Uniform Boundary Controllability and Homogenization of Wave Equations

We obtain sharp convergence rates, using Dirichlet correctors, for solutions of wave equations in a bounded domain with rapidly oscillating periodic coefficients. The results are used to prove the exact boundary controllability that is uniform in $\epsilon$ - the scale of the microstructure, for the projection of solutions to the subspace generated by the eigenfunctions with eigenvalues less that $C\epsilon^{-1/2}$.

Following [8], one may use Theorem 1.1 to prove the following result on the uniform boundary controllability. Let N ≤ δT −1 ε −1/2 and T ≥ T 0 , where δ = δ(d, A, Ω) > 0 is sufficiently small. Given (θ ε,0 , θ ε,1 ) ∈ L 2 (Ω) × H −1 (Ω), there exists g ε ∈ L 2 (S T ) such that the solution of (1.6) satisfies the conditions, (1.13) where P N denotes the projection operator from L 2 (Ω) or H −1 (Ω) to the space A N . Moreover, the control g ε satisfies the uniform estimates, where C > 0 and c > 0 are independent of ε. See Section 4. In the case d = 1, it was proved by C. Castro in [10] that the estimates (1.9) and (1.10) hold uniformly if the initial data are taken from A N × A N and N ≤ δε −2 , where δ > 0 is sufficiently small. Also see [9] for the case where the initial data are taken from a subspace generated by the eigenfunctions with eigenvalues greater than Cε −2−σ for some σ > 0. The approaches used in [10,9] do not extend to the multi-dimensional case. To the best of the authors' knowledge, the only results in the case d ≥ 2 are found in [3,15]. In [3] M. Avellaneda and the first author used the asymptotic expansion of the Poisson's kernel for the elliptic operator L ε in Ω to identify the weak limits of the controls. In [15] G. Lebeau considered the wave operator with oscillating density, ρ(x, x/ε)∂ 2 t − ∆ g , where ∆ g is the Laplace operator for some fixed smooth metric, and the function ρ(x, y) is periodic in y. Theorem 1.1 seems to be the first result on the observability inequalities (1.9) and (1.10) for wave operators with oscillating coefficients A(x/ε) in higher dimensions.
Counterexamples were constructed using eigenfunctions with eigenvalues λ ε,k ∼ ε −2 -the wave length of the solutions is of the order of the size of the microstructure. Also see related work in [12] by A. Hassell and T. Tao for Dirichlet eigenfunctions on a compact Riemannian manifold with boundary. In [13] C. Kenig and the present authors proved that for d ≥ 2, ∂Ω |∇ψ ε,k | 2 dσ ≤ Cλ ε,k (1 + ελ ε,k ), (1.16) if ε 2 λ ε,k ≤ 1, where C is independent of ε and k. This, in particular, implies that the upper bound in (1.15) holds if ελ ε,k ≤ 1. Furthermore, it is proved in [13] that if ελ ε,k ≤ δ, where δ > 0 depends only on A and Ω, then the lower bound in (1.15) also holds uniformly in ε and k. These results suggest that one may be able to extend Theorem 1.1 to the case N ≤ Cε −1 . But this remains unknown. In view of the one-dimensional results in [10,8], one may conjecture further that the main conclusion in Theorem 1.1 is valid when N ≤ δε −2 and δ is sufficiently small. We now describe our approach to Theorem 1.1, which is based on homogenization. Under the assumptions (1.3), (1.4) and (1.5) as well as suitable conditions on F , ϕ ε,0 and ϕ ε,1 , the solution u ε of the initial-Dirichlet problem, (1.17) converges strongly in L 2 (Ω T ) to the solution of the homogenized problem, (1.18) where L 0 is an elliptic operator with constant coefficients (see e.g. [6]). In the first part of this paper we shall investigate the problem of convergence rates.
where u 0 is the solution of (1.18). Then for any t ∈ (0, T ], where C depends only on d and µ.
Theorem 1.2, together with Rellich identities, allows us to control the boundary integral where the initial data (ϕ 0 , ϕ 1 ) in (1.18) is chosen so that L 0 (ϕ 0 ) = L ε (ϕ ε,0 ) and ϕ 1 = ϕ ε, 1 in Ω (see [18] for the case d = 1). Since |∇Φ ε | ≤ C [2] and |det(∇Φ ε )| ≥ c > 0 on ∂Ω [13], this reduces the problem to the estimates (1.9) and (1.10) for the homogenized operator ∂ 2 t + L 0 with constant coefficients. We remark that the Rellich identities, which use the Lipschitz condition (1.12), are applied to the function w ε in (1.20). We further point out that the power of ε in the condition N ≤ C 0 T −1 ε −1/2 is dictated by the highest-order term in the right-hand side of (1.21). Also, the C 3 condition on Ω is only used for estimates of the homogenized solutions.
The problem of convergence rates is of much interest in its own right in the theory of homogenization. Note that no smoothness condition on A is needed in Theorem 1.2. Let w ε be given by (1.20 and where C depends only on d and µ. As a result, Theorem 1. However, a better estimate with lower order derivatives required for u 0 is obtained at the end of Section 3 (see (3.14)). We mention that in the case Ω = R d , the following estimate was proved in [11] by M.A. Dorodnyi and T.A. Suslina, , and u ε and u 0 have the same initial data (ϕ 0 , ϕ 1 ). The results in [11] (also see [19]) are obtained by an operator-theoretic approach, using the Floquet-Bloch theory. In the case of bounded domains, for a periodic hyperbolic system, Yu. M. Meshkova obtained an O(ε) estimate for u ε (·, t) − u 0 (·, t) L 2 (Ω) , assuming the initial data (ϕ 0 , ϕ 1 ) belong to some subspace of H 4 (Ω) [17]. We note that the highest-order term in the right-hand side of (3.14) involves ϕ 0 H 2 (Ω) and ϕ 1 H 1 (Ω) .
We point out that the symmetry condition (1.4) is essential in the proofs of Theorems 1.1 and 1.2, but the assumption that equations are scalar is not. Theorem 1.1 continues to hold for elliptic systems ∂ t − div(A(x/ε)∇), if A(y) = (a αβ ij (y)), with 1 ≤ i, j ≤ d and 1 ≤ α, β ≤ m, satisfies the ellipticity condition (1.3) for ξ = (ξ α i ) ∈ R m×d , the periodicity condition (1.5), the Lipschitz condition (1.12), and the symmetry condition a αβ ij = a βα ji . In the case of Theorem 1.2, the estimate (1.21) holds in a C 1,η domain Ω, if A satisfies (1.3), (1.5), the symmetry condition above, and is Hölder continuous. The additional smoothness conditions on A and Ω are used for the estimates of correctors χ and Φ ε .
The summation convention that repeated indices are summed is used throughout the paper. Finally, we thank Mathias Schäffner, who pointed out a flaw in the previous version of this paper.

Preliminaries
Throughout this section we will assume that A = A(y) satisfies conditions (1.3), (1.4) and (1.5). A function u in R d is said to be 1-periodic if u(y + z) = u(y) for a.e. y ∈ R d and for Let χ(y) = (χ 1 (y), χ 2 (y), . . . , χ d (y)) denote the first-order corrector for L ε , where, for Note that χ j is 1-periodic and (the summation convention is used). Under the conditions (1.3), (1.4) and (1.5), one may show that the matrix A is symmetric and satisfies the ellipticity condition, with the same constant µ as in (1.3). It is well known that the homogenized operator for ∂ 2 t + L ε is given by ∂ 2 t + L 0 . In particular, if ϕ ε,0 = ϕ 0 and ϕ ε,1 = ϕ 1 , the solution u ε of the initial-Dirichlet problem (1.17) converges strongly in L 2 (Ω T ) to the solution u 0 of the homogenized problem (1.18).
It follows by the definitions of χ j and a ij that Let Φ ε (x) be the Dirichlet corrector for L ε in Ω, defined by (1.19). Since by the maximum principle, where C depends only on d and µ. If Ω is a bounded C 1,α domain in R d for some α > 0 and A is Hölder continous, by the boundary Lipschitz estimate for L ε [2], we also have where C depends only on d, A and Ω. (2.14) Proof. Note that by (2.12), where we have used (2.8) for the last step. Finally, in view of (2.9), we have This completes the proof.

Convergence rates
Throughout this section we assume that A = A(y) satisfies ( Lemma 3.1. Let u ε , u 0 , and w ε be the same as in Lemma 2.2. Also assume that u ε = u 0 on ∂Ω × [T 0 , T 1 ]. Then

2)
where C depends only on d and µ.
Proof. Using the symmetry condition (1.4), we obtain We will use the formula (2.14) for (∂ 2 t + L ε )w ε to bound the right-hand side of (3.3). The fact w ε = 0 on ∂Ω × [T 0 , T 1 ] is also used.
Let I 1 denote the first term in the right-hand side of (2.14). It follows from integration by parts (first in x and then in t) that By the Cauchy inequality this leads to where C depends only on d and µ. Let I 2 denote the second term in the right-hand side of (2.14). Since Φ ε,k − x k L ∞ (Ω) ≤ Cε, it is easy to see that (3.4) also holds with I 2 in the place of I 1 .
Next, let I 3 denote the third term in the right-hand side of (2.14). Using integration by parts in the t variable, we see that It follows from the Cauchy inequality that Since L ε (Φ ε − x − εχ(x/ε)) = 0 in Ω and w ε = 0 on ∂Ω, by Caccioppoli's inequality, we have for t ∈ [T 0 , T 1 ]. As a result, the estimate (3.4) continues to hold if we replace I 1 by I 3 .
Finally, let I 4 denote the last term in the right-hand side of (2.14). By the Cauchy inequality, we obtain This completes the proof of (3.2).
The next lemma gives an estimate of E ε (t; w ε ) for t = 0.

6)
where C depends only on d and µ.
Next, to bound ∇w ε (·, 0) L 2 (Ω) , we usê and the following formula, (3.8) The proof of (3.8) is similar to that of (2.14). It follows from (3.7) and (3.8) that where we have used the Caccioppoli's inequality (3.5) for the last step. This yields and completes the proof.
We are now in a position to give the proof of Theorem 1.2 Proof of Theorem 1.2. Let Let w ε be defined by (2.13). We will show that for any t ∈ [0, T ], where C depends only on d and µ. This, together with the estimate of E ε (0; w ε ) in Lemma 3.2, gives the inequality (1.21). It follows by Lemma 3.1 that for 0 ≤ t ≤ T , where C depends only on d and µ. This yields from which the estimate (3.10) follows.

Uniform boundary controllability
Throughout this section we will assume that A = A(y) satisfies conditions (1.  We are interested in the estimates (1.9) and (1.10) with positive constants C and c independent of ε > 0.
Let h = (h 1 , h 2 , . . . , h d ) be a vector field in C 1 (R d ; R d ) and n = (n 1 , n 2 , . . . , n d ) denote the outward unit normal to ∂Ω. We start with the following well known Rellich identity, where a ε ij = a ij (x/ε). The identity (4.2) follows from integration by parts (in the x variable). We remark that the symmetry condition (1.4), which is essential for (4.2) even in the case of constant coefficients, is used to obtain in the proof of (4.2). It also follows from integration by parts that with (4.3), we obtain Let Ω be a bounded Lipschitz domain in R d . Let u 0 be a weak solution of (3.12) for the homogenized operator

5)
where r 0 denotes the diameter of Ω. Moreover, if T ≥ C 0 r 0 , The constants C and C 0 depend only on d, µ and the Lipschitz character of Ω.
Proof. This is well known and follows readily from (4.4) (with a ij in the place of a ε ij ) (see e.g. [16]). We include a proof here for the reader's convenience. To see (4.5), we choose a vector field h ∈ C 1 (R d ; R d ) such that h, n ≥ c 0 > 0 on ∂Ω and |∇h| ≤ C/r 0 . It follows from (4.4), with a ij in the place of a ε ij , that where we have used the energy estimate (2.16) for the last step.
To prove (4.6), we choose h(x) = x − x 0 , where x 0 ∈ Ω. Note that div(h) = d. It follows from (4.4) that where Note that by the conservation of energy, and that , where we have used Poincaré's inequality and the energy estimates for the last step. By from which the inequality (4.6) follows if T ≥ C 0 r 0 .
Proof. Let h be a vector field in C 1 (R d ; R d ) such that h, n ≥ c 0 > 0 on ∂Ω and |∇h| ≤ Cr −1 0 . We apply the Rellich identity (4.4) with w ε in the place of u ε . This giveŝ where we have used the Cauchy inequality for the last step.
Since Ω is C 3 and A is Lipschitz, ∇Φ ε is bounded. Also, under the smoothness condition (1.12), the functions ∇χ j and ∇φ kij are bounded. Thus, in view of (2.14), we obtain This, together with (4.10) and Theorem 1.
from which the estimate (4.9) follows by using the energy estimates (2.19) and (2.20).
Proof. Let u 0 , w ε be the same as in Lemma 4.2. Note that (4.14) To bound the first term in the right-hand side of (4.14), we use (4.10) as well as the fact that ϕ 1 = ϕ ε,1 and L 0 (ϕ 0 ) = L ε (ϕ ε,0 ) in Ω. The second term in the right-hand side of (4.14) is handled by Lemma 4.1. Finally, to bound the third term, we use the inequalitŷ To see (4.15), one chooses a vector field h ∈ C 1 0 (R d ; R d ) such that h, n ≥ c 0 > 0 on ∂Ω, and applies the divergence theorem to the integral ∂Ω |∇ 2 u 0 | 2 h, n dσ.
Proof. The proof uses (4.13) and the fact that which was proved in [13]. Let u 0 , w ε be the same as in Lemma 4.2. It follows from (4.6), (4.13) and (4.17) that (4.18) The last two terms in the right-hand side of (4.18) are treated exactly as in the proof of Theorem 4.3.
By a duality argument [8] and (4.25), one may also show that We omit the details and refer the reader to [8] for the one-dimensional case.