\mathrm{\prime }}-2}\left(C\right(x^{\mathrm{\prime }})-s)\mathrm{d}s=-{\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^y\frac{\mathrm{d}}{\mathrm{d}s}\left(\frac{1}{p^{\mathrm{\prime }}}|C\left(x^{\mathrm{\prime }}\right)-s{|}^{p^{\mathrm{\prime }}}\right)\mathrm{d}s\\ & =\frac{1}{p^{\mathrm{\prime }}}\left(|C\right(x^{\mathrm{\prime }})-h^{-}(x^{\mathrm{\prime }}){|}^{p^{\mathrm{\prime }}}-|C(x^{\mathrm{\prime }})-y{|}^{p^{\mathrm{\prime }}}).\end{array}$(55) From the boundary condition $\psi (x^{\mathrm{\prime }},h^{-}(x^{\mathrm{\prime }}\left)\right)=0$ and $\psi (x^{\mathrm{\prime }},h^{+}(x^{\mathrm{\prime }}\left)\right)=0$, we see that $C\left(x^{\mathrm{\prime }}\right)=m\left(x^{\mathrm{\prime }}\right),$where $m\left(x^{\mathrm{\prime }}\right)$ is defined in (1(1) $\begin{array}{rl}& -\frac{1}{2}\le h^{-}(x_1,x_2)\le -\frac{1}{4}\mathrm{a}\mathrm{n}\mathrm{d}\frac{1}{4}\le h^{+}(x_1,x_2)\le \frac{1}{2}\\ & h\left(x^{\mathrm{\prime }}\right)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{h}^{+}(x_1,x_2)-h^{-}(x_1,x_2)\mathrm{a}\mathrm{n}\mathrm{d}m(x_1,x_2)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{h^{+}(x_1,x_2)+h^{-}(x_1,x_2)}{2},\end{array}$(1) ) and therefore we get (22(22) $\psi (x^{\mathrm{\prime }},y)=\frac{1}{p^{\mathrm{\prime }}}\left({\left|\frac{h\left(x^{\mathrm{\prime }}\right)}{2}\right|}^{p^{\mathrm{\prime }}}-|m\left(x^{\mathrm{\prime }}\right)-y{|}^{p^{\mathrm{\prime }}}\right),$(22) ) from (55(55) $\begin{array}{rl}\psi (x^{\mathrm{\prime }},y)& ={\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^y|C\left(x^{\mathrm{\prime }}\right)-s{|}^{p^{\mathrm{\prime }}-2}\left(C\right(x^{\mathrm{\prime }})-s)\mathrm{d}s=-{\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^y\frac{\mathrm{d}}{\mathrm{d}s}\left(\frac{1}{p^{\mathrm{\prime }}}|C\left(x^{\mathrm{\prime }}\right)-s{|}^{p^{\mathrm{\prime }}}\right)\mathrm{d}s\\ & =\frac{1}{p^{\mathrm{\prime }}}\left(|C\right(x^{\mathrm{\prime }})-h^{-}(x^{\mathrm{\prime }}){|}^{p^{\mathrm{\prime }}}-|C(x^{\mathrm{\prime }})-y{|}^{p^{\mathrm{\prime }}}).\end{array}$(55) ). To obtain the flow factor $\varrho \left(x^{\mathrm{\prime }}\right)$, we integrate the permeability function on $\left(h^{-}\right(x^{\mathrm{\prime }}),h^{+}(x^{\mathrm{\prime }}\left)\right)$, i.e. $\varrho \left(x^{\mathrm{\prime }}\right)={\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}\psi (x^{\mathrm{\prime }},y)\mathrm{d}y=\frac{2^{-p^{\mathrm{\prime }}}}{p^{\mathrm{\prime }}}|h\left(x^{\mathrm{\prime }}\right){|}^{p^{\mathrm{\prime }}+1}-\frac{1}{p^{\mathrm{\prime }}}{\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}|m\left(x^{\mathrm{\prime }}\right)-y{|}^{p^{\mathrm{\prime }}}\mathrm{d}y.$Observe that the last integral can be written as $\begin{array}{rl}& {\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}|m\left(x^{\mathrm{\prime }}\right)-y{|}^{p^{\mathrm{\prime }}}\mathrm{d}y={\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{m\left(x^{\mathrm{\prime }}\right)}|m\left(x^{\mathrm{\prime }}\right)-y{|}^{p^{\mathrm{\prime }}-1}\left(m\right(x^{\mathrm{\prime }})-y)\mathrm{d}y\\ & +{\int }_{m\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}|m\left(x^{\mathrm{\prime }}\right)-y{|}^{p^{\mathrm{\prime }}-1}(y-m(x^{\mathrm{\prime }}\left)\right)\mathrm{d}y=\frac{2}{p^{\mathrm{\prime }}+1}{\left|\frac{h^{+}\left(x^{\mathrm{\prime }}\right)-h^{-}\left(x^{\mathrm{\prime }}\right)}{2}\right|}^{p^{\mathrm{\prime }}+1}.\text{■}\end{array}$

Lemma 6.1

Let $\psi \in W^{1,\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ be the permeability function defined by (22(22) $\psi (x^{\mathrm{\prime }},y)=\frac{1}{p^{\mathrm{\prime }}}\left({\left|\frac{h\left(x^{\mathrm{\prime }}\right)}{2}\right|}^{p^{\mathrm{\prime }}}-|m\left(x^{\mathrm{\prime }}\right)-y{|}^{p^{\mathrm{\prime }}}\right),$(22) ) and $(u,{\pi }^0)\in W_y^p\left(\mathrm{\Omega }\right)\times L^{p^{\mathrm{\prime }}}\left(\omega \right)$ be any solution of the boundary value problem (48(48) $\left\{\begin{array}{ll}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}(-{\pi }^0+p_b)+{\mu }_0{\mathrm{\Delta }}_{p,y}{u}^{\mathrm{\prime }}=0& \mathrm{i}\mathrm{n}\mathrm{\Omega }\\ -\frac{\mathrm{\partial }{\pi }^0}{\mathrm{\partial }y}=0& \mathrm{i}\mathrm{n}\mathrm{\Omega }\\ u^{\mathrm{\prime }}=0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_D\\ \left(\begin{array}{ll}-{\pi }^0& {\mu }_0{\left|\frac{\mathrm{\partial }{u}^{\mathrm{\prime }}}{\mathrm{\partial }y}\right|}^{p-2}\frac{\mathrm{\partial }{u}^{\mathrm{\prime }}}{\mathrm{\partial }y}\\ 0& -{\pi }^0\end{array}\right)\hat{n}=0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_N.\end{array}\right.$(48) ). Then ${\pi }^0$ belongs to $W^{1,p^{\mathrm{\prime }}}\left(\omega \right)$, with ${\pi }^0=0$ on ${\gamma }_N$.

Proof.

To show the $W^{1,p^{\mathrm{\prime }}}$-regularity for ${\pi }^0$, in view of the Lemma 5.4, we take test functions $v(x^{\mathrm{\prime }},y)$ of the form $v^{\mathrm{\prime }}(x^{\mathrm{\prime }},y)=\psi (x^{\mathrm{\prime }},y)\mathrm{\Theta }\left(x^{\mathrm{\prime }}\right),v_3(x^{\mathrm{\prime }},y)=0$where Θ is any vector field in $W^{1,p}(\omega ;{\mathbb{R}}^2)$ such that $\mathrm{\Theta }\cdot \hat{n}=0$ on ${\gamma }_D$, in (54(54) $\begin{array}{rl}& {\int }_{\omega }\left(-{\pi }^0{\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}\left({\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}v\mathrm{d}y\right)\right.\\ & +\left.{\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}\left\{{\mu }_0{\left|\frac{\mathrm{\partial }{u}^{\mathrm{\prime }}}{\mathrm{\partial }y}\right|}^{p-2}\frac{\mathrm{\partial }{u}^{\mathrm{\prime }}}{\mathrm{\partial }y}\cdot \frac{\mathrm{\partial }{v}^{\mathrm{\prime }}}{\mathrm{\partial }y}+{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{p}_b\cdot v^{\mathrm{\prime }}\right\}\mathrm{d}y\right)\mathrm{d}{x}^{\mathrm{\prime }}=0,\end{array}$(54) ), we obtain ${\int }_{\omega }{\pi }^0{\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}\left(\varrho \mathrm{\Theta }\right)\mathrm{d}{x}^{\mathrm{\prime }}={\int }_{\omega }\left\{{\mu }_0{\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}\frac{\mathrm{\partial }\psi }{\mathrm{\partial }y}\left({\left|\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right|}^{p-2}\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\cdot \mathrm{\Theta }\right)\mathrm{d}y+\varrho {\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{p}_b\cdot \mathrm{\Theta }\right\}\mathrm{d}{x}^{\mathrm{\prime }},$where ϱ is the flow factor defined in (23(23) $\varrho \left(x^{\mathrm{\prime }}\right)=\frac{2^{-p^{\mathrm{\prime }}}}{p^{\mathrm{\prime }}+1}{h}^{p^{\mathrm{\prime }}+1}\left(x^{\mathrm{\prime }}\right),$(23) ). Since ${\pi }^0\in L^{p^{\mathrm{\prime }}}\left(\omega \right)$, we can define a linear functional with ${\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{\pi }^0\in W^{-1,p^{\mathrm{\prime }}}\left(\omega \right)$ determined by $〈{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{\pi }^0,\varrho \left(x^{\mathrm{\prime }}\right)\mathrm{\Theta }\left(x^{\mathrm{\prime }}\right)〉\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{\int }_{\omega }{\pi }^0{\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}\left(\varrho \mathrm{\Theta }\right)\mathrm{d}{x}^{\mathrm{\prime }},$i.e. $〈{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{\pi }^0,\varrho \left(x^{\mathrm{\prime }}\right)\mathrm{\Theta }\left(x^{\mathrm{\prime }}\right)〉={\int }_{\mathrm{\Omega }}{\mu }_0\frac{\mathrm{\partial }\psi }{\mathrm{\partial }y}\left({\left|\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right|}^{p-2}\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\cdot \mathrm{\Theta }\right)\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y+{\int }_{\omega }\varrho {\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{p}_b\cdot \mathrm{\Theta }\mathrm{d}{x}^{\mathrm{\prime }}.$Thus, (56) $|〈{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{\pi }^0,\varrho \left(x^{\mathrm{\prime }}\right)\mathrm{\Theta }\left(x^{\mathrm{\prime }}\right)〉|\le C\parallel \mathrm{\Theta }{\parallel }_{L^p\left(\omega \right)},$(56) where C is the a positive constant given by $C\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{\mu }_0{∥\frac{\mathrm{\partial }\psi }{\mathrm{\partial }y}∥}_{L^{\mathrm{\infty }}\left(\omega \right)}{∥\frac{\mathrm{\partial }u}{\mathrm{\partial }y}∥}_{L^p\left(\mathrm{\Omega }\right)}^{p-1}+\parallel \varrho {\parallel }_{L^{\mathrm{\infty }}\left(\omega \right)}\parallel {\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{p}_b{\parallel }_{L^{p^{\mathrm{\prime }}}\left(\omega \right)}.$Furthermore, since the functions ϱ and ${\varrho }^{-1}=1/\varrho$ belong to $W^{1,\mathrm{\infty }}\left(\omega \right)$, (see Remark 3.7) we can replace Θ with ${\varrho }^{-1}\mathrm{\Theta }$ in (56(56) $|〈{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{\pi }^0,\varrho \left(x^{\mathrm{\prime }}\right)\mathrm{\Theta }\left(x^{\mathrm{\prime }}\right)〉|\le C\parallel \mathrm{\Theta }{\parallel }_{L^p\left(\omega \right)},$(56) ) which gives $|〈{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{\pi }^0,\mathrm{\Theta }〉|\le C\parallel {\varrho }^{-1}\mathrm{\Theta }{\parallel }_{L^p\left(\omega \right)}\le C\parallel {\varrho }^{-1}{\parallel }_{L^{\mathrm{\infty }}\left(\omega \right)}\parallel \mathrm{\Theta }{\parallel }_{L^p\left(\omega \right)}\le \tilde{C}\parallel \mathrm{\Theta }{\parallel }_{L^p\left(\omega \right)},$for all $\mathrm{\Theta }\in W^{1,p}(\omega ;{\mathbb{R}}^2)$ such that $\mathrm{\Theta }\cdot \hat{n}=0$ on ${\gamma }_D$, and therefore, in view of Lemma A.5 in Appendix 4, we conclude that ${\pi }^0$ belongs to $W^{1,p^{\mathrm{\prime }}}\left(\omega \right)$ with ${\pi }^0=0$ on ${\gamma }_N$.

Remark 6.2

Note that $q={\pi }^0+p_b\in W^{1,p^{\mathrm{\prime }}}\left(\omega \right)$ with $q=p_b$ on ${\gamma }_N$.

Lemma 6.3

Let $(u,{\pi }^0)\in W_y^p\left(\mathrm{\Omega }\right)\times W^{1,p^{\mathrm{\prime }}}\left(\omega \right)$ be any solution of (48(48) $\left\{\begin{array}{ll}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}(-{\pi }^0+p_b)+{\mu }_0{\mathrm{\Delta }}_{p,y}{u}^{\mathrm{\prime }}=0& \mathrm{i}\mathrm{n}\mathrm{\Omega }\\ -\frac{\mathrm{\partial }{\pi }^0}{\mathrm{\partial }y}=0& \mathrm{i}\mathrm{n}\mathrm{\Omega }\\ u^{\mathrm{\prime }}=0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_D\\ \left(\begin{array}{ll}-{\pi }^0& {\mu }_0{\left|\frac{\mathrm{\partial }{u}^{\mathrm{\prime }}}{\mathrm{\partial }y}\right|}^{p-2}\frac{\mathrm{\partial }{u}^{\mathrm{\prime }}}{\mathrm{\partial }y}\\ 0& -{\pi }^0\end{array}\right)\hat{n}=0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_N.\end{array}\right.$(48) ). Then u is uniquely determined by ${\pi }^0$, more precisely (57) $u(x^{\mathrm{\prime }},y)=-\psi (x^{\mathrm{\prime }},y){\left|\frac{1}{{\mu }_0}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}q\left(x^{\mathrm{\prime }}\right)\right|}^{p^{\mathrm{\prime }}-2}\frac{1}{{\mu }_0}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}q\left(x^{\mathrm{\prime }}\right),$(57) where $q={\pi }^0+p_b$ and ψ must satisfy the scalar ordinary differential Equation (20(20) $\left\{\begin{array}{ll}-{\mathrm{\Delta }}_{p,y}\psi =1& \mathrm{i}\mathrm{n}\mathrm{\Omega }\\ \psi =0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_D^{\pm },\end{array}\right.$(20) ).

Proof.

In view of Lemma 6.1, the problem (48(48) $\left\{\begin{array}{ll}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}(-{\pi }^0+p_b)+{\mu }_0{\mathrm{\Delta }}_{p,y}{u}^{\mathrm{\prime }}=0& \mathrm{i}\mathrm{n}\mathrm{\Omega }\\ -\frac{\mathrm{\partial }{\pi }^0}{\mathrm{\partial }y}=0& \mathrm{i}\mathrm{n}\mathrm{\Omega }\\ u^{\mathrm{\prime }}=0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_D\\ \left(\begin{array}{ll}-{\pi }^0& {\mu }_0{\left|\frac{\mathrm{\partial }{u}^{\mathrm{\prime }}}{\mathrm{\partial }y}\right|}^{p-2}\frac{\mathrm{\partial }{u}^{\mathrm{\prime }}}{\mathrm{\partial }y}\\ 0& -{\pi }^0\end{array}\right)\hat{n}=0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_N.\end{array}\right.$(48) ) can be written as an ordinary differential equation in the variable y, i.e. (58) $\left\{\begin{array}{ll}-{\mathrm{\Delta }}_{p,y}u(x^{\mathrm{\prime }},\cdot )=\xi & \mathrm{i}\mathrm{n}\left(h^{-}\left(x^{\mathrm{\prime }}\right),h^{+}\left(x^{\mathrm{\prime }}\right)\right)\\ u(x^{\mathrm{\prime }},\cdot )=0& \mathrm{o}\mathrm{n}y=h^{\pm }\left(x^{\mathrm{\prime }}\right),\end{array}\right.$(58) where $x^{\mathrm{\prime }}\in \omega$ is a parameter and $\xi =\frac{1}{{\mu }_0}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}q\left(x^{\mathrm{\prime }}\right)$ can be regarded as a constant vector in ${\mathbb{R}}^2$.

We want to show that (57(57) $u(x^{\mathrm{\prime }},y)=-\psi (x^{\mathrm{\prime }},y){\left|\frac{1}{{\mu }_0}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}q\left(x^{\mathrm{\prime }}\right)\right|}^{p^{\mathrm{\prime }}-2}\frac{1}{{\mu }_0}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}q\left(x^{\mathrm{\prime }}\right),$(57) ) is a solution of the form $u(x^{\mathrm{\prime }},y)=-\psi (x^{\mathrm{\prime }},y){\left|\xi \right|}^{p^{\mathrm{\prime }}-2}\xi (\xi \in {\mathbb{R}}^2).$Indeed, $\frac{\mathrm{\partial }}{\mathrm{\partial }y}\left({\left|\frac{\mathrm{\partial }u}{\mathrm{\partial }y}(x^{\mathrm{\prime }},y)\right|}^{p-2}\frac{\mathrm{\partial }u}{\mathrm{\partial }y}(x^{\mathrm{\prime }},y)\right)=-\xi .$As in [11, Propostion 3.4], by formal integration, we see that there exist a vector function $C\left(x^{\mathrm{\prime }}\right)\in L^{p^{\mathrm{\prime }}}(\omega ;{\mathbb{R}}^2)$ such that ${\left|\frac{\mathrm{\partial }u}{\mathrm{\partial }y}(x^{\mathrm{\prime }},y)\right|}^{p-2}\frac{\mathrm{\partial }u}{\mathrm{\partial }y}(x^{\mathrm{\prime }},y)=C\left(x^{\mathrm{\prime }}\right)-y\xi .$Thus, $\frac{\mathrm{\partial }u}{\mathrm{\partial }y}(x^{\mathrm{\prime }},y)=|C\left(x^{\mathrm{\prime }}\right)-y\xi {|}^{p^{\mathrm{\prime }}-2}\left(C\right(x^{\mathrm{\prime }})-y\xi )$whence by integration once again and taking into account the no-slip condition at $y=h^{-}\left(x^{\mathrm{\prime }}\right)$ we obtain the velocity field (59) $u(x^{\mathrm{\prime }},y)={\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^y|C\left(x^{\mathrm{\prime }}\right)-z\xi {|}^{p^{\mathrm{\prime }}-2}\left(C\right(x^{\mathrm{\prime }})-z\xi )\mathrm{d}z.$(59) Since $u(x^{\mathrm{\prime }},y)=0$ on $y=h^{+}\left(x^{\mathrm{\prime }}\right)$, as well, we find that $g\left(C\right)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}|C\left(x^{\mathrm{\prime }}\right)-y\xi {|}^{p^{\mathrm{\prime }}-2}\left(C\right(x^{\mathrm{\prime }})-y\xi )\mathrm{d}y=0.$Observe that $g\left(C\right)$ is the gradient with respect to C of a the integral functional $J\left(C\right)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{\int }_{\omega }F(x^{\mathrm{\prime }},C)\mathrm{d}{x}^{\mathrm{\prime }},$with convex Lagrangian $F(x^{\mathrm{\prime }},C)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{1}{p^{\mathrm{\prime }}}{\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^{h^{+}\left(x^{\mathrm{\prime }}\right)}|C\left(x^{\mathrm{\prime }}\right)-y\xi {|}^{p^{\mathrm{\prime }}}\mathrm{d}y.$Consequently using analogue of Lemma 3.1 we see that C is uniquely determined and the unique solution to the Euler equation is $C\left(x^{\mathrm{\prime }}\right)=m\left(x^{\mathrm{\prime }}\right)\xi ,$where $m\left(x^{\mathrm{\prime }}\right)$ is defined as in (1(1) $\begin{array}{rl}& -\frac{1}{2}\le h^{-}(x_1,x_2)\le -\frac{1}{4}\mathrm{a}\mathrm{n}\mathrm{d}\frac{1}{4}\le h^{+}(x_1,x_2)\le \frac{1}{2}\\ & h\left(x^{\mathrm{\prime }}\right)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{h}^{+}(x_1,x_2)-h^{-}(x_1,x_2)\mathrm{a}\mathrm{n}\mathrm{d}m(x_1,x_2)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{h^{+}(x_1,x_2)+h^{-}(x_1,x_2)}{2},\end{array}$(1) ) and so $u$ is uniquely determined, and finally (57(57) $u(x^{\mathrm{\prime }},y)=-\psi (x^{\mathrm{\prime }},y){\left|\frac{1}{{\mu }_0}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}q\left(x^{\mathrm{\prime }}\right)\right|}^{p^{\mathrm{\prime }}-2}\frac{1}{{\mu }_0}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}q\left(x^{\mathrm{\prime }}\right),$(57) ) is deduced from (59(59) $u(x^{\mathrm{\prime }},y)={\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^y|C\left(x^{\mathrm{\prime }}\right)-z\xi {|}^{p^{\mathrm{\prime }}-2}\left(C\right(x^{\mathrm{\prime }})-z\xi )\mathrm{d}z.$(59) ), using (55(55) $\begin{array}{rl}\psi (x^{\mathrm{\prime }},y)& ={\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^y|C\left(x^{\mathrm{\prime }}\right)-s{|}^{p^{\mathrm{\prime }}-2}\left(C\right(x^{\mathrm{\prime }})-s)\mathrm{d}s=-{\int }_{h^{-}\left(x^{\mathrm{\prime }}\right)}^y\frac{\mathrm{d}}{\mathrm{d}s}\left(\frac{1}{p^{\mathrm{\prime }}}|C\left(x^{\mathrm{\prime }}\right)-s{|}^{p^{\mathrm{\prime }}}\right)\mathrm{d}s\\ & =\frac{1}{p^{\mathrm{\prime }}}\left(|C\right(x^{\mathrm{\prime }})-h^{-}(x^{\mathrm{\prime }}){|}^{p^{\mathrm{\prime }}}-|C(x^{\mathrm{\prime }})-y{|}^{p^{\mathrm{\prime }}}).\end{array}$(55) ).

To summarize this section, from Lemma 5.5, we deduce that q satisfies the boundary value problem (25(25) $\left\{\begin{array}{ll}{\mathrm{d}\mathrm{i}\mathrm{v}}_{x^{\mathrm{\prime }}}\left(\varrho |{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}q{|}^{p^{\mathrm{\prime }}-2}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}q\right)=0& \mathrm{i}\mathrm{n}\omega \\ q=p_b& \mathrm{o}\mathrm{n}{\gamma }_N\\ \varrho |{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}q{|}^{p^{\mathrm{\prime }}-2}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}q\cdot \hat{n}=0& \mathrm{o}\mathrm{n}{\gamma }_D,\end{array}\right.$(25) ), hence q in $W^{1,p^{\mathrm{\prime }}}\left(\omega \right)$ is uniquely determined by $p_b$. Thereafter it follows from Lemma 6.3 that u in $W_y^p\left(\mathrm{\Omega }\right)$ is also unique. Since all subsequential limits u and q are the same, we conclude that the whole sequence of solutions $({ϵ}^{-p^{\mathrm{\prime }}}{u}^{ϵ},q^{ϵ})$ to the problem (11(11) $\left\{\begin{array}{ll}\mathrm{d}\mathrm{i}\mathrm{v}{\sigma }^{ϵ}=0& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ \mathrm{d}\mathrm{i}\mathrm{v}{u}^{ϵ}=0& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ u^{ϵ}=0& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_D^{ϵ}\\ {\sigma }^{ϵ}\hat{n}=-p_b\hat{n}& \mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_N^{ϵ},\end{array}\right.$(11) ) two-scale converges to $(u,q)$.

Acknowledgements

We thank the anonymous referees for many valuable comments and suggestions which improved the final version of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Appendices

Appendix 1. Proof of Theorem 2.1

In this appendix, we generalize the Korn inequalities proved in [2], where the case p = 2 was considered. Let □ be the cube in ${\mathbb{R}}^3$ defined by $◻=(-1/2,1/2{)}^3$ and let ${◻}^{ϵ}$ be the cube defined by ${◻}^{ϵ}=(-ϵ/2,ϵ/2{)}^3$. Moreover we define the planes ${\mathrm{\Gamma }}_{\pm }=\left\{\right(x_1,x_2,x_3)\in {\mathbb{R}}^3:x_3=\pm 1/2\}$and ${\mathrm{\Gamma }}_{\pm }^{ϵ}=\left\{\right(x_1,x_2,x_3)\in {\mathbb{R}}^3:x_3=\pm ϵ/2\}.$

Lemma A.1

Let $K_1$ and $K_2$ denote the best constants in the inequalities (A1) $\begin{array}{r}||v|{|}_{L^p\left(◻\right)}\le K_1||e\left(\mathrm{\nabla }v\right)|{|}_{L^p\left(◻\right)}\end{array}$(A1) (A2) $\begin{array}{r}||\mathrm{\nabla }v|{|}_{L^p\left(◻\right)}\le K_2||e\left(\mathrm{\nabla }v\right)|{|}_{L^p\left(◻\right)}\end{array}$(A2) for all $v\in W^{1,p}(◻,{\mathbb{R}}^3)$ such that v = 0 on ${\mathrm{\Gamma }}_{+}\cup {\mathrm{\Gamma }}_{-}$. Then (A3) $\begin{array}{r}||v|{|}_{L^p\left({◻}^{ϵ}\right)}\le ϵK_1||e\left(\mathrm{\nabla }v\right)|{|}_{L^p\left({◻}^{ϵ}\right)}\end{array}$(A3) (A4) $\begin{array}{r}||\mathrm{\nabla }v|{|}_{L^p\left({◻}^{ϵ}\right)}\le K_2||e\left(\mathrm{\nabla }v\right)|{|}_{L^p\left({◻}^{ϵ}\right)}\end{array}$(A4) for all $v\in W^{1,p}({◻}^{ϵ},{\mathbb{R}}^3)$ such that v = 0 on ${\mathrm{\Gamma }}_{+}^{ϵ}\cup {\mathrm{\Gamma }}_{-}^{ϵ}$.

Proof.

Suppose $v\in W^p\left({◻}^{ϵ}\right)$ and define $\hat{v}$ in $W^p\left(◻\right)$ as the function which satisfies $v\left(x\right)=\hat{v}\left(x/ϵ\right),(x\in ◻).$Making the change of variable $\hat{x}=x/ϵ$ and applying (A1(A1) $\begin{array}{r}||v|{|}_{L^p\left(◻\right)}\le K_1||e\left(\mathrm{\nabla }v\right)|{|}_{L^p\left(◻\right)}\end{array}$(A1) ) to $\hat{v}$ we obtain ${\int }_{{◻}^{ϵ}}{∣v\left(x\right)∣}^{p}dx={\int }_{◻}|\hat{v}\left(\hat{x}\right){|}^p{ϵ}^3\mathrm{d}\hat{x}\le K_1^p{\int }_{◻}{∣e\left(\mathrm{\nabla }\hat{v}\right)\left(\hat{x}\right)∣}^p{ϵ}^3\mathrm{d}\hat{x}=K_1^p{\int }_{{◻}^{ϵ}}{∣e\left(ϵ\mathrm{\nabla }v\right)\left(x\right)∣}^p\mathrm{d}x,$which implies (A3(A3) $\begin{array}{r}||v|{|}_{L^p\left({◻}^{ϵ}\right)}\le ϵK_1||e\left(\mathrm{\nabla }v\right)|{|}_{L^p\left({◻}^{ϵ}\right)}\end{array}$(A3) ). Similarly, but using (A2(A2) $\begin{array}{r}||\mathrm{\nabla }v|{|}_{L^p\left(◻\right)}\le K_2||e\left(\mathrm{\nabla }v\right)|{|}_{L^p\left(◻\right)}\end{array}$(A2) ) instead of (A1(A1) $\begin{array}{r}||v|{|}_{L^p\left(◻\right)}\le K_1||e\left(\mathrm{\nabla }v\right)|{|}_{L^p\left(◻\right)}\end{array}$(A1) ), we obtain (A4(A4) $\begin{array}{r}||\mathrm{\nabla }v|{|}_{L^p\left({◻}^{ϵ}\right)}\le K_2||e\left(\mathrm{\nabla }v\right)|{|}_{L^p\left({◻}^{ϵ}\right)}\end{array}$(A4) ).

To estimate the constant in the Korn inequality under anisotropic scaling of the domain, we use an extension operator in combination with a covering argument. In view of assumptions in (1(1) $\begin{array}{rl}& -\frac{1}{2}\le h^{-}(x_1,x_2)\le -\frac{1}{4}\mathrm{a}\mathrm{n}\mathrm{d}\frac{1}{4}\le h^{+}(x_1,x_2)\le \frac{1}{2}\\ & h\left(x^{\mathrm{\prime }}\right)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{h}^{+}(x_1,x_2)-h^{-}(x_1,x_2)\mathrm{a}\mathrm{n}\mathrm{d}m(x_1,x_2)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{h^{+}(x_1,x_2)+h^{-}(x_1,x_2)}{2},\end{array}$(1) ) we can deduce that ${\mathrm{\Omega }}^{ϵ}\subset \omega \times (-ϵ/2,ϵ/2),$is also thin domain, so that we can consider the unscaled domain Ω defined as $\mathrm{\Omega }=\omega \times (-1/2,1/2).$Indeed, due to the Dirichlet condition on ${\mathrm{\Gamma }}_D^{\pm }$, we can extend any $v\in W^p\left(\mathrm{\Omega }\right)$ by zero to $\omega \times (-1/2,1/2)$, which is confined between the hyperplane ${\mathrm{\Gamma }}_{\pm }$.

Finally, we can assume a sufficiently large domain $\tilde{\mathrm{\Omega }}$ of Ω given by $\tilde{\mathrm{\Omega }}=(-L-1,L+1{)}^2\times (-1/2,1/2),$such that $\mathrm{\Omega }\subset \tilde{\mathrm{\Omega }}.$

A.1. Extension operator

By consideration above, we prove the result

Lemma A.2

There exists a bounded linear operator $E:W^p\left(\mathrm{\Omega }\right)\to W^p\left(\tilde{\mathrm{\Omega }}\right),$such that:

1. Ev = v, for all v in $W^p\left(\mathrm{\Omega }\right);$

2. for some positive constant $C_{\omega }$ which depends only on ω, it holds ${∣∣e\left(\mathrm{\nabla }\right(Ev\left)\right)∣∣}_{L^p\left(\tilde{\mathrm{\Omega }}\right)}\le C_{\omega }{∣∣e\left(\mathrm{\nabla }v\right)∣∣}_{L^p\left(\mathrm{\Omega }\right)},$for all v in $W^p\left(\mathrm{\Omega }\right)$.

Proof.

Since Ω is a bounded open subset of ${\mathbb{R}}^3$ with Lipschitz boundary, there exists a continuous linear extension operator E from $W^p\left(\mathrm{\Omega }\right)$ into $W^p\left(\tilde{\mathrm{\Omega }}\right)$, such that $Ev\left(x^{\mathrm{\prime }}y\right)=v(x^{\mathrm{\prime }},y),\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}.\mathrm{e}(x^{\mathrm{\prime }},y)\in \mathrm{\Omega }.$See for instance, [31, p. 25. Theorem 1.4.3.4], whence $\left(i\right)$ follows. To show $\left(ii\right)$, first we note that ${∣∣e\left(\mathrm{\nabla }v\right)∣∣}_{L^p\left(\mathrm{\Omega }\right)}\le {∣∣\mathrm{\nabla }v∣∣}_{L^p\left(\mathrm{\Omega }\right)}.$Applying this inequality to Ev yields ${∣∣e\left(\mathrm{\nabla }\right(Ev\left)\right)∣∣}_{L^p\left(\tilde{\mathrm{\Omega }}\right)}\le {∣∣\mathrm{\nabla }\left(Ev\right)∣∣}_{L^p\left(\tilde{\mathrm{\Omega }}\right)}\le {∣∣Ev∣∣}_{W^{1,p}\left(\tilde{\mathrm{\Omega }}\right)}.$By the continuity of the extension operator, there exists a constant $C_1>0$, which depends only on Ω, such that ${∣∣Ev∣∣}_{W^{1,p}\left(\tilde{\mathrm{\Omega }}\right)}\le C_1{∣∣v∣∣}_{W^{1,p}\left(\mathrm{\Omega }\right)}.$From the Korn inequality for $W^{1,p}\left(\mathrm{\Omega }\right)$ (see [24, Theorem 4.8]) there exists a positive constant $C_2$ depends only in Ω, $\parallel v{\parallel }_{W^{1,p}\left(\mathrm{\Omega }\right)}\le C_2\parallel e\left(\mathrm{\nabla }v\right){\parallel }_{L^p\left(\mathrm{\Omega }\right)}.$Hence, ${∣∣e\left(\mathrm{\nabla }\right(Ev\left)\right)∣∣}_{L^p\left(\tilde{\mathrm{\Omega }}\right)}\le C_{\omega }{∣∣e\left(\mathrm{\nabla }v\right)∣∣}_{L^p\left(\mathrm{\Omega }\right)},$where $C_{\omega }=C_1\cdot C_2$, and thus, we conclude the proof of the lemma.

A.2. Covering argument

For $z\in {\mathbb{R}}^2$, let $Q(z,ϵ)$ denote the square in ${\mathbb{R}}^2$ of length ε centered in z, i.e. $Q(z,ϵ)=\{y\in {\mathbb{R}}^2:|y_i-z_i|<ϵ/2,i=1,2\}.$For each $0<ϵ\le 1$, there exist a finite collection of points $\left\{z^i\right\}$ in $ϵ{\mathbb{Z}}^2$ such that $\omega \subset \overline{\bigcup _{i}Q(z^i,ϵ)}\subset (-L-1,L+1{)}^2.$Define the cube ${◻}_i^{ϵ}=Q(z^i,ϵ)\times (-ϵ/2,ϵ/2).$Then ${\mathrm{\Omega }}^{ϵ}\subset \overline{\bigcup _i{◻}_i^{ϵ}}\subset \tilde{\mathrm{\Omega }},$for all $0<ϵ\le 1$. Since each cube ${◻}_i^{ϵ}$ is a translation of the cube ${◻}^{ϵ}=(-ϵ/2,ϵ/2{)}^3$ we can use the inequality (A3(A3) $\begin{array}{r}||v|{|}_{L^p\left({◻}^{ϵ}\right)}\le ϵK_1||e\left(\mathrm{\nabla }v\right)|{|}_{L^p\left({◻}^{ϵ}\right)}\end{array}$(A3) ) locally in Lemma A.1.

Proof

Proof of Theorem 2.1

Suppose $v\in W^p\left({\mathrm{\Omega }}^{ϵ}\right)$. To use the extension operator in Lemma A.2, let $\overline{v}$ in $W^p\left(\mathrm{\Omega }\right)$ be defined by extension $\overline{v}=\left\{\begin{array}{ll}v,& \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}\\ 0,& \mathrm{i}\mathrm{n}\mathrm{\Omega }-{\mathrm{\Omega }}^{ϵ}.\end{array}\right.$From Lemma A.1$\left(i\right)$, the covering argument and inequality (A3(A3) $\begin{array}{r}||v|{|}_{L^p\left({◻}^{ϵ}\right)}\le ϵK_1||e\left(\mathrm{\nabla }v\right)|{|}_{L^p\left({◻}^{ϵ}\right)}\end{array}$(A3) ) of Lemma A.2 it follows that $\begin{array}{rl}{\int }_{{\mathrm{\Omega }}^{ϵ}}|v{|}^p\mathrm{d}x& ={\int }_{{\mathrm{\Omega }}^{ϵ}}{∣E\overline{v}∣}^p\mathrm{d}x\le \sum _{i=1}{\int }_{{◻}_i^{ϵ}}{∣E\overline{v}∣}^p\mathrm{d}x\\ & \le \sum _{i=1}{ϵ}^p{k}_1^p{\int }_{{◻}_i^{ϵ}}{∣e\left(\mathrm{\nabla }\right(E\overline{v}\left)\right)∣}^p\mathrm{d}x\le {ϵ}^p{k}_1^p{\int }_{\tilde{\mathrm{\Omega }}}{∣e\left(\mathrm{\nabla }\right(E\overline{v}\left)\right)∣}^p\mathrm{d}x,\end{array}$or ${∣∣v∣∣}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\le ϵK_1{∣∣e\left(\mathrm{\nabla }\right(E\overline{v}\left)\right)∣∣}_{L^p\left(\tilde{\mathrm{\Omega }}\right)}.$This together with Lemma A.2$\left(ii\right)$ yields ${∣∣v∣∣}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\le ϵK_1{C}_{\omega }{∣∣e\left(\mathrm{\nabla }\right(\overline{v}\left)\right)∣∣}_{L^p\left(\mathrm{\Omega }\right)}=ϵK_1{C}_{\omega }{∣∣e\left(\mathrm{\nabla }\right(v\left)\right)∣∣}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}.$This proves the first inequality in Theorem 2.1. The second inequality is proved in a similar way.

Appendix 2. Proof of Theorem 2.2

We will need a simple scaling  argument to prove the upper bound of norm of the Bogovski operator. To obtain a lower bound we consider an eigenvalue problem.

Simple scaling argument

For any $0<ϵ\le 1$, let $S^{ϵ}$ denote the linear scaling operator defined by $\begin{array}{ll}{S}^{ϵ}f\left(x\right)=f(x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3)& \left(f\mathrm{i}\mathrm{s}\mathrm{a}\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\right)\\ S^{ϵ}v\left(x\right)=\left(v^{\mathrm{\prime }}\right(x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3),ϵv_3(x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3\left)\right)& \left(v\mathrm{i}\mathrm{s}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\right),\end{array}$where $x=(x^{\mathrm{\prime }},x^3)\in {\mathbb{R}}^2\times \mathbb{R}$ and similarly for $v=(v^{\mathrm{\prime }},v_3)$. We note that $S^{ϵ}$ is invertible with $(S^{ϵ}{)}^{-1}=S^{1/ϵ}$. Using a simple change of variables we show that $S^{ϵ}$ maps $L^p\left(\mathrm{\Omega }\right)$ onto $L^p\left({\mathrm{\Omega }}^{ϵ}\right)$ with $\frac{1}{|{\mathrm{\Omega }}^{ϵ}{|}^{1/p}}\parallel S^{ϵ}f{\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}^p=\frac{1}{|\mathrm{\Omega }{|}^{1/p}}\parallel f{\parallel }_{L^p\left(\mathrm{\Omega }\right)}^p.$Moreover, $S^{ϵ}$ maps $W^p\left(\mathrm{\Omega }\right)$ onto $W^p\left({\mathrm{\Omega }}^{ϵ}\right)$ such that $\mathrm{d}\mathrm{i}\mathrm{v}\left(S^{ϵ}v\right)=S^{ϵ}\left(\mathrm{d}\mathrm{i}\mathrm{v}v\right)\mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}.$In particular, we define $S^{ϵ}$ acting on $\left\{v\right\}\in W^p\left(\mathrm{\Omega }\right)/V^p\left(\mathrm{\Omega }\right)$, as $S^{ϵ}\left\{v\right\}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\left\{S^{ϵ}v\right\}.$It is readily checked that $S^{ϵ}\left\{v\right\}$ is well defined in $W^p\left(\mathrm{\Omega }\right)/V^p\left(\mathrm{\Omega }\right)$.

Lemma A.3

Let $B:L^p\left(\mathrm{\Omega }\right)\to W^p\left(\mathrm{\Omega }\right)/V^p\left(\mathrm{\Omega }\right),B^{ϵ}:L^p\left({\mathrm{\Omega }}^{ϵ}\right)\to W^p\left({\mathrm{\Omega }}^{ϵ}\right)/V^p\left({\mathrm{\Omega }}^{ϵ}\right)$denote the Bogovski operators for Ω and ${\mathrm{\Omega }}^{ϵ}$ respectively. Then (A5) $\begin{array}{rl}{S}^{ϵ}B(S^{ϵ}{)}^{-1}f& =B^{ϵ}f\mathrm{\forall }f\in L^p\left({\mathrm{\Omega }}^{ϵ}\right)\end{array}$(A5) (A6) $\begin{array}{rl}(S^{ϵ}{)}^{-1}{B}^{ϵ}{S}^{ϵ}f& =Bf\mathrm{\forall }f\in L^p\left(\mathrm{\Omega }\right).\end{array}$(A6)

Proof.

The statements (A5(A5) $\begin{array}{rl}{S}^{ϵ}B(S^{ϵ}{)}^{-1}f& =B^{ϵ}f\mathrm{\forall }f\in L^p\left({\mathrm{\Omega }}^{ϵ}\right)\end{array}$(A5) ) and (A6(A6) $\begin{array}{rl}(S^{ϵ}{)}^{-1}{B}^{ϵ}{S}^{ϵ}f& =Bf\mathrm{\forall }f\in L^p\left(\mathrm{\Omega }\right).\end{array}$(A6) ) are equivalent, so it suffices to prove the first one. Given $f\in L^p\left({\mathrm{\Omega }}^{ϵ}\right)$, suppose that u in $W^p\left({\mathrm{\Omega }}^{ϵ}\right)$ satisfies $\mathrm{d}\mathrm{i}\mathrm{v}u=f\mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}⟺\left\{u\right\}=B^{ϵ}f.$Then $(S^{ϵ}{)}^{-1}u\in W^p(\mathrm{\Omega })$ satisfies $\mathrm{d}\mathrm{i}\mathrm{v}\left(\right(S^{ϵ}{)}^{-1}u)=(S^{ϵ}{)}^{-1}\left(\mathrm{d}\mathrm{i}\mathrm{v}u\right)=(S^{ϵ}{)}^{-1}f⟺(S^{ϵ}{)}^{-1}\left\{u\right\}=B(S^{ϵ}{)}^{-1}f.$Inverting, the last relation we see that $B^{ϵ}f=\left\{u\right\}=S^{ϵ}B(S^{ϵ}{)}^{-1}f.\text{■}$

Lemma A.4

For all v in $W^p\left(\mathrm{\Omega }\right)$ and all $0<ϵ\le 1$ we have $\frac{1}{|{\mathrm{\Omega }}^{ϵ}{|}^{1/p}}\parallel S^{ϵ}\left\{v\right\}{\parallel }_{W^p\left({\mathrm{\Omega }}^{ϵ}\right)/V^p\left({\mathrm{\Omega }}^{ϵ}\right)}\le {ϵ}^{-1}\frac{K_2}{|\mathrm{\Omega }{|}^{1/p}}\parallel \left\{v\right\}{\parallel }_{W^p\left(\mathrm{\Omega }\right)/V^p\left(\mathrm{\Omega }\right)},$where $K_2$ is the constant in Theorem 2.1.

Proof.

From $\mathrm{\nabla }\left(S^{ϵ}v\right)=\left(\begin{array}{ll}{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{v}^{\mathrm{\prime }}& {ϵ}^{-1}\frac{\mathrm{\partial }{v}^{\mathrm{\prime }}}{\mathrm{\partial }y}\\ ϵ{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{v}_3& \frac{\mathrm{\partial }{v}_3}{\mathrm{\partial }y}\end{array}\right)(x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3),$we obtain $|\mathrm{\nabla }\left(S^{ϵ}v\right)\left(x\right){|}^p\le {ϵ}^{-p}|\mathrm{\nabla }v(x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3){|}^p.$Integrating this inequality over ${\mathrm{\Omega }}^{ϵ}$ gives ${\int }_{{\mathrm{\Omega }}^{ϵ}}|\mathrm{\nabla }\left(S^{ϵ}v\right)\left(x\right){|}^p\mathrm{d}x\le {ϵ}^{-p}{\int }_{{\mathrm{\Omega }}^{ϵ}}|\mathrm{\nabla }v(x^{\mathrm{\prime }},{ϵ}^{-1}{x}_3){|}^p\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}{x}_3={ϵ}^{-p}\frac{|{\mathrm{\Omega }}^{ϵ}|}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }v(x^{\mathrm{\prime }},y){|}^p\mathrm{d}{x}^{\mathrm{\prime }}\mathrm{d}y.$By the Korn inequality (7(7) $\begin{array}{r}||\mathrm{\nabla }v|{|}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\le K_2||e\left(\mathrm{\nabla }v\right)|{|}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)},\end{array}$(7) ) and some rearrangement, we take infimum on both sides, the lemma is proved.

Proof

Proof of Theorem 2.2

Assume $f\in L^p\left({\mathrm{\Omega }}^{ϵ}\right)$. From Lemma A.3 and Lemma A.4 we deduce, $\begin{array}{rl}\frac{1}{|{\mathrm{\Omega }}^{ϵ}{|}^{1/p}}\parallel B^{ϵ}f{\parallel }_{W^p\left({\mathrm{\Omega }}^{ϵ}\right)/V^p\left({\mathrm{\Omega }}^{ϵ}\right)}& =\frac{1}{|{\mathrm{\Omega }}^{ϵ}{|}^{1/p}}\parallel S^{ϵ}B(S^{ϵ}{)}^{-1}f{\parallel }_{W^p\left({\mathrm{\Omega }}^{ϵ}\right)/V^p\left({\mathrm{\Omega }}^{ϵ}\right)}\\ & \le {ϵ}^{-1}\frac{K_2}{|\mathrm{\Omega }{|}^{1/p}}\parallel B(S^{ϵ}{)}^{-1}f{\parallel }_{W^p\left(\mathrm{\Omega }\right)/V^p\left(\mathrm{\Omega }\right)}\le {ϵ}^{-1}\parallel B\parallel \frac{K_2}{|\mathrm{\Omega }{|}^{1/p}}\parallel (S^{ϵ}{)}^{-1}f{\parallel }_{W^p\left(\mathrm{\Omega }\right)/V^p\left(\mathrm{\Omega }\right)}\\ & ={ϵ}^{-1}\parallel B\parallel \frac{K_2}{|{\mathrm{\Omega }}^{ϵ}{|}^{1/p}}\parallel f{\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)},\end{array}$which implies the upper bound in (8(8) $C_1\le \underset{0<ϵ\le 1}{inf}ϵ\parallel B_{{\mathrm{\Omega }}^{ϵ}}\parallel \le \underset{0<ϵ\le 1}{sup}ϵ\parallel B_{{\mathrm{\Omega }}^{ϵ}}\parallel \le C_2.$(8) ) $\parallel B_{{\mathrm{\Omega }}^{ϵ}}\parallel \le K_2{ϵ}^{-1}\parallel B_{\mathrm{\Omega }}\parallel .$To prove the lower bound, we consider the eigenfunction ψ in $W_0^{1,p}\left(\omega \right)$ corresponding to the eigenvalue problem $\left\{\begin{array}{ll}-{\mathrm{\Delta }}_p\psi =\lambda |\psi {|}^{p-2}\psi & \mathrm{i}\mathrm{n}\omega \\ \psi =0& \mathrm{o}\mathrm{n}\mathrm{\partial }\omega .\end{array}\right.$where $-{\mathrm{\Delta }}_p=\mathrm{d}\mathrm{i}\mathrm{v}\left(|\mathrm{\nabla }\psi {|}^{p-2}\mathrm{\nabla }\psi \right)$, and λ is the smallest eigenvalue of $-{\mathrm{\Delta }}_p$ on ω. It is well know that λ can be characterized as the minimum of the Rayleigh quotient $\lambda =\underset{\psi }{inf}\frac{\parallel \mathrm{\nabla }\psi {\parallel }_{L^p\left(\omega \right)}}{\parallel \psi {\parallel }_{L^p\left(\omega \right)}},$see [32] for more details. Now suppose u in $W^p\left({\mathrm{\Omega }}^{ϵ}\right)$ satisfies $\mathrm{d}\mathrm{i}\mathrm{v}u=\psi \mathrm{i}\mathrm{n}{\mathrm{\Omega }}^{ϵ}.$Since u = 0 on ${\mathrm{\Gamma }}_D^{ϵ}$ and $\psi =0$ on ${\mathrm{\Gamma }}_N^{ϵ}$ we have (A7) $\begin{array}{rl}{\int }_{{\mathrm{\Omega }}^{ϵ}}|\psi {|}^p\mathrm{d}x& ={\int }_{{\mathrm{\Omega }}^{ϵ}}\left(|\psi {|}^{p-2}\psi \right)\mathrm{d}\mathrm{i}\mathrm{v}u\mathrm{d}x=-{\int }_{{\mathrm{\Omega }}^{ϵ}}\mathrm{\nabla }\left(|\psi {|}^{p-2}\psi \right)\cdot u\mathrm{d}x\\ & \le {\left({\int }_{{\mathrm{\Omega }}^{ϵ}}|\mathrm{\nabla }\left(|\psi {|}^{p-2}\psi \right){|}^{p^{\mathrm{\prime }}}\mathrm{d}x\right)}^{1/p^{\mathrm{\prime }}}{\left({\int }_{{\mathrm{\Omega }}^{ϵ}}|u{|}^p\mathrm{d}x\right)}^{1/p}.\end{array}$(A7) Using ${\int }_{{\mathrm{\Omega }}^{ϵ}}|\mathrm{\nabla }\left(|\psi {|}^{p-2}\psi \right){|}^{p^{\mathrm{\prime }}}\mathrm{d}x=\lambda {\int }_{{\mathrm{\Omega }}^{ϵ}}|\psi {|}^p\mathrm{d}x,$and applying the Korn inequality (6(6) $\begin{array}{r}||v|{|}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\le ϵK_1||e\left(\mathrm{\nabla }v\right)|{|}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\end{array}$(6) ) in the last integral on right hand side of (A7(A7) $\begin{array}{rl}{\int }_{{\mathrm{\Omega }}^{ϵ}}|\psi {|}^p\mathrm{d}x& ={\int }_{{\mathrm{\Omega }}^{ϵ}}\left(|\psi {|}^{p-2}\psi \right)\mathrm{d}\mathrm{i}\mathrm{v}u\mathrm{d}x=-{\int }_{{\mathrm{\Omega }}^{ϵ}}\mathrm{\nabla }\left(|\psi {|}^{p-2}\psi \right)\cdot u\mathrm{d}x\\ & \le {\left({\int }_{{\mathrm{\Omega }}^{ϵ}}|\mathrm{\nabla }\left(|\psi {|}^{p-2}\psi \right){|}^{p^{\mathrm{\prime }}}\mathrm{d}x\right)}^{1/p^{\mathrm{\prime }}}{\left({\int }_{{\mathrm{\Omega }}^{ϵ}}|u{|}^p\mathrm{d}x\right)}^{1/p}.\end{array}$(A7) ) we deduce that ${\left({\int }_{{\mathrm{\Omega }}^{ϵ}}|\psi {|}^p\mathrm{d}x\right)}^{1/p}\le K_1ϵ{\lambda }^{1/p^{\mathrm{\prime }}}{\left({\int }_{{\mathrm{\Omega }}^{ϵ}}|e\left(\mathrm{\nabla }u\right){|}^p\mathrm{d}x\right)}^{1/p},$for all $u\in W^p\left({\mathrm{\Omega }}^{ϵ}\right)$ such that $\mathrm{d}\mathrm{i}\mathrm{v}u=\psi$. Consequently, taking the infimum on the last integral we obtain $\parallel \psi {\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\le K_1ϵ{\lambda }^{1/p^{\mathrm{\prime }}}\parallel \left\{u\right\}{\parallel }_{W^p\left({\mathrm{\Omega }}^{ϵ}\right)/V^p\left({\mathrm{\Omega }}^{ϵ}\right)}.$Estimating the right hand side with the norm of the Bogovski operator, we see that $\parallel \left\{u\right\}{\parallel }_{W^p\left({\mathrm{\Omega }}^{ϵ}\right)/V^p\left({\mathrm{\Omega }}^{ϵ}\right)}=\parallel B_{{\mathrm{\Omega }}^{ϵ}}\psi {\parallel }_{W^p\left({\mathrm{\Omega }}^{ϵ}\right)/V^p\left({\mathrm{\Omega }}^{ϵ}\right)}\le \parallel B_{{\mathrm{\Omega }}^{ϵ}}\parallel \parallel \psi {\parallel }_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)},$whence, we deduce, $\frac{1}{K_1{\lambda }^{1/p^{\mathrm{\prime }}}}\cdot {ϵ}^{-1}\le \parallel B_{{\mathrm{\Omega }}^{ϵ}}\parallel .$This proves the lower bound in (8(8) $C_1\le \underset{0<ϵ\le 1}{inf}ϵ\parallel B_{{\mathrm{\Omega }}^{ϵ}}\parallel \le \underset{0<ϵ\le 1}{sup}ϵ\parallel B_{{\mathrm{\Omega }}^{ϵ}}\parallel \le C_2.$(8) ) with $C_1=\frac{1}{K_1{\lambda }^{1/p^{\mathrm{\prime }}}}.\text{■}$

Appendix 3. Proof of Lemma 3.1

Proof

Proof of Lemma 3.1

For fixed $ϵ>0$, consider the minimization problem $\left\{\begin{array}{l}J\left(u^{ϵ}\right)=\underset{v\in V^p\left({\mathrm{\Omega }}^e\right)}{min}J\left(v\right)\\ J\left(v\right)={\int }_{{\mathrm{\Omega }}^{ϵ}}\left(\frac{2{\mu }_0}{p}{∣e\left(\mathrm{\nabla }v\right)∣}^p-f\cdot v\right)\mathrm{d}x-{\int }_{{\mathrm{\Gamma }}_N^{ϵ}}g\cdot v\mathrm{d}S\end{array}\right.$

(1) Let us begin to show that $J\left(v\right)$ is weakly coercive on $W^{1,p}\left({\mathrm{\Omega }}^{ϵ}\right)$. By the Young inequality with $x\cdot y\le \frac{r^{-p^{\mathrm{\prime }}}}{p^{\mathrm{\prime }}}|x{|}^{p^{\mathrm{\prime }}}+\frac{r^p}{p}|y{|}^p,$for appropriate r>0, the Trace theorem (see [23, p. 153, Sec. 2.5]) and the Korn inequality (6(6) $\begin{array}{r}||v|{|}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\le ϵK_1||e\left(\mathrm{\nabla }v\right)|{|}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\end{array}$(6) ), we obtain $J\left(v\right)\ge {\int }_{{\mathrm{\Omega }}^{ϵ}}\left(2{\mu }_0-r^p\frac{C+1}{ϵK_1}\right)\frac{|e\left(\mathrm{\nabla }v\right){|}^p}{p}\mathrm{d}x-\frac{r^{-p^{\mathrm{\prime }}}}{p^{\mathrm{\prime }}}\left({\int }_{{\mathrm{\Omega }}^{ϵ}}{∣f∣}^{p^{\mathrm{\prime }}}\mathrm{d}x+{\int }_{{\mathrm{\Gamma }}_N^{ϵ}}{∣g∣}^{p^{\mathrm{\prime }}}\mathrm{d}x\right),$where C is a constant from the Trace inequality in the Trace theorem. We choose $r^p=ϵK_1{\mu }_0/(C+1)$, to obtain that $J\left(v\right)\ge \frac{{\mu }_0}{p}||v|{|}_{V^p\left({\mathrm{\Omega }}^{ϵ}\right)}^p-\frac{K}{p^{\mathrm{\prime }}}\left(||f|{|}_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Omega }}^{ϵ}\right)}^{p^{\mathrm{\prime }}}+||g|{|}_{L^{p^{\mathrm{\prime }}}\left({\mathrm{\Gamma }}_N^{ϵ}\right)}^{p^{\mathrm{\prime }}}\right).$where K is a positive constant which depends on ${\mathrm{\Omega }}^{ϵ}$. Hence, $J\left(v\right)\to \mathrm{\infty }$ if $||v|{|}_{V^p\left({\mathrm{\Omega }}^{ϵ}\right)}\to \mathrm{\infty }$, this proves the weak coercivity of J.

(2) Choose any minimizing sequence $\{u_k{\}}_{k=1}^{\mathrm{\infty }}$ such that $\underset{k\to \mathrm{\infty }}{lim}J\left(u_k\right)=\underset{v\in V^p\left({\mathrm{\Omega }}^{ϵ}\right)}{inf}J\left(v\right).$Then the weak coercivity condition implies that $\{u_k{\}}_{k=1}^{\mathrm{\infty }}$ is bounded in $W^{1,p}\left({\mathrm{\Omega }}^{ϵ}\right)$, $1<p<\mathrm{\infty }$, and since, $W^{1,p}\left({\mathrm{\Omega }}^{ϵ}\right)$ is reflexive there exist a subsequence $\{u_{k_j}{\}}_{j=1}^{\mathrm{\infty }}$ and a function $u^{ϵ}$ in $W^{1,p}\left({\mathrm{\Omega }}^{ϵ}\right)$ such that $u_{k_j}\rightharpoonup u^{ϵ}\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}\mathrm{l}\mathrm{y}\mathrm{i}\mathrm{n}{W}^{1,p}\left({\mathrm{\Omega }}^{ϵ}\right).$Since $V^p\left({\mathrm{\Omega }}^{ϵ}\right)$ is a closed subspace of $W^{1,p}\left({\mathrm{\Omega }}^{ϵ}\right)$, then $V^p\left({\mathrm{\Omega }}^{ϵ}\right)$ is weakly closed i.e. $u_{k_j}\rightharpoonup u^{ϵ}\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}\mathrm{l}\mathrm{y}\mathrm{i}\mathrm{n}{V}^p\left({\mathrm{\Omega }}^{ϵ}\right),$and so $u^{ϵ}\in V^p\left({\mathrm{\Omega }}^{ϵ}\right).$

(3) The weak lower semi continuity of J implies that $J\left(u^{ϵ}\right)\le \underset{j\to \mathrm{\infty }}{lim}infJ\left(u_{k_j}\right)=\underset{v\in V^p\left({\mathrm{\Omega }}^{ϵ}\right)}{inf}J\left(v\right),$therefore $u^{ϵ}$ is a minimizer of J over $V^p\left({\mathrm{\Omega }}^{ϵ}\right)$.

(4) We turn to deriving the corresponding Euler–Lagrange equation. Let $u^{ϵ}$ in $V^p\left({\mathrm{\Omega }}^{ϵ}\right)$ be a minimizer of J and pick $v\in V^p\left({\mathrm{\Omega }}^{ϵ}\right)$. Then $u^{ϵ}+tv$ belongs the admissible class $V^p\left({\mathrm{\Omega }}^{ϵ}\right)$ for all $t\in \mathbb{R}$. Let $j\left(t\right)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}J(u^{ϵ}+tv)$, then (A8) $\frac{j\left(t\right)-j\left(0\right)}{t}={\int }_{{\mathrm{\Omega }}^{ϵ}}\left({\mathcal{L}}^t\left(x\right)-f\cdot v\right)\mathrm{d}x-{\int }_{{\mathrm{\Gamma }}_N^{ϵ}}g\cdot v\mathrm{d}S,$(A8) where, ${\mathcal{L}}^t\left(x\right)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{2{\mu }_0}{p}\frac{{∣e(\mathrm{\nabla }{u}^{ϵ}+t\mathrm{\nabla }v)∣}^p-{∣e\left(\mathrm{\nabla }{u}^{ϵ}\right)∣}^p}{t}.$Note that, ${\mathcal{L}}^t\left(x\right)\to 2{\mu }_0{∣e\left(\mathrm{\nabla }{u}^{ϵ}\right)∣}^{p-2}e\left(\mathrm{\nabla }{u}^{ϵ}\right):e\left(\mathrm{\nabla }v\right)\mathrm{a}\mathrm{s}t\to 0.$Consequently, by Dominated Convergence theorem, the limit (A8(A8) $\frac{j\left(t\right)-j\left(0\right)}{t}={\int }_{{\mathrm{\Omega }}^{ϵ}}\left({\mathcal{L}}^t\left(x\right)-f\cdot v\right)\mathrm{d}x-{\int }_{{\mathrm{\Gamma }}_N^{ϵ}}g\cdot v\mathrm{d}S,$(A8) ) exist as $t\to 0$, and thus, $j^{\mathrm{\prime }}\left(0\right)={\int }_{{\mathrm{\Omega }}^{ϵ}}\left(2{\mu }_0{∣e\left(\mathrm{\nabla }{u}^{ϵ}\right)∣}^{p-2}e\right(\mathrm{\nabla }{u}^{ϵ}):e(\mathrm{\nabla }v)-f\cdot v)\mathrm{d}x-{\int }_{{\mathrm{\Gamma }}_N^{ϵ}}g\cdot v\mathrm{d}S.$Since $j\left(0\right)\le j\left(t\right)$ for all t it must hold that $j^{\mathrm{\prime }}\left(0\right)=0$, hence the minimizer $u^{ϵ}$ satisfies the weak form of the Euler–Lagrange equation ${\int }_{{\mathrm{\Omega }}^{ϵ}}2{\mu }_0{∣e\left(\mathrm{\nabla }{u}^{ϵ}\right)∣}^{p-2}e\left(\mathrm{\nabla }{u}^{ϵ}\right):e\left(\mathrm{\nabla }v\right)\mathrm{d}x={\int }_{{\mathrm{\Omega }}^{ϵ}}f\cdot v\mathrm{d}x+{\int }_{{\mathrm{\Gamma }}_N^{ϵ}}g\cdot v\mathrm{d}S.$( 5) Finally, let us prove the uniqueness of the minimizer $u^{ϵ}$. Let $u_1^{ϵ}$ and $u_2^{ϵ}$ be two different minimizers, then by Korn's inequality (6(6) $\begin{array}{r}||v|{|}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\le ϵK_1||e\left(\mathrm{\nabla }v\right)|{|}_{L^p\left({\mathrm{\Omega }}^{ϵ}\right)}\end{array}$(6) ) we have $e\left(\mathrm{\nabla }{u}_1^{ϵ}\right)\ne e\left(\mathrm{\nabla }{u}_2^{ϵ}\right)$ on a set of positive measure. Since J is strictly convex it follows that $J\left(\frac{1}{2}{u}_1^{ϵ}+\frac{1}{2}{u}_2^{ϵ}\right)<\frac{1}{2}J\left(u_1^{ϵ}\right)+\frac{1}{2}J\left(u_2^{ϵ}\right)=J\left(u_1^{ϵ}\right),$which contradicts that $u_1^{ϵ}$ is a minimizer, hence $u_1^{ϵ}=u_2^{ϵ}$.

Appendix 4. Characterization of the normalized limit pressure ${\pi }^0$

Following [33, p. 153. Proposition IX.3], we present a characterization of the subspace of $W^{1,p^{\mathrm{\prime }}}\left(\omega \right)$ to which the normalized limit pressure ${\pi }^0$ belongs.

Lemma A.5

The following statements are equivalent

1. ${\pi }^0\in L^{p^{\mathrm{\prime }}}\left(\omega \right)$ and there exists a constant $\tilde{C}$ such that $|〈{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{\pi }^0,\mathrm{\Theta }〉|\le \tilde{C}\parallel \mathrm{\Theta }{\parallel }_{L^p\left(\omega \right)},$for all $\mathrm{\Theta }\in W^{1,p}(\omega ;{\mathbb{R}}^2)$ such that $\mathrm{\Theta }\cdot \hat{n}=0$ on ${\gamma }_D$;

2. ${\pi }^0\in W^{1,p^{\mathrm{\prime }}}\left(\omega \right)$ with ${\pi }^0=0$ on ${\gamma }_N$.

Proof.

If ${\pi }^0\in W^{1,p^{\mathrm{\prime }}}\left(\omega \right)$ with ${\pi }^0=0$ on ${\gamma }_N$, then by the Divergence theorem we have that ${\int }_{\mathrm{\partial }\omega }{\pi }^0\mathrm{\Theta }\cdot \hat{n}\mathrm{d}S={\int }_{\omega }({\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{\pi }^0\cdot \mathrm{\Theta }+{\pi }^0\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{\Theta })\mathrm{d}{x}^{\mathrm{\prime }},$for all $\mathrm{\Theta }\in W^{1,p}(\omega ;{\mathbb{R}}^2)$ such that $\mathrm{\Theta }\cdot \hat{n}=0$ on ${\gamma }_D$. And therefore $-{\int }_{\omega }{\pi }^0\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{\Theta }\mathrm{d}{x}^{\mathrm{\prime }}={\int }_{\omega }{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{\pi }^0\cdot \mathrm{\Theta }\mathrm{d}{x}^{\mathrm{\prime }}.$Hence, $|〈{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{\pi }^0,\mathrm{\Theta }〉|\le \tilde{C}\parallel \mathrm{\Theta }{\parallel }_{L^p\left(\omega \right)},$where $\tilde{C}=\parallel {\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{\pi }^0{\parallel }_{L^{p^{\mathrm{\prime }}}\left(\omega \right)}$. By density [22, Lemma 2] this holds for all $\mathrm{\Theta }\in L^p(\omega ;{\mathbb{R}}^2)$ such that $\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{\Theta }\in L^p\left(\omega \right)$ and $\mathrm{\Theta }\cdot \hat{n}=0$ on ${\gamma }_D$.

Conversely, assuming that ${\pi }^0\in L^{p^{\mathrm{\prime }}}\left(\omega \right)$, we can define the gradient of ${\pi }^0$ as the linear functional ${\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{\pi }^0\in W^{-1,p^{\mathrm{\prime }}}(\omega ;{\mathbb{R}}^2)$, given by $〈{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{\pi }^0,\mathrm{\Theta }〉=-{\int }_{\omega }{\pi }^0\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{\Theta }\mathrm{d}{x}^{\mathrm{\prime }},$defined on the subspace $\{\mathrm{\Theta }\in W^{1,p}(\omega ;{\mathbb{R}}^2):\mathrm{\Theta }\cdot \hat{n}=0\mathrm{o}\mathrm{n}{\gamma }_D\}\subset L^p(\omega ;{\mathbb{R}}^2)$. By assumption $|〈{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{\pi }^0,\mathrm{\Theta }〉|\le \tilde{C}\parallel \mathrm{\Theta }{\parallel }_{L^p\left(\omega \right)},$so, ${\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{\pi }^0$ is bounded. According to the Hahn–Banach theorem the ${\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{\pi }^0$ can be extended to a bounded linear functional on $L^p(\omega ;{\mathbb{R}}^2)$. Thus by the Riesz representation theorem, there exists $g\in L^{p^{\mathrm{\prime }}}(\omega ,{\mathbb{R}}^2)$ such that $〈{\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{\pi }^0,\mathrm{\Theta }〉={\int }_{\omega }g\cdot \mathrm{\Theta }\mathrm{d}{x}^{\mathrm{\prime }},\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{\Theta }\in L^p(\omega ;{\mathbb{R}}^2).$Therefore, $-{\int }_{\omega }{\pi }^0\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{\Theta }\mathrm{d}{x}^{\mathrm{\prime }}={\int }_{\omega }g\cdot \mathrm{\Theta }\mathrm{d}{x}^{\mathrm{\prime }},$for all $\mathrm{\Theta }\in W^{1,p}(\omega ;{\mathbb{R}}^2)$ such that $\mathrm{\Theta }\cdot \hat{n}=0$ on ${\gamma }_D$. This implies that $g=\mathrm{\nabla }{\pi }^0\in L^{p^{\mathrm{\prime }}}\left(\omega \right)$ and therefore ${\pi }^0\in W^{1,p^{\mathrm{\prime }}}\left(\omega \right)$. Now by Divergence theorem we have ${\int }_{{\gamma }_N}{\pi }^0\mathrm{\Theta }\cdot \hat{n}\mathrm{d}S={\int }_{\omega }({\mathrm{\nabla }}_{x^{\mathrm{\prime }}}{\pi }^0\cdot \mathrm{\Theta }+{\pi }^0\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{\Theta })\mathrm{d}{x}^{\mathrm{\prime }}=0,$for all $\mathrm{\Theta }\in W^{1,p}(\omega ;{\mathbb{R}}^2)$ such that $\mathrm{\Theta }\cdot \hat{n}=0$ on ${\gamma }_D$. We conclude that ${\pi }^0\in W^{1,p^{\mathrm{\prime }}}\left(\omega \right)$ with ${\pi }^0=0$ on ${\gamma }_N$.