Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 4, 953–975

Homogenization of the Poisson Equation in a Thick Periodic Junction

T. A. Mel’nyk

Abstract. A convergence theorem and asymptotic estimates as ε → 0 are proved for a solution to a mixed boundary-value problem for the Poisson equation in a junction Ω_{ε} of a domain Ω_{0} and a large number N^{2} of ε-periodically situated thin cylinders with thickness of order ε = O( \frac{1}{N}). For this junction, we construct an extension operator and study its properties.

Keywords: Homogenization, asymptotic estimates, extension operators AMS subject classification: Primary 35 B 27, secondary 35 B 40, 35 B 25

0. Introduction

Let D_{ε} be a domain in ℝ^{n} which depends on a small parameter ε > 0 and, by the limit as ε → 0, is transformed to a submanifold S of dimension m. The number m is called the limit dimension of the domain D_{ε}. If m < n, then the domain D_{ε} is called thin. Asymptotic methods for thin domains are well-known.

Some years ago several papers appeared that deal with the asymptotic investigation of boundary-value problems in junctions consisting of a finite number of domains with different limit dimensions [1, 5, 11, 12, 22, 23]. Boundary-value problems in thick periodic junctions whose number of components increases as ε → 0 have own specific difficulties (see below), and until recently, there were no full asymptotic investigations of these problems. For these junctions we give the following classification:

A thick periodic junction Ω_{ε} of type m : k : d is a domain in ℝ^{n} that is obtained by joining a large number of ε-periodically situated thin domains with limit dimensions d to an external part of the boundary (which is the contact zone with limit dimension k ≤ m) of a domain Ω_{0} (which is the junction’s “body” with limit dimension m ≤ n). Here ε is a small parameter which depends on the number of the joined thin domains. The junction can have two or more “bodies”.

These junctions are prototypes of widely adapted engineering constructions such as long bridges on supports, frameworks of houses, industrial installations, spaceship grids as well as other physical systems with very distinct characteristic scales. The objective of studying boundary-value problems in thick periodic junctions is to describe the asymptotic behaviour of solutions as ε → 0, i.e. when the number of joined thin domains increases and their thickness decreases.

T. A. Mel’nyk: Inst. Math. A Univ., PF 801140, D-70511 Stuttgart ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

In the papers [10,29] the asymptotic behaviour of Green’s function of the Neumann problem for the Helmholtz equation in the unbounded junction of type 3 : 2 : 1 was studied. The limit equations describing acoustic vibrations in a porous medium, made by narrow parallel channels or thin parallel sheets in a solid body (the junctions of the type 3:2:1 or 3:2:2, respectively) were obtained in the papers [2, 7, 27]. In these articles the authors established some new qualitative properties of the homogenized equations: The corresponding Helmholtz equation is no longer elliptic and the operator which corresponds to the spectral limit problem is non-compact. These properties were the main difficulties in the asymptotic investigation of boundary-value problems in thick periodic junctions. As a result of these difficulties convergence theorems and asymptotic estimates were not obtained.

In the articles [15 - 20], using some results on the spectrum of discontinuous operator functions (see [8, 20]) and constructing special extension operators, the asymptotic behaviour of eigenvalues and eigenfunctions of the Neumann problems for the Laplace operator in junctions of different types 2 : 1 : 1, 3 : 2 : 2, 3 : 2 : 1, 3 : 1 : 1 was studied. The problem in the junction 3 : 1 : 1 was examined in the case when the limiting process as ε → 0 is accompanied by a concentration of masses on the joined thin domains [17]. These papers show that the type of a thick periodic junction defines a resulting boundary-value problem and junction-layer problems in the contact zone of this junction. However, the junction layers behave as powers at infinity and do not decrease exponentially. Therefore, they influence directly the principal terms of the asymptotics of the solution to the initial problem. The cause of this effect is in the modification of the geometrical structure of the domain, where junction-layer problems are considered. The type of a thick periodic junction defines also the construction scheme of an extension operator P_{ε} : H^{1}(Ω_{ε}) → H^{1}(Ω_{0} ∪ D_{0}), where D_{0} is a domain that is filled up by the joined thin domains in the limit as ε → 0.

Extension operators play an important role in the proofs of convergence theorems for boundary-value problems in domains depending on a small parameter. They give us the possibility to pass from a domain depending on a small parameter to a fixed domain that does not depend on one. For many problems, these operators have to be uniformly bounded with respect to a small parameter in the Sobolev space H^{1}. The uniformly boundedness of extension operators is the necessary condition in the statement of some problems (see [30]). For other problems, such extension operators exist, for example, for domains that are ε−periodically perforated by holes with the diameter of order ε [6, 25, 30]. But for thick periodic junctions, there exist no extension operators that are bounded uniformly in ε. This is one more difficulty in the research.

It should be emphasized here the difference between the asymptotic investigation of boundary-value problems in thick periodic junctions and domains with rapidly oscillating boundaries [4] (those results were also referred in [26: Subsection 3.3]). The main difference is the following: the function h, which defines the oscillating boundary, must be a continuously differentiable periodic function, i.e., the boundary is smooth, and there must exist the reciprocal functions of h to construct an extension operator. These conditions do not hold for periodic thick junction: there does not exist any function, which would define the boundary of the junction Ω_{ε}. This junction has only the Lipschitz boundary and the periodical structure of joining of thin domains only (in our case thin cylinders, the cylinder can have various length). Therefore, the scheme of the construction of the extension operator in [4] is not applicable for thick periodic junctions. Furthermore, if the right-hand side of a boundary-value problem depends on the small parameter ε, then we need the special condition on this right-hand side (see (1.3)) to construct the bounded extension of the solution, and to obtain the convergence results. Also in this paper, using the method of matched asymptotic expansions, we construct the leading terms for the asymptotics of the solution and obtain the asymptotic estimates of the difference between the solution to the initial problem and the solution to the limiting problem.

In Section 1 we formulate the boundary-value problem (the initial problem) and prove auxiliary inequalities. Section 2 deals with the construction of an extension operator provided that the right-hand side of the initial problem satisfies a special condition. This operator is uniformly bounded in ε for the solution to the initial problem. The construction of the extension operator for junctions of type 3 : 2 : 1 is both most complex with respect to the constructions of extensions for other types of thick periodic junctions and most general for ones in conception. In Section 3 we prove the convergence theorem. If the right-hand side has a special form, then we can construct an approximation function and obtain asymptotic estimates. This is done in Section 4. In this section we use some symmetry in the structure of the thick periodic junction. It helps us to define more exactly asymptotic relations for junctions-layer solutions, to detect other properties of ones, and to obtain better estimates for residuals of the approximation function.

1. The initial problem

1.1 Statement of the problem. Let a thick periodic junction Ω_{ε} of type 3 : 2 : 1 consist of the “body”

Ω_{0} = {x ∈ ℝ^{3} : x′ ∈ K and 0 < x_{3} < γ(x′)}

and a large number of thin cylinders

G_{ε} = ⋃_{i,j=0}^{N−1} G_{ε}(i, j),

where

G_{ε}(i, j) = \{x ∈ ℝ^{3} : (\frac{x_{1}}{ε} − i, \frac{x_{2}}{ε} − j) ∈ ω and − l < x_{3} ≤ 0\},

i.e. Ω_{ε} = Ω_{0} ∪ G_{ε}. Here x′ = (x_{1}, x_{2}), K = (0, a)^{2}, γ is a smooth positive function on \overline{K} and γ > γ_{0} = const > 0, N is a large positive integer and, therefore, the value ε = \frac{a}{N} is a small parameter which characterizes the distance between thin cylinders and their thickness; the plane domain ω with smooth boundary belongs together with its closure to the disk \{x′ ∈ ℝ^{2} : (x_{1} − \frac{1}{2})^{2} + (x_{2} − \frac{1}{2})^{2} < ρ_{0}^{2} < \frac{1}{4}\}.

Define Γ_{ε} = ∂Ω_{ε} ∩ {x_{3} = −l} and consider the mixed boundary-value problem

\begin{aligned}Δ_{x}u_{ε}(x) & = f_{ε}(x) \\ ∂_{ν}u_{ε}(x) & = 0 \\ u_{ε}(x) & = 0 \end{aligned}  \begin{matrix}(x ∈ Ω_{ε}) \\ (x ∈ ∂Ω_{ε} \ Γ_{ε}) \\ (x ∈ Γ_{ε}) \end{matrix} \}  (1.1)

in the junction Ω_{ε} with given Dirichlet data on the bases Γ_{ε} of the joined thin cylinders and the Neumann boundary condition on the remaining part of the surface ∂Ω_{ε}. Here ∂_{ν} = \frac{∂}{∂ν} is the outward normal derivative.

We can regard without loss of generality that the right-hand side f_{ε} belongs to L_{2}(Ω) where Ω is the interior of the union \overline{Ω}_{0} ∪ \overline{D}, D = K × (−l, 0) being a parallelepiped that is filled up by the thin cylinders G_{ε}(i, j) (i, j = 0, …, N − 1) in the limit as ε → 0 (N → ∞). We also assume that the function f_{ε} satisfies the following two conditions:

f_{ε} → f_{0}  in L_{2}(Ω) as ε → 0  (1.2)

and there exist positive constants C_{i}, r_{0}, ε_{0} such that for all ε ∈ (0, ε_{0})

∫_{Ω_{ε,r_{0}}^{(i)}} (F_{ε}^{(i)}(x))^{2}dx ≤ C_{i}  (i = 1, 2)  (1.3)

where

F_{ε}^{(i)}(x) = ε^{−1}(f̂_{ε}^{(i)}(x + εē_{i}) − f̂_{ε}^{(i)}(x)) (ē_{1} = (1, 0, 0), ē_{2} = (0, 1, 0)).

Here and further we interpret the symbol Ŷ^{(i)} as follows: if Y is a set, then Ŷ^{(i)} is the union of Y and of its image, symmetric with respect to the plane {x_{i} = 0}, and if Y is a function, then Ŷ^{(i)} is its even extension into the relevant domain with respect to the plane {x_{i} = 0}. Further, Ω_{0,r_{0}} = K × (0, r_{0}) and Ω_{ε,r_{0}} is the interior of the union \overline{Ω}_{0,r_{0}} ∪ \overline{G_{ε}}. Condition (1.3) means that the function f_{ε} has not strong scattering of the values on the neighboring cylinders.

Consider some example: The function

ϕ_{0}(x) = \{ \begin{matrix}0 & if x ∈ Ω_{0} \\ (−1)^{i+j}x_{3} & if x ∈ G_{ε}(i, j) (i, j = 0, …, N − 1) \end{matrix}

does not satisfy condition (1.3). For this function, the variation of its values on the neighboring thin cylinders is very big, i.e., if we construct any extension of this function in the Sobolev space H^{1}, then the gradient of the extension will be of order \frac{1}{ε}.

Let for a function g_{ε} ∈ L_{2}(Ω) there exist constants C_{0} > 0 and ε_{0} > 0 such that, for any ε ∈ (0, ε_{0}) and x′, x′′ ∈ Ω, |g_{ε}(x′) − g_{ε}(x′′)| ≤ C_{0}|x′ − x′′|. Then condition (1.3) holds for g_{ε}.

So, our aim is to describe the asymptotic behaviour of the solution u_{ε} to problem (1.1) as ε → 0 (N → +∞).

1.2 Auxiliary inequalities. Consider a space H^{1}(Ω_{ε}, Γ_{ε}) formed by functions of the Sobolev space H^{1}(Ω_{ε}) whose traces vanish on Γ_{ε}. In this subspace we introduce along with the usual norm ∥u∥_{1} = ( ∫_{Ω_{ε}}(|∇u|^{2} + u^{2}) dx)^{\frac{1}{2}} a new norm ∥ · ∥_{ε} that is generated by the scalar product

⟨u, v⟩_{ε} = ∫_{Ω_{ε}} ∇u · ∇v dx.  (1.4)

Denote the space H^{1}(Ω_{ε}, Γ_{ε}) with scalar product (1.4) by H_{ε}.

Lemma 1.1. For ε small enough, the norms ∥ · ∥_{1} and ∥ · ∥_{ε} are equivalent, i.e. there exist constants c_{1} > 0 and ε_{0} > 0 such that for all ε ∈ (0, ε_{0}) the inequalities

∥u∥_{ε} ≤ ∥u∥_{1} ≤ c_{1}∥u∥_{ε}  (u ∈ H_{ε})  (1.5)

hold.

Proof. In (1.5), it is not obvious only that the second inequality holds. Suppose the contrary. Then there exist sequences {ε_{m}}_{m≥1}, {v_{m}}_{m≥1} ∈ H_{ε_{m}} such that

lim_{m→0} ε_{m} = 0
∥v_{m}∥_{1} = 1  (1.6)
∥v_{m}∥_{ε_{m}}^{2} = ∫_{Ω_{εm}} |∇v_{m}|^{2}dx < \frac{1}{m}.  (1.7)

Since the sequence {v_{m}}_{m≥1} is bounded in H^{1}(Ω_{0}), we may assume without loss of generality that it is a Cauchy sequence in L_{2}(Ω_{0}). From inequality (1.7) it follows that {v_{m}}_{m≥1} is a Cauchy sequence also in H^{1}(Ω_{0}):

∥v_{m} − v_{n}; H^{1}(Ω_{0})∥^{2} ≤ ∥v_{m} − v_{n}; L_{2}(Ω_{0})∥^{2} + \frac{1}{m} + \frac{1}{n}.

Hence, {v_{m}}_{m≥1} converges in this space to some element v_{0} ∈ H^{1}(Ω_{0}). By virtue of the Friedrich inequality and inequality (1.7) we have

∫_{G_{εm}} v_{m}^{2}dx ≤ ∫_{Ω_{εm}} (\frac{∂v_{m}}{∂x_{3}} )^{2}dx < \frac{1}{m}.

Granting this, from (1.6) and (1.7) we obtain that

1 = ∥v_{m}∥_{1}^{2} → ∫_{Ω_{0}} v_{0}^{2}dx as m → ∞  and  ∫_{Ω_{0}} |∇v_{0}|^{2}dx = 0.

This means that v_{0} = _{\frac{1}{|Ω_{0}| \frac{1}{2} 3}} in Ω_{0}, where |Υ|_{n} is the measure of a domain Υ in ℝ^{n}.

On the one hand, from the trace theorem for functions in Sobolev spaces and [24: Corollary 1.7] it follows that

∫_{Q_{εm}} v_{m}^{2}dx → |ω|_{2} |Ω_{0}|_{3}^{−1}a^{2} as m → ∞

where Q_{ε} = G_{ε} ∩ {x_{3} = 0}. On the other hand, we have

∫_{Q_{εm}} v_{m}^{2}dx ≤ ∫_{G_{εm}} (\frac{∂v_{m}}{∂x_{3}} )^{2}dx < \frac{1}{m} → 0 as m → ∞.

The lemma is proved

Remark 1.1. It should be noted that here and further all constants c_{i} and C_{i} in asymptotic inequalities are independent of the parameter ε.

Using Lemma 1.1 we can state that the right-hand side of the integral identity

∫_{Ω_{ε}} ∇u_{ε}(x) · ∇φ(x) dx = − ∫_{Ω_{ε}} f_{ε}(x)φ(x) dx  (φ ∈ H^{1}(Ω_{ε}, Γ_{ε}))  (1.8)

for problem (1.1) defines a linear continuous functional in H^{1}(Ω_{ε}, Γ_{ε}) and, therefore, there exists a weak solution u_{ε} that is unique and satisfies the inequality

∥u_{ε}; H^{1}(Ω_{ε}, Γ_{ε})∥ ≤ c_{2}∥f_{ε}; L_{2}(Ω)∥ ≤ c_{3}.  (1.9)

Show that u_{ε} has not strong scattering of values on the neighboring cylinders. Let χ_{r_{0}} be a smooth cut-off function on ℝ^{n},

χ_{r_{0}}(x_{3}) = \{ \begin{matrix}0 & if x_{3} ≥ r_{0} \\ 1 & if x_{3} ≤ \frac{r_{0}}{2} \end{matrix} (r_{0} taken from (1.3)).

Define Γ_{r_{0}} = {x : x′ ∈ K and x_{3} = r_{0}}. Then the function v_{ε} = χ_{r_{0}}u_{ε} is a weak solution to the problem

\begin{aligned}Δ_{x}v_{ε}(x) & = Φ_{ε}(x) \\ Δ_{x}v_{ε}(x) & = f_{ε}(x) \\ v_{ε}(x′, r_{0}) & = 0 \\ v_{ε}(x) & = 0 \\ ∂_{ν}v_{ε}(x) & = 0 \end{aligned}  \begin{matrix}(x ∈ Ω_{0,r_{0}}) \\ (x ∈ G_{ε}) \\ ((x′, r_{0}) ∈ Γ_{r_{0}}) \\ (x ∈ Γ_{ε}) \\ (x ∈ ∂Ω_{ε,r_{0}} \ (Γ_{ε} ∪ Γ_{r_{0}})) \end{matrix} \}  (1.10)

where Φ_{ε} = χ_{r_{0}}f_{ε} + 2(χ_{r_{0}})′∂_{x_{3}}u_{ε} + (χ_{r_{0}})′′u_{ε}.

We extend problem (1.10) in the even way through the plane {x_{i} = 0} (i = 1, 2) and set 2a−periodicity conditions on the corresponding side of the parallelepiped Ω̂_{0,r_{0}}^{(i)}. Since the extended problem is invariant with respect to shifts by ε along the axis x_{i}, the function

V_{ε}^{(i)}(x) = \frac{v_{ε}^{(i)}(x + εē_{i}) − v_{ε}^{(i)}(x)}{ε}  (ē_{1} = (1, 0, 0), ē_{2} = (0, 1, 0))  (1.11)

that is 2a-periodic in x_{i} satisfies the relations

−Δ_{x}V_{ε}^{(i)}(x) = \frac{Φ̂^{(i)}(x + εē_{i}) − Φ̂^{(i)}(x)}{ε} (x ∈ Ω̂_{0,r_{0}}^{(i)})
−Δ_{x}V_{ε}^{(i)}(x) = \frac{f̂^{(i)}(x + εē_{i}) − f̂^{(i)}(x)}{ε} (x ∈ Ĝ_{ε}^{(i)})
V_{ε}^{(i)}(x′, r_{0}) = 0  (x′ ∈ K̂^{(i)})
V_{ε}^{(i)}(x) = 0  (x ∈ Γ̂_{ε}^{(i)})
∂_{ν}V_{ε}^{(i)}(x) = 0  (x ∈ ∂Ω̂_{ε,r_{0}}^{(i)} \ (Γ̂_{ε}^{(i)} ∪ Γ̂_{r_{0}}^{(i)} ∪ {x_{i} = ±a})

whence, using condition (1.3), Lemma 1.1, (1.9) and the second energy inequality [13], we deduce the inequalities (i = 1,2)

∥V_{ε}^{(i)}; H^{1}(Ω̂_{ε,r_{0}}^{(i)}, Γ̂_{ε}^{(i)})∥

≤ c_{i} (∥F_{ε}^{(i)}; L_{2}(Ω̂_{ε,r_{0}}^{(i)})∥ + ∥∂_{x_{i}}u_{ε}; L_{2}(Ω_{ε})∥ + ∥(χ_{r_{0}})′∂_{x_{i}x_{3}}u_{ε}; L_{2}(Ω_{ε})∥)

≤ c_{i} (∥F_{ε}^{(i)}; L_{2}(Ω̂_{ε,r_{0}}^{(i)})∥ + ∥f_{ε}; L_{2}(Ω_{ε})∥ + ∥u_{ε}; H^{1}(Ω_{ε}, Γ_{ε})∥)

≤ C_{i}.

(1.12)

2. The extension operator

Now our aim is to pass to the limit in identity (1.8). To this end we construct an extension operator. As we mentioned above, there exists no extension operator bounded uniformly in ε. In order to verify this assertion, it suffices to make use of the function ϕ_{0} from Subsection 1.1. But we can prove the following

Theorem 2.1. For the solution to problem (1.1) there exists an extension operator

P_{ε} : H^{1}(Ω_{ε}, Γ_{ε}) → H^{1}(Ω, Γ_{−l})

such that

∥P_{ε}u_{ε}; H^{1}(Ω, Γ_{−l})∥ ≤ C_{0} (∑_{k=1}^{2} ∥F_{ε}^{(k)}; L_{2}(Ω̂_{ε,r_{0}}^{(k)})∥ + ∥f_{ε}; L_{2}(Ω_{ε})∥)  (2.1)

where the space H^{1}(Ω, Γ_{−l}) is formed by functions of the Sobolev space H^{1}(Ω) whose traces vanish on Γ_{−l} = {x : x′ ∈ K and x_{3} = −l}.

To prove this theorem, just as in [20] for thin junctions, first we use an extension from a thin cylinder into a thin beam. Since the proof of this result was not given in the mentioned paper, we give it here.

Lemma 2.1. Let

T_{ε} = {x : x′ ∈ Σ_{ε} = εΣ and x_{3} ∈ (−l, 0)}
B_{ε} = {x : x′ ∈ S_{ε} = εS_{1} and x_{3} ∈ (−l, 0)}

where Σ is a plane domain with Lipschitz boundary belonging with its closure to the square S_{1} = {x′ : |x_{1}| < 1 and |x_{2}| < 1}. Then for all u ∈ H^{1}(T_{ε}) there exists a linear extension operator P_{ε} : H^{1}(T_{ε}) ↦ H^{1}(B_{ε}) such that

∫_{B_{ε}}(P_{ε}u)^{2}dx ≤ c_{0} ∫_{T_{ε}} u^{2}dx  (2.2)
∫_{B_{ε}} |∇_{x}P_{ε}u|^{2}dx ≤ c_{0} ∫_{T_{ε}} |∇_{x}u|^{2}dx.  (2.3)

Proof. Obviously, it is sufficient to prove inequalities (2.2) and (2.3) for smooth functions. Expand the thin domains T_{ε} and B_{ε} with respect to x_{1} and x_{2} in \frac{1}{ε} times. By ũ(η′, x_{3}) we denote the function u(εη′, x_{3}) with η′ = \frac{x′}{ε}. There exists a linear extension operator E_{1} : H^{1}(Σ) ↦ H^{1}(S_{1}) (see [6, 14, 21, 25, 28, 30]) such that

∥E_{1}(ũ(·, x_{3})); H^{i}(S_{1})∥ ≤ c_{1}∥ũ(·, x_{3}); H^{i}(Σ)∥  (i = 0, 1)  (2.5)

where the constant c_{1} is independent of both the function u and the variable x_{3}. By repeating the construction of the operator E_{1} in [21, 28] one can see that

∂_{x_{3}}(E_{1}(ũ(η′, x_{3}))) = E_{1}(∂_{x_{3}} ũ(η′, x_{3})).  (2.6)

Define the desired extension by the formula

(P_{ε}u)(x) = t(x_{3}) + E_{1}(ũ(η′, x_{3}) − t(x_{3}))|_{η′= \frac{x′}{ε}}

where

t(x_{3}) = \frac{1}{|Σ|_{2}} ∫_{Σ} ũ(η′, x_{3}) dη′.

Using (2.5) and (2.6), the Poincare inequality and the elementary estimate

|t(x_{3})| ≤ \frac{∥ũ(·, x_{3}); L_{2}(Σ)∥}{|Σ|2^{\frac{1}{2}}}

we can prove inequality (2.3) :

∫_{B_{ε}} |∇_{x′}P_{ε}u|^{2}dx
= ∫_{−l}^{0} ∫_{S_{1}} |∇_{η′}E_{1}(ũ(η′, x_{3}) − t(x_{3}))|^{2}dη′dx_{3}
≤ c_{1} ∫_{−l}^{0} (∫_{Σ} (ũ(η′, x_{3}) − t(x_{3}))^{2}dη′ + ∫_{Σ} |∇_{η′} ũ(η′, x_{3})|^{2}dη′) dx_{3}
≤ c_{2} ∫_{−l}^{0} ∫_{Σ} |∇_{η′} ũ(η′, x_{3})|^{2}dη′dx_{3}
= c_{2} ∫_{T_{ε}} |∇_{x′}u|^{2}dx

and

∫_{B_{ε}} (∂_{x_{3}}(P_{ε}u))^{2}dx
= ε^{2} ∫_{−l}^{0} ∫_{S_{1}} (t′(x_{3}) + E_{1}(∂_{x_{3}} ũ(η′, x_{3}) − t′(x_{3})))^{2}dη′dx_{3}
≤ ε^{2} ∫_{−l}^{0} (\frac{1}{|Σ|} ∫_{Σ} (∂_{x_{3}} û(η′, x_{3}))^{2}dη′ + c_{1} ∫_{Σ} (∂_{x_{3}} ũ(η′, x_{3}) − t′(x_{3}))^{2}dη′) dx_{3}
≤ c_{3} ∫_{T_{ε}}(∂_{x_{3}}u)^{2}dx.

Inequality (2.2) can be proved by analogy with the previous inequality

Remark 2.1. In just the same way one can prove the existence of an extension P_{ε}^{(1)} : H^{1}(B_{ε} \ T_{ε}) ↦ H^{1}(B_{ε}) with the same properties as the operator P_{ε}.

Proof of Theorem 2.1. This construction will be carried out in several steps. At first we prove that there exists an extension P_{ε}^{(1)} from each thin cylinder G_{ε}(i, j) into the thin beam

B_{ε}(i, j) = \{x : (\frac{x_{1}}{ε} − i, \frac{x_{2}}{ε} − j) ∈ B^{(h)} and − l < x_{3} ≤ 0\}

such that

∥P_{ε}^{(1)}u; H^{1}(Ω_{0} ∪ ( ∪_{i,j=0}^{N−1} B_{ε}(i, j)))∥ ≤ c_{1}∥u; H^{1}(Ω_{ε})∥  (u ∈ H^{1}(Ω_{ε})).  (2.7)

Here B^{(h)} = \{x′ : |x_{1} − \frac{1}{2}| < h and |x_{2} − \frac{1}{2}| < h\} is a square containing the domain ω, ρ_{0} < h < \frac{1}{2}. To this end, we use Lemma 2.1 to extend a function u to each cut beam B_{ε}(i, j) ∩ \{x : −l < x_{3} < −\frac{ε}{2}\}.

Next, we use the operator 𝔓_{ε} from [14] to extend the function u from the domain

\begin{matrix}\{x : |\frac{x_{1}}{ε} − i − \frac{1}{2}| < h, and |\frac{x_{2}}{ε} − i − \frac{1}{2}| < h (−ε < x_{3} < ε\} \\ \((B_{ε}(i, j) \ G_{ε}(i, j)) ∩ \{x : −\frac{ε}{2} < x_{3} < 0\}) \end{matrix}

to the domain

\{x : |\frac{x_{1}}{ε} − i − \frac{1}{2}| < h and |\frac{x_{2}}{ε} − i − \frac{1}{2}| < h − ε < x_{3} < ε\}

(i, j = 0, 1, …, N − 1). In [14], there are given two-side estimates for the norm of the extension operator 𝔓_{ε} acting in Sobolev spaces on the exterior or interior of a domain with small diameter of order ε. In our case, as follows from this paper, the H^{1}-norm of 𝔓_{ε} is uniformly bounded in ε, and estimates of the type as in Lemma 2.1 are valid.

For the next steps it is important that we construct the extension for the solution u_{ε} to problem (1.1). By u_{ε} we denote again the extended function P_{ε}^{(1)}u_{ε} for which estimate (2.7) is valid and which vanishes when x_{3} = −l. For each i ∈ {0, 1, …, N − 1} we extend the function u_{ε} from the system of thin beams B_{ε}(i, j) (j = 0, 1, …, N − 1) to the thin plate

B_{ε}(i, ∗) = \{x : |\frac{x_{1}}{ε} − i − \frac{1}{2}| < h, 0 < x_{2} < a, −l < x_{3} ≤ 0\}.

At first we extend the function u_{ε} to each domain

D̃_{ε}(i, j) = \{x : |\frac{x_{1}}{ε} − i − \frac{1}{2}| < h, h < \frac{x_{2}}{ε} − j − \frac{1}{2} < 1 − h, −l < x_{3} < −ε\}

(j = −1, 0, 1, …, N), that is situated between two cut beams

B̃_{ε}(i, j) = B_{ε}(i, j) ∩ {x : −l < x_{3} < −ε}  and  B̃_{ε}(i, j + 1)

using the “linear matching”

ŭ_{ε}(x) := a_{ij}(ε, x_{1}, x_{3}) + b_{ij}(ε, x_{1}, x_{3})(x_{2} − ε(h + j + \frac{1}{2}))

where

\begin{aligned}a_{ij}(ε, x_{1}, x_{3}) & = u_{ε}(x_{1}, ε(j + h + \frac{1}{2}), x_{3}) \\ b_{ij}(ε, x_{1}, x_{3}) & = \frac{1}{ε(1 − 2h)} (u_{ε}(x_{1}, ε(j − h + \frac{3}{2}), x_{3}) − a_{ij}(ε, x_{1}, x_{3})) . \end{aligned}

In the case of extreme beams, we perform the even extension of problem (1.1) through the plane {x_{2} = 0} and {x_{2} = a}, respectively. Estimating the norm of the function ŭ_{ε}, we get

\begin{matrix}∥ŭ_{ε}; H^{1}(D̃_{ε}(i, j))∥^{2} ≤ \\ c_{1} ∫_{ε(i−h+ \frac{1}{2} )}^{ε(i+h+ \frac{1}{2} )} ∫_{−l}^{−ε} (ε(a_{ij}^{2} + |∇_{x_{1}x_{3}}a_{ij}|^{2}) + ε^{3}(b_{ij}^{2} + |∇_{x_{1}x_{3}}b_{ij}|^{2}) + εb_{ij}^{2})dx_{3}dx_{1}. \end{matrix} (2.8)

From Lemma 2.1 and the inequalities

\begin{aligned}u^{2}(0) & ≤ \frac{2}{ε} ∫_{0}^{ε} u^{2}(t) dt + 2ε ∫_{0}^{ε}(u′)^{2}(t) dt \\ (u(ε) − u(0))^{2} & ≤ ε ∫_{0}^{ε}(u′)^{2}(t) dt (u ∈ H^{1}([0, ε])) \end{aligned}

we obtain

ε ∫_{ε(i−h+ \frac{1}{2} )}^{ε(i+h+ \frac{1}{2} )} ∫_{−l}^{−ε} a_{ij}^{2} dx_{3}dx_{1}
≤ c_{2} (∥u_{ε}; L_{2}(B̃_{ε}(i, j))∥^{2} + ε^{2}∥∂_{x_{2}}u_{ε}; L_{2}(B̃_{ε}(i, j))∥^{2})
≤ c_{3}∥u_{ε}; H^{1}(G̃_{ε}(i, j))∥^{2},

and

ε ∫_{ε(i−h+ \frac{1}{2} )}^{ε(i+h+ \frac{1}{2} )} ∫_{−l}^{−ε} b_{ij}^{2} dx_{3}dx_{1}

≤ c_{4} (∥V_{ε}^{(2)}; L_{2}(B̃_{ε}(i, j))∥^{2} + ε^{2}∥∂_{x_{2}}V_{ε}^{(2)}; L_{2}(B̃_{ε}(i, j))∥^{2} + ∥∂_{x_{2}}u_{ε}; L_{2}(B̃_{ε}(i, j))∥^{2})

≤ c_{5} (∥V_{ε}^{(2)}; H^{1}(G̃_{ε}(i, j))∥^{2} + ∥u_{ε}; H^{1}(G̃_{ε}(i, j))∥^{2})

where the function V_{ε}^{(2)} is defined by (1.11) and G̃_{ε}(i, j) = G_{ε}(i, j)∩{x : −l < x_{3} < −ε}. Using the second energy inequality [12] with the smooth cut-off function χ satisfying

χ_{ε}(x_{3}) = \{ \begin{matrix}0 & if x_{3} ≥ −\frac{ε}{2} \\ 1 & if x_{3} ≤ −ε \end{matrix}

we analogously estimate the other values in (2.8) as

ε ∫_{ε(i−h+ \frac{1}{2} )}^{ε(i+h+ \frac{1}{2} )} ∫_{−1}^{−ε} |∇_{x_{1}x_{3}}a_{ij}|^{2}dx_{3}dx_{1}
≤ c_{1} (∥∇u_{ε}; L_{2}(G̃_{ε}(i, j))∥^{2} + ε^{2}∥u_{ε}; H^{2}(G̃_{ε}(i, j))∥^{2})
≤ c_{2}(∥u_{ε}; H^{1}(G_{ε}(i, j))∥^{2} + ∥f_{ε}; L_{2}(G_{ε}(i, j))∥^{2}),

and

ε^{3} ∫_{ε(i−h+ \frac{1}{2} )}^{ε(i+h+ \frac{1}{2} )} ∫_{−1}^{−ε} |∇_{x_{1}x_{3}}b_{ij}|^{2}dx_{3}dx_{1}

≤ c_{3}ε^{2}(∥∇V_{ε}^{(2)}; L_{2}(G̃_{ε}(i, j))∥^{2} + ε^{2}∥V_{ε}^{(2)}; H^{2}(G̃_{ε}(i, j))∥^{2} + ∥u_{ε}; H^{2}(G̃_{ε}(i, j))∥^{2})

≤ c_{4}(∥V_{ε}^{(2)}; H^{1}(G_{ε}(i, j))∥^{2} + ∥u_{ε}; H^{1}(G_{ε}(i, j))∥^{2}

+ ∥F_{ε}^{(2)}; L_{2}(G_{ε}(i, j))∥^{2} + ∥f_{ε}; L_{2}(G_{ε}(i, j))∥^{2}).

Thus, the right-hand side of (2.8) is estimated by the sum of the terms

∥V_{ε}^{(2)}; H^{1}(G_{ε}(i, j))∥^{2}, ∥u_{ε}; H^{1}(G_{ε}(i, j))∥^{2}, ∥F_{ε}^{(2)}; L_{2}(G_{ε}(i, j))∥^{2} ∥f_{ε}; L_{2}(G_{ε}(i, j))∥^{2}.

To extend the function u_{ε} to the whole domain

D_{ε}(i, j) = \{x : |\frac{x_{1}}{ε} − i − \frac{1}{2}| < h, h < \frac{x_{2}}{ε} − j − \frac{1}{2} < 1 − h, −l < x_{3} ≤ 0\}

it is sufficient to use the periodicity of the domains D_{ε}(i, j) (j = −1, 0, 1, …, N) and Remark 2.1 with the domains

\begin{aligned}T_{ε} & = \{x : |\frac{x_{1}}{ε} − \frac{1}{2}| < h, h < \frac{x_{2}}{ε} − \frac{1}{2} < 1 − h, −ε < x_{3} < 0\} \\ B_{ε} & = \{x : |\frac{x_{1}}{ε} − \frac{1}{2}| < h, \frac{1}{2}ε < x_{2} < \frac{3}{2}ε, −\frac{3}{2}ε < x_{3} < \frac{1}{2}ε\}. \end{aligned}

As a result, we obtain the extension to the thin plate B_{ε}(i, ∗) (i = 0, 1, …, N − 1). Similarly we extend the function u_{ε} to each thin plate

B_{ε}(∗, j) = \{x : 0 < x_{1} < a, |\frac{x_{2}}{ε} − j − \frac{1}{2}| < h, −l < x_{3} ≤ 0\}

along the direction of the x_{1}-axis (j = 0, 1, …, N − 1).

Thus, on the second step we have constructed the extension P_{ε}^{(2)}u_{ε} (P_{ε}^{(2)}u_{ε} = 0 when x_{3} = −l) for which on the basis of conditions (1.2), (1.3) and inequalities (1.9), (1.12), (2.8) the estimate

\begin{matrix}∥P_{ε}^{(2)}u_{ε}; H^{1}(Ω_{0} ∪ ( ∪_{i,j=0}^{N−1} (B_{ε}(i, ∗) ∪ B_{ε}(∗, j))))∥ \\ ≤ c_{9} (∑_{i=k}^{2} ∥F_{ε}^{(k)}; L_{2}(Ω̂_{ε,r_{0}}^{(k)})∥ + ∥f_{ε}; L_{2}(Ω_{ε})∥) \end{matrix}  (2.9)

holds.

Now it remains to construct an extension P_{ε}^{(3)} into the thin beams

T_{ε}(i, j) = \{x : h < \frac{x_{1}}{ε} − i − \frac{1}{2} < 1 − h, h < \frac{x_{2}}{ε} − j − \frac{1}{2} < 1 − h, −l < x_{3} ≤ 0\}

(i, j = −1, 0, 1, …, N) just as previously, we perform the even extension of problem (1.1) through the planes {x_{i} = 0} and {x_{i} = a} (i = 1, 2) for the extreme beams. To this end, we use at first Remark 2.1 with the domains

\begin{aligned}T_{ε} & = T_{ε}(0, 0) ∩ \{x : x_{3} ≤ −ε\} \\ B_{ε} & = \{x : \frac{1}{2}ε < x_{1} < \frac{3}{2}ε, \frac{1}{2}ε < x_{2} < \frac{3}{2}ε, −l < x_{3} ≤ −ε\} \end{aligned}

to extend P_{ε}^{(2)}u_{ε} into the cut beams T_{ε}(i, j) ∩ {x_{3} ≤ −ε} (i, j = −1, 0, 1, …, N). Next, the extension P_{ε}^{(3)} of the function P_{ε}^{(2)}u_{ε} to parallelepipeds with the diameter of order ε is constructed in the same way as for perforated domains [6, 25, 30]. As we mentioned in the introduction, this extension is bounded uniformly with respect to the parameter ε in the Sobolev space H^{1}. Thus, according to what has been said and estimate (2.9), the extension operator P_{ε} := P_{ε}^{(3)} ∘ P_{ε}^{(2)} ∘ P_{ε}^{(1)} was constructed, and it satisfies estimate (2.1)

3. The convergence theorem

Its proof consists of several steps. First, using the extension operator P_{ε}, we pass to the limit in the integral identity (1.8). Next, selecting test-functions, we find the weak limit of P_{ε}u_{ε}, and conclude that this limit is the solution of a homogenized problem.

Theorem 3.1. Let conditions (1.2) and (1.3) hold. Then for the extension P_{ε}u_{ε} of the solution u_{ε} to problem (1.1) we have

P_{ε}u_{ε} → v_{0} weakly in H^{1}(Ω, Γ_{−l}) as ε → 0

where the function

v_{0}(x) = \{ \begin{matrix}v_{0}^{+}(x) & for x ∈ Ω_{0} \\ v_{0}^{−}(x) & for x ∈ D \end{matrix}  (3.1)

is a weak solution to the problem

\begin{aligned}Δv_{0}^{+}(x) & = f_{0}(x) \\ ∂_{x_{3}}^{2}v_{0}^{−}(x) & = f_{0}(x) \\ ∂_{ν}v_{0}^{+}(x) & = 0 \\ v_{0}^{−}(x′, −l) & = 0 \\ v_{0}^{+}(x′, 0) & = v_{0}^{−}(x′, 0) \\ ∂_{3}v_{0}^{+}(x′, 0) & = |ω|_{2} ∂_{3}v_{0}^{−}(x′, 0) \end{aligned}  \begin{matrix}(x ∈ Ω_{0}) \\ (x ∈ D) \\ (x ∈ ∂Ω_{0} \ K) \\ ((x′, −l) ∈ Γ_{−l}) \\ (x′ ∈ K) \\ (x′ ∈ K). \end{matrix} \}  (3.2)

Proof. Using the extension operator constructed in Theorem 2.1, we can rewrite the integral identity (1.8) as

\begin{matrix}∫_{Ω_{0}} ∇u_{ε} · ∇φ dx + ∫_{D} χ_{ω}(\frac{x′}{ε} )∇P_{ε}u_{ε} · ∇φ dx \\ = − ∫_{Ω_{0}} f_{ε}φ dx + ∫_{D} χ_{ω}(\frac{x′}{ε} )f_{ε}φ dx \end{matrix}  (φ ∈ H^{1}(Ω_{ε}, Γ_{−l}))  (3.3)

where χ_{ω}(η′) (η′ = (η_{1}, η_{2}) ∈ ℝ^{2}) is a 1-periodic function in η′ satisfying

η_{ω}(η′) = \{ \begin{matrix}1 & on ω \\ 0 & on [0, 1]^{2} \ ω. \end{matrix}

Due to conditions (1.2), (1.3) and inequality (2.1), the sequences

\{χ_{ω}(\frac{x′}{ε} )∂_{x_{i}}(P_{ε}u_{ε})\}  (i = 1, 2, 3)  (3.4)

are bounded in L_{2}(D). Therefore, we can choose a subsequence of {ε} (still denoted by {ε}) and pass to the limit ε → 0 in (3.3). We obtain

∫_{Ω_{0}} ∇v_{0}^{+}(x) · ∇φ(x) dx + ∫_{D} γ_{i}(x)∂_{x_{i}}φ(x) dx = − ∫_{Ω_{0}} f_{0}φ dx − |ω|_{2} ∫_{D} f_{0}φ dx (3.5)

for all φ ∈ H^{1}(Ω, Γ_{−l}) where γ_{i} (i = 1, 2, 3) are weak limits of sequences (3.4) in L_{2}(D), respectively, and v_{0} = v_{0}^{±} (see (3.1)) is a weak limit of the sequence {P_{ε}u_{ε}} in H^{1}(Ω, Γ_{−l}).

Next, we select test functions to find these values. Since

∫_{D} χ_{ω}(\frac{x′}{ε} )∂_{x_{3}}P_{ε}(u_{ε}(x))ϕ dx = − ∫_{D} χ_{ω}(\frac{x′}{ε} )P_{ε}(u_{ε}(x))∂_{x_{3}}ϕ dx

for all ϕ ∈ C_{0}^{∞}(D), we have

γ_{3}(x) = |ω|_{2} ∂_{x_{3}}v_{0}^{−}(x) (x ∈ D).

In order to determine γ_{1} and γ_{2}, we consider the integral identity (1.8) for problem (1.1) with the test function ψ_{i} defined by

ψ_{i}(x) = \{ \begin{matrix}0 & for x ∈ Ω_{0} \\ εY_{i}(\frac{x_{i}}{ε} )ϕ(x) & for x ∈ G_{ε} \end{matrix}

where ϕ ∈ C_{0}^{∞}(D) and Y_{i}(η_{i}) = −η_{i} + [η_{i}] + \frac{1}{2} (i = 1, 2) ([x] is the entire part of x); it is obvious that ψ_{i} ∈ H^{1}(Ω_{ε}, Γ_{ε}). As a result, we get

∫_{D} χ_{ω}(\frac{x′}{ε} )∂_{x_{i}}P_{ε}(u_{ε}(x))ϕ dx = O(ε) (ε → 0)

whence γ_{1} ≡ γ_{2} ≡ 0. Thus we have

∫_{Ω_{0}} ∇v_{0}^{+}(x) · ∇φ(x) dx + |ω|_{2} ∫_{D} ∂_{x_{3}}v_{0}^{−}(x)∂_{x_{3}}φ(x) dx = − ∫_{Ω_{0}} f_{0}φ dx − |ω|_{2} ∫_{D} f_{0}φ dx

for all φ ∈ H^{1}(Ω, Γ_{−l}). This integral identity means that the function v_{0} is a weak solution to problem (3.2). This problem is called resulting for problem (1.1). The operator that corresponds to problem (3.2) is non-compact as the differentiation in D is taken only with respect to x_{3}. It should be noted there are no boundary conditions on the vertical sides of D.

By solving the ordinary equation of problem (3.2) in the parallelepiped D with regard to the boundary condition on Γ_{−l} and the first transmission condition in the contact zone K, we find that

v_{0}^{−}(x) = ∫_{−l}^{x_{3}}(x_{3} − t)f_{0}(x′, t) dt + \frac{x_{3} + l}{l} (v_{0}^{+}(x′, 0) + ∫_{−l}^{0} tf_{0}(x′, t) dt) .  (3.6)

Now, according to the second transmission condition in problem (3.2), we obtain the problem

\begin{aligned}Δv_{0}^{+}(x) & = f_{0}(x) \\ ∂_{ν}v_{0}^{+}(x) & = 0 \\ ∂_{3}v^{+}(x′, 0) − |ω|_{2}l^{−1}v^{+}(x′, 0) & = f̂_{0}(x′) \end{aligned}  \begin{matrix}(x ∈ Ω_{0}) \\ (x ∈ ∂Ω_{0} \ K) \\ (x′ ∈ K) \end{matrix} \}  (3.7)

where

f̂_{0}(x′) = \frac{|ω|_{2}}{l} ∫_{−l}^{0}(l + x_{3})f_{0}(x′, x_{3}) dx_{3}.

Obviously, problem (3.7) has a unique solution. It means that there is also a unique solution to problem (3.2).

Due to the uniqueness of the solution to problem (3.2), the above reasoning holds for any subsequence of {ε} chosen at the beginning of the proof. Therefore, the theorem is proved

4. Asymptotic estimates

In this section we suppose that the function f_{ε} satisfies only the condition

f_{ε}(x) = f_{0}(x) + εf_{1}(ε, x)  (x ∈ Ω)  (4.1)

where

f_{0}, f_{1} ∈ L_{2}(Ω)  and  ∥f_{1}(ε, ·); L_{2}(Ω)∥ = O(1) (ε → 0)

and the plane domain ω is symmetric with respect to the straight lines \{x_{i} = \frac{1}{2}\} (i = 1, 2).

4.1 Formal derivation of the resulting problem using asymptotic expansions. Combining the algorithm of constructing asymptotics in thin domains with methods of the homogenization theory (see [3] and others), we seek the main terms of the asymptotics for the solution u_{ε} restricted to Ω_{0} in the form

u_{ε} ≈ v_{0}^{+}(x) + ∑_{k=1}^{∞} ε^{k}v_{k}^{+}(x, ε)  (4.2)

and, restricted to G_{ε}(i, j), in the form

u_{ε} ≈ v_{0}^{−}(x) + ∑_{k=1}^{∞} ε^{k}v_{k}^{−}(x, η_{1}^{(i)}, η_{2}^{(j)})  (η_{1}^{(i)} = \frac{x_{1}}{ε} − i, η_{2}^{(j)} = \frac{x_{2}}{ε} − j).  (4.3)

Decomposing formally the function v_{k}^{−} in the Taylor series with respect to the variables x_{1} and x_{2} in a neighborhood of the pont x_{1}^{(i)} = ε(i + \frac{1}{2}) and x_{2}^{(j)} = ε(j + \frac{1}{2}), we rewrite (4.3) in the form

u_{ε} = v_{0}^{−}(i, j, x_{3}) + ∑_{k=1}^{2} ε^{k}V_{k}^{i,j}(x_{3}, η_{1}^{(i)}, η_{2}^{(j)}) + O(ε^{3})  (x ∈ G_{ε}(i, j))  (4.4)

where

V_{k}^{i,j} = ∑_{m=0}^{k} \frac{1}{m!} ((η_{1}^{(i)} − \frac{1}{2})\frac{∂}{∂x_{1}} + (η_{2}^{(j)} − \frac{1}{2})\frac{∂}{∂x_{2}} )^{m} v_{k−m}^{−}(i, j, x_{3}, η_{1}^{(i)}, η_{2}^{(j)}) (4.5)

and v_{k}^{−}(i, j, …) = v_{k}^{−}(x_{1}^{(i)}, x_{2}^{(j)}, …). Let us substitute (4.4) into (1.1) instead of u_{ε}. Collecting and equating to 0 the coefficients by equal degrees of ε, we obtain

\begin{aligned}Δ_{η′}V_{1}^{i,j}(x_{3}, η′) & = 0 \\ ∂_{ν_{η′}} V_{1}^{i,j}(x_{3}, η′) & = 0 \end{aligned}  \begin{matrix}(η′ ∈ ω) \\ (∂_{ν_{η′}} V_{1}^{i,j}(x_{3}, η′) = 0 \end{matrix} \}  (4.6)

and

\begin{aligned}Δ_{η′}V_{2}^{i,j}(x_{3}, η′) + ∂_{x_{3}}^{2}v_{0}^{−}(i, j, x_{3}) & = f_{0}(i, j, x_{3}) \\ ∂_{ν_{η′}} V_{2}^{i,j}(x_{3}, η′) & = 0 \end{aligned}  \begin{matrix}(η′ & ∈ ω) \\ (η′ & ∈ ∂ω). \end{matrix} \}  (4.7)

Here η′ = (η_{1}^{(i)}, η_{2}^{(j)}), the variable x_{3} ∈ (−l, 0) is regarded as a parameter in these problems. From (4.6), it follows that the function V_{1}^{i,j} does not depend on η′. We restrict ourselves to the leading term of the asymptotics, and thus set V_{1}^{i,j} = 0. Then, by virtue of (4.5), we have

v_{1}^{−}(i, j, x_{3}, η′) = −∂_{x_{1}}v_{0}^{−}(i, j, x_{3})(η_{1}^{(i)} − \frac{1}{2}) − ∂_{x_{2}}v_{0}^{−}(i, j, x_{3})(η_{2}^{(j)} − \frac{1}{2}).  (4.8)

The solvability condition for (4.7) is given by the ordinary differential equation with respect to x_{3} that is present in problem (3.2), and the points (x_{1}^{(i)}, x_{2}^{(j)}) (i, j = 0, 1, …, N − 1) are regarded as parameters in this equation. Since these points make up the ε-net in K, we can spread this equation in all points of the square K.

It is understandable what equation we will obtain for the function v_{0}^{+} from (4.2) if we substitute (4.2) into (1.1). It remains to provide the continuity of the asymptotic approximation and their gradients in the contact zone K. It is doubtless that

v_{0}^{+}(x′, 0) = v_{0}^{−}(x′, 0) (x′ ∈ K).

To get the second transmission condition in (3.2) we can use the method of matched asymptotic expansion: the outer expansions are given by (4.2) and (4.3); the main terms of the inner one takes the form

u_{ε} ≈ v_{0}^{+}(x′, 0) + ε ∑_{i=1}^{3} Z_{i}(η)∂_{x_{i}}v_{0}^{+}(x′, 0) + …  (4.9)

where η = \frac{x}{ε} and \{Z_{i}\} are junction-layer solutions which we will consider in the following section. Thus, the first term v_{0}^{±} in asymptotics (4.2) and (4.3) is a solution to the resulting problem (3.2).

4.2 Junction-layer problems. Let us introduce the “rapid” coordinates η = \frac{x}{ε} in problem (1.1). Passing to ε = 0 , we see that the cylinder G_{ε}(0, 0) transforms into the semi-infinite cylinder

Π^{−} = ω × (−∞, 0]

and the set Ω_{0} transforms into the first octant {η_{i} > 0 (i = 1, 2, 3)}. Taking into account the periodicity of the cylinders G_{ε}(i, j) (i, j = 0, …, N −1) we can regard that the union Π of the semi-cylinders Π^{−} and

Π^{+} = (0, 1) × (0, 1) × (0, +∞)

is a base domain in which the junction-layer problems have to be considered. Obviously, solutions of these junction-layer problems must be 1-periodic in η_{1} and η_{2}, i.e.

∂_{η_{τ}}^{k} Z(η)|_{η_{τ} =0} = ∂_{η_{τ}}^{k} Z(η)|_{η_{τ} =1}  (η ∈ ∂Π^{+}, η_{3} > 0; k = 0, 1; τ = 1, 2).  (4.10)

Let us investigate some properties of solutions to the following junction-layer problem:

\begin{aligned}−Δ_{ηη}Z(η) & = F (η) \\ ∂_{ν_{η}}Z(η) & = B(η) \\ ∂_{η_{3}}Z(η′, 0) & = 0 \\ ∂_{η_{τ}}^{k} Z(η)|_{η_{τ} =0} & = ∂_{η_{τ}}^{k} Z(η)|_{η_{τ} =1} \end{aligned}  \begin{matrix}(η ∈ Π) \\ (η ∈ ∂Π^{−} \ ω \\ (η′, 0) ∈ ∂Π^{+} \ ω) \\ (η ∈ ∂Π^{+}, η_{3} > 0; k = 0, 1; τ = 1, 2). \end{matrix} \} (4.11)

At first we study the solvability of this problem. In this connection we use the scheme given in [22]. Let Ĉ_{0}^{∞}(\overline{Π}) be a space of infinitely differentiable functions in \overline{Π} that satisfy the periodical conditions (4.10) and are finite in η_{3}, i.e.

∀ v ∈ Ĉ_{0}^{∞}(\overline{Π}) ∃ R > 0 ∀η ∈ \overline{Π} |η_{3}| ≥ R : v(η) = 0.

Let H be the completion of the space Ĉ_{0}^{∞}(\overline{Π}) by the norm

∥u∥_{H} = (∥∇_{η}u∥_{L_{2}(Π)}^{2} + ∥ρu∥_{L_{2}(Π)}^{2})^{\frac{1}{2}}

where ρ(η_{3}) = \frac{1}{1+|η_{3}|} (η_{3} ∈ ℝ). We will call a function Z a generalized solution to problem (4.11) if for all functions v ∈ H the integral identity

∫_{Π} ∇_{η}Z · ∇_{η}v dη = ∫_{Π} F v dη + ∫_{∂Π^{−}\ω} Bv dη  (4.12)

holds.

Lemma 4.1. Let \frac{1}{ρ}F ∈ L_{2}(Π) and \frac{1}{ρ}B ∈ L_{2}(∂Π^{−} \ ω), and let
∫_{Π} F (η) dη + ∫_{∂Π^{−}\ω} B(η) dσ_{η} = 0.  (4.13)

Then there exists a solution Z ∈ H to problem (4.11) that is defined up to an additive constant.

Proof. We rewrite identity (4.12) in the form

⟨Z, v⟩ − ∫_{Π_{−2,2}} Zv dη = ∫_{Π} F v dη + ∫_{∂Π^{−}\ω} Bv dη  (4.14)

where

Π_{α,β} = {η ∈ Π : α < η_{3} < β}

and

⟨u, v⟩ = ∫_{Π} ∇_{η}u · ∇_{η}v dη + ∫_{Π_{−2,2}} uv dη.  (4.15)

Then the new scalar product (4.15) generates an equivalent norm in H. It is obvious that ⟨u, u⟩ ≤ c_{1}∥u∥_{H}^{2} (u ∈ H). The inverse inequality with another constant follows from the Hardy inequality

∫_{0}^{+∞} \frac{ϕ^{2}(η_{3})}{(1 + η_{3})^{2}} dη_{3} ≤ 4 ∫_{0}^{+∞} |∂_{η_{3}}ϕ|^{2} dη_{3} (ϕ ∈ C^{1}([0, +∞)) with ϕ(0) = 0)

and the inequality

\begin{matrix}∫_{Π} ρ^{2}(η_{3})u^{2}(η) dη \\ ≤ ∫_{Π_{−2,2}} ρ^{2}u^{2} dη + ∫_{Π} ρ^{2}((1 − χ(η_{3}))u)^{2}dη \\ ≤ ∫_{Π_{−2,2}} ρ^{2}u^{2} dη + c_{χ} (∫_{Π}(∂_{η_{3}}u)^{2} dη + ∫_{Π_{−2,2}}(χ′(η_{3})u)^{2} dη) \\ ≤ c_{2}⟨u, u⟩ \end{matrix}  (4.16)

where χ ∈ C^{∞}(ℝ), 0 ≤ χ ≤ 1 and

χ(η_{3}) = \{ \begin{matrix}1 & if |η_{3}| ≤ 1 \\ 0 & if |η_{3}| ≥ 2. \end{matrix}  (4.17)

Due to the conditions of Lemma 4.1, inequality (4.16) and the inequality

\begin{matrix}∫_{∂Π^{−}\ω} ρ^{2}(η_{3})v^{2}(η) dσ_{η} & ≤ ∫_{−∞}^{0} ρ(η_{3}) ∫_{∂ω} v^{2} dσ_{η′} dη_{3} \\ & ≤ c ∫_{Π} (|∇_{η}v|^{2} + ρ(η_{3})v^{2})dη \end{matrix} (v ∈ H)

the right-hand side of identity (4.14) defines a linear continuous functional in H. As the embedding H ⊂ L_{2}(Π_{−2,2}) is compact, there exists a self-adjoint positive compact operator A : H ↦ H such that

⟨Au, v⟩ = ∫_{Π_{−2,2}} u(η)v(η) dη ({u, v} ∈ H).

Thus, we can rewrite identity (4.14) as operator equation

Z − AZ = f

and apply Fredholm theorems to it. It is obvious that every solution of the homogeneous problem (4.11) in the space H is constant (its Dirichlet integral is trivial). Therefore, equality (4.13) is the solvability condition for problem (4.11)

Remark 4.1. Let exp(δ_{0}|η_{3}|)F ∈ L_{2}(Π) and exp(−δ_{0}η_{3})B ∈ L_{2}(∂Π^{−}\ω) (δ_{0} > 0). Taking into account the properties of solutions to elliptic problems in semi-cylinders, we can select a solution Z̃ of problem (4.11) such that

exp(−δ_{1}η_{3})Z̃ ∈ H^{1}(Π^{−})  (4.18)

where δ_{1} is an arbitrary number that satisfies the inequalities 0 < δ_{1} < δ_{0} and δ_{1} < \sqrt{Λ(ω)}; Λ(ω) is the first positive eigenvalue of the Neumann problem in the plane domain ω. It is clear that the solution Z̃ has the following asymptotics in the semicylinder Π^{+}:

Z̃(η) = C + O( exp(−δ_{2}η_{3}))  (η_{3} → +∞).  (4.19)

Remark 4.2. If the functions F and B from Remark 4.1 are even or odd in any of the variables \{η_{1}, η_{2}\} with respect to \frac{1}{2}, then the solution Z̃ has the same symmetry. In fact, let for example F and B be even in η_{1} with respect to \frac{1}{2}, i.e.

F (η_{1}, η_{2}, η_{3}) = F (1 − η_{1}, η_{2}, η_{3}) and B(η_{1}, η_{2}, η_{3}) = B(1 − η_{1}, η_{2}, η_{3}).

Then, due to the symmetry of the domain ω and using the substitution η_{1} = 1 − η′_{1} in problem (4.11), we obtain that the difference Z̃(η_{1}, η_{2}, η_{3})−Z̃(1−η_{1}, η_{2}, η_{3}) is a solution of the homogeneous problem (4.11), and relation (4.18) is satisfied for it. By virtue of the uniqueness of such a solution, it follows that this difference vanishes.

Corollary 4.1. The homogeneous problem (4.11) has a solution Ξ_{0} ∉ H with the asymptotics

Ξ_{0}(η) = \{ \begin{matrix}C_{ω} + η_{3} + O(exp(−δ_{3}η_{3})) & as η_{3} → +∞ \\ |ω|_{2}^{−1}η_{3} + O(exp(δ_{3}η_{3})) & as η_{3} → −∞ \end{matrix}  (4.20)

and this solution is even in η_{1}, η_{2} with respect to 1/2; δ_{3} > 0.

Proof. The solution Ξ_{0} is sought in the form of a sum

Ξ_{0}(η) = χ_{+}(η_{3})η_{3} + \frac{η_{3}}{|ω|_{2}} χ_{−}(η_{3}) + Z̃_{0}

where Z̃_{0} ∈ H and Z̃_{0} is the solution to problem (4.11) with right-hand sides

F (η) = 2χ′_{+}(η_{3}) + χ′′_{+}(η_{3})η_{3} + \frac{2}{|ω|_{2}} χ′_{−}(η_{3}) + \frac{1}{|ω|_{2}} χ′′_{−}(η_{3})η_{3} =: F_{ω}(η_{3})

and B = 0. Here χ_{+} is a smooth cut-off function with

χ_{+} = \{ \begin{matrix}1 & when η_{3} ≥ 2 \\ 0 & when η_{3} ≤ 1 \end{matrix}

and χ_{−}(η_{3}) = χ_{+}(−η_{3}) (η_{3} < 0). By virtue of Lemma 4.1 and Remarks 4.1 and 4.2, there exists a unique solution Z̃_{0} ∈ H to such problem that is even in η_{1} and η_{2} with respect to \frac{1}{2} and has the asymptotics

Z̃_{0}(η) = \{ \begin{matrix}C_{ω} + O( exp(−δ_{3}η_{3})) & as η_{3} → +∞ \\ O( exp(δ_{3}η_{3})) & as η_{3} → −∞. \end{matrix}  (4.21)

In order to find the constant C_{ω} in (4.21), it is necessary to substitute the function Ξ_{0} and Z̃_{0} into the Green formula

∫_{Π_{−R,R}} (Ξ_{0}ΔZ̃_{0} − Z̃_{0}ΔΞ_{0}) dη = ∫_{∂Π_{−R,R}} (Ξ_{0}∂_{ν_{η}} Z̃_{0} − Z̃_{0}∂_{ν_{η}}Ξ_{0}) dη

and to pass to the limit as R → ∞. As a result, we obtain C_{ω} = ∫_{Π} Ξ_{0}(η)F_{ω}(η_{3}) dη

Remark 4.3. By analogy we can show that the constant C in (4.19) equals

C = ∫_{Π} Ξ_{0}(η)F (η) dη + ∫_{∂Π^{−}\ω} Ξ_{0}(η)B(η) dσ_{η}.  (4.22)

Let us substitute (4.9) into (1.1) instead of u_{ε}. Collecting and equating to 0 the coefficients by equal degrees of ε, we get the problems for the functions Z_{i}. The conditions at infinity for ones follow from the conditions of the matching of the outer and inner expansions (4.2), (4.3) and (4.9). As a result, we obtain that Z_{3} = Ξ_{0} and Z_{1}, Z_{2} are equal to

Z_{i}(η) = χ_{−}(η_{3})(−η_{i} + \frac{1}{2}) + Z̃_{i}(η) (η ∈ Π)

where Z̃_{i} (i = 1, 2) are solutions to problem (4.11) with right-hand sides

\begin{aligned}F_{i} & = χ′′_{−}(η_{3})(−η_{i} + \frac{1}{2})  (η ∈ Π) \\ B_{i} & = −(1 − χ_{−}(η_{3}))ν_{i}(η)  (η ∈ ∂Π^{−} \ ω). \end{aligned}

Remark 4.4. Since the functions F_{i} and B_{i} are odd in η_{i} and even in η_{i+(−1)^{i+1}} with respect to \frac{1}{2}, then, due to the above-mentioned results, it follows that, in the first place, the solvability condition for these problems is realized. In the second place, the correspondent constants in (4.19) for the solutions Z̃_{1} and Z̃_{2} are equal to zero. And, in the third place, the functions Z_{1} and Z_{2} have the same symmetry, i.e. with respect to \frac{1}{2} the function Z_{1} is odd in η_{1} and even in η_{2}, and the function Z_{2} is even in η_{1} and odd in η_{2}.

Remark 4.5. The main asymptotic relations for the functions Z_{i} can be obtained from general results about the asymptotic behaviour of solutions to elliptic problems in domains with different exits to infinity [9, 24]. In our case, using the symmetry of the domain ω and the existence theorem for the concrete problem, we can define more exactly the asymptotic relations and detect other properties of the junction-layer solutions. These properties help us to find residuals of an approximation function U_{ε} in the equation and in the boundary conditions of problem (1.1) and to obtain better estimates for ones.

4.3 The asymptotic approximation. Let v_{0} be a unique solution to the resulting problem (3.2), i.e. v_{0}^{−} is defined by (3.6) and v_{0}^{+} is a solution to problem (3.7). Matching the outer and inner expansions for the solution u_{ε}, we construct the global asymptotic approximation function U_{ε} ∈ H^{1}(Ω_{ε}, Γ_{ε}):

U_{ε}(x) = v_{0}^{+}(x) + εχ_{0}(x_{3}) ∑_{i=1}^{3} (Z_{i}(η) − δ_{i,3}η_{3})∂_{x_{i}}v_{0}^{+}(x′, 0)  (η = \frac{x}{ε}, x ∈ Ω_{0}) (4.23)

and

\begin{matrix}U_{ε}(x) & = v_{0}^{−}(x) + ε(Y_{1}(η_{1})∂_{x_{1}}v_{0}^{−}(x) + Y_{2}(η_{2})∂_{x_{2}}v_{0}^{−}(x) \\ & + χ_{0}(x_{3}) ∑_{i=1}^{3} (Z_{i}(η) − δ_{i,1}Y_{1}(η_{1}) − δ_{i,2}Y_{2}(η_{2}) \\ & − δ_{i,3}|\frac{η_{3}}{|ω|_{2}} )∂_{x_{i}}v_{0}^{+}(x′, 0)) \end{matrix} (η = \frac{x}{ε}, x ∈ G_{ε}). (4.24)

Here χ_{0}(x_{3}) = χ(\frac{2}{r_{0}} x_{3}) (x_{3} ∈ ℝ) (the function χ is defined by (4.17) and Y_{i}(η_{i}) = −η_{i} + [η_{i}] + \frac{1}{2} (see (4.8)). Substituting U_{ε} into problem (1.1) in place of u_{ε} we find that, for any ψ ∈ H^{1}(Ω_{ε}, Γ_{ε}),

∫_{Ω_{ε}} (∇U_{ε} · ∇ψ + f_{ε}ψ) dx = F_{ε}(ψ)  (4.25)

and

|F_{ε}(ψ)| ≤ c(δ) ε^{1−δ}∥ψ; H^{1}(Ω_{ε}, Γ_{ε})∥,  (4.26)

where δ is arbitrary positive number. Using Lemma 1.1 and the integral identity (1.8), we get from (4.25) and (4.26) the following

Homogenization of the Poisson Equation 973

Theorem 4.3. The difference between the solution u_{ε} to problem (1.1) with righthand side (4.1) and the approximation function U_{ε} satisfies the estimate

∥u_{ε} − U_{ε}; H^{1}(Ω_{ε}, Γ_{ε})∥ ≤ c_{1}(δ) ε^{1−δ}  (δ > 0).  (4.27)

Corollary 4.2. It follows from (4.27) that

∥u_{ε} − v_{0}; L_{2}(Ω_{ε})∥ ≤ c_{2}(δ) ε^{1−δ}  (δ > 0)  (4.28)

where v_{0} is the solution to the resulting problem.

4.4 Conclusion. The symmetry of the domain ω with respect to the straight lines \{x_{1} = \frac{1}{2}\} and \{x_{2} = \frac{1}{2}\} is only a technical condition. It helps us to avoid awkward calculation and to obtain better estimates for residuals of the approximation function U_{ε}. If the right-hand side has the form f_{ε} = ∑_{k=0}^{∞} ε^{k}f_{k}(x), then we can define the other terms in the asymptotic expansions (4.2), (4.3) and (4.9).

From asymptotic estimates (4.27) and (4.28) we come to the following conclusion: For applied problems or for numerical calculation in thick periodic junctions, we can use the resulting problem instead of the initial problem with sufficient probability. The similar boundary-value problem for the elasticity equations in thick periodic junction of type 3 : 2 : 1 is prepared for publication.

I would like to note that the thin cylinders G_{ε} in the junction Ω_{ε} have the same length. It was done only for simplification of the presentations. In the case when the cylinders have various length, and their bases describe some surface γ_{−}(x′), x′ ∈ K; we must perform the odd extension of the solution to problem (1.1); and then construct the extension P_{ε}u_{ε} in the domain Ω_{0} ∩ {x′ ∈ K, γ_{−}(x′) < x_{3} ≤ 0} as in Section 2. It should be also noted that extension operators for spectral boundary-value problems in thick periodic junction are uniformly bounded in ε only on finite linear combinations of eigenfunctions. For spectral problem we are not need of additional conditions like condition (1.3) (see [16, 20]).

Acknowledgment. The author is grateful to the Alexander von Humboldt Foundation and Prof. W. L. Wendland for the possibility to carry out this research at the University of Stuttgart.

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Received 22.01.1999