Stability of the determination of a coefficient for the wave equation in an infinite wave guide

We consider the stability in the inverse problem consisting in the determination of an electric potential $q$, appearing in a Dirichlet initial-boundary value problem for the wave equation $\partial_t^2u-\Delta u+q(x)u=0$ in an unbounded wave guide $\Omega=\omega\times\mathbb R$ with $\omega$ a bounded smooth domain of $\mathbb R^2$, from boundary observations. The observation is given by the Dirichlet to Neumann map associated to a wave equation. We prove a H\"older stability estimate in the determination of $q$ from the Dirichlet to Neumann map. Moreover, provided that the gap between two electric potentials rich its maximum in a fixed bounded subset of $\bar{\Omega}$, we extend this result to the same inverse problem with measurements on a bounded subset of the lateral boundary $(0,T)\times\partial\Omega$.

In particular the following operator, usually called the Dirichlet to Neumann (DN map in short), Λ q : L → L 2 (Σ), f → ∂ ν u q is bounded.
In the present paper, we consider the inverse problem which consists in determining the electric potential q from the DN map Λ q . We establish a stability estimate for this inverse problem. For 0 < α < 1 and h ∈ C(Ω), we set Theorem 0.1 Let M > 0, 0 < α < 1 and let B M be the ball centered at 0 and of radius M of C α b (Ω). Then, for T > Diam(ω) and q 1 , q 2 ∈ B M , we have min(2α,1)α 3(2α+2)(min(4α,2)+21) (0.2) with C depending of M , T and Ω. Here Λ q1 − Λ q2 is the norm of Λ q1 − Λ q2 with respect to B L, L 2 (Σ) .
Let us remark that in this result we consider the full DN map. This means that we determine the coefficient q from measurements on the whole lateral boundary Σ which is an unbounded set. This is due to the fact that we consider a large class of coefficients q without any restriction on their behavior outside a compact set (we only assume that the coefficients are uniformly bounded and Hölderian). In order to extend this result to the determination of q from measurements in a bounded subset of Σ, we need more informations about q. Namely, we need that the gap between two coefficients q 1 , q 2 reach its maximum in a fixed bounded subset of Ω. More precisely, let R > 0 and consider the spaces L R which consists of functions f ∈ L satisfying Let us introduce the partial DN map defined by Our second result is the following.
Theorem 0.2 Let M > 0, 0 < α < 1 and let B M be the ball centered at 0 and of radius M of C α b (Ω). Let T > Diam(ω), q 1 , q 2 ∈ B M and assume that there exists r > 0 such that Then, for all R > r we have Clearly condition (0.3) will be fulfilled if we assume that q 1 , q 2 are compactly supported. Let us remark that this condition can also be fulfilled in more general cases. For example, consider the condition (0.5) Let g : R → R be a non negative continuous even function which is decreasing in (0, +∞). Then, condition (0.3) will be fulfilled if we assume that q 1 , q 2 are lying in the set In recent years the problem of recovering time-independent coefficients for hyperbolic equations in a bounded domain from boundary measurements has attracted many attention. In [RS1], the authors proved that the DN map determines uniquely the time-independent electric potential in a wave equation and [RS2] has extended this result to the case of time-dependent potential. Isakov [Is] considered the determination of a coefficient of order zero and a damping coefficient. Note that all these results are concerned with measurements on the whole boundary. The uniqueness by local DN map has been considered by [E1] and [E2]. The stability estimate in the case where the DN map is considered on the whole lateral boundary were treated by Stefanov and Uhlmann [SU]. The uniqueness and Hölder stability estimate in a subdomain were established by Isakov and Sun [IS] and, assuming that the coefficients are known in a neighborhood of the boundary, Bellassoued, Choulli and Yamamoto [BCY] proved a log-type stability estimate in the case where the Neumann data are observed in an arbitrary subdomain of the boundary. In [BJY1], [BJY2] and [R90] the authors established results with a finite number of data of DN map.
Let us also mention that the method using Carleman inequalities was first considered by Bukhgeim and Klibanov [BK]. For the application of Carleman estimate to the problem of recovering time-independent coefficients for hyperbolic equations we refer to [B], [IY] and [K].
Let us observe that all these results are concerned with wave equations in a bounded domain. Several authors considered the problem of recovering timeindependent coefficients in an unbounded domain from boundary measurements. Most of them considered the half space or the infinite slab. In [R93], Rakesh considered the problem of recovering the electric potential for the wave equation in the half space from Neumann to Dirichlet map. Applying a unique continuation result for the timelike Cauchy problem for the constant speed wave equation and the result of X-ray transform obtained by Hamaker, Smith, Solmon, Wagner in [HSSW], he proved a uniqueness result provided that the electric potentials are constant outside a compact set. In [Nak], Nakamura extended this work to more general coefficients. In [E3], Eskin proved uniqueness modulo gauge invariance of magnetic and electric time-dependent potential with respect to the DN map for the Schrödinger equation in a simply-connected bounded or unbounded domain. In [Ik] and [SW], the authors considered the inverse problem of identifying an embedded object in an infinite slab. In [LU], the authors considered the problem of determining coefficients for a stationary Schrödinger equation in an infinite slab. Assuming that the coefficients are compactly supported, they proved uniqueness with respect to Dirichlet and Neumann data of the solution on parts of the boundary. This work was extended to the case of a magnetic stationary Schrödinger equation by [KLU]. In [CS], the authors considered the problem of determining the twisting for an elliptic equation in an infinite twisted wave guide. Assuming that the first derivative of the twisting is sufficiently close to some a priori fixed constant, they established a stability estimate of the twisting with respect to the DN map. To our best knowledge, with the one of [CS], this paper is the first where one establishes a stability estimate for the inverse problem of recovering a coefficient in an infinite domain with DN map without any assumption on the coefficient outside a compact set.
The main ingredient in the proof of the stability estimates (0.2) and (0.4) are geometric optic solutions. The novelty in our approach comes from the fact that we take into account the cylindrical form of the infinite wave guide and we use suitable geometric optic solutions for this geometry.
This paper is organized as follows. In Section 1 we introduce the geometric optic solutions for our problem and, in a similar way to [RS1] (see also Section 2.2.3 of [Ch]), we prove existence of such solutions. Using these geometric optic solutions, in Section 2 we prove Theorem 0.1 and in Section 3 we prove Theorem 0.2. In the appendix, we treat the direct problem. We prove existence of solutions and we define the DN map. Let us remark that in the case of a bounded domain Ω, applying some results of [LLT], [BCY] have treated the direct problem. Since in this paper we consider an unbounded domain Ω, it was necessary to treat this problem.

Geometric optic solutions
The goal of this section is to construct geometric optic solutions for the inverse problem. Let us recall that every variable x ∈ Ω take the form x = (x ′ , x 3 ) with x ′ ∈ ω and x 3 ∈ R. Using this representation we can split the differential operator Keeping in mind this decomposition, we will construct geometric optic solutions u ∈ H 2 (Q) which are solutions in Q of the equation ∂ 2 t u − ∆u + qu = 0 and take the form , ρ > 0 a parameter and Ψ ± a remainder term that satisfies Our result is the following.
Here Ψ ± satisfies where C depends only on T , Ω and M q L ∞ (Ω) .
Proof. We show existence of u + . The existence of u − follows from similar arguments.
Remark 1.2 In order to apply Theorem 2.1 in Chapter 5 of [LM2] we combine the arguments introduced in Remark 4.4 with the fact that the operator A = −∆ + q with Dirichlet boundary condition is a selfadjoint operator with domain D(A) = H 2 (Ω) ∩ H 1 0 (Ω). Then, we prove that Lemma 2.1 in Chapter 5 of [LM2] holds in our case and by the same way Theorem 2.1 in Chapter 5 of [LM2].
From the energy estimate associated to the solution of this problem, we get Combining this estimate with the energy estimate of (1.7) we deduce (1.6).

Stability estimate
The goal of this section is to prove Theorem 0.1. Without lost of generality, we can assume that 0 ∈ ω. From now on, we assume that T > Diam(ω), we fix and we set We shall need a stability estimate for the problem of recovering a function from X-ray transform.
Proof. In view of Lemma 1.1, we can set Combining this with the fact that suppΦ ⊂ ω ε , we deduce In view of (2.10) and Theorem 2.2 in Chapter 4 of [LM2], we have f ρ ∈ L and in Ω, u = 0, on Σ.
Applying (2.11) and integrating by parts, we find Using the fact that f ρ ∈ L, from this representation we get In view of (2.9), an application of the Cauchy-Schwarz inequality yields and from (2.12), we obtain . Combining this estimates with (2.13) and using the fact that suppq ⊂ Ω we get Then, using the fact that Repeating the above arguments with v j (t, x ′ , x 3 ; θ, ρ) = u j (t, x ′ , x 3 ; −θ, ρ), j = 1, 2, we get and we deduce (2.8).
Let δ < ε. From now on, we will set the following. First, we fix ϕ ∈ C ∞ 0 (R 2 ) real valued, supported on the unit ball centered at 0 and satisfying ϕ L 2 (R 2 ) = 1. We define We also define with h ∈ C ∞ 0 (R) real valued, supported on [−1, 1] and satisfying h L 2 (R) = 1. Finally, we set and we introduce the X-ray transform in R 2 defined for all f ∈ L 1 (R 2 ) by We set also Lemma 2.2 Let M > 0, 0 < α < 1 and let B M be the ball centered at 0 and of radius M of C α b (Ω). Let q 1 , q 2 ∈ B M and let q be equal to q 1 − q 2 extended by 0 outside of Ω. Then, for δ * = ε 4 , we have , 0 < δ < δ * , ρ > 1(2.14) with C > 0 depending only of ω, M and T .

It follows
and (2.15) implies In view of this estimate, (2.16) and the well known properties of the X-ray transform (see for example Theorem 5.1, p. 42 in [Nat]), for all y 3 ∈ R, we obtain (2.17) Here C depends only on ω, M and T . By interpolation we get, , y 3 ∈ R. (2.18) Consider the following estimate.
with C depending only of ω and M .
Proof. Note that
Then, applying the Fubini's theorem and making the substitution v = y ′ + δu, we get (2.20) Moreover, for all v ∈ R 2 , y 3 ∈ R, applying the Cauchy-Schwarz inequality, we obtain and, making the substitution z 3 = x3−y3 δ , we find Therefore, since q = 0 outside of Ω, we have and combining this estimate with (2.20), we obtain R δ [q](., y 3 ) L 2 (R 2 ) C, y 3 ∈ R (2.21) with C depending of ω and M . Note that with G(u) = ∂ u ϕ 2 (u). Therefore, repeating the above arguments, we get with C a constant depending of ω and M . By interpolation, we obtain which implies (2.14) since C is independent of y 3 ∈ R.
Lemma 2.4 Let M > 0, 0 < α < 1 and let B M be the ball centered at 0 and of radius M of C α b (Ω). Let q 1 , q 2 ∈ B M and let q be equal to q 1 − q 2 extended by 0 outside of Ω.
with C depending of ω, M andα = min(α, 1 2 ). Proof. Set Since q ∈ C α b (Ω)), one can easily check that for all y 3 ∈ R, q(., y 3 ) = y ′ → q(y ′ , y 3 ) ∈ C α (ω). Thus, in view of Lemma 2.40 in [Ch], we have Cδα q(., y 3 ) C α (ω) , y 3 ∈ R with C depending only of ω, M and α. It follows In view of this estimate, it only remains to prove Cδα. (2.25) Notice that Making the substitution u = x3−y3 δ we find In addition, for all y ′ ∈ R 2 , y 3 , u ∈ R, we have From this estimate we obtain with C depending only of ω, M and α. Finally, since This last estimate implies (2.25) and we deduce (2.23).
In addition, for γ γ * , we find Combining these two estimates we deduce (0.2).

Proof of Theorem 0.2
In this section we treat the special case introduced in Theorem 0.2 where condition (0.3) is fulfilled. Like in the previous section, we assume that 0 ∈ ω, T > Diam(ω) and we fix 0 < ε < min 1, T −Diam(ω)

3
. We start with the following.

Appendix
In this appendix we treat the direct problem. Our goal is to prove the following.
Theorem 4.1 Let q ∈ L ∞ (Ω) and f ∈ L. Then problem (0.1) admits a unique solution u ∈ C([0, T ]; H 1 (Ω)) ∩ C 1 ([0, T ]; L 2 (Ω)) such as ∂ ν u ∈ L 2 (Σ). Moreover, this solution u satisfies In the case of a bounded domain Ω, applying Theorem 2.1 of [LLT], [BCY] proved this result for f ∈ H 1 (Σ). Since Ω is an unbounded domain, we can not apply the analysis of [LLT]. Nevertheless, we can solve problem (0.1) by the classical argument which comprises in transforming this problem into a problem with an inhomogeneous equation and homogeneous boundary conditions. For this propose, we first need to establish a result of lifting for Sobolev spaces in a wave guide.

Result of lifting for Sobolev spaces
In this subsection we will show the following. and For this purpose, we will establish more general result of lifting for Sobolev spaces. Repeating the arguments of pages 38-40 in Chapter 1 of [LM1], by the mean of local coordinates we can replace ω by R 2 + , with R 2 + = {(x 1 , x 2 ) ∈ R 3 : x 1 > 0}, and ∂ω by R. Using the fact that Ω = ω × R and ∂Ω = ∂ω × R, with the same changes applied only to x ′ ∈ ω for any variable x = (x ′ , x 3 ) ∈ Ω, we can replace Ω by R 3 and, without lost of generality, we can assume T = ∞. Then, in our result we can replace H r,r (Q) (respectively H r (Ω) and H r,r (Σ)) by H r,r ((0, +∞) × R 3 + ) (respectively H r (R 3 + ) and H r,r ((0, +∞) × R 2 )). Let K 1 be the space of (g 0 , g 1 , u 0 , u 1 ) satisfying , k = 0, 1 and the compatibility conditions (4.36) Conditions (4.35) and (4.36) are global compatibility conditions (see subsection 2.4 in Chapter 4 of [LM2]). Let us also introduce the Hilbert space .
Clearly, for j + k = 1, we find and, from Theorem 2.2 in Chapter 4 of [LM2], we get .
From these two estimates we deduce that ( In the same way we show that Combining these estimates with Theorem 2.1 in Chapter 4 of [LM2] and (4.39), we deduce that In view of (4.39), (4.41) and (4.42), if we set w , conditions (4.37) and (4.38) will be fulfilled.
Proof of Theorem 4.2. Using local coordinate, in the same way as in the beginning of this subsection (see also Section 7.2 and the proof of Theorem 8.3 in Chapter 1 of [LM1]), we can replace the space L by K 2 . Then, we deduce Theorem 4.2 from the last part of Lemma 4.3.

Proof of Theorem 4.1
We will now go back to Theorem 4.1. First, using Theorem 4.2 we split u into two terms u = v[F ] + w with w ∈ H 2,2 (Q) satisfying ) C w H 2,2 (Q) .