A stability result for the diffusion coeﬀicient of the heat operator defined on an unbounded guide

In this article we consider the inverse problem of determining the diﬀusion coeﬃcient of the heat operator in an unbounded guide using a ﬁnite number of localized observations. For this problem, we prove a stability estimate in any ﬁnite portion of the guide using an adapted Carleman inequality. The boundary measurements are also realized on the boundary of a larger ﬁnite portion of the guide. A special care is required to avoid measurements on the cross-section boundaries which are inside the actual guide. This stability estimate uses a technical positivity assumption. Using arguments from control theory, we manage to remove this assumption for the inverse problem with non homogeneous given boundary conditions.

1 Introduction and main results

Introduction
Inverse problems associated with the heat operator have been frequently investigated as well on theoretical aspects as for applications, applications which cover large domains such medicine, ecology, biology, evolution of populations, physics . . . For such inverse problems there exist several methods involving different type of observations. In the case of finite number of observations we can consider for example boundary data, observation on all the domain at one time, spectral data, pointwise observation. On the other hand, nice results have been obtained using infinite number of measurements (e.g. methods using Dirichlet to Neumann map) but we don't consider such approaches here. In this paper, we focus on the method by the Carleman estimates which allows to derive stability inequality in the form: for a function f satisfying lim s→0 f (s) = 0. This kind of inequality links the distance between two sets of coefficients to be reconstructed with the distance between two sets of observations in appropriate norms. Such stability inequalities lead to the uniqueness of the coefficient to be reconstructed given the observation in a good norm. They are also useful to improve the numerical reconstruction of the coefficients using noise-free observations [12]. The paper by [3] has initiated this method and a lot of authors have been inspired by the Carleman estimates to solve inverse problems. There is a huge literature on the determination of nonlinear spatially homogeneous terms or source terms in reaction-difusion equations from boundary measurements,

Settings and hypotheses
Let ω be a bounded domain in R n−1 , n ≥ 2 with C 2 boundary. Denote Ω := R × ω and Q = Ω × (0, T ), Σ = ∂Ω × (0, T ). We consider the following problem    ∂ t u − ∇ · (c∇u) = 0 in Q, u = 0 on Σ, u(x, 0) = u 0 (x) in Ω, (1.1) where u 0 is a sufficiently smooth function and c is a bounded coefficient defined in Ω such that c > 0 and c ∈ C 1 (Ω). Our problem can be stated as follows. Let l > 0 and denote Ω * = (− * , * ) × ω. We want to determine the coefficient c on Ω l from a finite number of measurements of the solution u of the system (1.1) on a lateral subset of ∂Ω L for some L > l and from the knowledge of the solution at the time T 2 . We stress out that the required measurements are not performed on all the boundary ∂Ω L and that they avoid the cross-sections {±L} × ω.
Our proof will require the strong positivity condition (1.10). This condition, involved in almost all inverse problems dedicated to coefficients of the principal part of operators, is removed by the construction of an adapted control. The main difficulty is to prove a controllability result in H 1 norm on the unbounded domain Ω. The strategy is inspired by [1,Section 4] where the authors deal with a system of coupled equations but on a bounded domain.
We denote by Q * the set The cornerstone of Carleman estimates is to carry out special weight functions. In our inverse problem, the design of such weight function will allow us to eliminate observations on the cross-section of the wave guide. We follow some ideas from [8] mainly to eliminate the cross section observations since there is less constraints in the choice of the weight functions in the case of a parabolic operator. Nevertheless, we are going to detail the different steps which allow to get the main two inequalities (1.3). For this we choose a ∈ R n \ Ω L such that if From now on and for simplicity, we denote θ = T 2 . From [25,26] we consider classical form of weight functions as follows, t ∈ (0, T ), The constant λ > 0 will be set in Proposition 2.1. In view of the Carleman inequalities for the parabolic operators with regular weights, we need to use cut off functions in time. On the other hand, in view to manage our infinite wave guide we will need also to consider cut off functions in space but only in the infinite direction x 1 . These cut-off functions will make appear additive terms coming from the commutator between the evolution operator and these cut-off functions. These residual terms will be estimated thanks to the following crucial properties of the weight function.
Proposition 1.1. There exist T > 0, L > l and > 0 such that (1.2) holds and, setting These two estimates will be fruitful in Section 3 to solve our inverse problem.
As we have changed L we might have to push a further away from ω to ensure simultaneously the condition (1.2) and β 2 > 0. In the case n ≥ 3, we can push a in a specific direction that does not modify the value of β 1 . The case n = 2 can be dealt with explicit computations. Here we take advantage of the particular form of our weight function d(x).
With these definitions, we get Then, for all x ∈ Ω L , 3 As ψ(x, 0) = ψ(x, T ), we deduce that there exists > 0 such that < T 4 and: for all x ∈ Ω L and t ∈ ((0, 2 ) Now, we choose small enough such that l ≤ L − 2 and due to the symmetric role played by t − θ and x 1 in the formulation of ψ(x, t) by the same way we have: These two estimates end the proof of Proposition 1.1.
Let us remark that the weight function is bounded by below (away from 0) et by above on Q L . The fact that the weights do not explode will be convenient in various estimates.
Finally we define Therefore γ L doesn't contain any cross section. Here ·, · denotes the usual inner product in R n and ν(x) is the outward unit normal vector to ∂Ω L at x. Notice that, due to the geometry, ν 1 (x) the first component of ν(x) vanishes on γ L . Thus, we obtain This is one of the conditions required for the weight function in the Carleman inequality proved in [26] that we will use in Section 2. Let c * ∈ C 1 (Ω) ∩ L ∞ (Ω), c * > C min > 0 on Ω and M > 0 a given constant. We will consider the following admissible set of diffusion coefficients Notice that this means that the diffusion coefficient is supposed to be known in a neighbourhood of the lateral boundary of interest.
To ease the reading of Section 4, up to a restriction of V L , we assume that there exists r > 0 such that (1.8)

Regularity assumptions for the inverse problem
The method of Carleman estimate used in this paper requires solutions of the problem (1.1) that are sufficiently regular. Indeed the Buckgheim-Klibanov method [3] implies several time differentiations of system (1.1).
We assume in the following that c ∈ D and that u is an element of H = H 3 (0, T, H 3 (Ω)) satifying the a-priori bound u H < M for a given M > 0.

Main results
The first main result of this article is the following global stability estimate.
Theorem 1.1. Let l > 0. Let T > 0, L > l and a ∈ R n \Ω L satisfying the conditions of Proposition 1.1. Assume that u j for j = 1, 2 are solutions of (1.1) where c j and u 0,j are substituted respectively to c and u 0 . Assume also that c 1 , c 2 ∈ D. Then, if there exists C > 0 such that where Ω L is defined by (1.7), then, Here, K > 0 and κ ∈ (0, 1) are two constants depending only on ω, l, M , M , M 1 , T and a.
inf Ω c j > 0, c j = c * and ∇c j = ∇c * on ∂Ω ∩ ∂Ω L , and c j C 1 (Ω) < M and one replaces (1.10) by Remark 1.2. Notice that, as already highlighted, the boundary observation is on γ L and thus does not contain any cross-section in Ω L . Though the equation is set on the unbounded guide Ω, the stability estimate on Ω l is obtained with measurements on the finite portion Ω L . Remark 1.3. We look at the homogeneous Dirichlet case in order to simplify the regularity required for the solution u of (1.1) but we could obtain the same inverse result in the case of the inhomogeneous Dirichlet case (1.11) The proof of Theorem 1.1 follows the ideas of [3] and is proved in Section 3. It strongly relies on the Carleman estimate given in Section 2. This strategy is quite classical.
The drawback is that, to develop the Carleman machinery, one needs the technical (not so easy to verify) assumption (1.10). In the following we propose a strategy to remove this assumption: • we prove the existence of a boundary control such that the associated solution of (1.11) satisfies (1.10); • we fix this boundary condition for all the required measurements that is we deal only with system (1.11) for a fixed h.
More precisely, the second main result of this article reads as follows Let T > 0, L > l and a ∈ R n \Ω L satisfying the conditions of Proposition 1.1. For j = 1, 2 and h ∈ L 2 (0, T ; ∂Ω) we denote by u j the solutions of (1.11) where c j and u 0,j are substituted respectively to c and u 0 .
Let c 2 ∈ D be such that c 2 and for every j ∈ {1, . . . , n}, ∂ j c 2 , are uniformly continuous and bounded functions in Ω. Then there exists a control h ∈ L 2 (γ L × (0, T )) depending on c 2 such that for any c 1 ∈ D, Here, K > 0 and κ ∈ (0, 1) are two constants depending only on ω, l, M , M , M 1 , T and a.
As noticed the boundary condition h exerted for the measurements associated to c 1 only depends on c 2 . Morally, this means that the coefficient c 2 is known and that Theorem 1.2 is a local stability estimate around c 2 .
Remark 1.4. Notice that for any γ L ⊂ γ L the same statement holds true with a source-term h supported in γ L .
This result is proved in Section 4. The main difficulty is to prove the existence of a control h such that the associated solution satisfies (1.10). This part of the proof is inspired by the strategy developed in [1] for systems on bounded domains.

Global Carleman estimate for a parabolic equation in a cylindrical domain
We start from a global Carleman-type estimate proved by Yuan and Yamamoto [26, Theorem 2.1, (2)] in a bounded domain. Its validity is ensured by the estimates (1.2) and (1.5) satisfied 6 by our weight functions. For more informations about Carleman estimates in a parabolic setting we refer to Yamamoto [25].
Let s > 0 and denote In the following parts, C will be a generic positive constant.
Proposition 2.1. There exist a value of λ > 0 and positive constants s 0 and C = C(s 0 ) such that

1)
for all s > s 0 , and all u ∈ H 2,1 (Q L ) satisfying Let us remark that the Carleman inequality [26, Theorem 2.1, (2)] uses also λ as a second large parameter. As we will not use it, we now consider λ fixed in all the rest of the article such that Proposition 2.1 holds.
As in [4,Prop 4.2], we deduce the following Carleman inequality and we detail the proof for better understanding. The key difference with the Carleman inequality of Proposition 2.1 is to remove, on the cross-sections of Ω L , the boundary condition and the observation. for all s > s 0 , λ > λ 0 and all u ∈ H 2,1 (Q L ) satisfying Proof. Let χ, η be C ∞ cut-off functions defined by 0 ≤ χ ≤ 1, 0 ≤ η ≤ 1, Recall that ∂ t u − ∇ · (c∇u) = f. We consider y = ηχu and we get where R is the first order differential operator defined by R(u) = −∇ · (cu∇χ) − c∇χ · ∇u.
Then by the identities Then, from (2.4), we finish the proof.
3 Inverse Problem

Preliminary lemmas
We denote in the following b(θ) := b(·, θ) for any function b. As for the reconstruction of zeroth order coefficient (see [4, p.6]), we need to assume some hypothesis on the solution u at time θ on Ω L . For the diffusion coefficient, the assumption is more involved and is exactly (1.10). Now following an idea developed for example in [2, Lemma 2.4], we obtain the following result. Note that this key lemma is the one that requires the assumption (1.10). Thus, for the sake of completeness, we will give its proof. Lemma 3.1. Assume that (1.10) is satisfied and consider the first order partial differential operator P f = ∇ · (f ∇u 2 (θ)). Then there exist positive constants s 1 > 0 and C 1 > 0 such that for all s ≥ s 1 Proof. Let f ∈ H 1 0 (Ω L ) be such that f = 0 in V L . We denote w = e sφ(θ) f and Qw = e sφ(θ) P (e −sφ(θ) w). So we get Qw = P w − sw∇φ(θ) · ∇u 2 (θ). Therefore we have As, P w = ∇ · (w∇u 2 (θ)) = ∇w · ∇u 2 (θ) + w∆u 2 (θ), we obtain, Thus integrating by parts the second term of the right-hand side we obtain

Getting back to the original variables we obtain
Thus, using assumption (1.10), there exists a constant C > 0 depending on M such that where Ω L is defined by (1.7). As f = 0 in V L , We conclude taking s sufficiently large.
Moreover we recall the following classical result (see [8]). 9 Lemma 3.2. There exist positive constants s 2 and C such that for all s ≥ s 2 and z ∈ H 1 (0, T ; L 2 (Ω L )).
Proof. Recall that η is defined by (2.3). Consider any w ∈ H 1 (0, T ; L 2 (Ω L )). We have As 0 ≤ η ≤ 1, using Young's inequality, it comes that for any s > 0, Then we can conclude replacing w by e sφ z.

Proof of Theorem 1.1
Now we study the linearized inverse problem associated with (1.1) and we evaluate some Sobolev norm of the conductivity in terms of suitable observations of the solution of (1.1) on a part of the lateral boundary γ L . For this consider the following systems with c 1 , We obtain ∂ t y − ∇ · (c 1 ∇y) = ∇ · (c∇u 2 ) in Q.
Remark 3.1. In the rest of the proof we will deal with y only. Thus, even if we had started with (1.11), y would still enjoy homogeneous Dirichlet boundary condition. This remark will be crucial in Section 4 to prove Theorem 1.2.
We decompose the proof in three steps.
• First step: as in Proposition 2.2, we derive the equation satified by z = χηy where the cut-off functions are defined by (2.3). This will allow to work on the bounded domain Ω L × (0, T ).
As previously, the fact that c 1 , c 2 ∈ D and the definition of χ imply that we can apply again Lemma 3.1: there exists a positive constant C such that for s sufficiently large, Thus, expressing P (∂ xi (cχ)) from (3.8) we obtain So using Lemma 3.2 Moreover by the Carleman inequality (2.2), we have for j = 1, 2, The term in f j in the right-hand side is estimated as in the previous step. This leads to (3.10) From (3.9) and (3.10) we deduce (3.11) which ends this step.

13
Using inequalities (3.11) for 1 ≤ i ≤ n and gathering with (3.7), we get for s sufficiently large . As d 1 − d 0 < 0 and d 2 − d 0 > 0, optimizing this last inequality with respect to s, we complete the proof.

Removing the technical assumption by an adapted control
In Theorem 1.1, we assume (1.10) satisfied. However since this hypothesis concerns the gradient of one solution of (1.1) in Ω L , it is difficult to verify it in the case of real applications. Thus, we develop ideas based on control theory to ensure such strong hypothesis.
The goal of this section is to prove that for any c ∈ D sufficiently regular, there exists a control h such that the associated solution of (1.11) satisfies (1.10). Then, the proof of Theorem 1.2 will follow directly (see Section 4.4). As the condition (1.10) involves a property for the gradient, we will seek for controllability results in stronger norms, namely H 1 (Ω L ). The strategy is inspired by [1] and is as follows: • Assume that there is a function u b sufficiently smooth and satisfying (1.10) i.e. there exists δ > 0 such that |∇u b · ∇d| ≥ δ > 0 a.e. in Ω L .
The existence and regularity of such function is detailed in Section 4.3.
• Let β a positive constant be such that |∇d| ≤ β in Ω L . Then, if there exists h such that the associated solution of (1.11), denoted by u(x, t; h), satisfies it comes that u(·, ·; h) satisfies (1.10).
• To do so, we will prove the following result: for any u b sufficiently regular, any ε > 0 and any τ > 0, there exists h ∈ L 2 (γ L × (0, T )) such that A more precise statement is given in Proposition 4.5. This allows to construct a sequence of controls h n such that , for a.e. x ∈ Ω, and thus proves the previous item. To prove this result, first, we use classical results from control theory to establish this approximate controllability in L 2 (Ω)-norm. Then, using the regularity properties of system (1.1), we extend this result to the H 2 (Ω)-norm. To do so, we extend to unbounded domains the strategy given in [1]: after reaching approximately (in the weak norm) the target, let the system evolves freely to benefit from the regularizing properties but not too long to stay close to the target (in strong norms). We start this section recalling those regularization properties we will need.
14 Assume in the rest of this section that c satisfies the assumptions given in Theorem 1.2 for c 2 .

Analytic properties of the elliptic operator
First, notice that the H 2 (Ω)-norm we are interested in is related to the operator A 0 (see (1.9)). Indeed from classical elliptic results (see for instance [17, Theorem 3.1.1]), one has the following result.
Finally, from (4.1) and (4.2) we get that there exists a positive constant C such that for all t > 0 and for all u ∈ D(A 0 ) This is the key inequality we will use.

Approximate controllability
First we give a classical result of approximate controllability in L 2 (Ω).
Proposition 4.4. For any u b ∈ L 2 (Ω), any ε > 0 and any τ > 0, there exists h ∈ L 2 (γ L ×(0, τ )) such that Proof. This result is quite classical in control theory. Let us sketch its proof for the sake of completeness. Let Ω be an open set in R n satisfying Ω ⊂ Ω and ∂Ω ∩ R n \∂ Ω ⊂ γ L and letω be a domain such thatω ⊂⊂ Ω ∩ R n \Ω . From [24] and a classical duality argument, we have the approximate controllability in L 2 ( Ω × (0, τ )) with a localized control g ∈ L 2 (ω × (0, τ )) for the auxiliary problem We conclude the proof of this proposition by taking h as the trace of u on γ L .
We now turn to approximate controllability in more regular norms.

Construction of an appropriate target
In the above proof, to obtain approximate controllability in H 2 (Ω)-norms we need targets that are sufficiently regular, namely in D(A 2 0 ). To apply our strategy, we now prove that there exists some function u b ∈ D(A 2 0 ) and a constant δ > 0 such that |∇d(x) · ∇u b (x)| ≥ δ, for a.e. x ∈ Ω L .
Let ξ 1 ∈ C ∞ (R; R) such that 0 ≤ ξ 1 ≤ 1 and Let ξ ∈ C ∞ (R n−1 ; R) such that 0 ≤ ξ ≤ 1 and Thus, u b ∈ D(A 0 ) and ∇u b (x) = ∇d(x), for any x ∈ Ω L . Due to (1.2), this implies that (1.10) is satisfied. As u b ∈ C ∞ (Ω) and u b identically vanishes near the boundary we also obtain that u b ∈ D(A 2 0 ). Remark 4.1. Finding such u b in C ∞ (Ω) such that inf Ω L |∇u b · ∇d| > 0, can easily be done taking for instance u b = d. The main difficulty is to ensure all the boundary conditions so that u b ∈ D(A 2 0 ). This is precisely why we assumed that c is known in V L . With this assumption, there is no requirement on ∇u b near the boundary which allows to use cut-off functions to design u b .

Conclusion
We now have all the ingredients to prove Theorem 1.2. Let u b as constructed in Section 4.3. Then, from Proposition 4.5, there exists h depending on c 2 such that the solution u 2 satisfies (1.10).
As proved in Section 3, the stability derives from the Carleman inequality applied to y defined by (3.1). As, u 1 and u 2 are both solutions of (1.11) with the same boundary condition h it comes that y solves (1.1) that is with homogeneous Dirichlet boundary conditions. The rest of the proof remains unchanged.