Determining the waveguide conductivity in a hyperbolic equation from a single measurement on the lateral boundary

We consider the multidimensional inverse problem of determining the conductivity coefficient of a hyperbolic equation in an infinite cylindrical domain, from a single boundary observation of the solution. We prove H{\"o}lder stability with the aid of a Carleman estimate specifically designed for hyperbolic waveguides.

1 Statement of the problem and results

Introduction
The present paper deals with the inverse problem of determining the time-independent isotropic conductivity coefficient c : Ω → R appearing in the hyperbolic partial differential equation ∂ 2 t − ∇ · c∇ = 0, where Ω := ω × R is an infinite cylindrical domain whose cross section ω is a bounded open subset of R n−1 , n ≥ 2. Namely, ℓ > 0 being arbitrarily fixed, we seek Hölder stability in the identification of c in Ω ℓ := ω × (−ℓ, ℓ) from the observation of u on the lateral boundary Γ L := ∂ω × (−L, L) over the course of time (0, T ), for L > ℓ and T > 0 sufficiently large.
Several stability results in the inverse problem of determining one or several unknown coefficients of a hyperbolic equation from a finite number of measurements of the solution are available in the mathematics literature [1,5,6,7,17,18,26,29]. Their derivation relies on Bukhgeim-Klibanov's method [9], which is by means of a Carleman inequality specifically designed for hyperbolic systems. More precisely, [17,29] study the determination of the zero-th order term p : Ω → R appearing in ∂ t − ∆ + p = 0, while [1,7] deal with the identification of the speed c : Ω → R in the hyperbolic equation ∂ t − cA = 0 where A = A(x, D x ) is a second order differential operator. The case of a principal matrix term in the divergence form, arising from anisotropic media, was treated by Bellassoued, Jellali and Yamamoto in [5], using the full data (i.e. the measurements are performed on the whole boundary). Using the FBI transform Bellassoued and Yamamoto claimed logarithmic stability in [6] from arbitrarily small boundary observations. Imanuvilov and Yamamoto derived stability results in [18] by means of H −1 Carleman inequality, from data observation on subdomains fulfilling specific geometric assumptions. In [26] Klibanov and Yamamoto employed a different approach inspired by [25].
Similarly, numerous authors have used the Dirichlet-to-Neumann operator to claim stability in the determination of unknown coefficients of a hyperbolic equation. We refer to [4,19,28] for a non-exhaustive list of such references.
In all the above mentioned papers, the domain was bounded. Recently, the Bukhgeim-Klibanov method was adapted to the framework of infinite quantum cylindrical domains in [10,23,24]. Inverse boundary value problems stated in unbounded waveguides were also studied in [11,12,13] with the help of the Dirichlet-to-Neumann operator. In all the six previous articles, the observation is taken on the infinitely extended lateral boundary of the waveguide. The approach developed in this paper is rather different in the sense that we aim to retrieve any arbitrary bounded subpart of the non-compactly supported conductivity c from one data taken on a compact subset of the lateral boundary. This requires that suitable smoothness properties of the solution to (1.1) be preliminarily established in the context of the unbounded domain Ω.
The paper is organized as follows. Section 2 is devoted to the analysis of the direct problem associated with the hyperbolic system under study. In Section 3 we prove a global Carleman estimate specifically designed for hyperbolic systems in the cylindrical domain Ω. Finally Section 4 contains the analysis of the inverse problem and the proof of the main result.
For any open subset D of R m , m ∈ N * , we denote by H p (D) the p-th order Sobolev space on D for every p ∈ N, where H 0 (D) stands for L 2 (D). We write ·, · p,D (resp., · p,D ) for the usual scalar product (resp., norm) in H p (D) and we denote by

Statement of the problem
We examine the following initial boundary value problem (IBVP in short) with initial conditions (θ 0 , θ 1 ), where c is the unknown conductivity coefficient we aim to retrieve. This is by means of the Bukhgeim-Klibanov method imposing that the solution u to (1.1) be sufficiently smooth and appropriately bounded. Throughout the entire text we shall suppose that c fulfills the ellipticity condition c ≥ c m in Ω, (1.2) for some positive constant c m . Notice that we may assume, and this will be systematically the case in the sequel, without restricting the generality of the foregoing, that c m ∈ (0, 1). Let us now say a few words on the solution to (1.1). In order to exhibit sufficient conditions on the initial conditions (θ 0 , θ 1 ) (together with the cross section ω and the conductivity c) ensuring that the solution to (1.1) is within an appropriate functional class we shall make precise further, we need to introduce the self-adjoint operator A = A c , associated with c, generated in L 2 (Ω) by the closed sesquilinear form Evidently, A acts on its domain as −∇ ·c∇. Since A is positive in L 2 (Ω), by (1.2), the operator A 1/2 is well defined from the spectral theorem, and D(A 1/2 ) = D(q A ) = H 1 0 (Ω). For the sake of definiteness, we set A 0 := I and D(A 0 ) := L 2 (Ω), where I denotes the identity operator in L 2 (Ω), and we put is Hilbertian, and it is established in Proposition 2.6 that provided ∂ω is C 2m and c ∈ W 2m−1,∞ (Ω). As a matter of fact we know from Corollary 2.7 for any natural number m, that the system (1.1) admits a unique solution where C > 0 depends only on T , ω and c M .

Admissible conductivity coefficients and initial data
In order to solve the inverse problem associated with (1.1) we seek solutions belonging to ∩ 4 k=3 C k ([0, T ]; H 5−k (Ω)). Hence we chose m = 4 in (1.4) and impose on c to be in W 4,∞ (Ω; R) and satisfy (1.2). In what follows we note c M a positive constant fulfilling (1.6) Since our strategy is based on a Carleman estimate for the hyperbolic system (1.1), it is also required that the condition hold for some a ′ = (a 1 , . . . , a n−1 ) ∈ S n−1 and a 0 > 0. Hence, given O * , a neighborhood of Γ in R n−1 , and c * ∈ W 4,∞ (O * ∩ Ω; R) satisfying Notice that the above choice of m = 4 dictates that (θ 0 , θ 1 ) be taken in D(A 5/2 ) × D(A 2 ), which is embedded in H 5 (Ω) × H 4 (Ω) according to (1.3). Furthermore, it is required by the analysis of the inverse problem carried out in this article that θ 0 be in W 3,∞ (Ω) and satisfy for some η 0 > 0 and some open subset ω * in R n−1 , with C 2 boundary, satisfying Thus, for M 0 > 0 such that we define the set Θ ω * = Θ ω * (a ′ , M 0 , η 0 ) of admissible initial conditions (θ 0 , θ 1 ) as Having introduced all these notations we may now state the main result of this paper.
We stress out that the measurement of the observation data is performed on Γ L and not on the whole boundary ∂Ω L .

Analysis of the direct problem
In this section we establish existence and uniqueness results as well as regularity properties, for the solution to hyperbolic (1.1)-like IBVP systems. The corresponding results are similar to the ones obtained for hyperbolic equations in bounded domains (see e. g. [15, Sect. 7.2, Theorem 6]) but since Ω is unbounded here, they cannot be derived from them.

Existence and uniqueness result
With reference to (1.1) we consider the boundary value problem where f , g and h are suitable data, and we recall from 4 [27, Sect. 3, Theorem 8.2] the following existence and uniqueness result.
Moreover we have the estimate

Improved regularity
and we have the estimate Proof. By differentiating (2.1) with respect to t, we check that w := ∂ t v obeys by Proposition 2.1. Further, as Av = f − ∂ 2 t v from the first line in (2.1), we get that v ∈ C 0 ([0, T ]; D(A)), and that v(·, t) D(A) is majorized by the right hand side of (2.7), uniformly in t ∈ [0, T ]. This and (2.7) yield the desired result.

Higher regularity
Proof. a) The proof is by an induction on m, the case m = 0 following from Proposition 2.1. b) We assume that the theorem is valid for some m ∈ N and suppose that (2.10) We use the same strategy as in the proof of Proposotion 2.2. That is we differentiate (2.1) with respect to t and get that w : . . , m + 1, and the estimate: Here we used the identity v(·, t) 2 ) are majorized by the right hand side of (2.11), uniformly in t ∈ [0, T ], (2.11)-(2.12) yield the assertion of the theorem for m + 1.

Elliptic boundary regularity
In this subsection we extend the classical elliptic boundary regularity result for the operator ∇ · c∇ , which is well known in any sufficiently smooth bounded subdomain of R n (see e.g. Lemma 2.5. Let r be a nonnegative integer. We assume that ∂ω is C r+2 and that c ∈ W r+1,∞ (Ω) obeys (1.2). Then, for any ϕ ∈ H r (Ω), there exists a unique solution v ∈ H r+2 (Ω) to the boundary problem Moreover we have the estimate v r+2,Ω ≤ C r ϕ r,Ω , (2.14) where C r is a positive constant depending only on r, ω, the constant c m appearing in (1.2) and c W r+1,∞ (Ω) .
Proof. The proof is by induction on r. . This and (2.15) yield (2.14) with r = 0. b) Suppose that the statement of the lemma is true for r ∈ N fixed, and assume that ∂ω is C r+3 , c ∈ W r+2,∞ (Ω) and ϕ ∈ H r+1 (Ω where the constant C ′′ > 0 depends only on r, ω, c m and c W r+2,∞ (Ω) . Putting this together with (2.14), we obtain (2.14) where r is replaced by r + 1, proving that the statement of the lemma remains valid upon substituting r + 1 for r.
In view of Theorem 2.3 and Proposition 2.6 we obtain the following result.

Global Carleman estimate for hyperbolic equations in cylindrical domains
In this section we establish a global Carleman estimate for the system (1.1). To this purpose we start by time-symmetrizing the solution u of (1.1). Namely, we put u(x, t) := u(x, −t), x ∈ Ω, t ∈ (−T, 0).

The case of second order hyperbolic operators
In view of establishing a Carleman estimate for the operator where R is a first-order partial differential operator with L ∞ (Q) coefficients, we define for every δ > 0 and γ > 0 the following weight functions:

5)
holds for any s ≥ s 0 and v ∈ X L,T . Here C is a positive constant depending only on ω, a ′ , a 0 , δ 0 , γ 0 , s 0 , c m and c M . Moreover there exists a constant d ℓ > 0, depending only on ω, ℓ, δ 0 and γ 0 , such that the weight function ϕ defined by (3.3) satisfies

A Carleman estimate for the system (1.1)
In this subsection we derive from Proposition 3.1 a global Carleman estimate for the solution to the boundary value problem 20) where f ∈ L 2 (Q). To this purpose we introduce a cut-off function χ ∈ C 2 (R; [0, 1]), such that where ǫ is the same as in Proposition 3.1, and we set holds for any solution u to (3.20), uniformly in s ≥ s * .
4 Inverse problem

Linearized inverse problem and preliminary estimate
In this subsection we introduce the linearized inverse problem associated with (1.1) and relate the first Sobolev norm of the conductivity to some suitable initial condition of this boundary problem.
Having said that we may now upper bound, up to suitable additive and multiplicative constants, the e sϕ(·,0) -weighted first Sobolev norm of the conductivity c χ in Ω L , by the corresponding norm of the initial condition u (2) χ (·, 0). Lemma 4.1. Let u be the solution to the linearized problem (4.2) and let χ be defined by (3.21). Then there exist two constants s * > 0 and C > 0, depending only on ω, ε and the constant M 0 defined by (1.12), such that the estimate j=0,1 holds for all s ≥ s * .
Proof. Let Ω * be an open subset of R n with C 2 boundary, such that where ǫ is defined by Proposition 3.1. We notice from (3.21) and (1.11) that ∂ j i c χ ∈ H 1 0 (Ω * ) for all i ∈ N * n and j = 0, 1.

Completion of the proof
The proof is divided into three steps.
Therefore we have lim s→+∞ ρ s = 0, uniformly in Ω L , by the dominated convergence theorem, so we derive from (4.14)-(4.15) that  Here we used (1.5)-(1.12) and the embedding Ω ℓ ⊆ Ω L in order to substitute Ω ℓ for Ω L in the left hand side of (4.16). Now, taking into account thatd ℓ < d ℓ , we end up getting the desired result from (4.17).