Mourad Choulli
email: mourad.choulli@univ-lorraine.fr

AND Yavar Kian
email: yavar.kian@univ-amu.fr

Eric Soccorsi
email: eric.soccorsi@univ-amu.fr

DOUBLE LOGARITHMIC STABILITY IN THE IDENTIFICATION OF A SCALAR POTENTIAL BY A PARTIAL ELLIPTIC DIRICHLET-TO-NEUMANN MAP

We examine the stability issue in the inverse problem of determining a scalar potential appearing in the stationary Schrödinger equation in a bounded domain, from a partial elliptic Dirichlet-to-Neumann map. Namely, the Dirichlet data is imposed on the shadowed face of the boundary of the domain and the Neumann data is measured on its illuminated face. We establish a log log stability estimate for the L 2 -norm (resp. the H -1 -norm) of H t , for t > 0, and bounded (resp. L 2 ) potentials.

1. Introduction 1.1. Settings and main result. In the present paper Ω is a bounded domain of R n , n ≥ 3, with C 2 boundary Γ. We denote by ν(x) the outward unit normal to Γ computed at x ∈ Γ. For ξ ∈ S n-1 fixed, we introduce the two following subsets of Γ In what follows Γ + (ξ) (resp. Γ -(ξ)) will sometimes be referred to as the ξ-shadowed (resp., ξ-illuminated) face of Γ. Next, given q ∈ L ∞ (Ω), real-valued, we consider the unbounded self-adjoint operator A q in L 2 (Ω), acting on his domain D(A q ) = H 1 0 (Ω) ∩ H 2 (Ω), as A q = -∆ + q.

We assume throughout the entire text that 0 is in the resolvent set of A q (i.e. 0 is not in the spectrum of A q ) and put Q = {q ∈ L ∞ (Ω; R); 0 is not an eigenvalue of A q }.

Date: January 7, 2015.

1

We establish in Section 2 for any q ∈ Q and g ∈ H -1/2 (Γ) that the boundary value problem (abbreviated to BVP in the sequel)

(1.2) (-∆ + q)u = 0 in Ω, u = g on Γ, admits a unique transposition solution u ∈ H ∆ (Ω) = {w ∈ L 2 (Ω); ∆w ∈ L 2 (Ω)} and that the so-called Dirichlet-to-Neumann (DN in short) map

(1.3) Λ q : g → ∂ ν u is a bounded operator from H -1/2 (Γ) into H -3/2 (Γ). For q j ∈ Q, j = 1, 2, we denote by u j the solution to (1.2) where q j is substituted for q. Since u = u 1u 2 satisfies (-∆ + q 1 )u = (q 2q 1 )u 2 in Ω, u = 0 on Γ, and (q 2q 1 )u 2 ∈ L 2 (Ω), it holds true that u ∈ D (A q1 ). Therefore ∂ ν u ∈ H 1/2 (Γ) and

(1.4) Λ q1,q2 = Λ q1 -Λ q2 ∈ B(H -1/2 (Γ), H 1/2 (Γ)), hence the operator

(1.5) Λ q1,q2 : g ∈ H -1/2 (Γ) ∩ E ′ (F ) → Λ q1,q2 (g) |G ,
is bounded from H -1/2 (Γ) ∩ E ′ (F ), endowed with the norm of H -1/2 (Γ), into H 1/2 (G). We denote by Λ q1,q2 the norm of Λ q1,q2 in B(H -1/2 (Γ) ∩ E ′ (F ), H 1/2 (G)).

In the present paper we examine the stability issue in the inverse problem of determining the potential q ∈ Q appearing in (1.2) from the knowledge of Λ q0,q , where q 0 is an priori known suitable potential of Q.

Upon denoting by B X the unit ball of any Banach space X, we may now state the main result of this article as follows.

Theorem 1.1. For any δ > 0 and t > 0 we may find two constants c > 0 and c > 0, both of them depending only on δ and t, such that we have (1.6) q 1q 2 L 2 (Ω) ≤ c Λ q1,q2 + ln c ln Λ q1,q2 -t

, for any q 1 , q 2 ∈ Q ∩ δB L ∞ (Ω) satisfying (q 2q 1 )χ Ω ∈ δB H t (R n ) , and

(1.7) q 1 -q 2 H -1 (Ω) ≤ c Λ q1,q2 + ln c ln Λ q1,q2 -1 , for any q 1 , q 2 ∈ Q ∩ δB L 2 (Ω) .
Let us now briefly comment on Theorem 1.1.

Remarks 1.1. (a) We suppose in Theorem 1.1 that q j , j = 1, 2, are real-valued but it is not hard to see that the statement can be adapted at the expense of greater technical difficulties, to the case of complex-valued potentials. Nevertheless, for the sake of clarity, we shall restrict ourselves to real-valued potentials in the remaining part of this text. (b) For s > n/2 and ǫ ∈ (0, sn/2) we recall from the interpolation theorem [START_REF] Lions | Non-homogenuous boundary value problems and applications I[END_REF]Theorem 12.4,page 73] that

H n/2+ǫ (Ω) = [H s (Ω), H -1 (Ω)] θ with θ = (s -(n/2 + ǫ))/(s + 1
). Therefore we have

q 1 -q 2 L ∞ (Ω) ≤ C(s) q 1 -q 2 1-θ H s (Ω) q 1 -q 2 θ H -1 (Ω) ≤ C(s)δ 1-θ q 1 -q 2 θ H -1 (Ω)
, for any q 1 , q 2 ∈ Q such that q 2 ∈ q 1 + δB H s (Ω) , and some constant C(s) > 0, depending only on s. From this and (1.7) then follows that (1.8)

q 1 -q 2 L ∞ (Ω) ≤ c Λ q1,q2 + ln c ln Λ q1,q2 -1 θ .
(c) Fix t ∈ (0, +∞). Then, arguing as in the derivation of (1.8), we find two positive constants c and c such that the estimate

q 1 -q 2 L 2 (Ω) ≤ c Λ q1,q2 + ln c ln Λ q1,q2 -1 t t+1
, holds uniformly in q 1 , q 2 ∈ Q obeying q 2 ∈ q 1 + δB H t (Ω) . However in the particular case where ∂Ω is C [t]+1 , we point out that the above estimate is weaker than (1.6).

1.2. State of the art and comments. The celebrated inverse problem of determining q from the knowledge of Λ q was first proposed (in a slightly different setting) by Calderón in [C]. The uniqueness issue was treated by Sylvester and Uhlmann in [SU] and a log-type stability estimate was derived by Alessandrini in [A]. As shown by Mandache in [M], this log-type estimate is optimal.

All the above mentioned results were obtained with the full data, i.e. when measurements are performed on the whole boundary Γ. Taking the Neumann data on Γ -(ξ), while the Dirichlet data is imposed on the whole boundary Γ, Bukhgeim and Uhlmann proved in [BU] that partial information of Λ q still determines uniquely the potential. Their result was improved by Kenig, Sjöstrand and Uhlmann in [KSU] by measuring the Dirichlet data on the shadowed face of Γ and the Neumann data on the illuminated one. Moreover a reconstruction result was derived by Nachman and Street in [NS] from the same data as in [KSU].

Stability estimates with partial data go back to Heck and Wang's article [HW], where the L ∞ (Ω)-norm of q is log log stably recovered from Λ q with partial Neumann data. The same type of estimate was derived in [START_REF] Caro | Stability estimates for the Calderón problem with partial data[END_REF]. Both papers require that the Dirichlet data be known on the whole boundary. This constraint was weakened by Caro,Dos Santos Ferreira and Ruiz in [CDR1]. These authors proved log-log stability of q with respect to a partial DN map associated with Dirichlet (resp. Neumann) data measured on a neighborhood of ∪ ξ∈N Γ -(ξ) (resp. ∪ ξ∈N Γ + (ξ)) where N in a subset of S n-1 . Their result, which is similar to (1.6), is established for the L p -norm, p ∈ (1, +∞), of bounded and W λ,p -potentials q with λ ∈ (0, 1/p]. Therefore (1.7) is valid for a wider class of allowable potentials than in [START_REF] Caro | Stability estimates for the Radon transform with restricted data and applications[END_REF].

The derivation of Theorem 1.1 relies on complex geometrical optics (CGO in short) solutions to (1.2) and the Carleman inequality established by Bukhgeim and Uhlmann in [BU]. These are the two main ingredients of the analysis carried out in [HW], but in contrast to [HW], we use here this Carleman estimate to construct CGO solutions vanishing on a definite part of the boundary Γ.

Notice that usual stability estimates in the inverse problem of determining a potential from the full DN map are of log type, while they are of log log type for partial data. Indeed, it turns out that the low frequencies of the Fourier transform of the potential are bounded uniformly in all directions by the DN map, but that this is no longer the case with the partial data. This technical issue can be remedied by using the analytic properties of the Fourier transform. The additional log in the stability estimate for the potential may thus be seen as the price to pay for recovering this analytic function by its values in a subdomain, which is an ill-posed problem.

1.3. Outline. The paper is organized as follows. In Sect. 2 we introduce the transposition solution associated with the BVP (1.2) and rigorously define the various DN maps required by the analysis of the inverse problem. Sect. 3 is devoted to building CGO solutions that vanish on some part of the boundary Γ. These functions are useful for the proof of Theorem 1.1, given in Sect. 4.

Transposition solutions

In this section we define the transposition solution to the BVP associated with suitable data (f, g),

(2.1) (-∆ + q)u = f in Ω, u = g on Γ,
which play a pivotal role in the analysis of the inverse problem carried out in this paper. To this purpose we start by recalling two useful results for the Hilbert space

H ∆ (Ω) = {u ∈ L 2 (Ω); ∆u ∈ L 2 (Ω)} endowed with its natural norm u H∆(Ω) = u 2 L 2 (Ω) + ∆u 2 L 2 (Ω) 1/2
.

The first result is the following trace theorem, borrowed from [START_REF] Bukhgeim | Recovering a potential from partial Cauchy data[END_REF]Lemma 1.1].

Lemma 2.1. For j = 0, 1, the trace map

t j u = ∂ j ν u |Γ , u ∈ D(Ω)
, extends to a continuous operator, still denoted by t j , from H ∆ (Ω) into H -j-1/2 (Γ). Namely, there exists c j > 0, such that the estimate

t j u H -j-1/2 (Γ) ≤ c j u H∆(Ω) ,
holds for every u ∈ H ∆ (Ω).

Let us denote by •, • j+1/2 , j = 0, 1, the duality pairing between H j+1/2 (Γ) and H -j-1/2 (Γ), where the second argument is conjugated. Then we have the following generalized Green formula, which can be found in [START_REF] Bukhgeim | Recovering a potential from partial Cauchy data[END_REF]Corollary 1.2

]. Lemma 2.2. Let q be in L ∞ (Ω). Then, for any u ∈ H ∆ (Ω) and v ∈ H 2 (Ω), we have Ω (∆ -q)uvdx = Ω u(∆ -q)vdx + t 1 u, t 0 v 3/2 -t 0 u, t 1 v 1/2 .
Let q ∈ Q. By the usual H 2 -regularity property for elliptic BVPs (see e.g. [START_REF] Lions | Non-homogenuous boundary value problems and applications I[END_REF]Theorem 5.4,page 165]), we know that for each f ∈ L 2 (Ω) and g ∈ H 3/2 (Γ), there exists a unique solution S q (f, g) ∈ H 2 (Ω) to (2.1). Moreover the linear operator S q is bounded from L 2 (Ω) × H 3/2 (Γ) into H 2 (Ω), i.e. there exists a constant C > 0 such that we have

(2.2) S q (f, g) H 2 (Ω) ≤ C f L 2 (Ω) + g H 3/2 (Γ) .
For further reference we put S q,0 (f ) = S q (f, 0) and S q,1 (g) = S q (0, g), so we have S q (f, g) = S q,0 (f ) + S q,1 (g) from the linearity of S q . Next, applying Lemma 2.2, we get for all (f, g)

∈ L 2 (Ω) × H 3/2 (Γ) and v ∈ H = H 1 0 (Ω) ∩ H 2 (Ω) that (2.3) - Γ g∂ ν vdσ(x) + Ω f vdx = Ω u(-∆ + q)vdx, u = S q (f, g).
In view of the left hand side of (2.3) we introduce the following continuous anti-linear form on H

(2.4)

ℓ(v) = - Γ g∂ ν vdσ(x) + Ω f vdx, v ∈ H.
In light of (2.2), the operator L = ℓ • S q,0 is bounded in L 2 (Ω). Further, with reference to (2.4), we generalize the definition of the anti-linear form ℓ to (f, g) ∈ H * × H -1/2 (Γ), upon setting

ℓ(v) = -g, ∂ ν v 1/2 + f, v , v ∈ H,
where •, • denotes the duality pairing between H and H * , conjugate linear in its second argument. For any

h ∈ L 2 (Ω), L(h) = ℓ (S q,0 (h)) satisfies (2.5) |L(h)| ≤ g H -1/2 (Γ) ∂ ν S q,0 (h) H 1/2 (Γ) + f H * S q,0 (h) H ≤ C g H -1/2 (Γ) + f H * h L 2 (Ω) ,
according to (2.2). Hence L is a continuous anti-linear form on L 2 (Ω). By Riesz representation theorem, there is a unique vector S t q (f, g) ∈ L 2 (Ω) such that we have

-g, ∂ ν S q,0 (h) 1/2 + f, S q,0 (h) = Ω S t q (f, g)hdx, h ∈ L 2 (Ω).
Bearing in mind that A q is boundedly invertible in L 2 (Ω) (since 0 is in the resolvent set of A q ) and that S q,0 = A -1 q , we obtain upon taking h = A q v in above identity, where v is arbitrary in H, that

(2.6) -g, ∂ ν v 1/2 + f, v = Ω S t q (f, g)(-∆ + q)vdx, v ∈ H. Moreover (2.5) entails (2.7) S t q (f, g) L 2 (Ω) ≤ C g H -1/2 (Γ) + f H * .
For each (f, g) ∈ H * × H -1/2 (Γ), S t q (f, g) will be referred to as the transposition solution to the BVP (2.1). As a matter of fact we deduce from (2.6) that (2.8) (-∆ + q)S t q (f, g) = f in the distributional sense in Ω.

Let us establish now that the transposition solution S t q (f, g) coincides with the classical H 2 (Ω)-solution S q (f, g) to (2.1) in the particular case where (f, g)

∈ L 2 (Ω) × H 3/2 (Γ). Proposition 2.1. For any (f, g) ∈ L 2 (Ω) × H 3/2 (Γ) we have S t q (f, g) = S q (f, g).
Proof. Put u = S t q (f, g). We have u ∈ H ∆ (Ω) directly from (2.8) hence t 0 u ∈ H -1/2 (Γ). Further, given ϕ ∈ H 1/2 (Γ), we may find v ∈ H such that t 1 v = ϕ by the usual extension theorem (see e.g. [START_REF] Lions | Non-homogenuous boundary value problems and applications I[END_REF]Theorem 8.3,p. 39]). Applying Lemma 2.2 for such a test function v, we find that

Ω f vdx = Ω (-∆ + q)uvdx = Ω u(-∆ + q)vdx + t 0 u, ϕ 1/2 .
From this and (2.6) then follows that

Ω f vdx = Ω f vdx -g, ϕ 1/2 + t 0 u, t 1 v 1/2 , which entails g -t 0 u, ϕ 1/2 = 0.
Since the above identity holds for any ϕ ∈ H 1/2 (Γ) we obtain that t 0 u = g and hence u = g on Γ. This yields the desired result.

For g ∈ H -1/2 (Γ) we put S t q,1 (g) = S t q (0, g). We have S t q,1 (g) ∈ H ∆ (Ω) by (2.8), with

S t q,1 (g) H∆(Ω) ≤ C 1 + q L ∞ (Ω) g H -1/2 (Γ) , from (2.7). Hence S t q,1 ∈ B(H -1/2 (Γ), H ∆ (Ω)) so we get Λ q = t 1 • S t q,1 ∈ B(H -1/2 (Γ), H -3/2 (Γ))
, with the aid of Lemma 2.1.

Finally we have S t q (f, g) = S t q (f, 0) + S t q (0, g), by linearity of S t q . Put S t q,0 (f ) = S t q (f, 0) and S t q,1 (g) = S t q (0, g). Since S t q,0 ∈ B(H * , H ∆ (Ω)) and S t q,1 ∈ B(H -1/2 (Γ), H ∆ (Ω)), it follows from Lemma 2.1 that

Λ q = t 1 • S t q,1 B(H -1/2 (Γ), H -3/2 (Γ)).
Further, as S t q,1 (g) = S t 0,1 (g) + S q,0 (f ), with f = -qS t 0,1 (g), we get that Λ q = Λ 0 + R q , where the operatorR q

R q : g → t 1 S q,0 -qS t 0,1 (g) ∈ H 1/2 (Γ), is bounded from H -1/2 (Γ) into H 1/2 (Γ).
Remark 2.1. Let us denote by Σ q the (continuous) DN map

Σ q : g ∈ H 3/2 (Γ) → t 1 S q,1 (g) ∈ H 1/2 (Γ).
Then, since q is real-valued, we get from the Green formula that

Γ gΣ q (h)dσ(x) = Γ Σ q (g)hdσ(x), g, h ∈ H 3/2 (Γ),
which entails that Σ * q |H 3/2 (Γ) = Σ q . On the other hand, we deduce from Lemma 2.2 that Σ * q = Λ q .

3. CGO solutions vanishing on some part of the boundary

In this section we build CGO solutions to the Laplace equation appearing in (1.2), that vanish on a prescribed part of the boundary Γ. This is by means of a suitable Carleman estimate borrowed from [BU]. The corresponding result is as follows.

Proposition 3.1. For δ > 0 fixed, let q ∈ δB L ∞ (Ω) . Let ζ, η ∈ S n-1 satisfy ζ • η = 0 and fix ǫ > 0 so small that Γ ǫ -= Γ ǫ -(ζ) = {x ∈ Γ; ζ • ν(x) < -ǫ} = ∅.
Then there exists τ 0 = τ 0 (δ) > 0, such that for any τ ≥ τ 0 we may find ψ ∈ L 2 (Ω) obeying ψ L 2 (Ω) ≤ Cτ -1/2 for some constant C > 0 depending only on δ, Ω and ǫ, and such that the function u = e τ (ζ+iη

)•x (1 + ψ) ∈ H ∆ (Ω) is solution to the BVP (-∆ + q)u = 0 in Ω, u = 0 on Γ ǫ -. Proof.
The proof is made of three steps.

Step 1: A Carleman estimate. For notational simplicity we write Γ ± instead of Γ ± (ζ), which isdefined in (1.1), and recall from [BU] that we may find two constants τ 0 = τ 0 (δ) > 0 and C = C(δ) > 0 such the estimate

Cτ 2 Ω e -2τ x•ζ |v| 2 dx + τ Γ+ |ζ • ν(x)|e -2τ x•ζ |∂ ν v| 2 dσ(x) ≤ Ω e -2τ x•ζ |(∆ -q)v| 2 dx + τ Γ- |ζ • ν(x)|e -2τ x•ζ |∂ ν v| 2 dσ(x),
holds for all τ ≥ τ 0 and v ∈ H. Since Γ ± (-ζ) = Γ ∓ (ζ), the above inequality may be equivalently rewritten as

Cτ 2 Ω e 2τ x•ζ |v| 2 dx + τ Γ- |ζ • ν(x)|e 2τ x•ζ |∂ ν v| 2 dσ(x) (3.1) ≤ Ω e 2τ x•ζ |(∆ -q)v| 2 dx + τ Γ+ |ζ • ν(x)|e 2τ x•ζ |∂ ν v| 2 dσ(x), v ∈ H, τ ≥ τ 0 .
In view of more compact reformulation of (3.1) we introduce for each real number τ the two following scalar products:

(u, v) τ = Ω e 2τ x•ζ uvdx in L 2 (Ω), and φ, ψ τ,µ,± = Γ± µ(x)e 2τ x•ζ φψdσ(x), where µ(x) = |ζ • ν(x)|, in L 2 (Γ ± ).
We denote by L 2 τ (Ω) (resp., L 2 τ,µ (Γ ± )) the space L 2 (Ω) (resp., L 2 (Γ ± )) endowed with the norm • τ (resp. • τ,µ,± ) generated by the scalar product (•, •) τ (resp., •, • τ,µ,± ). With these notations, the estimate (3.1) simply reads

(3.2) Cτ 2 v 2 τ + τ ∂ ν v 2 τ,µ,-≤ (∆ -q)v 2 τ + τ ∂ ν v 2 τ,µ,+ , v ∈ H, τ ≥ τ 0 .
Step 2: Building suitable transposition solutions. Let us identify

L 2 0 (Ω) = L 2 (Ω) (resp., L 2 0,1 (Γ ± ) = L 2 (Γ ± )) with its dual space, so the space dual to L 2 τ (Ω) (resp., L 2 τ,µ (Γ ± ))can be identified with L 2 -τ (Ω) (resp. L 2 -τ,µ -1 (Γ ± )).
Next we consider the operator

(3.3) P : v ∈ H → ((∆ -q)v, ∂ ν v |Γ+ ) ∈ L 2 τ (Ω) × L 2 τ,µ (Γ + ),
which is injective by (3.1). Therefore, for each (f, g)

∈ L 2 -τ (Ω) × L 2 -τ,µ -1 (Γ -), the following anti-linear form (3.4) ℓ(w 1 , w 2 ) = (f, v) + g, ∂ ν v -, (w 1 , w 2 ) = P v, v ∈ H,
where (•, •) (resp., •, • ± ) denotes the usual scalar product in L 2 (Ω) (resp., L 2 (Γ ± )), is well defined on Ran(P ). Moreover it holds true for every (w 1 , w 2 ) = P v, where v is arbitrary in H, that

|ℓ(w 1 , w 2 )| ≤ f -τ v τ + g -τ,µ -1 ,-∂ ν v τ,µ,- (3.5) ≤ τ -1 f -τ + τ -1/2 g -τ,µ -1 ,-τ 2 v 2 τ + τ ∂ ν v 2 τ,µ,- 1/2 .
Thus, upon equipping the space

L 2 τ (Ω) × L 2 τ,µ (Γ + ) with the norm | (w 1 , w 2 ) | τ = w 1 2 τ + τ w 2 2 τ,µ,+ 1/2 ,
we derive from (3.2) and (3.5) that

|ℓ(w 1 , w 2 )| ≤ C τ -1 f -τ + τ -1/2 g -τ,µ -1 ,- (∆ -q)v 2 τ + τ ∂ ν v 2 τ,µ,+ 1/2 ≤ C τ -1 f -τ + τ -1/2 g -τ,µ -1 ,-| P v | τ ,
for some constant C > 0 depending only on δ. As a consequence we have

(3.6) |ℓ(w 1 , w 2 )| ≤ C τ -1 f -τ + τ -1/2 g -τ,µ -1 ,-| (w 1 , w 2 ) | τ , (w 1 , w 2 ) ∈ Ran(P ).
Let us identify the dual space of

L 2 τ (Ω) × L 2 τ,µ (Γ + ) with L 2 -τ (Ω) × L 2 -τ,µ -1 (Γ + ) endowed with the norm | (w 1 , w 2 ) | -τ = w 1 2 -τ + τ -1 w 2 2 -τ,µ-1,+
1/2 . Thus, with reference to (3.3)-(3.4), we deduce from (3.6) upon applying Hahn-Banach extension theorem that there exists

(v 1 , v 2 ) ∈ L 2 -τ (Ω) × L 2 -τ,µ -1 (Γ + ) obeying (3.7) | (v 1 , v 2 ) | -τ ≤ C τ -1 f -τ + τ -1/2 g -τ,µ -1 ,-,
where C > 0 is the same as in the right hand side of (3.6), such that we have

(v 1 , (∆ -q)v) + v 2 , ∂ ν v + = (f, v) + g, ∂ ν v -, v ∈ H.
Bearing in mind that f ∈ H * and gχ Γ-v 2 χ Γ+ ∈ H -1/2 (Γ), where χ Γ± is the characteristic function of Γ ± in Γ, the above identity reads

(-f ), v -gχ Γ--v 2 χ Γ+ , ∂ ν v 1/2 = Ω v 1 (∆ -q)vdx, v ∈ H.
Therefore, by (2.6), v 1 is the transposition solution to the following IBVP:

(3.8) (-∆ + q)v 1 = -f in Ω, v 1 = gχ Γ--v 2 χ Γ+ on Γ.
Step 3: End of the proof. Set ρ = τ (ζ + iη). The last step of the proof involves picking ϕ ∈ D(R n ; [0, 1]) such that ϕ(x) = 1 for x ∈ Γ ǫ -and supp(ϕ

) ∩ Γ ⊂ Γ ǫ/2
-, and considering the transposition solution v 1 to the BVP (3.8) associated with f = qe ρ•x and g = -ϕe ρ•x . Thus, putting ψ = e -ρ•x v 1 , we derive from (3.7) that

ψ 0 = v 1 -τ ≤ C τ -1 qe ρ•x -τ + τ -1/2 ϕe ρ•x -τ,µ -1 ,- ≤ Cτ -1/2 δτ 0 (δ) -1/2 + 2|Γ -| ǫ ,
where |Γ -| denotes the (n -1)-dimensional Lebesgue-measure of Γ -. Further, bearing in mind that ρ • ρ = 0, it easy to check that u = e ρ•x (1 + ψ) ∈ H ∆ (Ω) satisfies (-∆ + q)u = 0 in Ω. Finally, since v 1 = g on Γ -by (3.8), we obtain that ψ = -ϕ = -1 and consequently that u = 0 on Γ ǫ -. This proves the desired result.

Proof of Theorem 1.1

This section contains the proof of Theorem 1.1 which consists of a succession of five lemmas.

4.1.

A suitable set of Fourier variables. For ǫ > 0 we choose η = η(ǫ) ∈ (0, π/2) so small that

(4.1) (1 -sin θ) 2 + 4 cos 2 θ 1/2 < ǫ, θ ∈ (π/2 -η, π/2 + η),
and we define B ǫ as the set of vectors β = (β 1 , . . . , β n ) ∈ R n with the following spherical coordinates (4.2)

             β 1 = s cos θ 1 , β j = s cos θ j j-1 k=1 sin θ k , j = 2, . . . , n -2, β n-1 = s sin φ n-2 k=1 sin θ k , β n = s cos φ n-2
k=1 sin θ k , where s ∈ (0, 1), θ 1 ∈ (π/2η, π/2 + η), θ 2 (0, π/3), θ j ∈ (0, π) for j = 3, . . . , n -2, and φ ∈ (0, 2π). Notice that B ǫ has positive Lebesgue measure in R n given by:

(4.3) |B ǫ | = 1 0 π/2+η π/2-η π/3 0 π 0 . . . π 0 2π 0 s n-1 n-2 k=1 sin k θ n-1-k dsdθ 1 . . . dθ n-2 dφ > 0.
For further reference, we now establish the following result.

Lemma 4.1. Let T be any orthogonal transformation in R n that maps ξ onto e 1 = (1, 0, . . . , 0). Then for all ǫ > 0 and κ ∈ T * B ǫ , there exists ζ ∈ S n-1 satisfying κ • ζ = 0 and |ζ -ξ| < ǫ.

Proof. Let β = (β 1 , . . . , β n ) ∈ B ǫ be given by (4.1)-(4.2). Introduce ζ = ( ζ1 , ζ2 , 0, . . . , 0), where

ζ1 = sin θ 1 sin 2 θ 1 + cos 2 θ1 cos 2 θ2 1/2 , ζ2 = - cos θ1 cos θ2 sin 2 θ 1 + cos 2 θ1 cos 2 θ2 1/2 .
Evidently we have ζ ∈ S n-1 and β • ζ = 0. Moreover it follows from (4.1) that

| ζ -e 1 | 2 = (sin θ 1 -1) 2 + cos 2 θ1 cos 2 θ2 sin 2 θ 1 + cos 2 θ1 cos 2 θ2 ≤ (sin θ 1 -1) 2 + 4 cos 2 θ 1 < ǫ 2 .
Finally, bearing in mind that κ = T * β and e 1 = T ξ, we obtain the desired result upon taking ζ = T * ζ.

4.2. Alessandrini's identity and consequence. For ǫ > 0 put

F ǫ (ζ) = Γ \ Γ ǫ -(ξ) and G ǫ (ξ) = Γ \ Γ ǫ -(-ξ), where Γ ǫ -(±ξ) is the same as in Proposition 3.1. Since F ǫ (ξ) = {x ∈ Γ; ξ • ν(x) ≥ -ǫ} (resp., G ǫ (ξ) = {x ∈ Γ; ξ • ν(x) ≤ ǫ}), it holds true that ∩ ǫ>0 F ǫ (ξ) = {x ∈ Γ; ξ • ν(x) ≥ 0} = Γ + (ξ) (resp., ∩ ǫ>0 G ǫ (ξ) = {x ∈ Γ; ξ • ν(x) ≤ 0} = Γ -(ξ)
). Thus, from the very definitions of F and G, we may choose ǫ 0 = ǫ 0 (ξ, F, G) > 0 so small that (4.4)

F 2ǫ (ξ) ⊂ F and G 2ǫ (ξ) ⊂ G, ǫ ∈ (0, ǫ 0 ].
Having said that we turn now to proving the following statement.

Lemma 4.2. Let T be the same as in Lemma 4.1, let τ ∈ [τ 0 , +∞), where τ = τ 0 (δ) is defined in Proposition 3.1, and let ǫ ∈ [0, ǫ 0 ). Then, there exists a constant C > 0, depending only on Ω, δ and ǫ, such that the estimate

Ω (q 2 -q 1 )e -iκ•x dx ≤ C e 2dτ Λ q1,q2 + τ -1/2 , holds uniformly in κ ∈ rT * B ǫ and r ∈ (0, 2τ ). Here Λ q1,q2 is the B(H -1/2 (Γ) ∩ E ′ (F ), H 1/2 (G))-norm of the operator Λ q1,q2 defined in (1.3)-(1.5), and d = d(Ω) = max x∈Ω |x| < ∞.
Proof. Fix r ∈ (0, 2τ ), κ ∈ rT * B ǫ , and let ζ be given by Lemma 4.

1. Pick ℓ ∈ R n such that ℓ • κ = ℓ • ζ = 0 (which is possible since n ≥ 3) and |κ + ℓ| 2 = |κ| 2 + |ℓ| 2 = 4τ 2 . Set ρ j = (-1) j τ ζ -i κ + (-1) j ℓ 2 , j = 1, 2,
and let u j = e ρj •x (1 + ψ j ) ∈ H ∆ (Ω) be defined in accordance with Proposition 3.1, in such a way that we have (4.5) (-∆ + q j )u j = 0 in Ω, u j = 0 on Γ ǫ -((2j -3)ζ), and (4.6)

ψ j L 2 (Ω) ≤ Cτ -1/2 .
Put u = S t q1,1 (t 0 u 2 ) so we have w = uu 2 = S q1,0 ((q 1q 2 )u 2 ) ∈ H and (4.7)

t 1 w = Λ q1 (t 0 u 2 ) -Λ q2 (t 0 u 2 ).
Upon applying the generalized Green formula of Lemma 2.2 with u = u 1 and v = w, we derive from (4.5) that (4. 8)

Ω (q 2 -q 1 )u 1 u 2 dx = t 0 u 1 , t 1 w 1/2 .
Notice from the second line of (4.5) that the trace t 0 u 2 (resp.,

t 0 u 1 ) is supported in F ǫ (ζ) = Γ \ Γ ǫ -(ζ) (resp., G ǫ (ζ) = Γ \ Γ ǫ -(-ζ)), where Γ ǫ -(±ζ) is defined in Proposition 3.1. Otherwise stated we have F ǫ (ζ) = {x ∈ Γ; ζ • ν(x) ≥ -ǫ} (resp., G ǫ (ζ) = {x ∈ Γ; ζ • ν(x) ≤ ǫ}) and hence F ǫ (ζ) ⊂ F 2ǫ (ξ) (resp., G ǫ (ζ) ⊂ G 2ǫ (ξ))
since |ζ -ξ| < ǫ. From this and (4.4) then follows that supp(t 0 u 2 ) ⊂ F (resp., supp(t 0 u 1 ) ⊂ G), which together with (4.7)-(4.8) yields (4.9)

Ω (q 2 -q 1 )u 1 u 2 dx ≤ Λ q1,q2 t 0 u 1 H -1/2 (Γ) t 0 u 2 H -1/2 (Γ) .
Moreover, by (4.5)-(4.6) and the very definition of u j , j = 1, 2, we get that

t 0 u j H -1/2 (Γ) ≤ c j u j L 2 (Ω) + ∆u j L 2 (Ω) ≤ c j u j L 2 (Ω) + q j u j L 2 (Ω) (4.10) ≤ Ce dτ (1 + τ -1/2 ).
Upon possibly substituting max(1, τ 0 (δ)) for τ 0 (δ) (which does obviously not restrict the generality of the above reasoning) we deduce from (4.9)-(4.10) that (4.11) Ω (q 2q 1 )u 1 u 2 dx ≤ Ce 2dτ Λ q1,q2 . Now the desired result follows readily from (4.6) and (4.11) upon taking into account that u 1 u 2 = e -iκ•x (1 + ψ 1 )(1 + ψ 2 ). 4.3. Bounding the Fourier coefficients. We introduce the function q : R n → R by setting q = (q 2 -q 1 )χ Ω , where χ Ω denotes the characteristic function of Ω in R n . We aim to upper bound the Fourier transform q of q, on the unit ball B of R n , by means of the following direct generalization of [START_REF] Apraiz | Observability inequalities and measurable sets[END_REF]Theorem 4] for complex-valued real-analytic functions.

Theorem 4.1. Assume that the function F : 2B n → C is real-analytic and satisfies the condition

|∂ α F (κ)| ≤ K |α|! ρ |α| , κ ∈ 2B, α ∈ N n ,
for some (K, ρ) ∈ R * + × (0, 1]. Then for any measurable set E ⊂ B with positive Lebesgue measure, there exist two constants M = M (ρ, |E|) > 0 and θ = θ(ρ, |E|) ∈ (0, 1) such that we have

F L ∞ (B) ≤ M K 1-θ 1 |E| E |F (κ)|dκ θ .
The result is as follows.

Lemma 4.3. For all ǫ ∈ (0, ǫ 0 ] there exist two constants C = C(Ω, δ, ǫ) > 0 and θ = θ(Ω, ǫ) ∈ (0, 1), such that we have

| q(κ)| ≤ Ce (1-θ)r e dτ Λ q1,q2 + τ -1/2 θ , κ ∈ rB, r ∈ (0, 2τ ), τ ∈ [τ 0 , +∞).
Proof. Let τ ∈ [τ 0 , +∞) and r ∈ (0, 2τ ) be fixed. By Lemma 4.2, we have

(4.12) | q(rκ)| ≤ C e 2dτ Λ q1,q2 + τ -1/2 , κ ∈ T * B ǫ .
In view of (4.3) we apply Theorem 4.1 with E = T * B ǫ and F (κ) = q(rκ) for κ ∈ 2B n . Indeed, in this particular case it holds true for any α = (α 1 , . . . , α n ) ∈ N n and κ ∈ 2B n that

∂ α F (κ) = Ω q(x)(-ir) |α| n k=1 x α k k e -irκ•x dx 1 . . . dx n , whence (4.13) |∂ α F (κ)| ≤ q L 1 (Ω) r |α| d |α| ≤ q L 1 (Ω) r |α| |α|! |α|! (d -1 ) |α| ≤ q L 1 (Ω) e r |α|! (d -1 ) |α| ,
where we recall that d = d(Ω) = max x∈Ω |x|. Thus, with reference to (4.12)-(4.13), we obtain that

(4.14) | q(rκ)| ≤ Ce (1-θ)r e dτ Λ q1,q2 + τ -1/2 θ , κ ∈ B,
which immediately yields the result.

4.4. Stability inequalities. We turn now to proving the stability inequalities (1.6)-(1.7). We start with (1.6). For t > 0 fixed, we assume that q ∈ H t (R n ) and put M = q H t (R n ) . By Parseval inequality, it holds true for every r > 0 that

q 2 L 2 (Ω) = q 2 L 2 (R n ) = |κ|≤r | q(κ)| 2 dk + |κ|>r | q(κ)| 2 dk (4.15) ≤ |κ|≤r | q(κ)| 2 dk + 1 r 2t |κ|>r |(1 + |κ| 2 ) t | q(κ)| 2 dκ ≤ |κ|≤r | q(κ)| 2 dk + M 2 r 2t , since R n |(1 + |κ| 2 ) t | q(κ)| 2 dκ = q 2 H t (Ω)
. From this and Lemma 4.3 then follows that (4.16)

q 2 L 2 (Ω) ≤ Cr n e 2(1-θ)r e dτ Λ q1,q2 + τ -1/2 2θ + M 2 r 2t , r ∈ (0, 2τ ), τ ∈ [τ 0 , +∞). Let us suppose that (4.17) Λ q1,q2 ∈ (0, γ 0 ),
where γ 0 = υ(τ 0 ) and υ(τ ) = τ -1/2 e -dτ for τ ∈ (0, +∞). Notice that γ 0 ∈ (0, 1) since τ 0 ∈ [1, +∞). Moreover, υ being a strictly decreasing function on [τ 0 , +∞), there exists a unique τ * ∈ (τ 0 , +∞) satisfying 2d , upon recalling that ln x < x for all x ∈ (0, +∞). Further, taking τ = τ * in (4.16) we obtain for each r ∈ (0, 2τ * ) that q 2 L 2 (Ω) ≤ C2 2θ r n e 2r τ -θ * + M 2 /r 2t ≤ C ′ e (n+2)r (2dτ * ) -θ + M 2 /r 2t , where C ′ = C2 3θ d θ . This and (4.18) entail

υ(τ * ) = Λ q1,q2
(4.19) q 2 L 2 (Ω) ≤ C ′ e (n+2)r ln Λ q1,q2 -θ + M 2 r 2t , r ∈ (0, 2τ * ).
The next step of the derivation involves finding r * ∈ (0, 2τ * ) such that both terms C ′ e (n+2)r * ln Λ q1,q2

-θ and M 2 /r 2t * appearing in the right hand side of (4.19) are equal. This can be achieved upon assuming in addition to (4.17) that Λq1,q2 ∈ (0, γ 1 ), where

γ 1 = e - M 2 (n+2)C ′ 1 1+2t-θ
, in such a way that we have

(4.20) (n + 2) ln Λ q1,q2 1+2t-θ ≥ M 2 C ′ .
Indeed, we see from (4.18) that the function ι d : r → (dr) 2t e (n+2)dr satisfies

ι d (2τ * ) > ln Λ q1,q2 2t e (n+2)|ln Λq 1 ,q 2 | > ln Λ q1,q2 2t
+ (n + 2) ln Λ q1,q2 1+2t
,

and
2 L 2 (Ω) ≤ 2M 2 /r 2t * , yields (4.21) q L 2 (Ω) ≤ 2M 2t + n + 2 θ t ln M 2 C ′ 1/θ ln Λ q1,q2 -t
.

Summing up, we obtain (1.6) with c = 2M ((2t + n + 2)/θ) t and c = (M 2 /C ′ ) 1/θ provided Λ q1,q2 is smaller that γ = min j=0,1 γ j .

On the other hand, in the particular case where Λ q1,q2 ≥ γ, we have q L 2 (Ω) ≤ (M/γ) Λ q1,q2 . Putting this together with (4.21) we end up getting (1.6) upon possibly enlarging c.

Finally, we obtain (1.7) by arguing as in the derivation of (1.6) upon preliminarily substituting the estimate

q H -1 (Ω) = |κ|≤r |(1 + |κ| 2 ) -1 | q(κ)| 2 dκ + |κ|>r |(1 + |κ| 2 ) -1 | q(κ)| 2 dκ ≤ |κ|≤r | q(κ)| 2 dκ + 1 r 2 |κ|>r | q(κ)| 2 dκ ≤ |κ|≤r | q(κ)| 2 dκ + M 2 r 2
for (4.15). This completes the proof of Theorem 4.1.

Application to conductivity problem

In this section we examine the stability issue in the inverse problem of determining the conductivity coefficient σ appearing in the system (5.1)

-div(σ∇u) = f in Ω, u = g on Γ, from the partial DN map. The strategy is to link this inverse problem to the one studied in the first four sections of this paper and then apply Theorem 1.1 in order to derive a suitable stability estimate for σ.

Finally, taking into account that φ = σ 1/2 1 σ 1/2 2 is solution to the system (-∆ + q 1 )φ = σ 1/2 2 (q 2q 1 ) in Ω φ = 0 on Γ, we get φ L 2 (Ω) ≤ C q 2q 1 H -1 (Ω) upon taking f = σ 1/2 2 (q 2q 1 ) and g = 0 in (2.7). Thus, applying Theorem 1.1 and recalling (5.5), we obtain the: Corollary 5.1. Let F and G be the same as in section 1, and let δ > 0 and σ 0 > 0. Then for any σ j ∈ δB W 2,∞ (Ω) , j = 1, 2, obeying σ j ≥ σ 0 and the condition (5.6)

σ 1 = σ 2 on Γ and ∂ ν σ 1 = ∂ ν σ 2 on F ∩ G,
we may find a constant C > 0, independent of σ 1 and σ 2 , such that we have:

σ 1 -σ 2 L 2 (Ω) ≤ C Λ σ1,σ2 + ln C ln Λ σ1,σ2 -1
.

Remark 5.1. It is not clear how to weaken assumption (5.6) in Corollary 5.1. Indeed, to our knowledge, the best available result in the mathematical literature (this is a byproduct of [START_REF] Alessandrini | The local Calderòn problem and the determination at the boundary of the conductivity[END_REF]Theorem 2.2,p. 922 and Theorem 2.4,p. 923]) on the recovery of the conductivity at the boundary, claims for any non empty open subset Γ 0 of Γ, and for all σ m , m = 1, 2, taken as in Corollary 5.1 and satisfying the condition supp (σ 1σ 2 ) |Γ ⊂ Γ 0 instead (5.6), that

∂ j ν σ 1 -∂ j ν σ 2 L ∞ (Γ) ≤ C Λ 0 σ1,σ2
1/(1+j) , j = 0, 1.

Here Λ 0 σ1,σ2 denotes the operator g ∈ H -1/2 (Γ) ∩ E ′ (Γ 0 ) → (Λ σ1 -Λ σ2 )(g) |Γ0 ∈ H 1/2 (Γ 0 ), Λ 0 σ1,σ2 is the norm of Λ 0 σ1,σ2 in B(H -1/2 (Γ) ∩ E ′ (Γ 0 ), H 1/2 (Γ 0 )) and C > 0 is a constant that depends neither on σ 1 nor σ 2 .

  ξ) = {x ∈ Γ; ±ξ • ν(x) > 0},and denote by F (resp. G) an open neighborhood of Γ + (ξ) (resp. Γ -(ξ)) in Γ.

  . By elementary computation we find that τ * = 2 ln Λ q1,q2ln τ * /(2d), which entails τ * < ln Λ q1,q2 , since τ * ∈ (1, +∞) and d ∈ [1, +∞). As a consequence the real number τ * is greater than

	2 ln Λ q1,q2 -ln ln Λ q1,q2	/(2d), so we get that
	(4.18)	τ * >	ln Λ q1,q2

  hence ι d (2τ * ) > (M 2 /C ′ ) ln Λ q1,q2

θ according to (4.20). Thus there exists r * ∈ (0, 2τ * ) such that we

have ι d (r * ) = (M 2 /C ′ ) ln Λ q1,q2 θ

. This entails that (2t + n + 2)dr * ≥ 2t ln(dr * ) + (n + 2)dr * is greater that ln (M 2 /C ′ ) ln Λ q1,q2 θ , which combined with the estimate q

Assume that σ ∈ W 1,∞ + (Ω) = {c ∈ W 1,∞ (Ω; R); c(x) ≥ c 0 for some c 0 > 0}. Then for any (f, g) ∈ L 2 (Ω)× H 3/2 (Γ), we know from the standard elliptic theory that (5.1) admits a unique solution S σ (f, g) ∈ H 2 (Ω), and that the linear operator

is bounded. In the more general case where (f, g) ∈ H * × H -1/2 (Γ), we obtain by arguing in the exact same way as in Section 2 that there exists a unique u ∈ L 2 (Ω) obeying

Such a function u will be referred to as the transposition solution of (5.1) and will be denoted by S t σ (f, g). Let us introduce the Hilbert space

. By a slight modification of the proof of Lemma 2.1 (e.g. [BU]), the trace map t σ j u = σ j ∂ j ν u |Γ , u ∈ D(Ω), j = 0, 1, is extended to a linear continuous operator, still denoted by t σ j , from H div(σ∇) (Ω) into H -j-1/2 (Γ). Thus, bearing in mind that S t (0, g) ∈ H ∆ (Ω) for g ∈ H -1/2 (Γ), we see that the DN map

where q σ = σ -1/2 ∆σ 1/2 . As a consequence we have σ 1/2 S t σ (0, g) = S t qσ (0, σ 1/2 g), and hence σ 1/2 S t σ (0, σ -1/2 g) = S t qσ (0, g). From this and the identity t 1 (σ 1/2 w) = σ -1/2 t σ 1 w + 1 2 σ -1/2 (∂ ν σ)t 0 w, which is valid for every w ∈ H 2 (Ω), and generalizes to w ∈ H div(σ∇) (Ω) by duality, we get that

Thus, putting q j = q σj for j = 1, 2, we deduce from (5.3) that

We notice from (5.4) that Λ q1,q2 g = σ

In this case, we may find a constant C > 0, such that we have (5.5) Λ q1,q2 ≤ C Λ σ1,σ2 , where • still denotes the norm of B(H -1/2 (Γ)∩E ′ (F ), H 1/2 (G)). Here we used the fact that the multiplier by σ -1/2 1 is an isomorphism of H ±1/2 (Γ).


Michel Cristofol

Shumin Li

Eric Soccorsi

Determining the waveguide conductivity in a hyperbolic equation from a single measurement on the lateral boundary 1

Keywords: AMS 2010 Mathematics Subject Classification: 35R30 Inverse problem, hyperbolic equation, conductivity, Carleman estimate, infinite cylindrical domain

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 Hölder 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 [START_REF] Bellassoued | Global logarithmic stability in inverse hyperbolic problem by arbitrary boundary observation[END_REF][START_REF] Bellassoued | Lipschitz stability in in an inverse problem for a hyperbolic equation with a finite set of boundary data[END_REF][START_REF] Bellassoued | Logarithmic stability in determination of a coefficient in an acoustic equation by arbitrary boundary observation[END_REF][START_REF] Bellassoued | Determination of a coefficient in the wave equation with a single measurement[END_REF][START_REF] Imanuvilov | Global Lipschitz stability in an inverse hyperbolic problem by interior observations[END_REF][START_REF] Imanuvilov | Determination of a coefficient in an acoustic equation with single measurement[END_REF][START_REF] Klibanov | Lipschitz stability of an inverse problem for an accoustic equation[END_REF][START_REF] Yamamoto | Uniqueness and stability in multidimensional hyperbolic inverse problems[END_REF]. Their derivation relies on Bukhgeim-Klibanov's method [START_REF] Bukhgeim | Global uniqueness of class of multidimentional inverse problems[END_REF], which is by means of a Carleman inequality specifically designed for hyperbolic systems. More precisely, [START_REF] Imanuvilov | Global Lipschitz stability in an inverse hyperbolic problem by interior observations[END_REF][START_REF] Yamamoto | Uniqueness and stability in multidimensional hyperbolic inverse problems[END_REF] study the determination of the zero-th order term p : Ω → R appearing in ∂ t -∆ + p = 0, while [START_REF] Bellassoued | Global logarithmic stability in inverse hyperbolic problem by arbitrary boundary observation[END_REF][START_REF] Bellassoued | Determination of a coefficient in the wave equation with a single measurement[END_REF] 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 [START_REF] Bellassoued | Lipschitz stability in in an inverse problem for a hyperbolic equation with a finite set of boundary data[END_REF], 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 [START_REF] Bellassoued | Logarithmic stability in determination of a coefficient in an acoustic equation by arbitrary boundary observation[END_REF] from arbitrarily small boundary observations. Imanuvilov and Yamamoto derived stability results in [START_REF] Imanuvilov | Determination of a coefficient in an acoustic equation with single measurement[END_REF] by means of H -1 Carleman inequality, from data observation on subdomains fulfilling specific geometric assumptions. In [START_REF] Klibanov | Lipschitz stability of an inverse problem for an accoustic equation[END_REF] Klibanov and Yamamoto employed a different approach inspired by [START_REF] Klibanov | Newton-Kantorovich method for 3-dimensional potential inverse scattering problem and stability of the hyperbolic Cauchy problem with time dependent data[END_REF].

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 [START_REF] Bellassoued | Stability estimate for an inverse wave equation and a multidimensional Borg-Levinson theorem[END_REF][START_REF] Isakov | Stability estimates for hyperbolic inverse problems with local boundary data[END_REF][START_REF] Stefanov | Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media[END_REF] 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 [START_REF] Cardoulis | Inverse problem for the Schrödinger operator in an unbounded strip[END_REF][START_REF] Kian | Carleman estimate for infinite cylindrical quantum domains and application to inverse problems[END_REF][START_REF] Kian | Hölder stable determination of a quantum scalar potential in unbounded cylindrical domains[END_REF]. Inverse boundary value problems stated in unbounded waveguides were also studied in [START_REF] Choulli | Determining the scalar potential in a periodic quantum waveguide from the DN map[END_REF][START_REF] Choulli | Stable determination of time-dependent scalar potential from boundary measurements in a periodic quantum waveguide[END_REF][START_REF] Choulli | Recovering the twisting function in a twisted waveguide from the DN map, to appear in[END_REF] 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.

Settings

Notations

Throughout this text we write x = (x ′ , x n ) ∈ Ω for every x ′ := (x 1 , . . . , x n-1 ) ∈ ω and x n ∈ R.

:= x ′ = (x 1 , . . . , x n-1 ) ∈ R n-1 , |x ′ | = 1 .
For the sake of shortness we write ∂ j for ∂/∂x j , j ∈ N * n+1 := {m ∈ N * , m ≤ n + 1}. For convenience the time variable t is sometimes denoted by x n+1 so that

∂ t = ∂/∂t = ∂ n+1 . We set ∇ := (∂ 1 , . . . , ∂ n ) T , ∇ x ′ := (∂ 1 , . . . , ∂ n-1 ) T and ∇ x,t = (∂ 1 , . . . , ∂ n , ∂ t ) T .
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

H 1 0 (D) the closure of C ∞ 0 (D) in the topology of H 1 (D) Finally, for d > 0 we put Ω d := ω × (-d, d), Q d := Ω d × (0, T ), Γ d := ∂ω × (-d, d) and Σ d := ∂ω × (-d, d) × (0, T ).

Statement of the problem

We examine the following initial boundary value problem (IBVP in short)

   ∂ 2 t u -∇ • c∇u = 0 in Q := Ω × (0, T ), u(•, 0) = θ 0 , ∂ t u(•, 0) = θ 1 in Ω, u = 0 on Σ := Γ × (0, T ), (1.1)
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

q A [u] := c 1/2 u 2 0,Ω = Ω c(x)|u(x)| 2 dx, u ∈ D(q A ) := H 1 0 (Ω).
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

A m/2 v := A (m-1)/2 (A 1/2 v), v ∈ D(A m/2 ) := {v ∈ D(A (m-1)/2 ), A 1/2 v ∈ D(A (m-1)/2 )},
for each m ∈ N * . It turns out that the linear space D(A m/2 ) endowed with the scalar product

v, w D(A m/2 ) := m j=0 A j/2 v, A j/2 w 0,Ω ,
is Hilbertian, and it is established in Proposition 2.6 that

D(A m ) = {v ∈ H 2m (Ω); v, Av, . . . , A p-1 v ∈ H 1 0 (Ω)}, m ∈ {p -1/2, p}, p ∈ N * , (1.3)
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

u ∈ m+1 k=0 C k ([0, T ]; H m+1-k (Ω)), (1.4) provided the boundary ∂ω is C m+1 , the conductivity c ∈ W m,∞ (Ω; R) fulfills (1.2) and (θ 0 , θ 1 ) ∈ D(A (m+1)/2 ) × D(A m/2 ). Moreover, if c W m,∞ (Ω) ≤ c M for some constant c M > 0,
then the solution u to (1.1) satisfies the estimate

m+1 k=0 u C k ([0,T ];H m+1-k (Ω)) ≤ C ( θ 0 m+1,Ω + θ 1 m,Ω ) , (1.5)
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

c W 4,∞ (Ω) ≤ c M .
(1.6)

Since our strategy is based on a Carleman estimate for the hyperbolic system (1.1), it is also required that the condition

a ′ • ∇ x ′ c ≥ a 0 in Ω, (1.7)
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 (1.9)

c * ∈ W 4,∞ (O * ∩ Ω; R) satisfying c * ≥ c m and a ′ • ∇ x ′ c * ≥ a 0 in O * ∩ Ω, ( 1
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

-a ′ • ∇ x ′ θ 0 ≥ η 0 in ω * , (1.10)
for some η 0 > 0 and some open subset ω * in R n-1 , with C 2 boundary, satisfying

ω \ O * ⊂ ω * ⊂ ω. (1.11)
Thus, for M 0 > 0 such that

θ 0 W 3,∞ (Ω) + 1 j=0 θ j 5-j,Ω ≤ M 0 , (1.12)
we define the set Θ ω * = Θ ω * (a ′ , M 0 , η 0 ) of admissible initial conditions (θ 0 , θ 1 ) as

Θ ω * := (θ 0 , θ 1 ) ∈ D(A 5/2 ) ∩ W 3,∞ (Ω) × D(A 2
), fulfilling (1.10) and (1.12) .

Having introduced all these notations we may now state the main result of this paper.

Main result

The following result claims Hölder stability in the inverse problem of determining c in Ω ℓ , where ℓ > 0 is arbitrary, from the knowledge of one boundary measurement of the solution to (1.1), performed on Σ L for L > ℓ sufficiently large. The corresponding observation is viewed as a vector of the Hilbert space

H (Σ L ) := H 4 (0, T ; L 2 (Γ L )) ∩ H 3 (0, T ; H 1 (Γ L )),
endowed with the norm,

u 2 H (Σ L ) := u 2 H 4 (0,T ;L 2 (Γ L )) + u 2 H 3 (0,T ;H 1 (Γ L )) , u ∈ H (Σ L ). Theorem 1.1. Assume that ∂ω is C 5 and let O * be a neighborhood of Γ in R n-1 . For a ′ = (a 1 , . . . , a n-1 ) ∈ S n-1 , a 0 > 0, c m ∈ (0, 1), c M > c m and c * ∈ W 4,∞ (O * ∩ Ω; R) fulfilling (1.8), pick c j , j = 1, 2, in Λ O * (a ′ , a 0 , c * , c m , c M )
, defined by (1.9). Further, given M 0 > 0, η 0 > 0 and an open subset ω * ⊂ R n-1 obeying (1.11), chose θ 0 ∈ Θ ω * (a ′ , M 0 , η 0 ), defined in (1.13), and θ 1 = 0.

Then for any ℓ > 0 we may find L > ℓ and T > 0, such that the 5 k=0 C k ([0, T ], H 5-k (Ω))solution u j , j = 1, 2, to (1.1) associated with (θ 0 , θ 1 ), where c j is substituted for c, satisfies

c 1 -c 2 H 1 (Ω ℓ ) ≤ C u 1 -u 2 κ H (Σ L ) .
Here C > 0 and κ ∈ (0, 1) are two constants depending only on ω, ℓ, M 0 , η 0 , a ′ , a 0 , c ⋆ , c m and c M .

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

∂ 2 t v + Av = f in Q v(•, 0) = g, ∂ t v(•, 0) = h in Ω, (2.1)
where f , g and h are suitable data, and we recall from4 [27, Sect. 3, Theorem 8.2] the following existence and uniqueness result.

Proposition 2.1. Assume that f ∈ C 0 ([0, T ]; D(A 0 )), g ∈ D(A 1/2
) and h ∈ D(A 0 ). Then there exists a unique solution v to (2.1) such that

∂ k t v ∈ C 0 ([0, T ]; D(A (1-k)/2 )), k = 0, 1. (2.2)
Moreover we have the estimate

1 k=0 sup t∈[0,T ] ∂ k t v(•, t) D(A (1-k)/2 ) ≤ C f C 0 ([0,T ];D(A 0 )) + g D(A 1/2 ) + h D(A 0 ) . (2.3) 2.2 Improved regularity Proposition 2.2. Assume that f ∈ C 1 ([0, T ]; D(A 0 )), g ∈ D(A) and h ∈ D(A 1/2 ). Then the solution v to (2.1) satisfies ∂ k t v ∈ C 0 ([0, T ]; D(A (2-k)/2 )), k = 0, 1, 2, (2.4)
and we have the estimate

2 k=0 sup t∈[0,T ] ∂ k t v(•, t) D(A (2-k)/2 ) ≤ C f C 1 ([0,T ];D(A 0 )) + g D(A) + h D(A 1/2 ) . (2.5)
Proof. By differentiating (2.1) with respect to t, we check that w := ∂ t v obeys

∂ 2 t w + Aw = ∂ t f in Q w(•, 0) = h, ∂ t w(•, 0) = f (•, 0) -Ag in Ω. (2.6) Since f (•, 0) -Ag is lying in D(A 0 ) then we have ∂ k+1 t v = ∂ k t w ∈ C 0 ([0, T ]; D(A (1-k)/2 )) for k = 0, 1, with the estimate 1 k=0 sup t∈[0,T ] ∂ k+1 t v(•, t) D(A (1-k)/2 ) ≤ C ∂ t f C 0 ([0,T ];D(A 0 )) + h D(A 1/2 ) + f (•, 0) -Ag D(A 0 ) ≤ C f C 1 ([0,T ];D(A 0 )) + g D(A) + h D(A 1/2 ) (2.7)
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

Theorem 2.3. Let m be a nonnegative integer. We assume that g ∈ D(A (m+1)/2 ), h ∈ D(A m/2 ), and

∂ k t f ∈ C 0 ([0, T ]; D(A (m-k)/2
)), k = 0, . . . , m. Then there exists a unique solution v to (2.1), such that

∂ k t v ∈ C 0 ([0, T ]; D(A (m+1-k)/2 )), k = 0, 1, . . . , m + 1. (2.8)
Moreover we have the estimate

m+1 k=0 sup t∈[0,T ] ∂ k t v(•, t) D(A (m+1-k)/2 ) ≤ C m k=0 ∂ k t f C 0 ([0,T ];D(A (m-k)/2 )) + g D(A (m+1)/2 ) + h D(A m/2 ) .
(2.9)

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

g ∈ D(A (m+2)/2 ), h ∈ D(A (m+1)/2 ), ∂ k t f ∈ C 0 ([0, T ]; D(A (m+1-k)/2 )), k = 0, . . . , m + 1.
(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 := ∂ t v is solution to (2.6). Next, using that h ∈ D(A (m+1)/2 ), f (0) -Ag ∈ D(A m/2 ) and

∂ k t (∂ t f ) = ∂ k+1 t f ∈ C 0 ([0, T ]; D(A (m-k)/2 )), k = 0, . . . , m, from (2.10), we get that ∂ k+1 t v = ∂ k t w ∈ C 0 ([0, T ]; D(A (m+1-k)/2
)) for k = 0, 1, . . . , m + 1, and the estimate:

m+1 k=0 sup t∈[0,T ] ∂ k+1 t v(•, t) D(A (m+1-k)/2 ) ≤ C m k=0 ∂ k+1 t f C 0 ([0,T ];D(A (m-k)/2 )) + h D(A (m+1)/2 ) + f (0) -Ag D(A m/2 ) . This entails ∂ k t v ∈ C 0 ([0, T ]; D(A (m+2-k)/2
)) for k = 1, . . . , m + 2, and

m+2 k=1 sup t∈[0,T ] ∂ k t v(•, t) D(A (m+2-k)/2 ) ≤ C m+1 k=0 ∂ k t f C 0 ([0,T ];D(A (m+1-k)/2 )) + h D(A (m+1)/2 ) + g D(A (m+2)/2 ) . (2.11)
Further, as Av = f -∂ 2 t v from the first line in (2.1), we find out that

v(•, t) D(A (m+2)/2 ) ≤ C Av(•, t) D(A m/2 ) + v D(A 0 ) ≤ C f (•, t) D(A m/2 ) + ∂ 2 t v(•, t) D(A m/2 ) + v D(A 0 ) . (2.12)
Here we used the identity v(

•, t) 2 D(A (m+2)/2 ) = Av(•, t) 2 D(A m/2 ) + 1 k=0 A k/2 v(•, t) 2 D(A 0 ) and the estimate A 1/2 v(•, t) D(A 0 ) ≤ 1 k=0 A k v(•, t) D(A 0 ) . Since v D(A 0 ) and ∂ 2 t v(•, t) D(A m/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.

Remark 2.4. The result of Theorem 2.3 is similar to the one of [15, Sect. 7.2, Theorem 6], which holds for a bounded domain. This can be seen from the characterization of the D(A k/2 ) for k = 0, . . . , m, displayed in Subsection 2.4. Namely, it is worth noticing that the m th -order compatibility conditions [15, Sect. 7.2, Eq. ( 62)] imposed on f , g and h, are actually hidden in the operatorial formulation of Theorem 2.3.

Characterizing the domain of

A m/2 for m ∈ N *

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. [15, Sect. 6.3, Theorem 5]), to the case of the infinite waveguide Ω under study. The proof of this result boils down to [START_REF] Kian | Hölder stable determination of a quantum scalar potential in unbounded cylindrical domains[END_REF]Lemma 2.4] which claims elliptic boundary regularity for the Dirichlet Laplacian in Ω.

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

-∇ • c∇v = ϕ in Ω v = 0 on ∂Ω. (2.13)
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. a) We first consider the case r = 0. Due to (1.2) there is a unique solution v ∈ H 1 0 (Ω) to (2.13) by the Lax-Milgram theorem. Moreover v satisfies the energy estimate

v 1,Ω ≤ C ϕ 0,Ω , (2.15)
where the constant C > 0 depends only on ω and c m . Here we used (1.2) and the Poincaré inequality, which holds true in Ω since ω is bounded. Furthermore, v is solution to the boundary value problem

-∆v = f in Ω v = 0 on ∂Ω, (2.16)
where f := c -1 (ϕ + ∇c • ∇v) .

(2.17) [START_REF] Kian | Hölder stable determination of a quantum scalar potential in unbounded cylindrical domains[END_REF]Lemma 2.4], and v 2,Ω is upper bounded, up to some multiplicative constant depending only on ω, by f 0,Ω . As a consequence we have

Since f ∈ L 2 (Ω) then v ∈ H 2 (Ω) by
v 2,Ω ≤ C ′ ( ϕ 0,Ω + v 1,Ω ) ,
from (2.17), the constant C ′ > 0 depending only on ω, c m and c W 1,∞ (Ω) . 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). Hence the solution v to (2.16) belongs to H r+2 (Ω) and satisfies the estimate (2.14), by induction assumption, and we have f ∈ H r+1 (Ω) in virtue of (2.17). Further, v being solution to (2.16) where the boundary ∂ω is

C r+3 then v ∈ H r+3 (Ω)
by [START_REF] Kian | Hölder stable determination of a quantum scalar potential in unbounded cylindrical domains[END_REF]Lemma 2.4]. Moreover v r+3,Ω is upper bounded (up to some multiplicative constant depending only on r and ω) by f r+1,Ω . From this and (2.17) then follows that

v r+3,Ω ≤ C ′′ ( ϕ r+1,Ω + v r+2,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.

The domain of

A m/2 for m ∈ N *
In this subsection we characterize the domain of A m/2 for m ∈ N * .

Proposition 2.6. Let m ∈ N * and let k be either m -1/2 or m. Assume that ∂ω is C 2k and that c ∈ W 2k-1,∞ (Ω) fullfills (1.2). Then we have

D(A k ) = {u ∈ H 2k (Ω), u, Au, . . . , A m-1 u ∈ H 1 0 (Ω)}.
Moreover, the norm associated with D(A k ) is equivalent to the usual one in H 2k (Ω): we may find a constant c(k) > 1, depending only on k, ω, the constant c m defined in (1.2) and c W 2k-1,∞ (Ω) , such that we have

c(k) -1 u D(A k ) ≤ u 2k,Ω ≤ c(k) u D(A k ) , u ∈ D(A k ).
Proof. It suffices to show that

D(A k ) ⊂ {u ∈ H 2k (Ω), u, Au, . . . , A m-1 u ∈ H 1 0 (Ω)}, (2.18)
and

u 2k,Ω ≤ c(k) u D(A k ) , u ∈ D(A k ). (2.19)
The proof is by induction on m. a) We start with m = 1 and notice from the very definition of A 1/2 that D(A 1/2 ) = D(q A ) = H 1 0 (Ω). Moreover we have

A 1/2 u 2 0,Ω = q A [u] ≥ c m ∇u 2 0,Ω , u ∈ D(A 1/2 ), in virtue of (1.2). Bearing in mind that c m ∈ (0, 1), we obtain that u 1,Ω ≤ c -1/2 m u D(A 1/2 ) for every u ∈ D(A 1/2 ). This establishes (2.19) for k = 1/2.
Similarly, bearing in mind that D(A) = {u ∈ H 1 0 (Ω), Au ∈ L 2 (Ω)}, we apply Lemma 2.5 with r = 0 and ϕ = Au, where u ∈ D(A) is arbitrary. We find that u ∈ H 2 (Ω) satisfies u 2,Ω ≤ C 0 Au 0,Ω , which entails (2.18)-(2.19) for k = 1. b) Let us now suppose that the statement of the lemma is true for some m ∈ N * fixed. Pick k ∈ {m -1/2, m} and assume that ∂ω is C 2(k+1) and that c ∈ W 2k+1,∞ (Ω) satisfies (1.2). As D(A k+1 ) = {u ∈ D(A k ), Au ∈ D(A k )}, we deduce from the induction assumption that we have

D(A k+1 ) = {u ∈ H 2k (Ω), Au ∈ H 2k (Ω) and u, . . . , A m u ∈ H 1 0 (Ω)},
with A j u 2k,Ω ≤ c(k) A j u D(A k ) for j = 0, 1. Thus, applying Lemma 2.5, with r = 2k and ϕ = Au, for u ∈ D(A k+1 ), we get that u ∈ H 2(k+1) (Ω), proving (2.18) where (k + 1, m + 1) is substituted for (k, m). Moreover, it holds true that

u 2(k+1),Ω ≤ C 2k Au 2k,Ω ,
and since Au ∈ D(A k ), the induction assumption yields Au 2k,Ω ≤ c(k) Au D(A k ) . Therefore u 2(k+1),Ω is majorized, up to a multiplicative constant depending only on ω, c and m, by u D(A k+1 ) , which is (2.19) where k + 1 is substituted for k.

In view of Theorem 2.3 and Proposition 2.6 we obtain the following result.

Corollary 2.7. Let m be a natural number. Assume that ∂ω is C m+1 , that c ∈ W m,∞ (Ω) fulfills (1.2) and that (θ 0 , θ 1 ) ∈ D(A (m+1)/2 ) × D(A m/2 ). Then the initial boundary value problem (1.1) admits a unique solution

u ∈ m+1 k=0 C k ([0, T ]; H m+1-k (Ω)).
Moreover, we have

m+1 k=0 u C k ([0,T ];H m+1-k (Ω)) ≤ C ( θ 0 m+1,Ω + θ 1 m,Ω ) , (2.20)
for some constant C > 0 depending only on T , ω and c W m,∞ (Ω) .

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). (

Under the conditions of Theorem 1.1, and since θ 1 = 0, it is not hard to check that

u ∈ 4 k=3 C k ([-T, T ]; H 5-k (Ω)).
With a slight abuse of notations we put Q := Ω × (-T, T ), Σ := Γ × (-T, T ) and Q L := Ω L × (-T, T ), Σ L := ∂ω × (-L, L) × (-T, T ) for any L > 0, in the remaining part of this text.

The case of second order hyperbolic operators

In view of establishing a Carleman estimate for the operator

A := A(x, t, ∂) = ∂ 2 t -∇ • c∇ + R, (3.2)
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:

ψ(x, t) = ψ δ (x, t) := |x ′ -δa ′ | 2 -x 2 n -t 2 and ϕ(x, t) = ϕ δ,γ (x, t) := e γψ(x,t) , (x, t) ∈ Q. (3.3)
Further for L > 0 and T > 0 we introduce the space

X L,T := u ∈ H 2 (Q L ); ∂ j xn u(•, ±L, •) = 0 in ω × (-T, T ), ∂ j t u(•, ±T ) = 0 in Ω L , j = 0, 1 , (3.4
) and state the: Proposition 3.1. Let A be defined by (3.2), where c obeys (1.2)-(1.7), and let ℓ be positive. Then there exist δ 0 > 0 and γ 0 > 0 such that for all δ ≥ δ 0 and γ ≥ γ 0 , we may find L > ℓ, T > 0 and s 0 > 0 for which the estimate

s j=0,1 s 2(1-j) e sϕ ∇ j x,t v 2 0,Q L ≤ C   e sϕ Av 2 0,Q L + s j=0,1 s 2(1-j) e sϕ ∇ j x,t v 2 0,∂Q L   , (3.5)
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

ϕ(x ′ , x n , 0) ≥ d ℓ , (x ′ , x n ) ∈ ω × [-ℓ, ℓ], (3.6)
and we may find ǫ ∈ (0, (L -ℓ)/2) and ν > 0 so small that we have:

max x∈ω×[-L,L] ϕ(x ′ , x n , t) ≤ dℓ := d ℓ e -γν 2 , |t| ∈ [T -2ǫ, T ], (3.7)
max

(x ′ ,t)∈ω×[-T,T ] ϕ(x ′ , x n , t) ≤ dℓ , |x n | ∈ [L -2ǫ, L]. (3.8)
Proof. The proof is divided in three parts and essentially boils down to [20, theorem 3.2.1'].

First part: Definition of δ 0 , L and T . Bearing in mind that |x ′ -δa

′ | 2 -|y ′ -δa ′ | 2 = |x ′ |-|y ′ | 2 - 2δa ′ •(x ′ -y ′ ) for all x ′ , y ′ ∈ ω, we see that sup x ′ ∈ω |x ′ -δa ′ | 2 -inf x ′ ∈ω |x ′ -δa ′ | 2 ≤ |ω|(|ω|+4δ|a ′ |)
for every δ > 0, where |ω| := sup x ′ ∈ω |x ′ |. Hence the function

g ℓ (δ) = sup x ′ ∈ω |x ′ -δa ′ | 2 -inf x ′ ∈ω |x ′ -δa ′ | 2 + ℓ 2 1/2 (3.9)
scales at most like δ 1/2 , proving that there exists δ 0 > 0 so large that

δa 0 > 1 + 2 c m 1/2 g ℓ (δ) + |ω| + 2 c M + 2, δ ≥ δ 0 . (3.10)
Further, since ω is bounded and a ′ = 0 R n-1 by (1.7), we may as well assume upon possibly enlarging δ 0 , that we have in addition c

1/2 m inf x ′ ∈ω |x ′ -δa ′ | > g ℓ (δ
) for all δ ≥ δ 0 . This and (3.10) yield that there exists ϑ > 0 so small that the two following inequalities

δa 0 -L + |ω| + 2 1 + T c m 1/2 c M -2 > 0, (3.11) and c 1/2 m inf x ′ ∈ω |x ′ -δa ′ | > T, (3.12)
hold simultaneously for every L and T in (g ℓ (δ), g ℓ (δ) + ϑ), uniformly in δ ≥ δ 0 .

Second part: Proof of (3.5). We first introduce the following notations, we shall use in the remaining part of the proof. For notational simplicity we put x := (x, t) for (x, t) ∈ Q L and

∇ x = (∂ 1 , . . . , ∂ n , ∂ n+1 ) T . We also write ξ ′ = (ξ 1 , . . . , ξ n-1 ) T ∈ R n-1 , ξ = (ξ 1 , . . . , ξ n ) T ∈ R n and ξ = (ξ 1 , . . . , ξ n , ξ n+1 ) T ∈ R n+1 . We call A 2 the principal part of the operator A, that is A 2 = A 2 (x, ∂) = ∂ 2 t -c(x)∆, and denote its symbol by A 2 (x, ξ) = c(x)|ξ| 2 -ξ 2 n+1 , where |ξ| = n j=1 ξ 2 j 1/2 . Since A 2 (x, ∇ x ψ(x)) = 4 c(x)(|x ′ -δa ′ | 2 + x 2 n ) -x 2 n+1 for every x ∈ Q L , we have A 2 (x, ∇ x ψ(x)) > 0, x ∈ Q L , (3.13)
by (1.7) and (3.12). For all x ∈ Q L and ξ ∈ R n+1 , put

J(x, ξ) = J = n+1 j,k=1 ∂A 2 ∂ξ j ∂A 2 ∂ξ k ∂ j ∂ k ψ + n+1 j,k=1 ∂ k ∂A 2 ∂ξ j ∂A 2 ∂ξ k -(∂ k A 2 ) ∂ 2 A 2 ∂ξ j ∂ξ k ∂ j ψ, (3.14)
where, for the sake of shortness, we write ∂ j , j ∈ N * n+1 , instead of ∂/∂x j , and x n+1 stands for t. Assuming that

A 2 (x, ξ) = c(x)|ξ| 2 -ξ 2 n+1 = 0, x ∈ Ω, ξ ∈ R n+1 \{0}, (3.15)
and

∇ ξ A 2 (x, ξ) • ∇ x ψ(x) = 4 c(x)(ξ ′ • (x ′ -δa ′ ) -ξ n x n ) + ξ n+1 x n+1 = 0, x ∈ Q L , ξ ∈ R n+1 \{0}, (3.16
) we shall prove that J(x, ξ) > 0 for any (x, ξ) ∈ Q L × R n+1 . To this end we notice that the first sum in the rhs of (3.14)

reads Hess(ψ)∇ ξ A 2 , ∇ ξ A 2 = 8 c 2 (|ξ ′ | 2 -ξ 2 n ) -ξ 2 n+1
, and that n+1 j,k=1

∂ k ∂A 2 ∂ξ j ∂A 2 ∂ξ k -(∂ k A) ∂ 2 A 2 ∂ξ j ∂ξ k ∂ j ψ = n j,k=1 ∂ k ∂A 2 ∂ξ j ∂A 2 ∂ξ k -(∂ k A 2 ) ∂ 2 A 2 ∂ξ j ∂ξ k ∂ j ψ = 2c n j,k=1 2ξ j ξ k -|ξ| 2 ∂ξ j ∂ξ k ∂ k c ∂ j ψ = 2 2(∇c • ξ)(∇ψ • ξ) -(∇c • ∇ψ)|ξ| 2 , since ∂ k ∂A 2 ∂ξ j ∂A 2 ∂ξ k -(∂ k A 2 ) ∂ 2 A 2 ∂ξ j ∂ξ k = 0 if either j or k is equal to n + 1. Therefore we have J = 2 4 c 2 (|ξ ′ | 2 -ξ 2 n ) -ξ 2 n+1 + 2c(∇c • ξ)(∇ψ • ξ) -c(∇c • ∇ψ)|ξ| 2 = 4 2c 2 (|ξ ′ | 2 -ξ 2 n ) -2 + (x ′ -δa ′ ) • ∇ x ′ c -x n ∂ n c ξ 2 n+1 -2x n+1 ξ n+1 ∇c • ξ , from (3.15)-(3.16). Further, in view of (3.15) we have c 2 (|ξ ′ | 2 -ξ 2 n ) ≥ -c 2 |ξ| 2 ≥ -cξ 2 n+1 and |∇c • ξ| ≤ |∇c||ξ| ≤ (|∇c|/c 1/2 )|ξ n+1 |, whence J ≥ 4 δa ′ • ∇ x ′ c -x • ∇c + 2c + 2T |∇c| c 1/2 + 2 ξ 2 n+1 .
(3.17)

Here we used that fact that

x n+1 = t ∈ [0, T ]. Due to (1.6)-(1.7
), the rhs of (3.17) is lower bounded, up to the multiplicative constant 4ξ 2 n+1 , by the lhs of (3.11). Since ξ n+1 is non zero by (1.6) and (3.15), then we obtain J(x, ξ) > 0 for all (x, ξ) ∈ Q L × R n+1 . With reference to (3.13), we may apply [20, Theorem 3.2.1'], getting two constants s 0 = s 0 (γ) > 0 and C > 0 such that (3.5) holds for any s ≥ s 0 and v ∈ H 2 (Q L ).

Third part: Proof of (3.6)-(3.8). First, (3.6) follows readily from (3.3), with d ℓ := e γβ ℓ and

β ℓ := inf x ′ ∈ω |x ′ -δa ′ | 2 -ℓ 2 .
Next, for ν ∈ (0, ϑ) arbitrarily fixed, we put

L = T = g ℓ (δ) + ν. (3.18)
Notice for further reference from (3.9), (3.12) and (3.18), that we have

β ℓ ≥ g ℓ (δ) 2 c m -ℓ 2 ≥ 1 -c m c m ℓ 2 > 0, (3.19)
since c m ∈ (0, 1), by assumption. Similarly, as

T 2 ≥ g ℓ (δ) 2 +ν 2 = sup x ′ ∈ω |x ′ -δa ′ | 2 -(β 2 ℓ -ν 2 ), we dedude from (3.3) that ϕ(x ′ , x n , ±T ) ≤ e γ(sup x ′ ∈ω |x ′ -δa ′ | 2 -x 2 n -T 2 ) ≤ e γ(β ℓ -ν 2 ) e -γx 2 n , (x ′ , x n ) ∈ ω × [-L, L].
With reference to (3.19) we may thus chose ǫ ∈ (0, (L -ℓ)/2) so small that

ϕ(x ′ , x n , t) ≤ d ℓ e -γν 2 e -γx 2 n , (x ′ , x n ) ∈ ω × [-L, L], |t| ∈ [T -2ǫ, T ],
which entails (3.7). Finally, since t and x n play symmetric roles in (3.3), and since T = L, we obtain (3.8) by substituting (T, t) for (L, x n ) in (3.7).

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

∂ 2 t u -∇ • c∇u = f in Q u = 0 on Σ, (3.20)
where f ∈ L 2 (Q). To this purpose we introduce a cut-off function χ ∈ C 2 (R; [0, 1]), such that

χ(x n ) := 1 if |x n | < L -2ǫ, 0 if |x n | ≥ L -ǫ, (3.21)
where ǫ is the same as in Proposition 3.1, and we set

u χ (x, t) := χ(x)u(x, t) and f χ (x, t) := χ(x)f (x, t), (x, t) ∈ Q. Corollary 3.2. Let f ∈ L 2 (Q).
Then, under the conditions of Proposition 3.1, there exist two constants s * > 0 and C > 0, depending only on ω, ℓ, M 0 , η 0 , a ′ , a 0 , c m and c M , such that the estimate

s j=0,1 s 2(1-j) e sϕ ∇ j x,t u 2 0,Q L ≤ C   e sϕ f 2 0,Q L + s 3 e 2s dℓ u 2 1,Q L + s j=0,1 s 2(1-j) e sϕ ∇ j x,t u χ 2 0,Σ L   ,
holds for any solution u to (3.20), uniformly in s ≥ s * .

Proof. Since u is solution to (3.20) we have

∂ 2 t u χ -∇ • c∇u χ = f χ + R 1 u in Q, where R 1 = R 1 (x, ∂) := [χ, ∇ • c∇] = -(c∆χ + ∇c • ∇χ + 2c∇χ • ∇), (3.22)
is a first-order differential operator. Therefore, the function v(x, t)

:= η(t)u χ (x, t), where η ∈ C 2 (R; [0, 1]) is such that η(t) := 1 if |t| < T -2ǫ, 0 if |t| ≥ T -ǫ, satisfies ∂ 2 t v -∇ • c∇v = g := ηf χ + ηR 1 u + η ′′ u χ + 2η ′ ∂ t u χ in Q. Moreover, as v(•, ±L, •) = ∂ xn v(•, ±L, •) = 0 in ω × (-T, T ) and v(•, ±T ) = ∂ t v(•, ±T ) = 0 in Ω L , we may apply Proposition 3.1, getting s j=0,1 s 2(1-j) e sϕ ∇ j x,t v 2 0,Q L ≤ C   e sϕ g 2 0,Q L + s j=0,1 s 2(1-j) e sϕ ∇ j x,t v 2 0,∂Q L   . (3.23)
Further, bearing in mind that

∂Q L = Σ L ∪ (ω × {±L} × (-T, T )) ∪ (Ω L × {±T }), we deduce from the vanishing of v(•, ±L, •) and ∇ x,t v(•, ±L, •) in ω × (-T, T
), and the one of v(•, ±T ) and

∇ x,t v(•, ±T ) in Ω L , that e sϕ ∇ j x,t v 0,∂Q L = e sϕ ∇ j x,t v 0,Σ L , j = 0, 1. (3.24)
Next we know from (3.8) and (3.22) that

e sϕ ηR 1 u 0,Q L ≤ Ce s dℓ u L 2 (-T,T ;H 1 (Ω L )) , (3.25)
and from (3.7) that e sϕ (η

′′ u χ + 2η ′ ∂ t u χ ) 0,Q L ≤ Ce s dℓ u χ H 1 (-T,T ;L 2 (Ω L )) . ( 3
s 2(1-j) e sϕ ∇ j x,t v 2 0,Q L ≤ C   e sϕ f 2 0,Q L + s j=0,1 s 2(1-j) e sϕ ∇ j x,t u χ 2 0,Σ L + e 2s dℓ u 2 H 1 (Q L )   .
(3.27) The next step of the proof involves noticing from (3.7) that e sϕ (1-η)∇ j

x u χ 0,Q L ≤ e s dℓ ∇ j x u χ 0,Q L for j = 0, 1, hence

e sϕ ∇ j x u χ 0,Q L ≤ e sϕ (1 -η)∇ j x u χ 0,Q L + e sϕ ∇ j x v 0,Q L ≤ e s dℓ ∇ j x u χ 0,Q L + e sϕ ∇ j x v 0,Q L , j = 0, 1. (3.28)
Furthermore, by combining the identity

∂ t u χ = (1-η)∂ t u χ +η∂ t u χ = (1-η)∂ t u χ -η ′ ∂ t u χ +∂ t v with (3.7), we get that e sϕ ∂ t u χ 0,Q L ≤ e sϕ (1 -η)∂ t u χ 0,Q L + e sϕ ηu χ 0,Q L + e sϕ ∂ t v 0,Q L ≤ e s dℓ ∂ t u χ 0,Q L + η ′ L ∞ (-T,T ) u χ 0,Q L + e sϕ ∂ t v 0,Q L ,
which, together with (3.28), yields j=0,1

s 2(1-j) e sϕ ∇ j x,t u χ 2 0,Q L ≤ C j=0,1 s 2(1-j) e 2s dℓ ∇ j x,t u χ 2 0,Q L + e sϕ ∇ j x,t v 2 0,Q L . (3.29)
Similarly, using (3.8), we derive from the identity ∂ j t u = ∂ j t u χ + (1 -χ)∂ j t u for j = 0, 1, that e sϕ ∂ j t u 0,Q L ≤ e sϕ ∂ j t u χ 0,Q L + e s dℓ ∂ j t u 0,Q L , j = 0, 1, and from

∇ x u = ∇ x u χ + (1 -χ)∇ x u -χ ′ (0, . . . , 0, u) T , that e sϕ ∇ x u 0,Q L ≤ e sϕ ∇ x u χ 0,Q L + e s dℓ ∇ x u 0,Q L + χ ′ L ∞ (-L,L) u 0,Q L .
As a consequence we have j=0,1 4 Inverse problem

s 2(1-j) e sϕ ∇ j x,t u 2 0,Q L ≤ C j=0,1 s 2(1-j) e 2s dℓ ∇ j x,t u 2 0,Q L + e sϕ ∇ j x,t u χ 2 0,Q L . ( 3

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.

Namely, given c i ∈ Λ Γ for i = 1, 2, we note u i the solution to (1.1) where c i is substituted for c, suitably extended to (-T, 0) in accordance with (3.1). Thus, putting

c := c 1 -c 2 and f c := ∇ • (c∇u 2 ), (4.1)
it is clear from (1.1) that the function u := u 1 -u 2 is solution to the linearized system

   ∂ 2 t u -∇ • (c 1 ∇u) = f c in Q u = 0 on Σ u(•, 0) = ∂ t u(•, 0) = 0 in Ω. (4.2)
By differentiating k-times (4.2) with respect to t, for k ∈ N * fixed, we see that

u (k) := ∂ k t u is solution to ∂ 2 t u (k) -∇ • (c 1 ∇u (k) ) = f (k) c in Q u (k) = 0 on Σ, (4.3)
with f

(k) c := ∂ k t f c = ∇ • (c∇u (k)
2 ), where u

(k)
2 stands for ∂ k t u 2 . We stick with the notations of Corollary 3.2. In particular, for any function v, we denote χv by v χ , where χ is defined in (3.21). Upon multiplying both sides of the identity (4.3) by χ, we obtain that

∂ 2 t u (k) χ -∇ • (c 1 ∇u (k) χ ) = f (k) cχ -g k in Q u (k) χ = 0 on Σ, (4.4)
with

f cχ := ∇ • (c χ ∇u 2 ) and g k := ∇ • (c 1 (∇χ)u (k) ) + c 1 ∇χ • ∇u (k) + c∇χ • ∇u (k) 2 . (4.5)
Notice that g k is supported in Ωǫ := {x = (x ′ , x n ), x ′ ∈ ω and |x n | ∈ (L -2ǫ, L -ǫ)}.

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

e sϕ(•,0) ∇ j c χ 2 0,Ω L ≤ Cs -1   j=0,1 e sϕ(•,0) ∇ j u (2) χ (•, 0) 2 0,Ω L + e 2s dℓ   ,
holds for all s ≥ s * .

Proof. Let Ω * be an open subset of R n with C 2 boundary, such that

ω * × (-(L -ǫ), L -ǫ) ⊂ Ω * ⊂ ω * × (-L, L), (4.6)
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.

Further, with reference to (1.10) we may assume upon possibly enlarging δ ∈ [δ 0 , +∞), where δ 0 is the same as in Proposition 3.1, that we have

|∇θ 0 • (x 1 -δa 1 , . . . , x n-1 -δa n-1 , -x n )| ≥ µ 0 > 0, x ∈ Ω * . Thus applying [18, Proposition 2.2] 5 with D = Ω * , P (x, ∂)v = ∇ • ((∇θ 0 )v) and v = ∂ j i c χ ∈ H 1 0 (Ω * ) since χ(x n ) = 0 for x n ≥ L -ǫ, for i ∈ N * n and j = 0, 1, we obtain that s e sϕ(•,0) ∂ j i c χ 2 0,Ω * ≤ C e sϕ(•,0) ∇ • ((∂ j i c χ )∇θ 0 ) 2 0,Ω * , i ∈ N * n . j = 0, 1. (4.7)
Since c χ (x ′ , x n ) = 0 a.e. in ω * × ((-L, -(L -ǫ)) ∪ (L -ǫ, L)) by (3.21), we have e sϕ(•,0) ∂ j i c χ 0,Ω * = e sϕ(•,0) ∂ j i c χ 0,Ω L for each i ∈ N * n and j = 0, 1, from (4.6). We derive from this and (4.7) that

s e sϕ(•,0) ∂ j i c χ 2 0,Ω L ≤ C e sϕ(•,0) ∇ • ((∂ j i c χ )∇θ 0 ) 2 0,Ω L , i ∈ N * n , j = 0, 1. (4.8)
Further, taking t = 0 in the first line of (4.4) with j = 0, we get that

∇ • (c χ ∇θ 0 ) = u (2) χ (•, 0) + c∇χ • ∇θ 0 . (4.9)
From this, (4.8) and (3.8) then follows that

s e sϕ(•,0) c χ 2 0,Ω L ≤ C e sϕ(•,0) u (2) χ (•, 0) 2 0,Ω L + e sϕ(•,0) c∇χ • ∇θ 0 2 0,Ω L ≤ C e sϕ(•,0) u (2) χ (•, 0) 2 0,Ω L + e 2s dℓ . (4.10)
Similarly, since ∇

• ((∂ i c χ )∇θ 0 ) = ∂ i ∇ • (c χ θ 0 ) -∇ • (c χ ∇∂ i θ 0 ) for every i ∈ N * n , we derive from (4.9) that ∇ • ((∂ i c χ )∇θ 0 ) = ∂ i u (2) χ (•, 0) + ∂ i (c∇θ 0 • ∇χ) -∇c χ • ∇∂ i θ 0 -c χ ∆∂ i θ 0 . As a consequence we have, e sϕ(•,0) ∇ • ((∂ i c χ )∇θ 0 ) 2 0,Ω L ≤ C   e sϕ(•,0) ∂ i u (2) χ (•, 0) 2 0,Ω l + j=0,1 e sϕ(•,0) ∇ j c χ 2 0,Ω l + e 2s dℓ   , i ∈ N * n ,
according to (3.8). Summing up the above estimate over i in N * n , it follows from (4.8) that

s e sϕ(•,0) ∇c χ 2 0,Ω L ≤ C   e sϕ(•,0) ∇u (2) χ (•, 0) 2 0,Ω L + j=0,1 e sϕ(•,0) ∇ j c χ 2 0,Ω L + e 2s dℓ   .
This and (4.10) yield the desired result.

Completion of the proof

The proof is divided into three steps.

Step 1. The first step of the proof is to upper bound u

χ (•, 0) in the e sϕ(•,0) -weighted H 1 (Ω L )norm topology, by the corresponding norms of u Lemma 4.2. There exists a constant s * > 0 depending only on T such that we have z(•, 0) 2 0,Ω L ≤ 2 s z 2 0,Q L + s -1 ∂ t z 2 0,Q L , for all s ≥ s * and z ∈ H 1 (-T, T ; L 2 (Ω L )).

Namely, we apply Lemma 4.2 with z = e sϕ ∂ j i u

χ for i ∈ N * n and j = 0, 1, getting e sϕ(•,0) ∂ j i u (2) χ (•, 0) 2 0,Ω L ≤ 2 s e sϕ ∂ j i u (2) χ 2 0,Q L + s -1 e sϕ ∂ j i u (3) χ 2 0,Q L , s ≥ s * .

Summing up the above estimate over i ∈ N * n and j = 0, 1, we obtain for all s ≥ s * that j=0,1 e sϕ(•,0) ∇ j u (2) χ (•, 0) 2 0,Ω L ≤ 2 Further, recalling (4.5), we see from (1.5) and (1.12) (resp., from (1.5), (1.6), (1.12) and (3.8)) that the first (resp., second) term of the sum in the right hand side of (4.13) is upper bounded up to some multiplicative constant, by j=0,1 e sϕ ∇ j c χ 2 0,Q L (resp., e 2s dℓ ( u (k) 2 1,Q L + 1)). From this and Lemma 4.1 then follows for s sufficiently large that Step 3. Finally, we notice from (3.3) that e sϕ ∇ j c χ 0,Q L = ρ 1/2 s e sϕ(•,0) ∇ j c χ 0,Ω L , j = 0, 1, (

where ρ s (x) := T -T e 2s(ϕ(x,t)-ϕ(x,0)) dt = T -T e -2sϕ(x,0)(1-exp(-γt 2 )) dt for all x ∈ Ω L . Bearing in mind that ϕ(x, 0) ≥ κ := e γ(inf x ′ ∈ω |x ′ -δa ′ | 2 -L 2 ) > 0 for all x ∈ Ω L , we get that s 2 e -2s(d ℓ -dℓ ) + h k (s) . (4.17)

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 that dℓ < d ℓ , we end up getting the desired result from (4.17).

Further, we denote by |y| := m i=1 y 2 j 1 / 2 the

 212 Euclidian norm of y = (y 1 , . . . , y m ) ∈ R m , m ∈ N * , and we put S n-1

. 8 )

 8 we introduce the set Λ O * = Λ O * (a ′ , a 0 , c * , c m , c M ) of admissible conductivity coefficients as Λ O * := {c ∈ W 4,∞ (Ω; R) obeying (1.2) and (1.6) -(1.7); c = c * in O * ∩ Ω}.

  .30) Finally we obtain the desired result by gathering (3.27) and (3.29)-(3.30).

5

  Let D be a bounded open subset of R n , n ≥ 1, with C 2 boundary, and consider the first-order operator P (x, ∂) := n i=1 pi(x)∂i + p0(x), where p0 ∈ C 0 (D) and p := (p1, . . . , pn) ∈ C 1 (D) n . Assume that |p(x) • (x1 -δa1, . . . , xn-1δan-1, -xn)| ≥ pm, x ∈ D,for some pm > 0. Then for any pM ≥ max{ p0 C 0 (D) , pi C 1 (D) , i ∈ N * n }, there exist s * > 0 and C > 0, depending only on pM , such that the estimate e sϕ(•,0) v 2 0,D ≤ Cs -1 e sϕ(•,0) P v 2 0,D , holds for all s ≥ s * and v ∈ H 1 0 (D).

χ

  in Q L , with the aid of the following technical result, borrowed from [3][Lemma 3.2].

j=0, 1 sStep 2 . 1 s 2 ( 1 χ 2 0+ s 3 e 2 1 1 e 2 0+ s 3 e 2 1

 121212321232 e sϕ ∇ j u(2) χ 2 0,Q L + s -1 e sϕ ∇ j u (3) χ 2 0,Q L . (4.11) The next step involves majorizing the right hand side of (4.11) withh k (s) := j=0,-j) e sϕ ∇ j x,t u (k) , for k = 2,3, is solution to (3.20) with c = c 1 and f = f cχ -g k , according to (4.4), then Corollary 3.2 yields s j=0,1s 2(1-j) e sϕ ∇ j x,t u (k) χ ,Q L ≤ C e sϕ f (k) 2s dℓ u (k) χ ,Q L + sh k (s) ,for s large enough. In light of(4.11) this entails that j=0,sϕ(•,0) ∇ j u(2) χ (•, 0) 2s dℓ u (k) χ ,Q L + sh k (s) . (4.13)

1 e 3 s 3 e

 133 sϕ ∇ j c χ 2 0,Q L + e 2s dℓ + k=2,2s dℓ u (k) 2 1,Q L + sh k (s)   . (4.14)

1 e 3 s 3 e 1 ∇ j c χ 2 0

 13312 ρ s L ∞ (Ω L ) ≤ T -T e -2sκ(1-exp(-γt 2 )) dt, s > 0.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) thats j=0,sϕ(•,0) ∇ j c χ 2 0,Ω L ≤ C   e 2s dℓ + k=2,2s dℓ u (k) 2 1,Q L + sh k (s)taking s sufficiently large. With reference to (3.6)-(3.8), this entails that j=0,,Ω L ≤ C k=2,3

Aix-Marseille Université, CNRS, I2M UMR 7373, École Centrale. E-mail: michel.cristofol@univ-amu.fr.

University of Science and Technology of China, School of Mathematical Sciences, P. O. Box 4, Hefei, Anhui 230026, People's Republic of China. E-mail: shuminli@ustc.edu.cn.

Aix-Marseille Université, CNRS, CPT UMR 7332, 13288 Marseille, France & Université de Toulon, CNRS, CPT UMR 7332, 83957 La Garde, France. E-mail: eric.soccorsi@univ-amu.fr.

Upon takingH := D(A 0 ) = L 2 (Ω), V := D(A 1/2 ) = H 1 0 (Ω) and a(t; u, v) := A 1/2 u, A 1/2 v 0,Ω = Ω c(x)∇u(x) • ∇v(x)dx for u, v ∈ V and all t ∈ [0, T ].