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On Some Geometric Inverse Problems for Nonscalar Elliptic Systems

In this paper, we consider several geometric inverse problems for linear elliptic systems. We prove uniqueness and stability results. In particular, we show the way that the observation depends on the perturbations of the domain. In some particular situations, this provides a strategy that could be used to compute approximations to the solution of the inverse problem. In the proofs, we use techniques related to (local) Carleman estimates and differentiation with respect to the domain.


Introduction
Let Ω ⊂ R N be a simply connected bounded domain whose boundary ∂Ω is of class W 2,∞ , let D * be a fixed nonempty open set with D * ⊂⊂ Ω and let γ ⊂ ∂Ω be a nonempty open set. In what follows, the symbols C, C 1 , C 2 , . . . will be used to denote generic positive constants. Sometimes, we will indicate the data on which they depend by writting (for example) C(Ω, D * ).
Let us consider the following family of subsets of D * : D = D ⊂ Ω : D is a nonempty simply connected domain, D ⊂ D * and ∂D is of class W 2,∞ and let us denote by A the set of all (a, b, A, B) such that a, b, A, B ∈ L ∞ (Ω) and for some λ with 0 < λ < µ 1 (Ω) −1 , where µ 1 (Ω) is the smallest positive constant such that ∀u ∈ H 1 0 (Ω).
In many physical phenomena, in order to predict the result of a measurement, we need a model of the system under investigation (typically a PDE system) and an explanation or interpretation of the observed quantities. If we are able to compute the solution to the model and quantify relevant observations, we say that we have solved the forward or direct problem. Contrarily, the inverse problem consists of using the observations to recover unknown data that characterize the model. For details about the main questions concerning inverse problems for PDEs from the theoretical and numerical viewpoints, see for instance the book [11].
A motivation of problems of this kind can be found, for instance, when one tries to compute the stationary temperature of a chemically reacting plate whose shape is unknown. More precisely, (2)-(3) has the following interpretation: assume that a chemical product, sensible to temperature effects, fills an unknown domain Ω\D; its concentration y = y(x) and its temperature z = z(x) are imposed on the whole outer boundary ∂Ω, the associated normal fluxes are measured on γ ⊂ ∂Ω and both y and z vanish on the boundary of the non-reacting unknown set D; what we pretend to do is to determine D from these data and measurements.
• Reconstruction: Find an iterative algorithm to compute the unknown domain D from the observation (α, β).
In this paper, we focus on the uniqueness and the stability of the inverse problem (2)-(3). Specifically, our first main result is the following: Theorem 1 Assume that (ϕ, ψ) ∈ H 1/2 (∂Ω)×H 1/2 (∂Ω) is nonzero. For i = 0, 1, let (y i , z i ) be the unique weak solution to (2) with D replaced by D i and let α i and β i be given by the corresponding equalities (3). Then one has the following: The proof is given in Section 2. It relies on some ideas from [9]; more precisely, we use two well known properties of (2): unique continuation and well-posedness in the Sobolev space H 1 .

Remark 2
In the one-dimensional case, if one consider (2) with only one boundary observation, uniqueness may not hold. Indeed, suppose that Ω = (0, 1), L ∈ (0, 1) and consider the system where A, b, η, ζ ∈ R (all them different from zero) and |A| + |b| < 2|η||ζ|. Then, (η 2 , b, A, ζ 2 ) ∈ A and, using the parameter variation method, we get that a solution (y, z) to (4) is given by for each K ∈ R. Therefore, if K = 0 we have z x (0) = 0 and this implies non-uniqueness. For N ≥ 2, the uniqueness in the N-dimensional case with only one information on γ is, to our knowledge, an open question.
In order to state our main stability result, let us introduce some notation. Thus, let D 0 ∈ D be a fixed subdomain, let µ ∈ W 1,∞ (R N ; R N ) satisfy µ W 1,∞ ≤ < 1, µ = 0 in Ω\D * and, for any σ ∈ (−1, 1), let us denote by m σ , D σ and (y σ , z σ ) respectively the mapping m σ := I + σµ, the open set D σ := m σ (D 0 ) and the solution to (2) with D replaced by D σ . For simplicity, it will be assumed that the coefficients a, b, A, B are constant. The following holds: Theorem 2 There exists σ 0 > 0 with the following properties: is well defined and analytic in (−σ 0 , σ 0 ), with values in H −1/2 (γ) 2 .
Note that, in the applications to fluid mechanics, the goal is to identify the shape of a body around which a fluid flows from measurements performed far from the body. In other contexts, the domain D can represent a rigid body immersed in an elastic medium. Thus, related inverse problems with relevant applications in Elastography have been analyzed in [5] for the wave equation and [6] for the Lamé system. This paper is organized as follows. In Section 2, we prove a unique continuation property for the solutions to (2) and, then, we prove the uniqueness result (Theorem 1). Section 3 is devoted to proof of the stability result (Theorem 2). Finally, in Section 4, we present some additional comments and open questions.

Unique continuation and uniqueness
In this section, we analyze a unique continuation property for (2). More precisely, we have the following result: Theorem 3 Let G ⊂ R N be a bounded domain whose boundary ∂G is of class W 1,∞ , let ω ⊂ G be a nonempty open set and assume that a, b, A, B ∈ L ∞ (G). Then, any solution (y, z) ∈ H 1 (G) × H 1 (G) to the linear system is zero everywhere.
As already mentioned, the proof relies on some ideas from [9]. In fact, we will divide the proof in two parts: (i) the proof of Theorem 3 when ω and G are open balls and (ii) the proof for general domains ω and G, using a compactness argument.

A unique continuation property for balls
In this Section, we prove a very particular result concerning the unique continuation of the solutions to (6): ) to the linear system (6) in B 2R (x 0 ), the following property holds: Before proving this lemma, let us introduce ϕ ∈ C ∞ 0 (R N ) and let us define Let us also recall that the Poisson bracket of a 0 and b 0 is given by A crucial result in the proof of Lemma 1 is the following: nonempty compact set and assume that ϕ ∈ C ∞ 0 (R N ). Suppose that ϕ is bi-convex in U with respect to the characteristics of a 0 and b 0 , i.e. ϕ satisfies the following: (8) Then, there exist C 1 > 0 and h 1 > 0 such that, for all 0 < h < h 1 and any function u ∈ H 2 0 (K), one has: Proof of Lemma 1: Without loss of generality we may assume that (6) such that y = 0 and z = 0 in B R .
Consequently, we can apply Theorem 4 toỹ and deduce that there exist C 2 > 0 and h 2 > 0 such that for all h ∈ (0, h 2 ). Here, we have absorbed the lower order term forỹ from the right hand side by taking h 2 small enough. Analogously, there exist positive constants C 3 > 0 and h 3 > 0 such that, for all h ∈ (0, h 3 ). Next, adding (12) and (13), taking h 4 sufficiently small and C 4 sufficiently large and absorbing again the lower order terms forỹ andz from the right hand side, we have for all h ∈ (0, h 4 ).
To conclude the proof, we note that (11), the fact that y = z = 0 in B R and ∇ζ = ∆ζ = 0 for R − ε ≤ |x| ≤ 2R − 2ε imply that H 1 and H 2 vanish in B 2R−2ε . Now, we have from (9) that ϕ is positive and radially decreasing in U . Thus, one has On the other hand, It follows from (14)- (16) that Since H 1 and H 2 are independent of h and ϕ(2R − 2ε) < ϕ(2R − 3ε), we can let h → 0 and get that and, consequently, y and z vanish in B 2R−3ε . Since ε > 0 is arbitrarily small, we conclude that y and z vanish identically in B 2R .

Unique continuation for general domains
The goal of this section is to prove Theorem 3 in the general case. Let (y, z) ∈ H 1 (G)×H 1 (G) be a solution to (6) satisfying (7) and let us assume that B ρ0 (x 0 ) ⊂ ω. Let x 1 be a point of G and let us see that y = 0 and z = 0 in a neighborhood of x 1 .
Now, we fix x 0 ∈ γ and we choose r > 0 such that B r (x 0 )∩∂Ω ⊂ γ. Let us set O := O ∪B r (x 0 ) and consider the extension by zero (ỹ,z) of (y, z) to the whole set O .
From (17) To conclude, let us prove that D 1 \D 0 and D 0 \D 1 must be empty. Thus, let us suppose that  Figure 1).

Preliminary results
Let us introduce some basic notation. Let m = (m 1 , . . . , m N ) ∈ W 1,∞ (R N ; R N ) be given and let us set In the sequel, we will consider the set where 0 < < 1. We will work with mappings of the form m := I + µ, where I : R N → R N is the identity and µ ∈ W . For any µ ∈ W , I + µ is obviously bijective, (I + µ) −1 ∈ W 1,∞ (R N ; R N ) (see [10], p. 193) and (I + µ)(D) ∈ D ∀D ∈ D.
Theorem 5 The mapping µ → (u 0 , v 0 ) is analytic in a neighbourhood of the origin in W .
Proof: We note first that (F 0 , G 0 ) does not depend of µ, because (ϕ 0 , ψ 0 ) = 0 where µ = 0. Then, since the mapping T is a isomorphism we have by (23) that From the results in [10,15], we know that the mapping µ → T is analytic in a neighbourhood of 0. Consequently, this is also the case for µ → (u 0 , v 0 ) and the proof is done.
2. Now, let us assume that m σ (D 0 ) = D 0 for some σ ∈ (−σ 0 , σ 0 ). In view of Theorem 1, not all the F j can be zero. Let k be the smallest j such that F j = 0. It is then clear that there Accordingly, for these σ, one must also have which allows to achieve the proof.

A similar inverse problem with internal observation
Let ω ⊂⊂ Ω\D * be a nonempty open set. Consider the following geometric inverse problem, where the observation is performed on ω: Given (ϕ, ψ) ∈ H 1/2 (∂Ω) × H 1/2 (∂Ω) and α ∈ H 1 (ω), find a set D ∈ D such that the solution (y, z) to the linear system (2) satisfies the following additional condition: We have the following uniqueness result: Consequently, Theorem 3 guarantees that (y, z) = (0, 0) in O. Arguing as in the proof of Theorem 1, we deduce that D 0 \D 1 and D 1 \D 0 are empty sets and, consequently, D 0 = D 1 .
We also have a stability result similar to Theorem 2. Thus, let us fix D 0 ∈ D and µ ∈ W with µ = 0 in D 0 , let us take m σ = I + σµ and D σ = m σ (D 0 ) and let (y σ , z σ ) be the solution to (24). The following holds: Theorem 7 Under the assumptions in Theorem 6 on (ϕ, ψ) and b, there exists σ 0 > 0 with the following properties: is well defined and analytic in (−σ 0 , σ 0 ), with values in L 2 (ω).
2. Either m σ (D 0 ) = D 0 for all σ ∈ (−σ 0 , σ 0 ) (and then the mapping in (27) is constant), or there exist σ * ∈ (0, σ 0 ), C > 0 and k ≥ 1 (an integer) such that Proof: Again, the proof is very similar to the proof of stability in the boundary observation case (Theorem 2). In fact, the unique difference appears in the last part of the argument, when we write instead of (25). For brevity, we omit the details.

Remark 3
Recall that, in the case of problem (2)-(3), we need two boundary observations, the normal derivatives of y and z on γ, to deduce uniqueness and stability. The last two results show that, with internal observations, this holds with the information supplied by just one variable.

A geometric inverse problem for a parabolic system
Let us present some ideas that allow to extend Theorems 1 and 6 to time-dependent parabolic systems. For brevity, we will only consider the boundary observation case. Thus, let T > 0 be given and let us consider the following inverse problem: Given (ϕ, ψ) and (α, β) in appropriate spaces and a nonempty open set γ ⊂ ∂Ω, find an open set D ∈ D such that the solution (y, z) to the linear evolution system: satisfies the additional conditions: ∂y ∂n γ×(0,T ) = α and ∂z ∂n γ×(0,T ) = β.
Indeed, the first step is to deduce a unique continuation property: Proposition 1 Let G ⊂ R N be a bounded domain whose boundary is of class W 2,∞ and let us set Q := G × (0, T ). Suppose that a, b, A, B ∈ L ∞ (Q) and let O be a nonempty open subset of Q.
Then, any solution (y, z) ∈ L 2 (0, T ; H 2 (G) × H 2 (G)) to satisfies the following property: The proof of this result is similar to the proof of Theorem 1.4 in [9].
Remark 5 If, in (28), we impose nonzero initial conditions on y and/or z, the situation is much more complex. In particular, the prevuious argument does not work. A detailed analysis will be the objective of a forthcoming paper.
Remark 6 Notice that, in this time-dependent case, no assumption of the kind (1) is needed.
Stability results like Theorems 2 and 7 can also be established in this framework. We will not give the details for brevity, since the arguments are not very different and can easily be completed by the reader.

Additional comments on stability
In the context of the stability problem, we can adopt another (more geometrical) viewpoint. To clarify the situation, let us consider the scalar systems where Ω, is as before, D 0 and D 1 are convex and have nonempty intersection and the following regularity properties hold: Let us assume that Then, it can be proved that the Haussdorf distance d H (D 0 , D 1 ) satisfies the estimate where C only depends on Ω, M and m; see [4]. The proof is based on the following well-posedness results, where C is as above: • Let us set again G = Ω \ D 0 ∪ D 1 and let us assume that |y 0 (x) − y 1 (x)| = max x∈G |y 0 (x) − y 1 (x)|.
Under additional properties for the ϕ i ,α i and y i , the previous estimates can be improved; see [4] for more details.
It would be interesting to extend this approach to the inverse problems (2), (3) and (2), (26). At present, to our knowledge, whether or not this is possible is an open question.

Reconstruction
As already said, reconstruction algorithms for the solution of problems of the kind (2)-(3) have been considered in several papers. In all them, the main idea is to reduce to finite dimension and reformulate the search of the unknown D as a constrained (maybe numerically ill-conditionned) extremal problem. Then, usual gradient, quasi-Newton or even Newton methods can be used to compute approximate solutions; see for instance [1,2,3].
Let us present in this section a different approach that relies on the domain variation techniques introduced in [13,14,15].