Periodic unfolding for anisotropically bounded sequences

Abstract

This paper is focused on the asymptotic behavior of sequences of functions, whose partial derivatives estimates in one or more directions are highly contrasted with respect to the periodic parameter ε. In particular, a direct application for the homogenization of a homogeneous Dirichlet problem defined on an anisotropic structure is presented. In general, the obtained results can be applied to thin structures where the behavior is different according to the observed direction.

Keywords

Periodic unfolding method homogenization anisotropic Sobolev spaces Dirichlet problem

1. Introduction

The periodic unfolding method has been presented the first time by D. Cioranescu and al. in [2], with further development in [3,4,7,8] and extensively in [5]. Such method, as an equivalent to the two-scale convergence, has largely found application in the homogenization of periodically perforated domains (see e.g. [1,6,9–12,23]) and of thin structures with periodic pattern, like periodically perforated shells (see [15]), textiles made of long woven beams in strong contact (see [21,22]) and 3D lattice structures made of either beams or segments (see [13,17,18]) in a stable configuration.

In all these cases, the classical unfolding theory can be directly applied: given a bounded domain $Ω \subset R^{N}$ with Lipschitz boundary and periodically paved with cells of size ε, a small parameter, the structures models (and the displacements over them) can be represented by sequences ${ϕ_{ε}}_{ε} \subset W^{1, p} (Ω)$ such that either $\begin{matrix} ‖ ϕ_{ε} ‖_{L^{p} (Ω)} + ε ‖ \nabla ϕ_{ε} ‖_{L^{p} (Ω)} ⩽ C or ‖ ϕ_{ε} ‖_{L^{p} (Ω)} + ‖ \nabla ϕ_{ε} ‖_{L^{p} (Ω)} ⩽ C, \end{matrix}$ whose asymptotic behavior is investigated in detail in [5, Section 1.4].

However, recent application of the unfolding method to a wider set of structure models that involve more complex setting in order to describe more phenomena and thus be closer to reality, finds its limitation in the investigation by the classical unfolding theorems. One of the encountered limitations concerns the anisotropic behavior of the sequences describing the structure model, i.e. the fact that these sequences present better estimates in some privileged directions with respect to others.

We can mention two representative examples: in the continuation of the work made in [22], the lack of strong contact between fibers and the partial clamp on a textile made of long beams leads to different gradient estimates for the displacement fields with respect to the observed direction (see Fig. 1 left), while in the extension of the works made in [18] to unstable lattice configurations, the same contrast on the gradient estimates appear on the unstable oscillating thin straits (see Fig. 1 right).

Fig. 1.

On the left, a sketch of the textile analyzed in [14]: as one can notice, in the unsupported part in $Ω_{2}$ the yarns in direction $e_{1}$ and $e_{2}$ behave differently. On the right, an unstable lattice structure analyzed in [16]: in many of these structures, the sequences describing the displacement have privileged directions.

In this context, an extension of the know unfolding theorems already developed in [2,4,5] to this new classes of functions, so-called “anisotropically bounded”, was needed to furnish the rigorous mathematical tools in order to apply the periodic unfolding to these types of structures. This is exactly the motivation and part of the purpose of this paper.

Following closely the classical results achieved in [5, Section 1.4], after presenting some standard notation for the periodic unfolding method in Section 2, in Section 3 we consider a decomposition of the Euclidean space $R^{N}$ by setting $x = (x^{'}, x^{″})$ , where the variable $x^{'}$ corresponds to the first $N_{1} < N$ directions and from [5, Chap. 7], we recall the unfolding with parameters and prove the main properties. The core of our work is contained in Section 4, where we show the asymptotic behavior of these anisoptropically bounded sequences ${ϕ_{ε}}_{ε}$ in the following cases:

$‖ ϕ_{ε} ‖_{L^{p} (Ω)} + ε ‖ \nabla_{x^{'}} ϕ_{ε} ‖_{L^{p} (Ω)} ⩽ C$ ;

$‖ ϕ_{ε} ‖_{L^{p} (Ω)} + ‖ \nabla_{x^{'}} ϕ_{ε} ‖_{L^{p} (Ω)} ⩽ C$ ;

$‖ ϕ_{ε} ‖_{L^{p} (Ω)} + ‖ \nabla_{x^{'}} ϕ_{ε} ‖_{L^{p} (Ω)} + ε ‖ \nabla_{x^{″}} ϕ_{ε} ‖_{L^{p} (Ω)} ⩽ C$ ;

$‖ ϕ_{ε} ‖_{L^{p} (Ω)} + ‖ \nabla_{x^{'}} ϕ_{ε} ‖_{L^{p} (Ω)} + ε ‖ \nabla_{x^{″}} (\nabla_{x^{'}} ϕ_{ε}) ‖_{L^{p} (Ω)} ⩽ C$ .

These results will later be used in Section 5 to extend some more complex lemmas concerning the periodic unfolding (see for an instance [5, Lemma 11.11]) to this new classes of functions, showing that different anisotropic assumptions arise different regularity of the limit fields. In the last Section 6, a direct application of the developed tools for the homogenization via unfolding of the following homogeneous Dirichlet problem

\begin{matrix} (1.1) & \{\begin{matrix} Find u_{ε} \in H_{0}^{1} (Ω) such that: \\ \int_{Ω} A_{ε} [\begin{matrix} \nabla_{x^{'}} u_{ε} \\ ε \nabla_{x^{″}} u_{ε} \end{matrix}] \cdot [\begin{matrix} \nabla_{x^{'}} ϕ \\ ε \nabla_{x^{″}} ϕ \end{matrix}] d x = \int_{Ω} f ϕ d x, \forall ϕ \in H_{0}^{1} (Ω), \end{matrix} \end{matrix}

whose nature is anisotropic. Existence and uniqueness of the limit problem are proven, the cell problems and the macroscopic limit problem are found.

The tools presented in this paper serves of multiple purposes.

In first place, the are the breakthrough for the investigation of above mentioned structures: in [14], the results are largely employed in [14, Section 6], where we deal with asymptotic behavior of sequences belonging to subspaces of $L^{2} {(Ω, \partial_{1})}^{2} \times L^{2} {(Ω, \partial_{2})}^{2}$ and need an equivalent anisotropic formulation of [5, Lemma 11.11] (already applied in the same fashion of [15,22]). In [16], further adaptation of the results made in this paper to lattices (rigorously made in [13]) provided the sufficient ingredients to find the asymptotic behavior of sequences that model such unstable structures (one can look at problem [16, Eq. (5.1)] for structures of type $S_{0}$ ).

A second important result is given by the investigated problem. It is a representative example of which results can be expected by the homogenization of anisotropic structures (one can also find its similar formulation in for the stretching fields in [16, Eq. (13.1)]) and it can be applied to partial differential equations on domains involving periodic grids, lattices, thin frames and fiber structures.

Moreover, even though the presented formulation assumes an isotropy of the material and anisotropy of the displacement, the variational problem (1.1) can be reformulated by shifting the anisotropy to the material law: $\begin{matrix} (1.2) & \{\begin{matrix} Find u_{ε} \in H_{0}^{1} (Ω) such that: \\ \int_{Ω} (\begin{matrix} A_{1, ε} & ε A_{2, ε} \\ ε A_{3, ε} & ε^{2} A_{4, ε} \end{matrix}) [\nabla u_{ε}] \cdot [\nabla ϕ] d x = \int_{Ω} f ϕ d x, \forall ϕ \in H_{0}^{1} (Ω), \end{matrix} \end{matrix}$ In this sense, the investigation of such problem applies even by starting by a different design, especially related to layered composites with contrast in the coefficients, e.g. made of stiff and soft materials. Among them, we would like to mention [20], where this effect is reached by a layer of thin vertical beams, and [19], where the same peculiarity is obtained by a composite weak layer, i.e., the layer of the stiffness of bulk domains, scaled by ε.

2. Preliminaries and notation

We start by introducing the standard notation for the periodic unfolding method already used in [5, Section 1.4].

Let $R^{N}$ be the Euclidean space with usual basis $(e_{1}, \dots, e_{N})$ and $Y = {(0, 1)}^{N}$ the open unit parallelotope associated with this basis. For a.e. $z \in R^{N}$ , we set the unique decomposition $z = {[z]}_{Y} + {z}_{Y}$ such that $\begin{matrix} {[z]}_{Y} ≐ \sum_{i = 1}^{N} k_{i} e_{i}, k_{i} \in Z^{N} and {z}_{Y} ≐ z - {[z]}_{Y} \in Y . \end{matrix}$ Let ${ε}$ be a sequence of strictly positive parameters going to 0. We scale our paving by ε writing $\begin{matrix} (2.1) & x = ε {[\frac{x}{ε}]}_{Y} + ε {\frac{x}{ε}}_{Y} for a.e. x \in R^{N} . \end{matrix}$ Let now Ω be a bounded domain in $R^{N}$ with Lipschitz boundary. We consider the covering $\begin{matrix} Ξ_{ε} ≐ {ξ \in Z^{N} ∣ ε (ξ + Y) \subset Ω} \end{matrix}$ and set $\begin{matrix} {\hat{Ω}}_{ε} ≐ int {⋃_{ξ \in Ξ_{ε}} ε (ξ + \overline{Y})}, Λ_{ε} ≐ Ω ∖ {\hat{Ω}}_{ε} . \end{matrix}$ We recall the definitions of classical unfolding operator and mean value operator.

Definition 2.1 (see [5, Definition 1.2]).

For every measurable function ϕ on ${\hat{Ω}}_{ε}$ , the unfolding operator $T_{ε}$ is defined as follows: $\begin{matrix} T_{ε} (ϕ) ≐ \{\begin{matrix} ϕ (ε {[\frac{x}{ε}]}_{Y} + ε y) & for a.e. (x, y) \in {\hat{Ω}}_{ε} \times Y, \\ 0 & for a.e. (x, y) \in Λ_{ε} \times Y . \end{matrix} \end{matrix}$

Note that such an operator acts on functions defined in Ω by operating on their restriction to ${\hat{Ω}}_{ε}$ .

Definition 2.2 (see [5, Definition 1.10]).

For every measurable function $\hat{ϕ}$ on $L^{1} (Ω \times Y)$ , the mean value operator $M_{Y}$ is defined as follows: $\begin{matrix} M_{Y} (\hat{ϕ}) (x) ≐ \frac{1}{| Y |} \int_{Y} \hat{ϕ} (x, y) d y, for a.e. x \in Ω . \end{matrix}$

Let $p \in [1, + \infty]$ . From [5, Propositions 1.8 and 1.11], we recall the properties of these operators: $\begin{matrix} {‖ T_{ε} (ϕ) ‖}_{L^{p} (Ω \times Y)} ⩽ | Y |^{\frac{1}{p}} ‖ ϕ ‖_{L^{p} (Ω)} for every ϕ \in L^{p} (Ω), \\ {‖ M_{Y} (\hat{ϕ}) ‖}_{L^{p} (Ω)} ⩽ | Y |^{- \frac{1}{p}} ‖ \hat{ϕ} ‖_{L^{p} (Ω \times Y)} for every \hat{ϕ} \in L^{p} (Ω \times Y) . \end{matrix}$ Since we will deal with Sobolev spaces, we give hereafter some definitions: $\begin{matrix} (2.2) & \begin{matrix} W_{per}^{1, p} (Y) ≐ {ϕ \in W^{1, p} (Y) ∣ ϕ is periodic with respect to y_{i}, i \in {1, \dots, N}}, \\ W_{per, 0}^{1, p} (Y) ≐ {ϕ \in W_{per}^{1, p} (Y) ∣ M_{Y} (ϕ) = 0}, \\ L^{p} (Ω; W^{1, p} (Y)) ≐ {ϕ \in L^{p} (Ω \times Y) ∣ \nabla_{y} ϕ \in L^{p} {(Ω \times Y)}^{N}} . \end{matrix} \end{matrix}$

3. The unfolding with parameters

In this section we prepare the ground for the investigation of the asymptotically bounded functions.

In order to show the contrast in behavior of the anisotropically bounded sequences, we find convenient to set a decomposition of the Euclidean space. Let $(N_{1}, N_{2})$ be in $N \times N^{*}$ and such that $N = N_{1} + N_{2}$ . Denote $\begin{array}{l} R^{N_{1}} = {x^{'} \in R^{N} | x^{'} = \sum_{i = 1}^{N_{1}} x_{i} e_{i}, x_{i} \in R}, \\ R^{N_{2}} = {x^{″} \in R^{N} | x^{″} = \sum_{i = N_{1} + 1}^{N} x_{i} e_{i}, x_{i} \in R}, \\ Y^{'} = {y^{'} \in R^{N} | y^{'} = \sum_{i = 1}^{N_{1}} y_{i} e_{i}, y_{i} \in (0, 1)}, \\ Y^{″} = {y^{″} \in R^{N} | y^{″} = \sum_{i = N_{1} + 1}^{N} y_{i} e_{i}, y_{i} \in (0, 1)} \end{array}$ and $\begin{matrix} Z^{N_{1}} = Z e_{1} \oplus \dots \oplus Z e_{N_{1}}, Z^{N_{2}} = Z e_{N_{1} + 1} \oplus \dots \oplus Z e_{N} . \end{matrix}$ One has $\begin{matrix} R^{N} = R^{N_{1}} \oplus R^{N_{2}}, Y = Y^{'} \oplus Y^{″}, Z^{N} = Z^{N_{1}} \oplus Z^{N_{2}} . \end{matrix}$ For every $x \in R^{N}$ and $y \in Y$ , we write $\begin{matrix} x = x^{'} + x^{″} \in R^{N_{1}} \oplus R^{N_{2}}, y = y^{'} + y^{″} \in Y^{'} \oplus Y^{″} . \end{matrix}$ From now on, however, we find easier to refer to such decomposition with the vectorial notation $\begin{matrix} x = (x^{'}, x^{″}) \in R^{N_{1}} \times R^{N_{2}}, y = (y^{'}, y^{″}) \in Y^{'} \times Y^{″} . \end{matrix}$ Similarly to (2.1), we apply the paving to a.e. $x^{'} \in R^{N_{1}}$ and $x^{″} \in R^{N_{2}}$ setting $\begin{matrix} x^{'} = ε {[\frac{x^{'}}{ε}]}_{Y^{'}} + ε {\frac{x^{'}}{ε}}_{Y^{'}}, with {[\frac{x^{'}}{ε}]}_{Y^{'}} \in Z^{N_{1}}, {\frac{x^{'}}{ε}}_{Y^{'}} \in Y^{'}, \\ x^{″} = ε {[\frac{x^{″}}{ε}]}_{Y^{″}} + ε {\frac{x^{″}}{ε}}_{Y^{″}}, with {[\frac{x^{″}}{ε}]}_{Y^{″}} \in Z^{N_{2}}, {\frac{x^{″}}{ε}}_{Y^{″}} \in Y^{″} . \end{matrix}$ We give now the definition of partial mean value operators.

Definition 3.1.
For every $\hat{ϕ} \in L^{1} (Ω \times Y)$ , the partial mean value operators are defined as follows: $\begin{array}{l} M_{Y^{'}} (\hat{ϕ}) (x, y^{″}) ≐ \frac{1}{| Y^{'} |} \int_{Y^{'}} \hat{ϕ} (x, y^{'}, y^{″}) d y^{'}, for a.e. (x, y^{″}) \in Ω \times Y^{″}, \\ M_{Y^{″}} (\hat{ϕ}) (x, y^{'}) ≐ \frac{1}{| Y^{″} |} \int_{Y^{″}} \hat{ϕ} (x, y^{'}, y^{″}) d y^{″}, for a.e. (x, y^{'}) \in Ω \times Y^{'} . \end{array}$

Denote $\begin{array}{l} L^{p} (Ω, \nabla_{x^{'}}) ≐ {ϕ \in L^{p} (Ω) ∣ \nabla_{x^{'}} ϕ \in L^{p} {(Ω)}^{N_{1}}}, \\ L^{p} (Ω, \nabla_{x^{″}}) ≐ {ϕ \in L^{p} (Ω) ∣ \nabla_{x^{″}} ϕ \in L^{p} {(Ω)}^{N_{2}}}, \\ \begin{matrix} L^{p} (Ω, \nabla_{x^{'}}; W^{1, p} (Y^{″})) ≐ & {\tilde{ϕ} \in L^{p} (Ω \times Y^{″}) ∣ \nabla_{x^{'}} \tilde{ϕ} \in L^{p} {(Ω \times Y^{″})}^{N_{1}}, \\ \nabla_{y^{″}} \tilde{ϕ} \in L^{p} {(Ω \times Y^{″})}^{N_{2}}}, \end{matrix} \\ \begin{matrix} L^{p} (Ω, \nabla_{x^{″}}; W^{1, p} (Y^{'})) ≐ & {\tilde{ϕ} \in L^{p} (Ω \times Y^{'}) ∣ \nabla_{x^{″}} \tilde{ϕ} \in L^{p} {(Ω \times Y^{'})}^{N_{2}}, \\ \nabla_{y^{'}} \tilde{ϕ} \in L^{p} {(Ω \times Y^{'})}^{N_{1}}}, \end{matrix} \\ L^{p} (Ω \times Y^{″}; W^{1, p} (Y^{'})) ≐ {\hat{ϕ} \in L^{p} (Ω \times Y) ∣ \nabla_{y^{'}} \hat{ϕ} \in L^{p} {(Ω \times Y)}^{N_{1}}}, \\ L^{p} (Ω \times Y^{'}; W^{1, p} (Y^{″})) ≐ {\hat{ϕ} \in L^{p} (Ω \times Y) ∣ \nabla_{y^{″}} \hat{ϕ} \in L^{p} {(Ω \times Y)}^{N_{2}}} . \end{array}$ We endow these spaces with the respective norms: $\begin{array}{l} ‖ \cdot ‖_{L^{p} (Ω, \nabla_{x^{'}})} ≐ ‖ \cdot ‖_{L^{p} (Ω)} + {‖ \nabla_{x^{'}} (\cdot) ‖}_{L^{p} (Ω)}, \\ ‖ \cdot ‖_{L^{p} (Ω, \nabla_{x^{″}})} ≐ ‖ \cdot ‖_{L^{p} (Ω)} + {‖ \nabla_{x^{″}} (\cdot) ‖}_{L^{p} (Ω)}, \\ ‖ \cdot ‖_{L^{p} (Ω, \nabla_{x^{'}}; W^{1, p} (Y^{″}))} ≐ ‖ \cdot ‖_{L^{p} (Ω \times Y^{″})} + {‖ \nabla_{x^{'}} (\cdot) ‖}_{L^{p} (Ω \times Y^{″})} + {‖ \nabla_{y^{″}} (\cdot) ‖}_{L^{p} (Ω \times Y^{″})}, \\ ‖ \cdot ‖_{L^{p} (Ω, \nabla_{x^{″}}; W^{1, p} (Y^{'}))} ≐ ‖ \cdot ‖_{L^{p} (Ω \times Y^{'})} + {‖ \nabla_{x^{″}} (\cdot) ‖}_{L^{p} (Ω \times Y^{'})} + {‖ \nabla_{y^{'}} (\cdot) ‖}_{L^{p} (Ω \times Y^{'})}, \\ ‖ \cdot ‖_{L^{p} (Ω \times Y^{″}; W^{1, p} (Y^{'}))} ≐ ‖ \cdot ‖_{L^{p} (Ω \times Y)} + {‖ \nabla_{y^{'}} (\cdot) ‖}_{L^{p} (Ω \times Y)}, \\ ‖ \cdot ‖_{L^{p} (Ω \times Y^{'}; W^{1, p} (Y^{″}))} ≐ ‖ \cdot ‖_{L^{p} (Ω \times Y)} + {‖ \nabla_{y^{″}} (\cdot) ‖}_{L^{p} (Ω \times Y)} . \end{array}$ We recall the unfolding with parameters tools already developed in [5, Chap. 7] and we define two partial unfolding operators. These operators are built in order to apply the unfolding only to their respective half of the domain and such that the composition of both gives the unfolding on the whole domain. As we will see, this “two-steps unfolding” proves to be the winning strategy to deal with anisotropically bounded sequences.
Definition 3.2.
For every measurable function ϕ on Ω, the unfolding operator $T_{ε}^{″}$ is defined as follows: $\begin{matrix} T_{ε}^{″} (ϕ) (x^{'}, x^{″}, y^{″}) = \{\begin{matrix} ϕ (x^{'}, ε {[\frac{x^{″}}{ε}]}_{Y^{″}} + ε y^{″}) & for a.e. (x^{'}, x^{″}, y^{″}) \in {\hat{Ω}}_{ε} \times Y^{″}, \\ 0 & for a.e. (x^{'}, x^{″}, y^{″}) \in Λ_{ε} \times Y^{″} . \end{matrix} \end{matrix}$ For every measurable function ψ on $Ω \times Y^{″}$ , the unfolding operator $T_{ε}^{'}$ is defined as follows: $\begin{matrix} T_{ε}^{'} (ψ) (x^{'}, x^{″}, y^{'}, y^{″}) = \{\begin{matrix} ψ (ε {[\frac{x^{'}}{ε}]}_{Y^{'}} + ε y^{'}, x^{″}, y^{″}) & for a.e. (x^{'}, x^{″}, y^{'}, y^{″}) \in {\hat{Ω}}_{ε} \times Y, \\ 0 & for a.e. (x^{'}, x^{″}, y^{'}, y^{″}) \in Λ_{ε} \times Y . \end{matrix} \end{matrix}$

Note that in the partial unfolding operator $T_{ε}^{″} (ϕ)$ the variable $x^{'}$ plays the role of a parameter, while in $T_{ε}^{'} (ψ)$ the role of parameters is played by the variables $(x^{″}, y^{″})$ . Lemma 3.3.
One has $\begin{matrix} (3.1) & T_{ε} = T_{ε}^{'} \circ T_{ε}^{″}, M_{Y} = M_{Y^{'}} \circ M_{Y^{″}} . \end{matrix}$ Moreover, for every $ϕ \in L^{1} (Ω, \nabla_{x^{'}})$ one has $\begin{matrix} (3.2) & \nabla_{x^{'}} T_{ε}^{″} (ϕ) = T_{ε}^{″} (\nabla_{x^{'}} ϕ) a.e. in {\hat{Ω}}_{ε} \times Y^{″} . \end{matrix}$
Proof.
Let ϕ be measurable on Ω. We have that $\begin{array}{l} T_{ε}^{'} \circ T_{ε}^{″} (ϕ) (x, y) & = T_{ε}^{'} (ϕ (x^{'}, ε {[\frac{x^{″}}{ε}]}_{Y^{″}} + ε y^{″})) = ϕ (ε {[\frac{x^{'}}{ε}]}_{Y^{'}} + ε y^{'}, ε {[\frac{x^{″}}{ε}]}_{Y^{″}} + ε y^{″}) \\ = ϕ (ε {[\frac{x}{ε}]}_{Y} + ε y) = T_{ε} (ϕ) (x, y) for a.e. (x, y) \in {\hat{Ω}}_{ε} \times Y . \end{array}$ For $(x, y) \in Λ_{ε} \times Y$ the result is obvious.

Let $\hat{ϕ}$ be in $L^{1} (Ω \times Y)$ . We have $\begin{array}{l} M_{Y^{'}} \circ M_{Y^{″}} (\hat{ϕ}) (x) & = M_{Y^{'}} (\frac{1}{| Y^{″} |} \int_{Y^{″}} \hat{ϕ} (x, y^{'}, y^{″}) d y^{″}) \\ = \frac{1}{| Y^{'} | | Y^{″} |} \int_{Y^{'}} \int_{Y^{″}} \hat{ϕ} (x, y^{'}, y^{″}) d y^{″} d y^{'} = \frac{1}{| Y |} \int_{Y} \hat{ϕ} (x, y) d y \\ = M_{Y} (\hat{ϕ}) (x) for a.e. x \in Ω . \end{array}$ Let now ϕ be in $L^{1} (Ω, \nabla_{x^{'}})$ . We have $\begin{array}{l} \nabla_{x^{'}} T_{ε}^{″} (ϕ) (x, y^{″}) & = \nabla_{x^{'}} (ϕ (x^{'}, ε {[\frac{x^{″}}{ε}]}_{Y^{″}} + ε y^{″})) = \nabla_{x^{'}} ϕ (x^{'}, ε {[\frac{x^{″}}{ε}]}_{Y^{″}} + ε y^{″}) \\ = T_{ε}^{″} (\nabla_{x^{'}} ϕ) (x, y^{″}) for a.e. (x, y^{″}) \in {\hat{Ω}}_{ε} \times Y^{″} . \end{array}$ □

4. Asymptotic behavior of anisotropically bounded sequences

In this section all the tools are ready to investigate the asymptotic behavior of anisotropically bounded sequences. Since the definition of “anisotropic behavior” only denotes a contrast in the estimates with respect to the direction observed, the different amount of information we have on the sequences arises a different asymptotic behavior.

We start with sequences bounded in $L^{p} (Ω, \nabla_{x^{'}})$ , $p \in [1, + \infty]$ , whose gradient is bounded with order $ε^{- 1}$ in $L^{p} (Ω)$ in the first $N_{1}$ directions.

Lemma 4.1.
Let ${ϕ_{ε}}_{ε}$ be a sequence in $L^{p} (Ω, \nabla_{x^{'}})$ , $p \in (1, + \infty)$ , satisfying $\begin{matrix} ‖ ϕ_{ε} ‖_{L^{p} (Ω)} + ε ‖ \nabla_{x^{'}} ϕ_{ε} ‖_{L^{p} (Ω)} ⩽ C, \end{matrix}$ where the constant does not depend on ε.

Then, there exist a subsequence of ${ε}$ , still denoted ${ε}$ , and a function $\hat{ϕ}$ in the space $L^{p} (Ω \times Y^{″}; W_{per}^{1, p} (Y^{'}))$ such that $\begin{matrix} ϕ_{ε} ⇀ ϕ weakly in L^{p} (Ω), \\ T_{ε} (ϕ_{ε}) ⇀ \hat{ϕ} weakly in L^{p} (Ω \times Y^{″}; W^{1, p} (Y^{'})), \end{matrix}$ where $ϕ = M_{Y} (\hat{ϕ})$ .

The same results hold for $p = + \infty$ with weak topology replaced by weak- topology in the corresponding spaces.*
Proof.
The proof is similar to [5, Theorem 1.36]. □

An analogous result holds for sequences uniformly bounded in $L^{p} (Ω, \nabla_{x^{'}})$ , $p \in [1, + \infty]$ .
Lemma 4.2.
Let ${ϕ_{ε}}_{ε}$ be a sequence in $L^{p} (Ω, \nabla_{x^{'}})$ , $p \in (1, + \infty)$ , such that $\begin{matrix} ‖ ϕ_{ε} ‖_{L^{p} (Ω, \nabla_{x^{'}})} ⩽ C . \end{matrix}$ Then, there exist a subsequence of ${ε}$ , still denoted ${ε}$ , and $\tilde{ϕ} \in L^{p} (Ω \times Y^{″}, \nabla_{x^{'}})$ , $\hat{ϕ} \in L^{p} (Ω \times Y^{″}; W_{per, 0}^{1, p} (Y^{'}))$ such that $\begin{matrix} ϕ_{ε} ⇀ ϕ weakly in L^{p} (Ω, \nabla_{x^{'}}), \\ T_{ε} (ϕ_{ε}) ⇀ \tilde{ϕ} weakly in L^{p} (Ω \times Y^{″}; W^{1, p} (Y^{'})), \\ T_{ε} (\nabla_{x^{'}} ϕ_{ε}) ⇀ \nabla_{x^{'}} \tilde{ϕ} + \nabla_{y^{'}} \hat{ϕ} weakly in L^{p} {(Ω \times Y)}^{N_{1}}, \\ \frac{1}{ε} (T_{ε} (ϕ_{ε}) - M_{Y^{'}} \circ T_{ε} (ϕ_{ε})) ⇀ \nabla_{x^{'}} \tilde{ϕ} \cdot y^{' c} + \hat{ϕ} weakly in L^{p} {(Ω \times Y)}^{N_{1}}, \end{matrix}$ where $ϕ = M_{Y^{″}} (\tilde{ϕ})$ and $y^{' c} ≐ y^{'} - M_{Y^{'}} (y^{'})$ .

The same results hold for $p = + \infty$ with weak topology replaced by weak- topology in the corresponding spaces.*
Proof.
The proof is similar to [5, Corollary 1.37] and [5, Theorem 1.41]. □

Now, we consider the sequences in $W^{1, p} (Ω)$ , $p \in [1, + \infty]$ , whose gradient is estimated with different order according to the considered direction.
Lemma 4.3.
Let ${ϕ_{ε}}_{ε}$ be a sequence in $W^{1, p} (Ω)$ , $p \in (1, + \infty)$ , satisfying $\begin{matrix} (4.1) & ‖ ϕ_{ε} ‖_{L^{p} (Ω, \nabla_{x^{'}})} + ε ‖ \nabla_{x^{″}} ϕ_{ε} ‖_{L^{p} (Ω)} ⩽ C, \end{matrix}$ where the constant does not depend on ε.

Then, there exist a subsequence of ${ε}$ , still denoted ${ε}$ , and functions $\begin{matrix} \tilde{ϕ} \in L^{p} (Ω, \nabla_{x^{'}}; W_{per}^{1, p} (Y^{″})) and \hat{ϕ} \in L^{p} (Ω \times Y^{″}; W_{per, 0}^{1, p} (Y^{'})) \end{matrix}$ such that $\begin{matrix} (4.2) & \begin{matrix} ϕ_{ε} ⇀ ϕ weakly in L^{p} (Ω, \nabla_{x^{'}}), \\ T_{ε} (ϕ_{ε}) ⇀ \tilde{ϕ} weakly in L^{p} (Ω; W^{1, p} (Y)), \\ T_{ε} (\nabla_{x^{'}} ϕ_{ε}) ⇀ \nabla_{x^{'}} \tilde{ϕ} + \nabla_{y^{'}} \hat{ϕ} weakly in L^{p} {(Ω \times Y)}^{N_{1}}, \\ ε T_{ε} (\nabla_{x^{″}} ϕ_{ε}) ⇀ \nabla_{y^{″}} \tilde{ϕ} weakly in L^{p} {(Ω \times Y)}^{N_{2}}, \\ \frac{1}{ε} (T_{ε} (ϕ_{ε}) - M_{Y^{'}} \circ T_{ε} (ϕ_{ε})) ⇀ \nabla_{x^{'}} \tilde{ϕ} \cdot y^{' c} + \hat{ϕ} weakly in L^{p} {(Ω \times Y)}^{N_{1}}, \end{matrix} \end{matrix}$ where $ϕ = M_{Y^{″}} (\tilde{ϕ})$ and $y^{' c} ≐ y^{'} - M_{Y^{'}} (y^{'})$ .

The same results hold for $p = + \infty$ with weak topology replaced by weak- topology in the corresponding spaces.*
Proof.
From (4.1), up to a subsequence of ${ε}$ , still denoted ${ε}$ , one has the existence of $ϕ \in L^{p} (Ω, \nabla_{x^{'}})$ such that (4.2)₁ holds.

Set ${Φ_{ε}}_{ε} = {T_{ε}^{″} (ϕ_{ε})}_{ε}$ . This sequence belongs to $L^{p} ({\hat{Ω}}_{ε}, \nabla_{x^{'}}; W^{1, p} (Y^{″}))$ and from estimate (4.1) and equality (3.2), it satisfies $\begin{matrix} (4.3) & ‖ Φ_{ε} ‖_{L^{p} ({\hat{Ω}}_{ε}, \nabla_{x^{'}}; W^{1, p} (Y^{″}))} ⩽ C, \end{matrix}$ where the constant does not depend on ε.

Up to a subsequence of ${ε}$ , still denoted ${ε}$ , there exists $\tilde{ϕ} \in L^{p} (Ω; W_{per}^{1, p} (Y^{″}))$ and $\tilde{Φ} \in L^{p} {(Ω \times Y^{″})}^{N_{1}}$ (the periodicity of $\tilde{ϕ}$ is proved as in [5, Theorem 1.36]) such that $\begin{matrix} Φ_{ε} 1_{{\hat{Ω}}_{ε} \times Y^{″}} ⇀ \tilde{ϕ} weakly in L^{p} (Ω; W^{1, p} (Y^{″})), \\ \nabla_{x^{'}} Φ_{ε} 1_{{\hat{Ω}}_{ε} \times Y^{″}} ⇀ \tilde{Φ} weakly in L^{p} {(Ω \times Y^{″})}^{N_{1}}, \end{matrix}$ where $1_{{\hat{Ω}}_{ε} \times Y^{″}}$ denotes the characteristic function of the domain ${\hat{Ω}}_{ε} \times Y^{″}$ .

Let g be in $C_{c}^{\infty} {(Ω \times Y^{″})}^{N_{1}}$ . For ε sufficiently small such that $supp (g) \subset {\hat{Ω}}_{ε} \times Y^{″}$ , we have $\begin{array}{l} \int_{Ω \times Y^{″}} \nabla_{x^{'}} Φ_{ε} 1_{{\hat{Ω}}_{ε} \times Y^{″}} \cdot g d x d y^{″} & = \int_{{\hat{Ω}}_{ε} \times Y^{″}} \nabla_{x^{'}} Φ_{ε} \cdot g d x d y^{″} \\ = - \int_{{\hat{Ω}}_{ε} \times Y^{″}} Φ_{ε} \nabla_{x^{'}} g d x d y^{″} = - \int_{Ω \times Y^{″}} Φ_{ε} 1_{{\hat{Ω}}_{ε} \times Y^{″}} \nabla_{x^{'}} g d x d y^{″} . \end{array}$ Then, passing to the limit yields $\begin{matrix} \int_{Ω \times Y^{″}} \tilde{Φ} \cdot g d x d y^{″} = - \int_{Ω \times Y^{″}} \tilde{ϕ} \cdot \nabla_{x^{'}} g d x d y^{″}, \forall g \in C_{c}^{\infty} {(Ω \times Y^{″})}^{N_{1}} . \end{matrix}$ This means that $\tilde{Φ} = \nabla_{x^{'}} \tilde{ϕ}$ a.e. in $Ω \times Y^{″}$ , thus $\nabla_{x^{'}} \tilde{ϕ} \in L^{p} {(Ω \times Y^{″})}^{N_{1}}$ and therefore $\tilde{ϕ} \in L^{p} (Ω, \nabla_{x^{'}}; W_{per}^{1, p} (Y^{″}))$ .

Now, we transform the sequence ${Φ_{ε}}_{ε}$ using the unfolding operator $T_{ε}^{'}$ , $Y^{″}$ being a set of parameters.

From the above convergence and estimate (4.3), up to a subsequence of ${ε}$ , still denoted ${ε}$ , [5, Corollary 1.37] and [5, Theorem 1.41] give $\hat{ϕ} \in L^{p} (Ω \times Y^{″}; W_{per, 0}^{1, p} (Y^{'}))$ such that (using the rule (3.1)₁) $\begin{matrix} T_{ε} (ϕ_{ε}) = T_{ε}^{'} (Φ_{ε}) ⇀ \tilde{ϕ} weakly in L^{p} (Ω; W^{1, p} (Y^{'} \times Y^{″})), \\ T_{ε} (\nabla_{x^{'}} ϕ_{ε}) = T_{ε}^{'} (\nabla_{x^{'}} Φ_{ε}) ⇀ \nabla_{x^{'}} \tilde{ϕ} + \nabla_{y^{'}} \hat{ϕ} weakly in L^{p} {(Ω \times Y^{'} \times Y^{″})}^{N_{1}}, \\ \frac{1}{ε} (T_{ε} (ϕ_{ε}) - M_{Y^{'}} (T_{ε} (ϕ_{ε}))) = \frac{1}{ε} (T_{ε}^{'} (Φ_{ε}) - M_{Y^{'}} (T_{ε}^{'} (Φ_{ε}))) ⇀ \nabla_{x^{'}} \tilde{ϕ} \cdot y^{' c} + \hat{ϕ} \\ weakly in L^{p} (Ω \times Y^{'} \times Y^{″}) . \end{matrix}$ This proves convergences (4.2)_2,3,5. Moreover, from convergence (4.2)₂ and the unfolding properties of $T_{ε}$ we get that $\begin{matrix} ε T_{ε} (\nabla_{x^{″}} ϕ_{ε}) = \nabla_{y^{″}} T_{ε} (ϕ_{ε}) ⇀ \nabla_{y^{″}} \tilde{ϕ} weakly in L^{p} {(Ω \times Y^{″})}^{N_{2}}, \end{matrix}$ which proves convergence (4.2)₄. □

As the lemma below shows, an analogous asymptotic behavior is achieved starting from sequences uniformly bounded in $L^{p} (Ω, \nabla_{x^{'}})$ , $p \in [1, + \infty]$ , with some assumptions on the gradient derivatives. Lemma 4.4.
Let ${ϕ_{ε}}_{ε}$ be a sequence in $L^{p} (Ω, \nabla_{x^{'}})$ , $p \in (1, + \infty)$ , satisfying $\begin{matrix} (4.4) & ‖ ϕ_{ε} ‖_{L^{p} (Ω, \nabla_{x^{'}})} + ε {‖ \nabla_{x^{″}} (\nabla_{x^{'}} ϕ_{ε}) ‖}_{L^{p} (Ω)} ⩽ C, \end{matrix}$ where the constant does not depend on ε.

Then, there exist a subsequence of ${ε}$ , still denoted ${ε}$ , functions $\begin{matrix} \tilde{ϕ} \in L^{p} (Ω, \nabla_{x^{'}}; W_{per}^{1, p} (Y^{″})) and \hat{Φ} \in L^{p} (Ω; W_{per}^{1, p} (Y)) \end{matrix}$ such that $M_{Y^{'}} (\hat{Φ}) = 0$ a.e. in $Ω \times Y^{″}$ , $\begin{matrix} \nabla_{x^{'}} \tilde{ϕ} \in L^{p} {(Ω; W_{per}^{1, p} (Y^{″}))}^{N_{1}}, \nabla_{y^{'}} \hat{Φ} \in L^{p} {(Ω \times Y^{'}; W_{per}^{1, p} (Y^{″}))}^{N_{1}} \end{matrix}$ and we have $\begin{matrix} (4.5) & \begin{matrix} ϕ_{ε} ⇀ ϕ weakly in L^{p} (Ω, \nabla_{x^{'}}), \\ T_{ε} (ϕ_{ε}) ⇀ \tilde{ϕ} weakly in L^{p} (Ω; W^{1, p} (Y)), \\ T_{ε} (\nabla_{x^{'}} ϕ_{ε}) ⇀ \nabla_{x^{'}} \tilde{ϕ} + \nabla_{y^{'}} \hat{Φ} weakly in L^{p} {(Ω \times Y^{'}; W^{1, p} (Y^{″}))}^{N_{1}}, \end{matrix} \end{matrix}$ where $ϕ = M_{Y^{″}} (\tilde{ϕ})$ .

The same results hold for $p = + \infty$ with weak topology replaced by weak- topology in the corresponding spaces.*
Proof.
By estimate (4.4)₁ and Lemma 4.2, there exists a subsequence of ${ε}$ , still denoted ${ε}$ , and functions $\tilde{ϕ} \in L^{p} (Ω \times Y^{″}, \nabla_{x^{'}})$ , $\hat{ϕ} \in L^{p} (Ω \times Y^{″}; W_{per, 0}^{1, p} (Y^{'}))$ such that $\begin{matrix} (4.6) & \begin{matrix} ϕ_{ε} ⇀ ϕ weakly in L^{p} (Ω, \nabla_{x^{'}}), \\ T_{ε} (ϕ_{ε}) ⇀ \tilde{ϕ} weakly in L^{p} (Ω \times Y^{″}; W^{1, p} (Y^{'})), \\ T_{ε} (\nabla_{x^{'}} ϕ_{ε}) ⇀ \nabla_{x^{'}} \tilde{ϕ} + \nabla_{y^{'}} \hat{ϕ} weakly in L^{p} {(Ω \times Y)}^{N_{1}} . \end{matrix} \end{matrix}$ Set ${ψ_{ε}}_{ε} ≐ {\nabla_{x^{'}} ϕ_{ε}}_{ε}$ . By estimate (4.4), this sequence satisfies $\begin{matrix} ‖ ψ_{ε} ‖_{L^{p} (Ω)} + ε ‖ \nabla_{x^{″}} ψ_{ε} ‖_{L^{p} (Ω)} ⩽ C, \end{matrix}$ where the constant does not depend on ε.

Hence, applying Lemma 4.1 to the above sequence (but swapping $Y^{'}$ and $Y^{″}$ ), there exists a function $\hat{ψ} \in L^{p} {(Ω \times Y^{'}; W_{per}^{1, p} (Y^{″}))}^{N_{1}}$ such that $\begin{matrix} T_{ε} (\nabla_{x^{'}} ϕ_{ε}) = T_{ε} (ψ_{ε}) ⇀ \hat{ψ} weakly in L^{p} {(Ω \times Y^{'}; W^{1, p} (Y^{″}))}^{N_{1}} . \end{matrix}$ This, together with convergence (4.6)₃ implies that the quantity $\nabla_{x^{'}} \tilde{ϕ} + \nabla_{y^{'}} \hat{ϕ}$ belongs to $L^{p} {(Ω \times Y^{'}; W_{per}^{1, p} (Y^{″}))}^{N_{1}}$ . Since $\tilde{ϕ}$ does not depend on $y^{'}$ and $\hat{ϕ}$ is periodic with respect to $y^{'}$ , we have that $\begin{matrix} \nabla_{x^{'}} \tilde{ϕ} = M_{Y^{'}} (\nabla_{x^{'}} \tilde{ϕ}) + M_{Y^{'}} (\nabla_{y^{'}} \hat{ϕ}) = M_{Y^{'}} (\hat{ψ}), \end{matrix}$ thus $\nabla_{x^{'}} \tilde{ϕ} \in L^{p} {(Ω; W_{per}^{1, p} (Y^{″}))}^{N_{1}}$ and therefore $\tilde{ϕ} \in L^{p} (Ω, \nabla_{x^{'}}; W_{per}^{1, p} (Y^{″}))$ .

Moreover, the quantity $\nabla_{y^{'}} \hat{ϕ}$ belongs to $L^{p} {(Ω \times Y^{'}; W_{per}^{1, p} (Y^{″}))}^{N_{1}}$ and thus Lemma A.2 implies that there exists $\hat{ϕ} \in L^{p} (Ω; W_{per}^{1, p} (Y))$ with $\nabla_{y^{'}} \hat{ϕ} = \nabla_{y^{'}} \hat{ϕ}$ such that (4.5)₃ hold. The proof follows by replacing $\hat{ϕ}$ by the function $\hat{Φ} = \hat{ϕ} - M_{Y^{'}} (\hat{ϕ})$ , which belongs to the space $L^{p} (Ω; W_{per}^{1, p} (Y))$ and satisfies $M_{Y^{'}} (\hat{Φ}) = 0$ a.e. in $Ω \times Y^{″}$ . □

5. Other unfolding results for anisotropically bounded sequences

In this section we use the achieved results for the unfolding of sequences in the anisotropic case to prove a more complex lemma involving multiple sequences. Even if it might seem a specific task, this lemma has proven to be a key component in the homogenization via unfolding of perforated structures (see [15]) and textiles (see [22]) and its extension can be applied to anisotropically structures, such as the textiles with loose contact investigated in [14].

We start by giving the original formulation.

Lemma 5.1 (see [5, Lemma 11.11]).

Let ${(u_{ε}, v_{ε})}_{ε}$ be a sequence converging weakly to $(u, v)$ in $W^{1, p} (Ω) \times W^{1, p} {(Ω)}^{N}$ , $p \in (1, + \infty)$ . Moreover, assume that there exist $Z \in L^{p} {(Ω)}^{N}$ and $\hat{v} \in L^{p} {(Ω; W_{per, 0}^{1, p} (Y))}^{N}$ such that $\begin{matrix} \frac{1}{ε} (\nabla u_{ε} + v_{ε}) ⇀ Z weakly in L^{p} {(Ω)}^{N}, \\ T_{ε} (\nabla v_{ε}) ⇀ \nabla v + \nabla_{y} \hat{v} weakly in L^{p} {(Ω \times Y)}^{N \times N} . \end{matrix}$ Then, u belongs to $W^{2, p} (Ω)$ . Moreover, there exist a subsequence of ${ε}$ , still denoted ${ε}$ , and $u \in L^{p} (Ω; W_{per, 0}^{1, p} (Y))$ such that $\begin{matrix} \frac{1}{ε} T_{ε} (\nabla u_{ε} + v_{ε}) ⇀ Z + \nabla_{y} u + \hat{v} weakly in L^{p} {(Ω \times Y)}^{N} . \end{matrix}$

As an immediate consequence, one has the following.

Corollary 5.2.
Let $O$ be an open set in $R^{k}$ , $k ⩾ 1$ . Let ${(u_{ε}, v_{ε})}_{ε}$ be a sequence converging weakly to $(u, v)$ in $L^{p} (O; W^{1, p} (Ω)) \times L^{p} {(O; W^{1, p} (Ω))}^{N}$ , $p \in (1, + \infty)$ . Moreover, assume that there exist $Z \in L^{p} {(O \times Ω)}^{N}$ and $\hat{v} \in L^{p} {(O \times Ω; W_{per, 0}^{1, p} (Y))}^{N}$ such that $\begin{matrix} \frac{1}{ε} (\nabla u_{ε} + v_{ε}) ⇀ Z weakly in L^{p} {(O \times Ω)}^{N}, \\ T_{ε} (\nabla v_{ε}) ⇀ \nabla v + \nabla_{y} \hat{v} weakly in L^{p} {(O \times Ω \times Y)}^{N \times N} . \end{matrix}$ Then, u belongs to $L^{p} (O; W^{2, p} (Ω))$ . Furthermore, there exist a subsequence of ${ε}$ , still denoted ${ε}$ , and $u \in L^{p} (O \times Ω; W_{per, 0}^{1, p} (Y))$ such that: $\begin{matrix} \frac{1}{ε} T_{ε} (\nabla u_{ε} + v_{ε}) ⇀ Z + \nabla_{y} u + \hat{v} weakly in L^{p} {(O \times Ω \times Y)}^{N} . \end{matrix}$

Define the spaces $\begin{array}{l} \begin{matrix} L^{p} (Ω \times Y^{″}, D_{x^{'}}^{2}) ≐ & {\tilde{ϕ} \in L^{p} (Ω \times Y^{″}) ∣ \nabla_{x^{'}} \tilde{ϕ} \in L^{p} {(Ω \times Y^{″})}^{N_{1}} \\ and (\nabla_{x^{'}} \otimes \nabla_{x^{'}}) \tilde{ϕ} \in L^{p} {(Ω \times Y^{″})}^{N_{1} \times N_{1}}}, \end{matrix} \\ \begin{matrix} L^{p} (Ω, D_{x^{'}}^{2}; W_{per}^{1, p} (Y^{″})) ≐ & {\tilde{ϕ} \in L^{p} (Ω; W_{per}^{1, p} (Y^{″})) ∣ \nabla_{x^{'}} \tilde{ϕ} \in L^{p} {(Ω; W_{per}^{1, p} (Y^{″}))}^{N_{1}} \\ and (\nabla_{x^{'}} \otimes \nabla_{x^{'}}) \tilde{ϕ} \in L^{p} {(Ω; W_{per}^{1, p} (Y^{″}))}^{N_{1} \times N_{1}}}, \end{matrix} \end{array}$ where $(\nabla_{x^{'}} \otimes \nabla_{x^{'}}) \tilde{ϕ}$ denotes the first $N_{1} \times N_{1}$ entries of the Hessian matrix of ϕ.

We endow such spaces with the respective norms: $\begin{array}{l} ‖ \cdot ‖_{L^{p} (Ω \times Y^{″}, D_{x^{'}}^{2})} ≐ ‖ \cdot ‖_{L^{p} (Ω \times Y^{″})} + {‖ \nabla_{x^{'}} (\cdot) ‖}_{L^{p} (Ω \times Y^{″})} + {‖ D_{x^{'}}^{2} (\cdot) ‖}_{L^{p} (Ω \times Y^{″})}, \\ ‖ \cdot ‖_{L^{p} (Ω, D_{x^{'}}^{2}; W^{1, p} (Y^{″}))} ≐ ‖ \cdot ‖_{L^{p} (Ω \times Y^{″}, D_{x^{'}}^{2})} + {‖ \nabla_{y^{″}} (\cdot) ‖}_{L^{p} (Ω \times Y^{″})} . \end{array}$ We are ready to extend Lemma 5.1 to the class of anisotropically bounded sequences.
Lemma 5.3.
Let ${(u_{ε}, v_{ε})}_{ε}$ be a sequence in the space $L^{p} (Ω, \nabla_{x^{'}}) \times L^{p} {(Ω, \nabla_{x^{'}})}^{N_{1}}$ , $p \in (1, + \infty)$ , satisfying $\begin{matrix} (5.1) & ‖ u_{ε} ‖_{L^{p} (Ω, \nabla_{x^{'}})} ⩽ C, ‖ v_{ε} ‖_{L^{p} (Ω, \nabla_{x^{'}})} ⩽ C, \end{matrix}$ where the constant does not depend on ε.

Moreover, assume that there exist $Z \in L^{p} {(Ω)}^{N_{1}}$ such that $\begin{matrix} (5.2) & \frac{1}{ε} (\nabla_{x^{'}} u_{ε} + v_{ε}) ⇀ Z weakly in L^{p} {(Ω)}^{N_{1}} . \end{matrix}$ Then, there exist a subsequence of ${ε}$ , still denoted ${ε}$ , and $\tilde{Z} \in L^{p} {(Ω \times Y^{″})}^{N_{1}}$ with $M_{Y^{″}} (\tilde{Z}) = Z$ , $u \in L^{p} (Ω \times Y^{″}; W_{per, 0}^{1, p} (Y^{'}))$ , $\tilde{u} \in L^{p} (Ω \times Y^{″}, D_{x^{'}}^{2})$ and a function $\hat{v} \in L^{p} {(Ω \times Y^{″}; W_{per, 0}^{1, p} (Y^{'}))}^{N_{1}}$ such that $\begin{matrix} (5.3) & \begin{matrix} T_{ε} (\nabla_{x^{'}} v_{ε}) ⇀ - D_{x^{'}}^{2} \tilde{u} + \nabla_{y^{'}} \hat{v} weakly in L^{p} {(Ω \times Y)}^{N_{1} \times N_{1}}, \\ \frac{1}{ε} T_{ε} (\nabla_{x^{'}} u_{ε} + v_{ε}) ⇀ \tilde{Z} + \nabla_{y^{'}} u + \hat{v} weakly in L^{p} {(Ω \times Y)}^{N_{1}} . \end{matrix} \end{matrix}$
Proof.
We first apply the unfolding operator $T_{ε}$ to both sequences ${u_{ε}}$ and ${v_{ε}}$ . By Lemma 4.2 and estimates (5.1), there exist a subsequence of ${ε}$ , still denoted ${ε}$ , $\tilde{u} \in L^{p} (Ω \times Y^{″}, \nabla_{x^{'}})$ , $\tilde{v} \in L^{p} {(Ω \times Y^{″}, \nabla_{x^{'}})}^{N_{1}}$ , $\hat{u} \in L^{p} (Ω \times Y^{″}; W_{per, 0}^{1, p} (Y^{'}))$ , $\hat{v} \in L^{p} {(Ω \times Y^{″}; W_{per, 0}^{1, p} (Y^{'}))}^{N_{1}}$ such that $\begin{matrix} (5.4) & \begin{matrix} T_{ε} (u_{ε}) ⇀ \tilde{u} weakly in L^{p} (Ω \times Y^{″}; W^{1, p} (Y^{'})), \\ T_{ε} (\nabla_{x^{'}} u_{ε}) ⇀ \nabla_{x^{'}} \tilde{u} + \nabla_{y^{'}} \hat{u} weakly in L^{p} {(Ω \times Y)}^{N_{1}}, \\ T_{ε} (v_{ε}) ⇀ \tilde{v} weakly in L^{p} {(Ω \times Y^{″}; W^{1, p} (Y^{'}))}^{N_{1}}, \\ T_{ε} (\nabla_{x^{'}} v_{ε}) ⇀ \nabla_{x^{'}} \tilde{v} + \nabla_{y^{'}} \hat{v} weakly in L^{p} {(Ω \times Y)}^{N_{1} \times N_{1}} . \end{matrix} \end{matrix}$ By convergence (5.2), there exist a subsequence of ${ε}$ , still denoted ${ε}$ , and functions $\hat{Z} \in L^{p} {(Ω \times Y)}^{N_{1}}$ with $M_{Y} (\hat{Z}) = Z$ such that $\begin{matrix} (5.5) & \frac{1}{ε} T_{ε} (\nabla_{x^{'}} u_{ε} + v_{ε}) ⇀ \hat{Z} weakly in L^{p} {(Ω \times Y)}^{N_{1}} . \end{matrix}$ From convergences (5.4)_2,3 and (5.5) we get $\begin{matrix} \nabla_{x^{'}} \tilde{u} + \nabla_{y^{'}} \hat{u} + \tilde{v} = 0 a.e. in Ω \times Y . \end{matrix}$ Applying $M_{Y^{'}}$ to the above equality and since $\hat{u} \in L^{p} (Ω \times Y^{″}; W_{per, 0}^{1, p} (Y^{'}))$ , while $\tilde{u} \in L^{p} (Ω \times Y^{″}, \nabla_{x^{'}})$ , $\tilde{v} \in L^{p} {(Ω \times Y^{″}, \nabla_{x^{'}})}^{N_{1}}$ , we get that $\nabla_{x^{'}} \tilde{u} + \tilde{v} = 0$ a.e. in $Ω \times Y^{″}$ . Hence, $\nabla_{y^{'}} \hat{u} = 0$ and thus $\hat{u} = 0$ because it belongs to $L^{p} (Ω \times Y^{″}; W_{per, 0}^{1, p} (Y^{'}))$ . As a consequence, one has $\begin{matrix} \tilde{u} \in L^{p} (Ω \times Y^{″}, D_{x^{'}}^{2}) . \end{matrix}$ Set $U_{ε} = T_{ε}^{″} (u_{ε})$ , $V_{ε} = T_{ε}^{″} (v_{ε})$ . Again by convergence (5.2), there exist a subsequence of ${ε}$ , still denoted ${ε}$ , and $\tilde{Z} \in L^{p} {(Ω \times Y^{″})}^{N_{1}}$ such that $\begin{matrix} \frac{1}{ε} \nabla_{x^{'}} U_{ε} + V_{ε} ⇀ \tilde{Z} weakly in L^{p} {(Ω \times Y^{″})}^{N_{1}} . \end{matrix}$ Then, due to convergence (5.5) we have $\tilde{Z} = M_{Y^{'}} (\hat{Z})$ .

Now, let $ω^{'}$ and $ω^{″}$ be two open sets such that $\begin{matrix} (5.6) & ω^{'} \subset R^{N_{1}}, ω^{″} \subset R^{N_{2}} and \overline{ω^{'} \times ω^{″}} \subset Ω . \end{matrix}$ First, observe that $\begin{matrix} U_{ε} \in L^{p} (ω^{″} \times Y^{″}; W^{1, p} (ω^{'})), V_{ε} \in L^{p} {(ω^{″} \times Y^{″}; W^{1, p} (ω^{'}))}^{N_{1}} . \end{matrix}$ By the above convergence and (5.4)₄, one has $\begin{matrix} \frac{1}{ε} \nabla_{x^{'}} U_{ε} + V_{ε} ⇀ \tilde{Z} weakly in L^{p} {(ω^{'} \times ω^{″} \times Y^{″})}^{N_{1}}, \\ T_{ε} (\nabla_{x^{'}} v_{ε}) = T_{ε}^{'} (\nabla_{x^{'}} V_{ε}) ⇀ \nabla_{x^{'}} \tilde{v} + \nabla_{y^{'}} \hat{v} weakly in L^{p} {(ω^{'} \times ω^{″} \times Y^{'} \times Y^{″})}^{N_{1} \times N_{1}} . \end{matrix}$ Lemma 5.2 claims that up to a subsequence, there exists $u_{ω^{'} \times ω^{″}}$ , which belongs to $L^{p} (ω^{'} \times ω^{″} \times Y^{″}; W_{per, 0}^{1, p} (Y^{'}))$ , such that the following convergence holds: $\begin{matrix} \frac{1}{ε} T_{ε}^{'} (\nabla_{x^{'}} U_{ε} + V_{ε}) ⇀ \tilde{Z} + \nabla_{y^{'}} u_{ω^{'} \times ω^{″}} + \hat{v} weakly in L^{p} {(ω^{'} \times ω^{″} \times Y^{'} \times Y^{″})}^{N_{1}} . \end{matrix}$ Taking into account convergence (5.5) we get $\begin{matrix} \hat{Z} = \tilde{Z} + \nabla_{y^{'}} u_{ω^{'} \times ω^{″}} + \hat{v} in ω^{'} \times ω^{″} \times Y . \end{matrix}$ Since one can cover Ω by a countable family of open subsets $ω^{'} \times ω^{″}$ satisfying (5.6), there exists $u$ in $L^{p} (Ω \times Y^{″}; W_{per, 0}^{1, p} (Y^{'}))$ such that $\hat{Z} - \tilde{Z} - \hat{v} = \nabla_{y^{'}} u$ . This completes the proof of (5.3). □

With some more assumptions, we can improve the regularity of the limit functions. Lemma 5.4.
Let ${(u_{ε}, v_{ε})}_{ε}$ be a sequence in $L^{p} (Ω, \nabla_{x^{'}}) \times L^{p} {(Ω, \nabla_{x^{'}})}^{N_{1}}$ , with $p \in (1, + \infty)$ , satisfying the assumptions in Lemma 5.3 . Moreover, assume that $\begin{matrix} (5.7) & {‖ \nabla_{x^{″}} (\nabla_{x^{'}} u_{ε} + v_{ε}) ‖}_{L^{p} (Ω)} + ε {‖ \nabla_{x^{″}} (\nabla_{x^{'}} v_{ε}) ‖}_{L^{p} (Ω)} ⩽ C, \end{matrix}$ where the constant does not depend on ε.

Then, there exist a subsequence of ${ε}$ , still denoted ${ε}$ , $\tilde{Z} \in L^{p} {(Ω; W_{per}^{1, p} (Y^{″}))}^{N_{1}}$ , $\hat{v} \in L^{p} {(Ω; W_{per, 0}^{1, p} (Y))}^{N_{1}}$ , $U \in L^{p} (Ω; W_{per, 0}^{1, p} (Y))$ and $\tilde{u} \in L^{p} (Ω, D_{x^{'}}^{2}; W_{per}^{1, p} (Y^{″}))$ such that $\begin{matrix} T_{ε} (\nabla_{x^{'}} v_{ε}) ⇀ - D_{x^{'}}^{2} \tilde{u} + \nabla_{y^{'}} \hat{v} weakly in L^{p} {(Ω \times Y)}^{N_{1} \times N_{1}}, \\ \frac{1}{ε} T_{ε} (\nabla_{x^{'}} u_{ε} + v_{ε}) ⇀ \tilde{Z} + \nabla_{y^{'}} U + \hat{v} weakly in L^{p} {(Ω \times Y)}^{N_{1}} . \end{matrix}$
Proof.
From Lemma 5.3, there exist a subsequence of ${ε}$ , still denoted ${ε}$ , and $\tilde{Z} \in L^{p} {(Ω \times Y^{″})}^{N_{1}}$ , $u \in L^{p} (Ω \times Y^{″}; W_{per, 0}^{1, p} (Y^{'}))$ , $\tilde{u} \in L^{p} (Ω \times Y^{″}, D_{x^{'}}^{2})$ and a function $\hat{v} \in L^{p} {(Ω \times Y^{″}; W_{per, 0}^{1, p} (Y^{'}))}^{N_{1}}$ such that $\begin{matrix} T_{ε} (\nabla_{x^{'}} v_{ε}) ⇀ - D_{x^{'}}^{2} \tilde{u} + \nabla_{y^{'}} \hat{v} weakly in L^{p} {(Ω \times Y)}^{N_{1} \times N_{1}}, \\ \frac{1}{ε} T_{ε} (\nabla_{x^{'}} u_{ε} + v_{ε}) ⇀ \tilde{Z} + \nabla_{y^{'}} u + \hat{v} weakly in L^{p} {(Ω \times Y)}^{N_{1}} . \end{matrix}$ By hypothesis (5.7), Lemma 4.1 (swapping $Y^{'}$ and $Y^{″}$ ) and the proof of Lemma 4.4 one has $\begin{matrix} T_{ε} (\nabla_{x^{'}} v_{ε}) ⇀ - D_{x^{'}}^{2} \tilde{u} + \nabla_{y^{'}} \hat{v} weakly in L^{p} {(Ω \times Y)}^{N_{1} \times N_{1}}, \\ \frac{1}{ε} T_{ε} (\nabla_{x^{'}} u_{ε} + v_{ε}) ⇀ \tilde{Z} + \nabla_{y^{'}} u + \hat{v} \in L^{p} {(Ω \times Y^{'}; W_{per}^{1, p} (Y^{″}))}^{N_{1}} \end{matrix}$ with $\tilde{Z} \in L^{p} {(Ω \times Y^{″})}^{N_{1}}$ , $\tilde{u} \in L^{p} (Ω, D_{x^{'}}^{2}; W_{per}^{1, p} (Y^{″}))$ and $\hat{v} \in L^{p} {(Ω; W_{per}^{1, p} (Y))}^{N_{1}}$ satisfying $M_{Y^{'}} (\hat{v}) = 0$ a.e. in $Ω \times Y^{″}$ .

Since, $\hat{v}$ satisfies $M_{Y^{'}} (\hat{v}) = 0$ a.e. in $Ω \times Y^{″}$ and $M_{Y^{'}} (\nabla_{y^{'}} u) = 0$ a.e. in $Ω \times Y^{″}$ by periodicity of $u$ , we obtain $\begin{matrix} \tilde{Z} = M_{Y^{'}} (\tilde{Z}) \in L^{p} {(Ω; W_{per}^{1, p} (Y^{″}))}^{N_{1}} . \end{matrix}$ Hence $\nabla_{y^{'}} u$ lies in $L^{p} {(Ω \times Y^{'}; W_{per}^{1, p} (Y^{″}))}^{N_{1}}$ . Lemma A.2 in the Appendix gives a function $U \in L^{p} (Ω; W_{per, 0}^{1, p} (Y))$ such that $\nabla_{y^{'}} U = \nabla_{y^{'}} u$ . The proof is complete. □

6. Application: Homogenization of an anisotropic diffusion problem

In this last section we want to give a direct application of the periodic unfolding for anisotropically bounded sequences to a diffusion problem.

Let $O$ be an open subset of $R^{N}$ and let $α, β \in R$ with $0 < α < β$ . Denote $M (α, β, O)$ the set of $N \times N$ matrices $A = {(a_{i j})}_{1 ⩽ i, j ⩽ N}$ with coefficients in $L^{\infty} (O)$ such that for every $λ \in R^{N}$ and for a.e. $x \in O$ , the following inequalities hold:

$(A (x) λ, λ) ⩾ α | λ |^{2}$ ;

$| A (x) λ |^{2} ⩽ β (A (x) λ, λ)$ .

Let A be in

M (α, β, Y)

and let

{A_{ε}}_{ε}

be the sequence of matrices belonging to

M (α, β, Ω)

defined by

\begin{matrix} (6.1) & A_{ε} ≐ A ({\frac{x}{ε}}_{Y}) a.e. x \in Ω . \end{matrix}

From (2.2) for

p = 2

, we recall the definition of the Hilbert spaces

\begin{matrix} H_{per}^{1} (Y) ≐ {ϕ \in H^{1} (Y) ∣ ϕ is periodic with respect to y_{i}, i \in {1, \dots, N}}, \\ H_{per, 0}^{1} (Y) ≐ {ϕ \in H_{per}^{1} (Y) ∣ M_{Y} (ϕ) = 0} . \end{matrix}

Let f be a function in

L^{2} (Ω)

Consider the following Dirichlet problem in variational formulation (see also (1.1) and (1.2) where it is written differently): $\begin{matrix} (6.2) & \{\begin{matrix} Find u_{ε} \in H_{0}^{1} (Ω) such that: \\ \int_{Ω} A_{ε} [\begin{matrix} \nabla_{x^{'}} u_{ε} \\ ε \nabla_{x^{″}} u_{ε} \end{matrix}] \cdot [\begin{matrix} \nabla_{x^{'}} ϕ \\ ε \nabla_{x^{″}} ϕ \end{matrix}] d x = \int_{Ω} f ϕ d x, \forall ϕ \in H_{0}^{1} (Ω), \end{matrix} \end{matrix}$ where · denotes the dot product by the column vectors $A_{ε} [\begin{matrix} \nabla_{x^{'}} u_{ε} \\ ε \nabla_{x^{″}} u_{ε} \end{matrix}]$ and $[\begin{matrix} \nabla_{x^{'}} ϕ_{ε} \\ ε \nabla_{x^{″}} ϕ_{ε} \end{matrix}]$ .

By the Poincaré inequality and the fact that $u_{ε} \in H_{0}^{1} (Ω)$ , we have that $\begin{matrix} ‖ u_{ε} ‖_{L^{2} (Ω)} ⩽ C ‖ \nabla_{x^{'}} u_{ε} ‖_{L^{2} (Ω)} . \end{matrix}$ Thus, problem (6.2) admits a unique solution by the Lax–Milgram theorem and the following inequality holds: $\begin{matrix} α (‖ \nabla_{x^{'}} u_{ε} ‖_{L^{2} (Ω)}^{2} + ε^{2} ‖ \nabla_{x^{″}} u_{ε} ‖_{L^{2} (Ω)}^{2}) ⩽ ‖ f ‖_{L^{2} (Ω)} ‖ u_{ε} ‖_{L^{2} (Ω)} ⩽ C ‖ f ‖_{L^{2} (Ω)} ‖ \nabla_{x^{'}} u_{ε} ‖_{L^{2} (Ω)} . \end{matrix}$ Hence $\begin{matrix} (6.3) & ‖ u_{ε} ‖_{L^{2} (Ω)} + ‖ \nabla_{x^{'}} u_{ε} ‖_{L^{2} (Ω)} + ε ‖ \nabla_{x^{″}} u_{ε} ‖_{L^{2} (Ω)} ⩽ C ‖ f ‖_{L^{2} (Ω)}, \end{matrix}$ where the constant does not depend on ε.

Set $\begin{matrix} H_{0, per}^{1} (Ω \times Y^{″}) = & {ϕ \in H^{1} (Ω \times Y^{″}) ∣ ϕ (x, y^{″}) = 0 for a.e. (x, y^{″}) \in \partial Ω \times Y^{″} \\ and ϕ (x, \cdot) is Y^{″} periodic for a.e. x \in Ω} . \end{matrix}$ Denote $L_{0}^{2} (Ω, \nabla_{x^{'}})$ (resp. $L_{0}^{2} (Ω, \nabla_{x^{'}}; H_{per}^{1} (Y^{″}))$ ) the closure of $H_{0}^{1} (Ω)$ (resp. of $H_{0, per}^{1} (Ω \times Y^{″})$ ) in $L^{2} (Ω)$ (resp. $L^{2} (Ω \times Y^{″})$ ) for the norm of $L^{2} (Ω, \nabla_{x^{'}})$ (resp. $L^{2} (Ω, \nabla_{x^{'}}; H_{per}^{1} (Y^{″}))$ ), see Section 3).

Below, we give the periodic homogenization via unfolding.

Theorem 6.1.

Let $u_{ε}$ be the solution of problem ( 6.2 ).

There exist $\tilde{u} \in L_{0}^{2} (Ω, \nabla_{x^{'}}; H_{per}^{1} (Y^{″}))$ and $\hat{u} \in L^{2} (Ω \times Y^{″}; H_{per, 0}^{1} (Y^{'}))$ such that $\begin{matrix} (6.4) & \begin{matrix} u_{ε} ⇀ M_{Y} (\tilde{u}) weakly in L_{0}^{2} (Ω, \nabla_{x^{'}}), \\ T_{ε} (u_{ε}) ⇀ \tilde{u} weakly in L^{2} (Ω; H^{1} (Y)), \\ T_{ε} (\nabla_{x^{'}} u_{ε}) \to \nabla_{x^{'}} \tilde{u} + \nabla_{y^{'}} \hat{u} strongly in L^{2} {(Ω \times Y)}^{N_{1}}, \\ ε T_{ε} (\nabla_{x^{″}} u_{ε}) \to \nabla_{y^{″}} \tilde{u} strongly in L^{2} {(Ω \times Y)}^{N_{2}} . \end{matrix} \end{matrix}$ The couple $(\tilde{u}, \hat{u})$ is the unique solution of problem $\begin{matrix} (6.5) & \{\begin{matrix} \int_{Ω \times Y} A (y) [\begin{matrix} \nabla_{x^{'}} \tilde{u} (x, y^{″}) + \nabla_{y^{'}} \hat{u} (x, y) \\ \nabla_{y^{″}} \tilde{u} (x, y^{″}) \end{matrix}] \cdot [\begin{matrix} \nabla_{x^{'}} \tilde{ϕ} (x, y^{″}) + \nabla_{y^{'}} \hat{ϕ} (x, y) \\ \nabla_{y^{″}} \tilde{ϕ} (x, y^{″}) \end{matrix}] d x d y \\ = | Y^{'} | \int_{Ω \times Y^{″}} f (x) \tilde{ϕ} (x, y^{″}) d x d y^{″}, \\ \forall \tilde{ϕ} \in L_{0}^{2} (Ω, \nabla_{x^{'}}; H_{per, 0}^{1} (Y^{″})) and \forall \hat{ϕ} \in L^{2} (Ω \times Y^{″}; H_{per, 0}^{1} (Y^{'})) . \end{matrix} \end{matrix}$

Proof.

Step 1. We show (6.5) and the weak convergences (6.4).

First, since $A \in M (α, β, Y)$ , by definition (6.1) and the unfolding operator properties we immediately get that $T_{ε} (A_{ε}) (x, y) = A (y)$ for a.e. $(x, y) \in {\hat{Ω}}_{ε} \times Y$ .

Now, note that the solution $u_{ε}$ of (6.2) satisfies (6.3). Hence, up to a subsequence of ${ε}$ , still denoted ${ε}$ , Lemma 4.3 gives $\tilde{u} \in L_{0}^{2} (Ω, \nabla_{x^{'}}; H_{per}^{1} (Y^{″}))$ and $\hat{u} \in L^{2} (Ω \times Y^{″}; H_{per, 0}^{1} (Y^{'}))$ such that $\begin{matrix} (6.6) & \begin{matrix} u_{ε} ⇀ M_{Y} (\tilde{u}) weakly in L_{0}^{2} (Ω, \nabla_{x^{'}}), \\ T_{ε} (u_{ε}) ⇀ \tilde{u} weakly in L^{2} (Ω; H^{1} (Y)), \\ T_{ε} (\nabla_{x^{'}} u_{ε}) ⇀ \nabla_{x^{'}} \tilde{u} + \nabla_{y^{'}} \hat{u} weakly in L^{2} {(Ω \times Y)}^{N_{1}}, \\ ε T_{ε} (\nabla_{x^{″}} u_{ε}) ⇀ \nabla_{y^{″}} \tilde{u} weakly in L^{2} {(Ω \times Y^{″})}^{N_{2}} . \end{matrix} \end{matrix}$ Now, we choose the test functions

$\tilde{Φ}$ in $H_{0}^{1} (Ω)$ , $\tilde{ϕ}$ in $H_{per}^{1} (Y^{″})$ ,

Φ in $C_{c}^{1} (Ω \times Y^{″})$ ,

$\hat{ϕ}$ in $H_{per, 0}^{1} (Y^{'})$ .

Set

ϕ_{ε} (x) ≐ \tilde{Φ} (x) \tilde{ϕ} (\frac{x^{″}}{ε}) + ε Φ (x, \frac{x^{″}}{ε}) \hat{ϕ} (\frac{x^{'}}{ε})

for a.e.

x \in Ω

Applying the unfolding operator to the sequence ${ϕ_{ε}}_{ε}$ , we get that $\begin{matrix} T_{ε} (ϕ_{ε}) \to \tilde{Φ} \tilde{ϕ} strongly in L^{2} (Ω; H^{1} (Y)), \\ T_{ε} (\nabla_{x^{'}} ϕ_{ε}) \to (\nabla_{x^{'}} \tilde{Φ}) \tilde{ϕ} + Φ \nabla_{y^{'}} \hat{ϕ} strongly in L^{2} {(Ω \times Y)}^{N_{1}}, \\ ε T_{ε} (\nabla_{x^{″}} ϕ_{ε}) \to \nabla_{y^{″}} \tilde{ϕ} strongly in L^{2} {(Ω \times Y)}^{N_{2}} . \end{matrix}$ Taking $ϕ_{ε}$ as test function in (6.2), then transforming by unfolding and passing to the limit give (6.5) with $(\tilde{Φ} \tilde{ϕ}, Φ \hat{ϕ})$ . By density argumentation, we extend such results for all $\tilde{ϕ} \in L_{0}^{2} (Ω, \nabla_{x^{'}}; H_{per}^{1} (Y^{″}))$ and all $\hat{ϕ} \in L^{2} (Ω \times Y^{″}; H_{per, 0}^{1} (Y^{'}))$ . Since the solution is unique the whole sequences converge to their limit.

Step 2. We prove that convergences (6.4)_3,4 are strong.

First, setting $ϕ = u_{ε}$ in (6.2), then transforming by unfolding and using the weak lower semicontinuity yield $\begin{matrix} \int_{Ω \times Y} A [\begin{matrix} \nabla_{x^{'}} \tilde{u} + \nabla_{y^{'}} \hat{u} \\ \nabla_{y^{″}} \tilde{u} \end{matrix}] \cdot [\begin{matrix} \nabla_{x^{'}} \tilde{u} + \nabla_{y^{'}} \hat{u} \\ \nabla_{y^{″}} \tilde{u} \end{matrix}] d x d y \\ ⩽ \underset{ε \to 0}{lim inf} \int_{Ω \times Y} T_{ε} (A_{ε}) [\begin{matrix} T_{ε} (\nabla_{x^{'}} u_{ε}) \\ ε T_{ε} (\nabla_{x^{″}} u_{ε}) \end{matrix}] \cdot [\begin{matrix} T_{ε} (\nabla_{x^{'}} u_{ε}) \\ ε T_{ε} (\nabla_{x^{″}} u_{ε}) \end{matrix}] d x d y \\ = \underset{ε \to 0}{lim inf} \int_{{\hat{Ω}}_{ε}} A_{ε} [\begin{matrix} \nabla_{x^{'}} u_{ε} \\ ε \nabla_{x^{″}} u_{ε} \end{matrix}] \cdot [\begin{matrix} \nabla_{x^{'}} u_{ε} \\ ε \nabla_{x^{″}} u_{ε} \end{matrix}] d x d y \\ ⩽ \underset{ε \to 0}{lim sup} \int_{{\hat{Ω}}_{ε}} A_{ε} [\begin{matrix} \nabla_{x^{'}} u_{ε} \\ ε \nabla_{x^{″}} u_{ε} \end{matrix}] \cdot [\begin{matrix} \nabla_{x^{'}} u_{ε} \\ ε \nabla_{x^{″}} u_{ε} \end{matrix}] d x d y \\ ⩽ \underset{ε \to 0}{lim sup} \int_{Ω} A_{ε} [\begin{matrix} \nabla_{x^{'}} u_{ε} \\ ε \nabla_{x^{″}} u_{ε} \end{matrix}] \cdot [\begin{matrix} \nabla_{x^{'}} u_{ε} \\ ε \nabla_{x^{″}} u_{ε} \end{matrix}] d x d y = \underset{ε \to 0}{lim sup} \int_{Ω} f u_{ε} d x \\ = lim_{ε \to 0} \int_{Ω \times Y} T_{ε} (f) T_{ε} (u_{ε}) d x = \int_{Ω \times Y} f \tilde{u} d x d y \\ = \int_{Ω \times Y} A [\begin{matrix} \nabla_{x^{'}} \tilde{u} + \nabla_{y^{'}} \hat{u} \\ \nabla_{y^{″}} \tilde{u} \end{matrix}] \cdot [\begin{matrix} \nabla_{x^{'}} \tilde{u} + \nabla_{y^{'}} \hat{u} \\ \nabla_{y^{″}} \tilde{u} \end{matrix}] d x d y, \end{matrix}$ from which it follows that all the above inequalities are in fact equalities. Hence $\begin{matrix} \int_{Λ_{ε}} A_{ε} [\begin{matrix} \nabla_{x^{'}} u_{ε} \\ ε \nabla_{x^{″}} u_{ε} \end{matrix}] \cdot [\begin{matrix} \nabla_{x^{'}} u_{ε} \\ ε \nabla_{x^{″}} u_{ε} \end{matrix}] d x d y = 0 \end{matrix}$ and $\begin{matrix} lim_{ε \to 0} \int_{Ω \times Y} A [\begin{matrix} T_{ε} (\nabla_{x^{'}} u_{ε}) \\ ε T_{ε} (\nabla_{x^{″}} u_{ε}) \end{matrix}] \cdot [\begin{matrix} T_{ε} (\nabla_{x^{'}} u_{ε}) \\ ε T_{ε} (\nabla_{x^{″}} u_{ε}) \end{matrix}] d x d y \\ = lim_{ε \to 0} \int_{Ω} A_{ε} [\begin{matrix} \nabla_{x^{'}} u_{ε} \\ ε \nabla_{x^{″}} u_{ε} \end{matrix}] \cdot [\begin{matrix} \nabla_{x^{'}} u_{ε} \\ ε \nabla_{x^{″}} u_{ε} \end{matrix}] d x d y \\ = \int_{Ω \times Y} A [\begin{matrix} \nabla_{x^{'}} \tilde{u} + \nabla_{y^{'}} \hat{u} \\ \nabla_{y^{″}} \tilde{u} \end{matrix}] [\begin{matrix} \nabla_{x^{'}} \tilde{u} + \nabla_{y^{'}} \hat{u} \\ \nabla_{y^{″}} \tilde{u} \end{matrix}] d x d y . \end{matrix}$ Since the map $Ψ \in L^{2} {(Ω \times Y)}^{N} ⟼ \sqrt{\int_{Ω \times Y} A Ψ \cdot Ψ d x d y}$ is a norm equivalent to the usual norm of $L^{2} {(Ω \times Y)}^{N}$ , we get $\begin{matrix} lim_{ε \to 0} \int_{Ω \times Y} {|[\begin{matrix} T_{ε} (\nabla_{x^{'}} u_{ε}) \\ ε T_{ε} (\nabla_{x^{″}} u_{ε}) \end{matrix}]|}^{2} d x d y = \int_{Ω \times Y} {|[\begin{matrix} \nabla_{x^{'}} \tilde{u} + \nabla_{y^{'}} \hat{u} \\ \nabla_{y^{″}} \tilde{u} \end{matrix}]|}^{2} d x d y . \end{matrix}$ This, together with the fact that (6.6)_3,4 already converge weakly, ensures the strong convergences (6.4)_3,4. The proof is therefore complete. □

Now, consider the following partition of A into blocks $\begin{matrix} A = (\begin{matrix} A_{1} & A_{2} \\ A_{3} & A_{4} \end{matrix}), \end{matrix}$ where

$A_{1}$ is a $N_{1} \times N_{1}$ matrix with entries in $L^{\infty} (Y)$ ,

$A_{2}$ is a $N_{1} \times N_{2}$ matrix with entries in $L^{\infty} (Y)$ ,

$A_{3}$ is a $N_{2} \times N_{1}$ matrix with entries in $L^{\infty} (Y)$ ,

$A_{4}$ is a $N_{2} \times N_{2}$ matrix with entries in $L^{\infty} (Y)$ .

We define the correctors

{\hat{χ}}_{k}

k \in {1, \dots, N}

, as the unique solutions in the space

L^{\infty} (Y^{″}, H_{per, 0}^{1} (Y^{'}))

of the cell problems

\begin{matrix} (6.7) & \{\begin{matrix} \int_{Y^{'}} A_{1} (y) [\nabla_{y^{'}} {\hat{χ}}_{k} (y^{'}, y^{″})] \cdot [\nabla_{y^{'}} \hat{w} (y^{'})] d y^{'} = - \int_{Y^{'}} A (y) e_{k} \cdot [\begin{matrix} \nabla_{y^{'}} \hat{w} (y^{'}) \\ 0 \end{matrix}] d y^{'}, \\ \forall \hat{w} \in H_{per, 0}^{1} (Y^{'}) . \end{matrix} \end{matrix}

By the Lax–Milgram theorem applied in Hilbert space

L^{2} (Y^{″}, H_{per, 0}^{1} (Y^{'}))

, we obtain the existence and uniqueness of the solution of (6.7) for every

k \in {1, \dots, N}

Since A belongs to $M (α, β, Y)$ we get for every $k \in {1, \dots, N}$ : $\begin{matrix} {‖ \nabla_{y^{'}} {\hat{χ}}_{k} (\cdot, y^{″}) ‖}_{H^{1} {(Y^{'})}^{N_{1}}} ⩽ \frac{β}{α} for a.e. y^{″} \in Y^{″} . \end{matrix}$ As a consequence ${\hat{χ}}_{k} \in L^{\infty} (Y^{″}, H_{per, 0}^{1} (Y^{'}))$ 1

One can prove that ${\hat{χ}}_{k}$ also belongs to $L^{\infty} (Y)$ .

for every

k \in {1, \dots, N}

and

\begin{matrix} ‖ {\hat{χ}}_{k} ‖_{L^{\infty} (Y^{″}; H^{1} (Y^{'}))} ⩽ C . \end{matrix}

We can finally give the form of the homogenized problem.

Proposition 6.2.

The function ${\tilde{u}}_{0} \in L_{0}^{2} (Ω, \nabla_{x^{'}}; H_{per}^{1} (Y^{″}))$ is the unique solution of the following homogenized problem: $\begin{matrix} (6.8) & \{\begin{matrix} \int_{Ω \times Y^{″}} A^{hom} (y^{″}) [\begin{matrix} \nabla_{x^{'}} {\tilde{u}}_{0} (x, y^{″}) \\ \nabla_{y^{″}} {\tilde{u}}_{0} (x, y^{″}) \end{matrix}] \cdot [\begin{matrix} \nabla_{x^{'}} \tilde{ϕ} (x, y^{″}) \\ \nabla_{y^{″}} \tilde{ϕ} (x, y^{″}) \end{matrix}] d x d y^{″} \\ = \int_{Ω \times Y^{″}} f (x) \tilde{ϕ} (x, y^{″}) d x d y^{″}, \forall \tilde{ϕ} \in L_{0}^{2} (Ω, \nabla_{x^{'}}; H_{per}^{1} (Y^{″})) . \end{matrix} \end{matrix}$ The homogenizing operator $A^{hom} \in L^{\infty} {(Y^{″})}^{N \times N}$ is the matrix defined by $\begin{matrix} (6.9) & A^{hom} (y^{″}) ≐ \frac{1}{| Y^{'} |} \int_{Y^{'}} (A + (\begin{matrix} A_{1} \\ A_{3} \end{matrix}) \nabla_{y^{'}} χ) (y^{'}, y^{″}) d y^{'}, \end{matrix}$ where $\hat{χ} = ({\hat{χ}}_{1} {\hat{χ}}_{2} \dots {\hat{χ}}_{N_{1}} {\hat{χ}}_{N_{1} + 1} \dots {\hat{χ}}_{N})$ and thus $\nabla_{y^{'}} \hat{χ}$ is the $N_{1} \times N$ matrix $\begin{matrix} \nabla_{y^{'}} \hat{χ} ≐ (\begin{matrix} \nabla_{y^{'}} {\hat{χ}}_{1} & \nabla_{y^{'}} {\hat{χ}}_{2} & \dots & \nabla_{y^{'}} {\hat{χ}}_{N_{1}} & \nabla_{y^{'}} {\hat{χ}}_{N_{1} + 1} & \dots & \nabla_{y^{'}} {\hat{χ}}_{N} \end{matrix}) . \end{matrix}$

Note, that in such a formulation the problem mixes the macroscopic $x^{'}$ and microscopic variables that correspond to $x^{″}$ . Nevertheless, the homogenization is considered to be concluded since all the involved functions depend on such variables.

Before proceeding to the proof, we find convenient to clarify the boundary conditions for the solutions of problem (6.8) in a simplified case.

Remark 6.3.

Assume that $Ω ≐ {(0, 1)}^{2} \subset R^{2}$ . Then, the function of ${\tilde{u}}_{0}$ belongs to the space $\begin{array}{l} {ϕ \in L^{2} (Ω, \partial_{1}; H_{per}^{1} (Y^{″})) ∣ ϕ (0, x_{2}, y_{2}) = ϕ (1, x_{2}, y_{2}) = 0 for a.e. (x_{2}, y_{2}) \in (0, 1) \times Y^{″}} . \end{array}$

Proof of Proposition 6.2.

Equation (6.5) with $\tilde{ϕ} = 0$ leads to: $\begin{matrix} (6.10) & \{\begin{matrix} \int_{Ω \times Y} (\begin{matrix} A_{1} & A_{2} \\ A_{3} & A_{4} \end{matrix}) (y^{'}, y^{″}) [\begin{matrix} \nabla_{y^{'}} \hat{u} (x, y^{'}, y^{″}) \\ 0 \end{matrix}] \cdot [\begin{matrix} \nabla_{y^{'}} \hat{ϕ} (x, y^{'}, y^{″}) \\ 0 \end{matrix}] d x d y^{'} d y^{″} \\ = - \int_{Ω \times Y} (\begin{matrix} A_{1} & A_{2} \\ A_{3} & A_{4} \end{matrix}) (y^{'}, y^{″}) [\begin{matrix} \nabla_{x^{'}} \tilde{u} (x, y^{″}) \\ \nabla_{y^{″}} \tilde{u} (x, y^{″}) \end{matrix}] \cdot [\begin{matrix} \nabla_{y^{'}} \hat{ϕ} (x, y^{'}, y^{″}) \\ 0 \end{matrix}] d x d y^{'} d y^{″}, \\ \forall \hat{ϕ} \in L^{2} (Ω \times Y^{″}; H_{per, 0}^{1} (Y^{'})), \end{matrix} \end{matrix}$ from which the form of the cell problems (6.7) follows.

By (6.10), we can write $\hat{u}$ as $\begin{matrix} \hat{u} (x, y^{'}, y^{″}) = \sum_{k = 1}^{N_{1}} {\hat{χ}}_{k} (y^{'}, y^{″}) \partial_{x_{k}} \tilde{u} (x, y^{″}) + \sum_{k = N_{1} + 1}^{N} {\hat{χ}}_{k} (y^{'}, y^{″}) \partial_{y_{k}} \tilde{u} (x, y^{″}) \\ for a.e. (x, y^{'}, y^{″}) \in Ω \times Y^{'} \times Y^{″} . \end{matrix}$ Replacing $\hat{u}$ by the above equality in (6.5) (note that $\hat{ϕ}$ is set to be zero since the correctors have been found) we first get $\begin{matrix} \int_{Ω \times Y} (\begin{matrix} A_{1} & A_{2} \\ A_{3} & A_{4} \end{matrix}) [\begin{matrix} \nabla_{x^{'}} \tilde{u} + \nabla_{y^{'}} \hat{u} \\ \nabla_{y^{″}} \tilde{u} \end{matrix}] \cdot [\begin{matrix} \nabla_{x^{'}} \tilde{ϕ} \\ \nabla_{y^{″}} \tilde{ϕ} \end{matrix}] d x d y \\ = \int_{Ω \times Y} (\begin{matrix} A_{1} & A_{2} \\ A_{3} & A_{4} \end{matrix}) [\begin{matrix} \nabla_{x^{'}} \tilde{u} \\ \nabla_{y^{″}} \tilde{u} \end{matrix}] \cdot [\begin{matrix} \nabla_{x^{'}} \tilde{ϕ} \\ \nabla_{y^{″}} \tilde{ϕ} \end{matrix}] d x d y \\ + \int_{Ω \times Y} (\begin{matrix} A_{1} & A_{2} \\ A_{3} & A_{4} \end{matrix}) [\begin{matrix} \sum_{k = 1}^{N_{1}} \nabla_{y^{'}} {\hat{χ}}_{k} \partial_{x_{k}} \tilde{u} + \sum_{k = N_{1} + 1}^{N} \nabla_{y^{'}} {\hat{χ}}_{k} \partial_{y_{k}} \tilde{u} \\ 0 \end{matrix}] \cdot [\begin{matrix} \nabla_{x^{'}} \tilde{ϕ} \\ \nabla_{y^{″}} \tilde{ϕ} \end{matrix}] d x d y . \end{matrix}$ Concerning the second term, straightforward calculations lead to $\begin{matrix} \int_{Ω \times Y} (\begin{matrix} A_{1} & A_{2} \\ A_{3} & A_{4} \end{matrix}) [\begin{matrix} \sum_{k = 1}^{N_{1}} \nabla_{y^{'}} {\hat{χ}}_{k} \partial_{x_{k}} \tilde{u} + \sum_{k = N_{1} + 1}^{N} \nabla_{y^{'}} {\hat{χ}}_{k} \partial_{y_{k}} \tilde{u} \\ 0 \end{matrix}] \cdot [\begin{matrix} \nabla_{x^{'}} \tilde{ϕ} \\ \nabla_{y^{″}} \tilde{ϕ} \end{matrix}] d x d y \\ = \int_{Ω \times Y} (\begin{matrix} A_{1} & A_{2} \\ A_{3} & A_{4} \end{matrix}) [(\begin{matrix} \nabla_{y^{'}} \hat{χ} \\ 0 \end{matrix}) [\begin{matrix} \nabla_{x^{'}} \tilde{u} \\ \nabla_{y^{″}} \tilde{u} \end{matrix}]] \cdot [\begin{matrix} \nabla_{x^{'}} \tilde{ϕ} \\ \nabla_{y^{″}} \tilde{ϕ} \end{matrix}] d x d y \\ = \int_{Ω \times Y} ((\begin{matrix} A_{1} \\ A_{3} \end{matrix}) \nabla_{y^{'}} \hat{χ}) [\begin{matrix} \nabla_{x^{'}} \tilde{u} \\ \nabla_{y^{″}} \tilde{u} \end{matrix}] \cdot [\begin{matrix} \nabla_{x^{'}} \tilde{ϕ} \\ \nabla_{y^{″}} \tilde{ϕ} \end{matrix}] d x d y, \end{matrix}$ where we denoted $(\begin{matrix} \nabla_{y^{'}} \hat{χ} \\ 0 \end{matrix})$ the $N \times N$ matrix partitioned into the upper $N_{1} \times N$ block $\nabla_{y^{'}} \hat{χ}$ and the lower $N_{2} \times N$ block with zero entrances. Hence, we get that $\begin{matrix} \int_{Ω \times Y} (\begin{matrix} A_{1} & A_{2} \\ A_{3} & A_{4} \end{matrix}) [\begin{matrix} \nabla_{x^{'}} \tilde{u} + \nabla_{y^{'}} \hat{u} \\ \nabla_{y^{″}} \tilde{u} \end{matrix}] \cdot [\begin{matrix} \nabla_{x^{'}} \tilde{ϕ} \\ \nabla_{y^{″}} \tilde{ϕ} \end{matrix}] d x d y \\ = \int_{Ω \times Y} ((\begin{matrix} A_{1} & A_{2} \\ A_{3} & A_{4} \end{matrix}) + (\begin{matrix} A_{1} \\ A_{3} \end{matrix}) \nabla_{y^{'}} \hat{χ}) [\begin{matrix} \nabla_{x^{'}} \tilde{u} \\ \nabla_{y^{″}} \tilde{u} \end{matrix}] \cdot [\begin{matrix} \nabla_{x^{'}} \tilde{ϕ} \\ \nabla_{y^{″}} \tilde{ϕ} \end{matrix}] d x d y . \end{matrix}$ Gathering all the $y^{'}$ dependent terms, we get the form (6.9) of the homogenizing operator $A^{hom}$ .

Since $A \in L^{\infty} {(Y)}^{N \times N}$ and the ${\hat{χ}}_{k}$ ’s are in $L^{\infty} (Y^{″}; H^{1} (Y^{'}))$ , it is clear that $A^{hom}$ belongs to $L^{\infty} {(Y^{″})}^{N \times N}$ .

We prove now that $A^{hom}$ is coercive. Let $ξ ≐ (ξ_{1}, ξ_{2})$ be a vector with fixed entries in $R^{N} = R^{N_{1}} \times R^{N_{2}}$ . By the construction of the homogenizing operator, straightforward calculation imply that $\begin{array}{l} A^{hom} [ξ] \cdot [ξ] & = \frac{1}{| Y^{'} |} \int_{Y^{'}} ((\begin{matrix} A_{1} & A_{2} \\ A_{3} & A_{4} \end{matrix}) + (\begin{matrix} A_{1} \\ A_{3} \end{matrix}) \nabla_{y^{'}} \hat{χ}) [\begin{matrix} ξ_{1} \\ ξ_{2} \end{matrix}] \cdot [\begin{matrix} ξ_{1} \\ ξ_{2} \end{matrix}] d y^{'} \\ = \frac{1}{| Y^{'} |} \int_{Y^{'}} (\begin{matrix} A_{1} & A_{2} \\ A_{3} & A_{4} \end{matrix}) [\begin{matrix} ξ_{1} + \nabla_{y^{'}} {\hat{χ}}_{ξ} \\ ξ_{2} \end{matrix}] \cdot [\begin{matrix} ξ_{1} \\ ξ_{2} \end{matrix}] d y^{'} \\ = \frac{1}{| Y^{'} |} \int_{Y^{'}} (\begin{matrix} A_{1} & A_{2} \\ A_{3} & A_{4} \end{matrix}) [\begin{matrix} ξ_{1} + \nabla_{y^{'}} {\hat{χ}}_{ξ} \\ ξ_{2} \end{matrix}] \cdot [\begin{matrix} ξ_{1} + \nabla_{y^{'}} {\hat{χ}}_{ξ} \\ ξ_{2} \end{matrix}] d y^{'} \\ - \frac{1}{| Y^{'} |} \int_{Y^{'}} (\begin{matrix} A_{1} & A_{2} \\ A_{3} & A_{4} \end{matrix}) [\begin{matrix} ξ_{1} + \nabla_{y^{'}} {\hat{χ}}_{ξ} \\ ξ_{2} \end{matrix}] \cdot [\begin{matrix} \nabla_{y^{'}} {\hat{χ}}_{ξ} \\ 0 \end{matrix}] d y^{'}, \end{array}$ where ${\hat{χ}}_{ξ} ≐ \sum_{k = 1}^{N} {\hat{χ}}_{k} ξ_{k}$ . Observe that by the cell problems (6.7), the second term in the last equality is equal to zero.

Now, the coercivity of the matrix A and the fact that ${\hat{χ}}_{ξ} \in L^{\infty} (Y^{″}; H_{per, 0}^{1} (Y^{'}))$ imply that $\begin{array}{l} A^{hom} (y^{″}) [ξ] \cdot [ξ] \\ = \frac{1}{| Y^{'} |} \int_{Y^{'}} (\begin{matrix} A_{1} & A_{2} \\ A_{3} & A_{4} \end{matrix}) (y^{'}, y^{″}) [\begin{matrix} ξ_{1} + \nabla_{y^{'}} {\hat{χ}}_{ξ} (y^{'}, y^{″}) \\ ξ_{2} \end{matrix}] \cdot [\begin{matrix} ξ_{1} + \nabla_{y^{'}} {\hat{χ}}_{ξ} (y^{'}, y^{″}) \\ ξ_{2} \end{matrix}] d y^{'} \\ ⩾ α ({‖ ξ_{1} + \nabla_{y^{'}} {\hat{χ}}_{ξ} (\cdot, y^{″}) ‖}_{L^{2} {(Y^{'})}^{N_{1}}}^{2} + | ξ_{2} |^{2}) \\ = α (| ξ_{1} |^{2} + | ξ_{2} |^{2} + {‖ \nabla_{y^{'}} {\hat{χ}}_{ξ} (\cdot, y^{″}) ‖}_{L^{2} {(Y^{'})}^{N_{1}}}^{2}) ⩾ α | ξ |^{2} for a.e. y^{″} \in Y^{″}, \end{array}$ which proves that $A^{hom}$ is coercive.

Replacing the form of $A^{hom}$ on the original problem (6.5), we get (6.8). By the boundedness and coercivity of $A^{hom}$ and by the fact that the function $\tilde{u}$ belongs to $L_{0}^{2} (Ω, \nabla_{x^{'}}; H_{per}^{1} (Y^{″}))$ , the above problem admits a unique solution ${\tilde{u}}_{0}$ by the Poincaré inequality and the Lax–Milgram theorem. □

Footnotes

Appendix

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