Asymptotic analysis for a weakly damped wave equation with application to a problem arising in elasticity

The present work is devoted to the study of homogenization of the weakly damped wave equation ∫ Ω ρ ∂uε ∂t2 (t) · vdx+ 2εμ ∫ Ωε f Eij ( ∂uε ∂t (t) ) Eij(v)dx +ελ ∫ Ωε f div ( ∂uε ∂t (t) ) divvdx+ θ ∫ Ωε f div (uε(t)) divvdx

Convergence homogenization results are achieved using the two-scale convergence theory.

Introduction
Our main objective in the present work is to prove a homogenization theorem for the weakly damped wave equation where Ω is a smooth bounded open set in the numerical space R 3 x of variables x = (x 1 , x 2 , x 3 ), T is a fixed positive real, the dot denotes the usual Euclidean inner product in R 3 , f = (f 1 , f 2 , f 3 ) is given in L 2 (0, T ; L 2 (Ω; R)) (with L 2 (Ω; R) = L 2 (Ω; R) 3 ), dx denotes the Lebesgue measure on R 3 x , the functions ρ ε , a ε , b ε , and the set Ω ε s are defined farther on and In what follows, all vector spaces F are assumed to be complex vector spaces, and F denotes the product F × F × F (three times).If X denotes a locally compact space, we will write C(X) = C(X; C) for the space of continuous mappings from X into C (complex numbers) and B(X) = B(X; C) for that of bounded continuous mappings from X into C. Likewise the usual spaces L 2 (X; C), L 2 oc (X; C), H n (X; C) and H n oc (X; C) (integer n ≥ 1, X provided with a positive Radon measure β ) will be denoted by L 2 (X), L 2 oc (X), H n (X) and H n oc (X).The corresponding subspaces of real functions will be denoted by L 2 (X; R), L 2 oc (X; R), H n (X; R) and H n oc (X; R).In the definitions of other function spaces, the functions are assumed to be measurable.If A is any Banach space, C(X; A) will denote the space of all continuous functions from X into A, and L 2 (X; A) will denote the space of all functions u : X → A with X u (x) 2 A dβ (x) < ∞, • A being the norm on A. Furthermore, we systematically use the convention of Einstein.For basic concepts and notations concerning integration theory we refer to [3,4].
Let us now give a precise description of the problem.Let α , λ, μ, ρ s , ρ f , ϑ, c 0 , b ijkh be real numbers with (1.4) b ijkh = λδ ij δ kh + μ(δ ik δ jh + δ ih δ jk ) for i, j, k, h = 1, 2, 3 (δ ij being the Kronecker symbol), and Then the symmetry condition follow immediately.Let Y s be a smooth open set in R 3 y (the 3 -dimensional numerical space of variables y = (y 1 , y 2 , y 3 )) such that and let Y f be the open set Y \Y s .Let ρ ∈ L ∞ (R 3 y ) be given by (1.9) where, denoting by Z the set of integers, we denote For fixed ε > 0 , let ρ ε be the function in L ∞ (R 3 x ) defined by Let Ω be as above.We endow H 1 (Ω; R) = H 1 (Ω; R) 3 with the norm Thanks to Korn's inequality (see [14]), the norm |•| is equivalent to the usual H 1 (Ω)-norm; and the seminorm is a norm in H 1 0 (Ω; R) equivalent to the H 1 (Ω)-norm.
For ε > 0, we set which defines two open subsets of Ω.Let b ε and a ε be the two continuous bilinear forms on H 1 (Ω; R) × H 1 (Ω; R) defined by and where divu = ∂u 1 ∂x1 + ∂u 2 ∂x2 + ∂u 3 ∂x3 for u = (u i ).It is clear that the space V ε in (1.3) is a closed vector subspace of H 1 0 (Ω; R), and the seminorm By the Galerkin method one can prove the following existence and uniqueness result (see [2] or [13]) : Theorem 1.For each given ε > 0 , there exists one and only one function u ε from [0, T ] into V ε defined by (1.1)-(1.2) and satisfying The periodic homogenization of (1.1)-(1.2),together with many other hyperbolic problems, is studied in Sanchez-Palencia [14].Here, the homogenization method is the pioneering approach using multiple-scale asymptotic expansions.We also refer to [11] for the homogenization, by means of two-scale convergence, of a mixture of solid and fluid.Another study of interest on this topic is presented in [8], where the homogenization of hyperbolic-parabolic equations in perforated domains is worked out.
The paper is organized as follows.Section 2 is devoted to some fundamentals of homogenization.In section 3 we study the homogenization problem for (1.2)-(1.2) in the periodic setting when Section 4 is concerned with the study of the periodic homogenization problem for (1.1)-(1.2) in the general case f ∈ L 2 (0, T ; L 2 (Ω; R)).

Fundamentals of homogenization
The aim is to study the homogenization of (1.1)-(1.2) by using the twoscale convergence method (see [1,8,12]).We will need the notion of weak two-scale convergence in , where We recall the following fundamental results : Lemma 1.Let E be a fundamental sequence, i.e., E is an ordinary sequence of positive real numbers tending to 0 ; and suppose the sequence Proof.See, e.g., [7].
Proof.Based on Lemma 1, there exist a subsequence E ⊂ E and two Let us prove (2.2).This will be do if we show that v 0 in (2.4) Taking into account the two limits in (2.4), it follows that Then, for almost all (x, t, y) per (Y )) and we deduce (2.2).Let us now prove (2.3).By taking in (2.4)

But we have
Then, there exist a subsequence E ⊂ E and two functions v 0 ∈ H n (0, T ; L 2 (Ω;

The homogenization result in the case
The preliminary results below are based on the following general estimates, which will be easily proved by using a Gronwall lemma.Lemma 5. Let (u ε ) ε>0 be the sequence of functions defined in Theorem 1.Then, Remark 1.The condition f ∈ C 1 ([0, T ]; L 2 (Ω; R)) with f (0) = ω allows us to prove that the solution u ε of (1.2)-(1.2) is also a solution of the problem (see [13]) Then, the last estimate becomes obvious.Further, the choice of f as an element of C 1 ([0, T ]; L 2 (Ω; R)) with f (0) = ω , will permits us to study the weakly two-scale convergence of the sequence of second derivative (u ε ) ε>0 .
3.1 Preliminaries.In the sequel E denotes a fundamental sequence (see Lemma 1).Lemma 6.There exist a sequence E ⊂ E and two functions where p ε = −ϑdivu ε .Furthermore, we have where e ij (u 0 ) = 1 2 Proof.We proceed in three steps.
Step 2. Let us prove (3.3).By Lemma 5 we have Then according to (3.6), we conclude by Lemma 2 that Let us bear in mind that the preceding two limits give for all ϕ ∈ D(Q), and (3.5) is proved.
Let us verify that u 0 satisfies the homogeneous initial conditions.Precisely we have the following lemma.
Proof.By Lemma 6, u 0 and ∂u0 ∂t belong to C([0, T ]; L 2 (Ω; L 2 per (Y ))), so that u 0 (0) and ∂u0 ∂t (0) are well defined.Now, denoting by On the other hand, recalling that u ε and ∂uε ∂t lie in L 2 (0, T ; The passage to the limit, as E ε → 0, permits to deduce Using the density of . Therefore, we have u 0 (0) = ω.The same arguments give The following lemma will permit to achieve one part of the proof of Lemma 9.
Proof.We proceed in two steps.
Since a 1 b 2 − a 2 b 1 = 0, then S 1 and S 2 are uniquely determined by ) being well defined.
Finally let ξ = (λ 3 , μ 3 , 1).There is We put Then, (3.15) According to (3.9), there is By substituting (3.14), we have Thanks to (3.15), S 3 is determined as a linear combination of q 1 , q 2 , This being so, we will need the following lemma to derive the homogenized problem.

Lemma 9. By letting
where and the function u Proof.Let us prove (3.17).We take in (1.2) test functions of the form [11]; next we integrate from 0 to T and it follows that Passing to the limit, as E ε → 0 , it follows by (3.6)-(3.7)that This yields  3 , div y w = 0 and θ ∈ C per (Θ); next we integrate from 0 to T and obtain Further we pass to the limit, as E ε → 0 .Using Lemma 6 and property (3.17), we are led to Hence, taking into account Lemma 8, we have ∂p ∂xi ∈ L 2 (Q), i = 1, 2, 3; and (3.18) follows. Consequently , w constant in Y and θ ∈ C per (Θ); and we obtain, at the limit when E ε → 0 and let , the usual trace operator on Ω 0 .Then there exist two continuous linear mappings

Proof.
By noting v Ω 0 the restriction of v on Ω 0 , we construct a surjective mapping Γ • by Γ It is an obvious matter to verify that Γ • belongs to L(H 1 0 (Ω 1 ); H 1 2 (∂Ω 0 )), where H 1 0 (Ω 1 ) and H 1 2 (∂Ω 0 ) are endowed with the usual H 1 (Ω 1 )-norm and H 1 2 (∂Ω 0 )norm respectively.Using (3.20) and the construction of Γ • , we prove easily the equality )) which makes it a Banach space.By a classical result of minimization of functionals, for fixed g ∈ H ≤ g ; and by virtue of the open mapping theorem (see, e.g., [5]), it follows that both the norms and • are equivalent on H 1 2 (∂Ω 0 ).Finally, we apply the usual canonical decomposition of Γ • to define R • as a linear mapping from H 1 2 (∂Ω 0 ) into H 1 0 (Ω 1 ) with (3.21) and there are two constants c Lemma 10 leads us to the following Lemma 11.Let Ω 0 and Ω 1 be two smooth open bounded sets in R N (N ≥ 1 ) with the property (3.20), let Φ • be a linear form on H 1 0 (Ω 1 ), and let V be a fixed vector subspace of H 1 0 (Ω 1 ).If Proof.
Under the above assumptions, for any (3.21) for the second equality).
We are now in position to prove that the solution u ε of (1.1)-(1.2) can be inserted in an appropriate variational furmulation.Indeed, it is not easy to take test functions in (1.1) The following proposition is then fundamental in the rest of this section.
Step 1.Let us introduce Ω 1 = {x ∈ R 3 : d(x, Ω) < 1} , where d designates the Euclidean metric and Ω the closure of Ω in R 3 x .For the bounded sets Ω 1 and Ω ε 0 are opened and at least of class C 1 in R 3 x with Ω ε 0 ⊂ Ω 1 .Let 0 < ε ≤ ε 0 be fixed, and let u ε be the function defined in Theorem 1.For each 0 ≤ t ≤ T , the linear form Φ(ε, t) on Step 2. We want to decompose Φ(ε, t), further we will apply Lemma 11.

and let R ε
• be defined in the sense of Lemma 10.We put Φ(ε, , where Φ • (ε, t) and Φ 1 (ε, t) are the continuous linear forms on Step 3. We want to solve the variational problem (for fixed t ∈ [0, T ] and 0 < ε ≤ ε 0 ) : We begin by noting that the mapping v → . By virtue of (3.28), there exists a unique function But, according to the definition of W ε , we may replace the space of test functions

27). Step 4. Let us prove that the function z
such that t + h ∈ (0, T ).According to (3.29) we have We take v = z ε (t + h) − z ε (t), and it follows where c does not depend to t, h and 0 Thus, there is a constant c > 0 independent to t, h and 0 < ε ≤ ε 0 such that Since f, u ε , divu ε , u ε and u ε are measurable, then the function z ε : t → z ε (t) from [0, T ] into H 1 (Ω ε s ) is measurable.Furthermore, we are quickly drived to z ε (t) H 1 (Ω ε s ) ≤ c for almost all t ∈ (0, T ) and all 0 < ε ≤ ε 0 , which proves that z ε belongs to L ∞ (0, T ; H Proof.There exists exactly one function Extending by periodicity, we obtain a unique function (still denoted u 1 ) . Hence, by extension, there exists a function and

Convergence theorem for the homogenization problem.
According to Lemma 6, we need to define some vector spaces.It is well- is one of its closed vector subspace (cf.[15]), where γ n is the continuous linear operator from E(Ω) into H − 1 2 (∂Ω) such that γ n u(x) = u i (x)n i (x) for all x ∈ ∂Ω and all u = (u i ) ∈ D(Ω) 3 , n = (n i ) being the outer unit normal to ∂Ω.

Now, the set
, and it is obvious that the mapping where u = ( u i ), u i being the function on Ω defined by u i (x) = Y u i (x, y)dy (x ∈ Ω).Clearly, E 0 (Ω; W ) is a vector subspace of L 2 (Ω; W ). Furthermore, E 0 (Ω; W ) is a Hilbert space for the norm Theorem 2. Let (u ε ) ε>0 be the sequence defined by Theorem 1.Then, as ε → 0 , we have where u 0 is the function from and the variational wave equation where Proof.
Step 3. We will pass to the limit in (3.22) for a suitable test function.We take in (3.22) ), the last two terms at the right hand side are equal to zero.Next, we observe that χ s E ij (Ψ 0 )θ and χ s e ij (Ψ 1n )θ belong to C(Q; L ∞ per (Z)).Thus, by using (3.39),Lemmas 6 and 9, at the limit (when E ε → 0 ), we conclude that the right-hand side converges to zero, and the left-hand side converges to Letting z = Θ rdτ and Step 4. Derivation of (3.37).Let us consider the function ξ = (ξ i ) from Y s into R 3 defined by : for According to (3.30) and (3.32) it follows that, as n → ∞, On the other hand, according to the convergence in (3.32), it follows that, up to a subsequence, still denoted (Ψ 1n ) n ,

Physical description in brief of the local problem.
Let u 0 be the function defined in Theorem 2. There exists a negligible set N 0 ⊂ Ω such that for x ∈ Ω\N 0 the function t → u 0 (x, t, •) from [0, T ] into L 2 per (Y ), denoted by u 0 (x), belongs to L 2 (0, T ; L 2 per (Y )).Thanks to the isometric isomorphism between the spaces L 2 (0, T ; L 2 (Ω; L 2 per (Y ))) and L 2 (Ω; L 2 (0, T ; L 2 per (Y ))), we show easily that u 0 (x), Theorem 3. Let u 0 be the function in Theorem 2. For almost all x ∈ Ω, u 0 (x) is uniquely determined by u 0 for all w ∈ H 1 per (Y f ), w = ω on ∂Y s and div y w = 0 .Then, for almost all (x, t) ∈ Q there exists p per (Y f ) : Y f vdy = 0 such that (see, e.g., [9]) ) denoting the dual of the space {w ∈ H 1 per (Y f ) : w = 0 on ∂Y s } ]; so that p1(x,t) ∂yi = p1(x) ∂yi (t).On letting we have , and the partial differential equations which govern this motion are (for almost all x ∈ Ω) : for all w ∈ W , we add − Y f p 1 (x)div y wdy to both sides and, by using Green formula combining with the first equation in (3.42), we obtain where γ s • is the trace operator on ∂Y s , n = (n i ) is the outer unit normal to ∂Y s with respect to Y s , [ , ] s denotes the scalar product in the duality between H − 1 2 (∂Y s ) and it follows that with I = (1, 1, 1) , where dσ is the superficial measure on ∂Y s .

The homogenization result in the case
f ∈ L 2 (0, T ; L 2 (Ω)) 4.1 Preliminaries.In the sequel, we assume that f is fixed in L 2 (0, T ; L 2 (Ω)).Then, it results just both the first and second estimates of Lemma 5. Therefore we have the following lemma, where we recall that (u ε ) ε>0 is the sequence obtained in Theorem 1.In what follows, E designates a fundamental sequence.

Lemma 13.
A subsequence E can be extracted from E such that, as E ε → 0 , we have , and u 0 (0) = ω.
Proof.As in the proof of Lemmas 6 and 7, all those properties follow to first and second estimates of Lemma 5.

Now, let (f
According to Theorem 1, there exists exactly one sequence (u nε ) n∈N * ,ε>0 such that, given n ∈ N * and ε > 0, u nε is the unique function from [0, T ] into V ε defined by : and Lemma 14.Let u 0 and E be defined in Lemma 13.Then we have Proof.
Step 1.Let us prove the first equality.Let ε ∈ E be fixed.We show easily that (u nε ) n∈N * is a Cauchy sequence in L 2 (0, T ; L 2 (Ω)), and where u ε = (u i ε ) is a term of the sequence (u ε ) ε∈E of Lemma 13.The first equality of Lemma 14 result to (4.1).
Step 2. Now, let us prove the second equality.It follows by Theorem 2 that, for each n ∈ N * , (4.5) lim and and the variational problem Since n is arbitrarily fixed, we obtain the a priori estimate for all m, n ∈ N * , m = n, where the constant c does not depend on m and n.
Remark 5.It results to (4.7) that ∂un0 ∂t n∈N * is a Cauchy sequence in L 2 (Q; W ), so that there is a function v ∞ ∈ L 2 (Q; W ) such that According to the continuity of the derivation operator in in the distributional space D ((0, T ) ; L 2 (Ω; W )), one proves that v ∞ = ∂u∞ ∂t = ∂u0 ∂t .
Proof.By virtue of Lemma 13, it suffices to prove that the weak twoscale limit (4.1) satisfies (4.11)-(4.12),further it is the unique solution of this variational problem.We will proceed in three steps.
The associated scalar product will be still denoted [•, •].We will assume that H(Ω; W ) is identified to its dual.Since D(Ω) ⊗ W is dense in L 2 (Ω; W ), one has the following continuous canonical injections (with density) where W and E 0 (Ω; W ) are the duals of W and E 0 (Ω; W ) respectively.Then, we will also denote by [•, •] the duality pairing between E 0 (Ω; W ) and E 0 (Ω; W ). Let A be a continuous linear operator from E 0 (Ω; W ) into E 0 (Ω; W ) and B a continuous linear operator from L 2 (Ω; W ) into E 0 (Ω; W ) defined by   Step 3. Let us derive (4.8).The uniqueness of the solution u 0 in the problem (4.11)-(4.12) is proved in the above section, by assuming that f = ω (then f ∈ C 1 ([0, T ]; L 2 (Ω)) with f (0) = ω ).Consequently, we can replace the fundamental sequence E in Lemma 13 by all the sequence 0 < ε ≤ 1.The proof follows.