A Corrector Result for the Homogenization of a Class of ... - IAENG

Proceedings of the World Congress on Engineering 2016 Vol I WCE 2016, June 29 - July 1, 2016, London, U.K.

A Corrector Result for the Homogenization of a Class of Nonlinear PDE in Perforated Domains Bituin Cabarrubias

Abstract—This paper is devoted to the corrector of the homogenization of a quasilinear elliptic problem with oscillating coefficients in a periodically perforated domain. A nonlinear Robin condition is prescribed on the boundary of the holes, which depends on a real parameter γ ≥ 1. We suppose that the data satisfy some suitable assumptions to ensure the existence and uniqueness of a weak solution of the problem. The periodic unfolding method is used to prove the result. Index Terms—correctors, homogenization, nonlinear, unfolding method, quasilinear.

I. I NTRODUCTION

I

N this paper, we study the corrector of the homogenization of a quasilinear elliptic problem with oscillating coefficients posed in a periodically perforated domain. We assume that the holes are of the same size as the period and on the boundary of the holes, we prescribe a nonlinear Robin condition, which depend on a real parameter γ ≥ 1. Some suitable growth conditions are also assumed on the nonlinear boundary term while a weaker than a Lipschitz condition is prescribed on the quasilinear term. The assumptions used here are the same as those considered in [9] and [16] (see also [4]) for the existence and uniqueness of the weak solution of the problem to hold. The physical motivation is that, in several composites the thermal conductivity depends in a nonlinear way from the temperature itself like the case of a glass or wood, where the conductivity is nonlinearly increasing with the temperature, as well as ceramics, where it is decreasing, or aluminium and semi-conductors, where the dependence is not even monotone (see [1], [2] and [23] for details). On the other hand, nonlinear Robin conditions appear in several physical situations like in some chemical reactions (see for instance [17]) or climatization (see [24]). To prove the main result in this work, we apply the Periodic Unfolding Method (PUM), a method of homogenization recently formed for the periodic case. It was first introduced in [11] for fixed domains (see also [12] for a general setting and detailed proofs and [20] for more simple approach), extended to perforated domains in [14] (see also [15] for complete proofs and [16] for more applications), to more general situations and comprehensive presentation in [10] and to time-dependent functions in [22]. Some nice properties of this method are: it only deals with the classical notion of convergences in Lp -spaces; any function in the perforated domain is mapped to the unfolded function in a fixed domain; and one do not need any extensions operators anymore when dealing with nonhomogeneous boundary conditions, like the case in this work. Manuscript received: March 12, 2016. B. Cabarrubias is with the Institute of Mathematics, University of the Philippines Diliman, 1101 Diliman, Quezon City, PHILIPPINES e-mail: [email protected].

ISBN: 978-988-19253-0-5 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

The correctors for the homogenizations of a class of linear elliptic problem in a periodically perforated domain when the oscillating matrix field depends on a weakly converging sequence, a linear elliptic problem with Dirichlet condition in a fixed domain and a linear elliptic problem with Robin boundary conditions in a perforated domain have been done in [8], [12] and [15], respectively, via PUM. Some corrector results were also obtained by applying mainly some lemmas, for composites with imperfect interface in [21] (see also the references therein). One can also refer to [18] for a corrector result for H-converging parabolic problems with time-dependent coefficients via Tartar’s oscillating test functions. The reader can also see [6] and the references therein, for the corrector of some wave equation with discontinuous coefficients in time. For the correctors for linear Dirichlet problems with simultaneously varying operators and domains obtained by using some special test function depending on the varying matrices and domains, see [19]. The reader is also referred to [3] and the references therein, for the correctors for the homogenizations of the wave and heat equations and to [7] for a corrector result for the wave equation with high oscillating periodic coefficients. Let us also mention here, that the existence and uniqueness of a solution of the problem as well as the homogenization were already studied in [4] and [5], respectively. This paper is organized as follows: Section 2 gives the geometric setting of the problem as well as the assumptions on the data to ensure existence and uniqueness of the weak solution of our problem; Section 3 contains a short discussion on PUM together with the operators and the corresponding properties that we need to prove the main result; homogenization results for the problem obtained in [5], for which the corrector result is based, are also recalled in Section 4; and the main results for this paper, which completes the study of the asymptotic bahavior in [5], are presented in Section 5. II. S ETTING OF THE PROBLEM Let us recall the geometric framework for the perforated domain (see e.g. [14]). Let b = (b1 , b2 , . . . , bN ) be a basis of RN (the set of reference periods) and Y a subset of RN such that,   N X X X Y + RN = kj bj  = (Y + ξ) , j=1 k∈ZN ξ∈G where ( G=

ξ ∈ RN | ξ =

N X

) ki bi , (k1 , . . . , kN ) ∈ ZN

,

i=1

and Ξε = {ξ ∈ G, ε(ξ + Y ) ⊂ Ω} .

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Proceedings of the World Congress on Engineering 2016 Vol I WCE 2016, June 29 - July 1, 2016, London, U.K. N X

For z ∈ RN , we let [z]Y =

kj bj , be the unique integer

j=1

combination of periods such that z − [z]Y is in Y , and {z}Y = z − [z]Y . That is, z = {z}Y + [z]Y ,

z ∈ RN .

Fig. 2. The perforated domain Ω∗ε and its boundary ∂Ω∗ε = Γε0 ∪ Γε1 .

Fig. 1. The numbers {z}Y and [z]Y . The set Y = (0, 1)N is called the reference cell, that is, ( ) N X N Y = y ∈ R |y = yi bi , (y1 , . . . , yN ) ∈ Y . i=1

Let {ε} be a positive sequence converging to zero and for each positive ε, one can write hxi n x o + , x=ε ε Y ε Y for all x ∈ RN . Denote a bounded open set on RN by Ω. Let also S, a compact subset of Y , be the reference hole and suppose that S has a Lipschitz continuous boundary with a finite number of connected components. Let us also define Y ∗ = Y \S the perforated reference cell. The perforated domain Ω∗ε is then given by [ Ω∗ε = Ω \ Sε , where Sε = ε(ξ + S). ξ∈G

As introduced in [10] (see also [12]), we set   [  b ε = interior b ε, Ω ε(ξ + Y¯ ) and Λε = Ω\Ω  

Lastly, let α, β ∈ R with 0 < α < β, and denote by M (α, β, Y ) the set of N × N matrix fields A = (aij )1≤i,j≤N ∈ (L∞ (Y ))N ×N , satisfying (A(y)λ, λ) ≥ α|λ|2

for all λ ∈ RN and a.e. in Y . The goal of this paper is to provide a corrector result for the homogenization of the following quasilinear elliptic problem (P):  ε in Ω∗ε ,   − div(A (x, uε )∇uε ) = f uε = 0 on Γε0 ,   ε A (x, uε )∇uε · n + εγ τε (x)h(uε ) = gε (x) on Γε1 , where n is the unit exterior normal to Ω∗ε and γ is a real parameter, with γ ≥ 1. Let us denote by MO the mean value of an integrable function on O, given by Z 1 MO (Φ) = Φ(y)dy, ∀Φ ∈ L1 (O). |O| O We also set

x τε (x) = τ , ε ( g xε if M∂S (g) = 0, gε (x) = εg xε if M∂S (g) 6= 0,

ξ∈Ξε

b ε is the interior of the largest union of ε(ξ + Y¯ ) that is, Ω cells fully contained in Ω, and Λε contains the parts from the ε(ξ + Y¯ ) cells that intersects the boundary ∂Ω. The corresponding perforated sets are then, given by, b ∗ε = Ω b ε \ Sε Ω

b ∗ε . and Λ∗ε = Ω∗ε \Ω

The boundary of the perforated domain Ω∗ε is, ∂Ω∗ε = Γε0 ∪ Γε1 ,

where

b ∗ε ∩ ∂Sε Γε1 = ∂ Ω

and Γε0 = ∂Ω∗ε \ Γε1 . b ε. Thus, Γε1 is the boundary of the set of holes contained in Ω In the perforated domain in Figure 2 below, the dark b ∗ and the boundary of the holes perforated part is the set Ω ε contained is Γε1 while the remaining part is the set Λ∗ε , the boundary of the holes contained being the Γε0 .


and |A(y)λ| ≤ β|λ|,

and Aε (x, t) = A

x ,t , ε

for every (x, t) ∈ RN × R.

Suppose that the data satisfy the following assumptions: A1. f, g, τ are functions such that for every ε, (i) f ∈ L2 (Ω∗ε ), (ii) g is a Y -periodic function in L2 (Γε1 ), (iii) τ is a positive Y -periodic function in L∞ (∂S); A2. h is a function from R to R such that (i) h is an increasing function in C 1 (R) such that h(0) = 0, (ii) there exists a constant C > 0 and an exponent q N with 1 ≤ q ≤ ∞ if N = 2 and 1 ≤ q ≤ if N −2 N > 2 such that ∀s ∈ R, |h0 (s)| ≤ C(1 + |s|q−1 );

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Proceedings of the World Congress on Engineering 2016 Vol I WCE 2016, June 29 - July 1, 2016, London, U.K. 2

A3. A : Y × R 7−→ RN , is a matrix field satisfying the following conditions: (i) A is a Caratheodory function and Y −periodic for every t (ii) for every t ∈ R, A(·, t) ∈ M (α, β, Y ), (iii) there exists a function ω : R 7→ R such that a. ω is continuous, nondecreasing and ω(t) > 0 for all t > 0, b. |A(y, t) − A(y, t1 )| ≤ ω(|t − t1 |) a.e. y ∈ Y, for t 6= t1 ∈ R, Z r dt = +∞. c. for any r > 0, lim+ ω(t) s→0 s Now, for p ∈ [1, +∞), we define Vεp = φ ∈ W 1,p (Ω∗ε ) | φ = 0 on Γε0

6) Let φε ∈ Lp (Ω) such that φε → φ

strongly in Lp (Ω).

Then Tε∗ (φε ) → φ strongly in Lp (Ω × Y ∗ ). D EFINITION 3. For p ∈ [1, +∞], the averaging operator Uε∗ : Lp (Ω × Y ∗ ) 7−→ Lp (Ω∗ε ) is defined as Z nxo hxi 1 Uε∗ (Φ)(x) = ∗ + εz, dz, Φ ε |Y | Y ∗ ε Y ε Y b ∗ and, a.e. for x ∈ Ω ε Uε∗ (Φ)(x) = 0, a.e. for x ∈ Λ∗ε . One has

and

Uε∗ Φ|Ω×Y ∗ = Uε (Φ)|Ω∗ε ,

Vε = Vε2 ,

when Φ belongs to Lp (Ω × Y ∗ ). Some of the properties of the averaging operator are given in the next two propositions.

which is a Banach space for the norm kukVεp = k∇ukLp (Ω∗ε ) ∀u ∈ Vεp . The variational formulation (VF) of problem P is then given by  Find uε ∈ Vε such that    Z Z    Aε (x, uε )∇uε ∇v dx + εγ τε (x)h(uε )v dσx ∗ ΩZ Γε1 ε Z      f v dx + gε (x)v dσx , ∀ v ∈ Vε .  = Γε1

Ω∗ ε

Under assumptions A1 − A3, the existence and uniqueness of a solution for problem VF has been proved in [4].

P ROPOSITION 4. [10], [12], [14], [15] Let p ∈ [1, +∞[. 1) Uε∗ is linear and continuous. 2) Uε∗ is almost a left inverse of Tε∗ on Ω∗ε . 3) For any φ in Lp (Ω), kφ − Uε∗ (φ)kLp (Ω∗ε ) → 0. 4) Let wε be in Lp (Ω∗ε ). Then the following are equivalent: b strongly in Lp (Ω × Y ∗ ) (i) Tε∗ (w Zε ) → w |wε |p dx → 0,

and Λ∗ ε

III. A SHORT REVIEW OF THE UNFOLDING METHOD

b Lp (Ω∗ ) → 0. (ii) kwε − Uε∗ (w)k ε

In this section, we briefly recall the main definitions and properties of the unfolding operators under PUM, that we need.

P ROPOSITION 5. [10] For p ∈ [1, +∞), suppose that ρ is in Lp (Ω) and θ in L∞ (Ω; Lp (Y )). Then the product Uε (ρ)Uε (θ) belongs to Lp (Ω) and

D EFINITION 1. For any Lebesgue-measurable function φ on Ω∗ε , the unfolding operator Tε∗ is a function from Lp (Ω∗ε ) to Lp (Ω × Y ∗ ), and is defined by  h i  φ ε x b ε × Y ∗, + εy , a.e. in Ω ε Y Tε∗ (φ)(x, y) =  0, a.e. in Λ × Y ∗ .

Uε (ρ θ) − Uε (ρ)Uε (θ) → 0 strongly in Lp (Ω).

ε

P ROPOSITION 2. [10], [12], [14], [15] Let p ∈ [1, +∞). 1) Tε∗ is linear and continuous. 2) Tε∗ (φψ) = Tε∗ (φ)Tε∗ (ψ) for every φ, ψ ∈ Lp (Ω∗ε ). 3) For w ∈ Lp (Ω), Tε∗ (w) → w

strongly in Lp (Ω × Y ∗ ).

4) For all φ ∈ L1 (Ω∗ε ) one has Z Z Z φ(x) dx = φ(x) dx − φ(x) dx b∗ Ω Ω∗ Λ∗ ε ε ε Z 1 T ∗ (φ)(x, y) dx dy. = |Y | Ω×Y ∗ ε 5) ∇y Tε∗ (φ)(x, y) = εTε∗ (∇x φ)(x, y) for every (x, y) in RN × Y ∗ .


Let us now define the boundary unfolding operator. Here, we assume that p ∈]1, +∞[ and that ∂S has a finite number of connected components. D EFINITION 6. For any Lebesgue-measurable function ϕ b ∗ε ∩ ∂Sε , the boundary unfolding operator is defined on ∂ Ω by ( hxi b ε × ∂S, ϕ ε + εy , a.e. in Ω b Tε (ϕ)(x, y) = ε Y 0 a.e. in Λε × ∂S. P ROPOSITION 7. [10], [15] Let p ∈ [1, ∞[. Then 1) Tεb is a linear operator. 2) Tεb (φψ) = Tεb (φ)Tεb (ψ) for every φ, ψ ∈ Lp (∂Sε ). 3) Let φ ∈ Lp (∂S) be a Y -periodic function. Set φε (x) = φ xε . Then Tεb (φε )(x, y) = φ(y). 4) For all φ ∈ L1 (∂Sε ), the integration formula is given by Z Z 1 φ(x) dσx = T b (φ)(x, y) dxdσy . ε|Y | Ω×∂S ε Γε1

WCE 2016

Proceedings of the World Congress on Engineering 2016 Vol I WCE 2016, June 29 - July 1, 2016, London, U.K. 5) Let φ ∈ Lp (∂Sε ). Then Tεb (φ) → φ

strongly in Lp (Ω × ∂S).

IV. S OME KNOWN RESULTS In this section, we recall the homogenization results as obtained in [5], on which the corrector result will be based upon. T HEOREM 8. Under assumptions A1 − A3, let uε be the unique solution of VF and γ ≥ 1. Then, there exists (u0 , u b) 1 in H01 (Ω) × L2 (Ω; Hper (Y ∗ )) with MY ∗ (ˆ u) = 0, such that:  (i) Tε∗ (uε ) → u0 strongly in L2 (Ω; H 1 (Y ∗ )),     (ii) Tε∗ (∇uε ) * ∇u0 + ∇y u b weakly in L2 (Ω × Y ∗ ),    1  (iii)Tεb (h(uε )) * h(u0 ) weakly in Lt (Ω; W 1− t ,t (∂S)). Case 1. If γ = 1, the couple (u0 , u b) is the unique solution 1 in the space H01 (Ω) × L2 (Ω; Hper (Y ∗ )) with MY ∗ (ˆ u) = 0, of the limit equation Z A(y, u0 )(∇u0 + ∇y u b)(∇φ(x) + ∇y Ψ(x, y)) dx dy Ω×Y ∗ Z h(u0 )φ dx +|∂S|M∂S (τ ) Ω Z Z = |Y ∗ | f φ dx + |∂S|M∂S (g) φ dx Ω

Ω

1 (Y ∗ )). for all φ ∈ and Ψ ∈ L (Ω; Hper Case 2. If γ > 1, the couple (u0 , u b) is the unique solution 1 (Y ∗ )) with MY ∗ (ˆ u) = 0, in the space H01 (Ω) × L2 (Ω; Hper of the limit equation Z A(y, u0 )(∇u0 + ∇y u b)(∇φ(x) + ∇y Ψ(x, y)) dx dy Ω×Y ∗ Z Z = |Y ∗ | f φ dx + |∂S|M∂S (g) φ dx

H01 (Ω)

R EMARK 10. One has (see e.g. [13] for the details of the computation), u b(x, y) = −

Ω

1 for all φ ∈ H01 (Ω) and Ψ ∈ L2 (Ω; Hper (Y ∗ )).

C OROLLARY 9. Under assumptions A1 − A3, let uε be the unique solution of VF. Then, if γ ≥ 1, |Y ∗ | u0 |Y |

weakly in L2 (Ω),

where e denotes the extension by 0 to Ω. If γ = 1, the function u0 is the unique solution of the limit problem  |∂S|   − div(A0 (u)∇u0 ) + M∂S (τ )h(u0 )   |Y |   |Y ∗ | |∂S| = f+ M∂S (g) in Ω   |Y | |Y |     u0 = 0 on ∂Ω.

N X

χe (y, u0 (x)) i

i=1

∂u0 (x), ∂xi

(1)

where u0 is the one given in Theorem 8. V. M AIN R ESULT First we recall a classical result that we need to prove the main result given in Theorem 14. L EMMA 11. [10] Let {Dε } be a sequence of n × n matrix fields in M (α, β, O) for some open set O, such that Dε → D a.e. on O, (or more generally, in measure in O). If the sequence {ζε } converges weakly to ζ in [L2 (O)]N , then Z Z lim inf Dε ζε ζε dx ≥ Dζζ dx. ε→0

2

Ω

u fε *

where wλ (y, t) = −χλ (y, t) + λ · y a.e. in Y ∗ and χλ (·, t) is, for every t, the solution of the cell problem  − div(A(·, t)∇χλ (·, t) = − div(A(·, t)λ in Y ∗ ,      A(·, t)∇χλ (·, t) · n = 0 on ∂S,  χλ (·, t) Y − periodic,     MY ∗ (χλ (·, t)) = 0.

O

O

Furthermore, if Z ε→0

then

Z Dε ζε ζε dx ≤

lim sup O

Dζζ dx, O

Z

Z Dζζ dx = lim

ε→0

O

Dε ζε ζε dx, O

and ζε → ζ

strongly in [L2 (O)]N .

Next, we present a proposition which is also essential in proving the corrector result. P ROPOSITION 12. Let γ ≥ 1 and suppose M∂S (g) 6= 0. Under the assumptions in Theorem 8, one has Tε∗ (∇uε ) → ∇u0 + ∇y u b strongly in L2 (Ω × Y ∗ ), and

Z lim

ε→0

|∇uε |2 dx = 0.

(2) (3)

Λε

Proof: We prove the case γ = 1. When γ > 1, the proof is similar but the the nonlinear term in the boundary approaches 0 at the limit. By applying PUM, Z Z 1 f uε dx = Tε∗ (f )Tε∗ (uε ) dx dy, |Y | ∗ ∗ Ωε Ω×Y

If γ > 1, the function u0 is the unique solution of the problem  ∗   − div(A0 (u)∇u0 ) = |Y | f + |∂S| M∂S (g) in Ω so that from Proposition 2-3) and Theorem 8 convergence |Y | |Y |  (i), Z  Z u0 = 0 on ∂Ω. 1 lim f uε dx = lim Tε∗ (f )Tε∗ (uε ) dx dy 0 ε→0 Ω∗ ε→0 Ω×Y ∗ |Y | The homogenized matrix field A (t) is given by ε Z Z 1 1 = f u0 dx dy. A0 (t)λ = A(y, t)∇wλ (t, y)dy, ∀λ ∈ RN , |Y | Ω×Y ∗ |Y | Y ∗


WCE 2016

Proceedings of the World Congress on Engineering 2016 Vol I WCE 2016, June 29 - July 1, 2016, London, U.K. Since f and u0 are just functions of x, one gets Z Z |Y ∗ | f uε dx = lim f u0 dx. ε→0 Ω∗ |Y | Ω ε Also, from Proposition 3.5 of [10], Z Z |∂S| gε (x)uε dσx = M∂S (g) u0 dx. |Y | Γε1 Ω Moreover, Z ε

(4)

(5)

(∇u0 + ∇y u b) dx dy which implies, using Lemma 11, the convergence given in (2). On the other hand, from Proposition 2-4) and the ellipticity of Aε , Z Z 2 α |∇uε | dx ≤ Aε (x, uε )∇uε ∇uε dx

τε (x)h(uε )uε dσx

Γε1

=

1 |Y |

Z

τ (y)Tεb (h(uε ))Tε∗ (uε ) dx dσy ,

Ω×∂S

from Proposition 7. Thus, one obtains from Theorem 8 (i) and (iii) that, Z 1 lim τ (y)Tεb (h(uε ))Tε∗ (uε ) dx dσy |Y | ε→0 Ω×∂S |∂S| M∂S (τ ) = |Y |

Z h(u0 )u0 dx, Ω

!

Z lim

ε

τε (x)h(uε )uε dσx Γε1

(6)

|∂S| M∂S (τ ) = |Y |

Z h(u0 )u0 dx. Ω

Now, using Lemma 11 with Dε = Tε∗ (Aε (x, uε )) and ζε = Tε∗ (∇uε ), together with (4)-(6), one obtains Z 1 A(y, u0 )(∇u0 + ∇y u b)(∇u0 + ∇y u b) dx dy |Y | Ω×Y ∗ Z 1 ≤ lim inf Tε∗ (Aε (x, uε )∇uε ∇uε ) dx dy ε→0 |Y | Ω×Y ∗ Z 1 ≤ lim sup Tε∗ (Aε (x, uε )∇uε ∇uε ) dx dy |Y | Ω×Y ∗ ε→0 Z ≤ lim sup Aε (x, uε )∇uε ∇uε dx ε→0

Ω∗ ε

Z

Z f v dx +

ε→0

γ

gε (x)v dσx Γε1

Ω∗ ε

−ε

!

Z τε (x)h(uε )v dσx Γε1

|Y ∗ | |Y |

=

1 |Y |

Λε

Z

Aε (x, uε )∇uε ∇uε dx

= Ω∗ ε

1 − |Y |

Z Ω×Y

∗

Tε∗ (Aε (x, uε ))Tε∗ (∇uε )Tε∗ (∇uε ) dx dy.

R EMARK 13. From the computations above, we have the following convergence of the energy: Z lim Aε (x, uε )∇uε ∇uε dx ε→0

Ω∗ ε

1 = |Y |

Z Ω×Y ∗

A(y, u0 )(∇u0 + ∇y u b)(∇u0 + ∇y u b) dx dy.

Let us now have the main result in this paper, the corrector result. T HEOREM 14. Let γ ≥ 1 and suppose M∂S (g) 6= 0. Under the assumptions in Theorem 8, we have

N

X ∂u0

∗ ∗ Uε (∇y χei (y, u0 (x)))Uε

∇uε − ∇u0 +

∂xi 2 ∗ i=1

L (Ωε )

converges to 0. Proof: From Proposition 12, and Proposition 4-4), one has k∇uε − Uε∗ (∇u0 + ∇y u b)kL2 (Ω∗ ) → 0. ε

This together with Proposition 4-1),3) gives

= lim sup

=

Λε

This, together with (7) gives the second convergence in (3).

which yields,

ε→0

by choosing φ = u0 and Ψ = u b in the limit equation for Case 1 in Theorem 8. Thus, Z 1 Tε∗ (Aε (x, uε )∇uε ∇uε ) dx dy lim ε→0 |Y | Ω×Y ∗ Z 1 (7) = A(y, u0 )(∇u0 + ∇y u b) |Y | Ω×Y ∗

Z |∂S| M∂S (g) u0 dx |Y | Ω Ω Z |∂S| − M∂S (τ ) h(u0 )u0 dx |Y | Ω

Z

f u0 dx +

Z Ω×Y ∗

A(y, u0 )(∇u0 + ∇y u b)

(∇u0 + ∇y u b) dx dy,


k∇uε − ∇u0 − Uε∗ (∇u0 ) + Uε∗ (∇u0 ) − Uε∗ (∇y u b)kL2 (Ω∗ ) ε

≤ k∇uε − Uε∗ (∇u0 ) − Uε∗ (∇y u b)kL2 (Ω∗ ) ε

+ k∇u0 − Uε∗ (∇u0 )kL2 (Ω∗ε ) → 0. Thus, from (1), Proposition 5 and the computations above, one obtains

! N

X ∂u0

∗ ∇y χei (y, u0 (x))

∇uε − ∇u0 − Uε −

∂xi i=1 ! N X ∂u0 +Uε∗ − ∇y χei (y, u0 (x)) ∂xi i=1

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Proceedings of the World Congress on Engineering 2016 Vol I WCE 2016, June 29 - July 1, 2016, London, U.K.

−

−

N X

Uε∗ (∇y χei (y, u0 (x)))Uε∗

i=1

≤ Uε∗

−

N X

∂u0 ∂xi

N X

!

∂u0 − ∇y χei (y, u0 (x)) ∂xi i=1

Uε∗ (∇y χei (y, u0 (x)))Uε∗

i=1

+ ∇uε − ∇u0 − Uε∗

∂u0 ∂xi

L2 (Ω∗ ε)

! −

!

N X

L2 (Ω∗ ε)

∂u0 − ∇y χei (y, u0 (x)) ∂xi i=1

!

L2 (Ω∗ ε)

→ 0,

[18] A. Dall’Aglio and F. Murat, “A corrector result for H-converging parabolic problems with time-dependent coefficients”, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e série, tome 25, n0 1-2, pp. 329-373, 1997. [19] G. Dal Maso and F. Murat, “Asymptotic behavior and correctors for linear Dirichlet problem with simultaneously varying operators and domains”, Ann. I. H. Poincaré, AN 21, pp. 445-486, 2004. [20] A. Damalamian, “An elementary introduction to periodic unfolding”, Gakuto International Series, Math.Sci. Appl. Vol. 24, pp. 119-136, 2005. [21] P. Donato, “Some corrector results for composites with imperfect interface”, Rendiconti di Matematica, Serie VII, Vol 26, pp. 189-209, 2006. [22] P. Donato and Z. Yang, “ The Periodic unfolding method for the wave equation in domains with holes”, Advances in Mathematical Sciences and Applications, (22)2, pp. 521-551, 2012. [23] Encyclopedia of Material Science and Engineering, edited by Michael B. Bever, Pergamon Press, Oxford, New York, tome 3 (1988), p. 2109, tome 7 (1986), p. 4917-4919, p. 4950, p. 5455. [24] C. Timofte, “On the homogenization of a climatization problem”, Stud. Univ. Babes-Bolyai Math., 52 (2), pp. 117-125, 2007.

which yields the desired result. Remark: The case M∂S (g) = 0 has been recently done in [8]. R EFERENCES [1] S. Bendib, “Homogenization of some class of nonlinear problem with Fourier condition in the holes” (Homogénéisation d’une classe de problèmes non linéaires avec des conditions de Fourier dans des ouverts perforés), Thèse, Institut National Polytechnique de Lorraine, 2004. [2] S. Bendib and R.L. Tcheugoué Tébou, “Homogenization of a class of nonlinear problem in perforated domains” (Homogénéisation d’une classe de problèmes non linéaires dans des domaines perforés), C.R. Acad. Sci. Paris, t. 328, Serie 1, pp. 1145-1149, 1999. [3] S. Brahim-Otsmane, G.A. Francfort, and F. Murat, “Correctors for the homogenization of the wave and heat equations”, J. Math. Pures Appl., 71, pp. 197-231, 1992. [4] B. Cabarrubias and P. Donato, “Existence and uniqueness for a quasilinear elliptic problem with nonlinear Robin conditions”, Carpathian J. Math. Vol 27, No.2, pp. 173-184, 2011. [5] B. Cabarrubias and P. Donato, “Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions”, Applicable Analysis, DOI:10.1080/00036811.2011.619982, 2011. [6] J. Casado-Diaz, et al., “Homogenization and corrector for the wave equation with discontinuous coefficients in time”, J. Math. Anal. Appl., 379, pp. 664-681, 2011. [7] J. Casado-Diaz, et al., “A corrector result for the wave equation with hgh oscillating periodic coefficients”, Advances in Differential Equations and Applicatgions SEMA SIMAI Springer Series 4, pp. 23-30, Springer International Publishing Switzerland 2014, DOI: 10.1007/978-3-31906953-1. [8] I. Chourabi and P. Donato, “Homogenization and correctors of a class of elliptic problems in perforated domains”, Asymptotic Analysis, 92 (1-2), pp. 1-43, 2015. [9] M. Chipot, Elliptic Equations: An Introductory Course. Germany: Birkh¨ auser Verlag AG, 2009. [10] D. Cioranescu, A. Damlamian, P. Donato, G. Griso and R. Zaki, “The Periodic Unfolding Method in Domains With Holes”, SIAM J. Math. Anal., Vol. 44, No. 2, pp. 718-760, 2012. [11] D. Cioranescu, A. Damlamian and G. Griso, “Periodic unfolding and homogenization”, C.R. Acad. Sci. Paris, Ser. I 335, pp. 99-104, 2002. [12] D. Cioranescu, A. Damlamian and G. Griso, “The periodic unfolding and homogenization”, SIAM J. Math. Anal., Vol. 40, No. 4, pp. 15851620, 2008. [13] D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford University Press, 1999. [14] D. Cioranescu, P. Donato and R. Zaki, “The Periodic Unfolding and Robin problems in perforated domains”, C.R.A.S. Paris, Série 1, 342, pp. 467-474, 2006. [15] D. Cioranescu, P. Donato and R. Zaki, “The Periodic Unfolding Method in Perforated Domains”,Portugal Math 63(4), pp. 476-496, 2006. [16] D. Cioranescu, P. Donato and R. Zaki, “Asymptotic behavior of elliptic problems in perforated domains with nonlinear boundary conditions”, Asymptotic Analysis 53, pp. 209-235, 2007. [17] C. Conca, J.I. Diaz, A. Linan and C. Timofte, “Homogenization in Chemical Reactive Flows”, Electronic Journal of Differential Equations, Vol. 40, pp. 1-22, 2004.


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