Uniqueness and Lipschitz stability for the identification of Lam\'e parameters from boundary measurements

In this paper we consider the problem of determining an unknown pair $\lambda$, $\mu$ of piecewise constant Lam\'{e} parameters inside a three dimensional body from the Dirichlet to Neumann map. We prove uniqueness and Lipschitz continuous dependence of $\lambda$ and $\mu$ from the Dirichlet to Neumann map.


Introduction
A relevant inverse problem arising in nondestructive testing of materials is the one of determining, within an isotropic, linearly elastic three dimensional body Ω, the elastic properties of the body from measurements of traction and displacement taken on the exterior boundary of the domain Ω. This leads mathematically to the formulation of the following boundary value problem for the system of linearized elasticity div (C ∇u where Ω is an open and bounded domain, ∇u denotes the strain tensor ∇u := 1 2 ∇u + (∇u) T , ψ ∈ H 1/2 (∂Ω) is the boundary displacement field, and C ∈ L ∞ (Ω) denotes the isotropic elasticity tensor with Lamé coefficients λ, µ: C = λI 3 ⊗ I 3 + 2µI sym , a.e. in Ω where I 3 is 3 × 3 identity matrix and I Sym is the fourth order tensor such that I Sym A =Â, The strong convexity condition is assumed µ ≥ α 0 > 0, 2µ + 3λ ≥ β 0 > 0 a.e. in Ω. * Dipartimento di Matematica "G. Castelnuovo" Università di Roma "La Sapienza" (beretta@mat.uniroma1.it). † Dipartimento di Matematica e Informatica "U. Dini", Università di Firenze (francini@math.unifi.it) ‡ Dipartimento di Matematica e Informatica "U. Dini", Università di Firenze (sergio.vessella@dmd.unifi.it) Under the above assumptions problem (1) has a unique weak solution u ∈ H 1 (Ω) and the Dirichlet-to-Neumann linear map (DN map), Λ C , is well defined where ν is the exterior unit normal to ∂Ω. The inverse problem consists in determining C, i.e. λ and µ, from knowledge of the DN map Λ C . This problem is closely related to the conductivity inverse problem arising in the modelling of EIT (Electrical Impedence Tomography). For the mathematical treatment of this problem we refer to the fundamental papers [Al1], [AP], [Na] and [SU]. Unfortunately, the mathematical approach used to investigate the conductivity inverse problem fails partly when dealing in the elasticity framework. This is due to the fact that we have to deal with an elliptic system instead that with a scalar equation and we have to recover two parameters λ and µ instead of the sole conductivity parameter. As a consequence of these difficulties only partial results to this inverse problem are known and mainly concern the uniqueness issue. More precisely, the study of the problem was initiated in the 90's by Ikeata in [Ik] who considered a linearized version of it. In two dimensions Akamatsu, Nakamura and Steinberg in [ANS] and for dimension n ≥ 3 Nakamura and Uhlmann in [NU] showed that one can determine uniquely and in a stable way C ∞ (Ω) Lamé parameters λ and µ and their derivatives on the boundary of a smooth domain Ω from the DN map. Local uniqueness has been proved in dimensions two by Nakamura and Uhlmann in [NU1] for C ∞ (Ω) Lamé parameters assuming that they are both close to positive constants. In three dimensions and higher Nakamura and Uhlmann in [NU2] and Eskin and Ralston in [ER] proved local uniqueness for smooth Lamé parameters whenever µ is close to a constant. To our knowledge no result concerning stability estimates is known. Based on the results obtained by Alessandrini in [Al1] who proved, logarithmic stability estimates for the conductivity inverse problem in the case of smooth conductivities and the example of Mandache in [M] who proved the optimality of this estimate also for the inverse elasticity problem in the case of smooth Lamé parameters logarithmic stability estimates or even worse ones are expected. These considerations lead in recent years to look for different a priori assumptions on the unknown parameters which take into account the applied context from which the problem arises and give rise to better stability estimates ( [V], [ABV], [BF], [BFV], [ABF], [BdHQ]). An attempt in this direction has been done by Alessandrini and Vessella in [AV] with for the conductivity inverse problem for unknown conductivities depending only on a finite number of parameters. In [AV] they proved Lipschitz continuous dependence from the DN map for conductivities that are constant on known subdomains, assuming ellipticity and C 1,α regularity at the interfaces joining contiguous domains.
In this paper we propose to consider L ∞ (Ω) elasticity tensors of the form where the D j ' s, j = 1, · · · , N , are known disjoint Lipschitz domains representing a partition of Ω and λ j , µ j , j = 1, · · · , N , are unknown constants. We will prove that if C 1 and C 2 are of this form, assuming that the boundaries of D j 's and of Ω contain flat portions, we have where the stability constant C appearing in the estimate depends on various parameters like α 0 , β 0 , on the regularity bounds on Ω and on the D j ' s and on their number N . In particular, the constructive character of the proof allows us to establish an estimate from above of the constant C.
We want now to emphasize that several significant examples fit in our analysis. Polyhedral partitions of Ω, appearing in any finite-element scheme used for effective reconstruction of the Lamé parameters (see for example [BJK]; and a layered configuration of the sets D j 's arising in the study of composite laminates [Mi] and in geophysical prospection, [BC]. Our approach is based on the use of the following key ingredients: existence of singular solutions and study of their behaviour close to the flat discontinuity interfaces of the D j 's, regularity estimates and quantitative estimates of unique continuation of solutions to system (1). As already pointed out in [BF] a relevant difference with respect to the scalar case treated in [AV] is the issue of existence of singular solutions. In fact, in the case of strongly elliptic systems with L ∞ coefficients in dimension n ≥ 3, existence of the fundamental solution and of the Green's function cannot be inferred without additional assumptions. In [HK] Hofmann and Kim prove their existence under the additional information that weak solutions of the system satisfy De Giorgi-Nash type local Hölder estimates. It is clear that in the case of a polyhedral partition of Ω solutions might not enjoy Hölder regularity at edges. On the other hand, in order to obtain our result, it is enough to construct singular solutions and analyze their behavior in a Lipschitz subsetK at a given positive distance from edges. Nevertheless, while for the scalar (even for the complex valued treated in [BF]) conductivity equation is fairly easy to get a solution of the equation close to a flat interface by using the fundamental solution for the Laplace equation and suitable reflection arguments, this seems not to be possible for the Lamé system. In order to construct singular solutions, we make use of special fundamental solutions constructed by Rongved in [R] for isotropic biphase laminates. Furthermore, looking at solutions of the elasticity system inK we can use the results of [CKVC] and of [LN] deriving regularity estimates for the solutions which allow us to obtain Hölder estimates of unique continuation inK.
We would like to point out that our proof is based on the use of solutions having boundary displacement fields supported in the flat portion Σ 1 of ∂Ω. Hence, our stability result also holds replacing the full DN map with the local DN map that we will define in Section 2. Moreover, we expect to derive similar stability estimates also the case of domains D j 's with C 1,α portions of interfaces and this analysis will be object of a forthcoming publication. Finally, we would like to emphasize that Lipschitz stability estimates have become crucial also for the effective reconstruction of the unknown coefficients. In fact, recently, in [dHQS1] and [dHQS2], de Hoop, Qiu and Scherzer have shown that Lipschitz type stability estimates imply local convergence of iterative reconstruction algorithms and the radius of convergence of the iterates depends on the stability constant and hence an explicit determination of the dependence of such constant from the a-priori parameters, in particular from the partition number N , is crucial.
The plan of the paper is the following: section 2 contains the description of the main result. In paragraph 2.1 we introduce the notation and main definitions. In paragraph 2.2 we state the main a priori assumptions and the main result. We also reformulate the inverse elasticity problem in terms of the nonlinear forward map F acting on a finite-dimensional subset of R 2N and recall a general result (Proposition 2.5) that will let us show that F has a Lipschitz continuous inverse.
Section 3 contains some auxiliary results. In particular, in paragraph 3.2 we collect some properties concerning the fundamental solution in biphase elastic isotropic materials introduced by Rongved in [R] and we prove existence of singular solutions for our Lamé system. In paragraph 3.3 we state some regularity results and estimates of unique continuation concerning solutions of piecewise constant Lamé systems.
In section 4 we give the proof of our main result by verifying that the forward map F corresponding to our inverse problem, satisfies all the assumptions of Proposition 2.5 thus concluding the proof.
Finally, in the Appendix we recall some known results concerning solutions to Lamé systems with constant coefficients and the proof of the estimate of unique continuation stated in section 3.

Aknowledgements
We want to thank Antonino Morassi for pointing out to us the paper by Rongved on biphase fundamental solution and for stimulating discussions. This work has been supported by MIUR within the project PRIN 20089PWTPS003.

Main result 2.1 Notation and main definitions
For every x ∈ R 3 we set x = (x ′ , x 3 ) where x ′ ∈ R 2 and x 3 ∈ R. For every x ∈ R 3 , r and L positive real numbers we will denote by B r (x), B ′ r (x ′ ) and Q r,L (x) the open ball in R 3 centered at x of radius r, the open ball in R 2 centered at x ′ of radius r and the cylinder B ′ r (x ′ ) × (x 3 − Lr, x 3 + Lr), respectively; in the sequel B r (0), B ′ r (0) and Q r,L (0) will be denoted by B r , B ′ r and Q r,L , respectively. We will also denote by R 3 For any subset D of R 3 and any h > 0, we denote by Definition 2.1. Let Ω be a bounded domain in R 3 . We shall say that a portion Σ ⊂ ∂Ω is of Lipschitz class with constants r 0 > 0, L ≥ 1 if for any point P ∈ Σ, there exists a rigid transformation of coordinates under which P = 0 and where ψ is a Lipschitz continuous function in B ′ r0 such that ψ(0) = 0 and ψ C 0,1 (B ′ r 0 ) ≤ Lr 0 . We say that Ω is of Lipschitz class with constants r 0 and L if ∂Ω is of Lipschitz class with the same constants.
Remark 2.2. We use the convention of normalizing all norms in such a way that all their terms are dimensionally homogeneous. For example: Similarly, denoting by D i u the vector which components are the derivatives of order i of the function u, , and so on for boundary and trace norms such as · H 1 2 (∂Ω) , , where Ω is a bounded subset of R 3 whose boundary is smooth enough.
We will also make use of the following notations for matrices and tensors: for any 3 × 3 matrices A and B we set A : B = 3 i,j=1 A ij B ij andÂ = 1 2 (A + A T ). By I 3 we denote the 3 × 3 identity matrix and by I Sym we denote the fourth order tensor such that I Sym A =Â.
In the whole paper we are going to consider isotropic elastic materials, hence the elasticity tensor C is a fourth order tensor given by where Ω is a bounded domain in R 3 of Lipschitz class, and the real valued functions λ = λ(x) and µ = µ(x) ∈ L ∞ (Ω) are the Lamé moduli. We will also use Poisson's ratio ν(x) = λ(x) 2(λ(x)+µ(x)) .
An elasticity tensor C is strongly convex if there is a positive number ξ 0 such that, for almost every x in Ω, In the isotropic case (2), the strong convexity condition takes the form In these case, the Poisson's ratio has values in an compact subset of (−1, 1 2 ). More precisely we can estimate In the sequel we will make use of the following norm in the linear space of isotropic tensors: This norm is equivalent to the usual L ∞ norm for tensors in the space of isotropic tensors.
Our boundary measurements are represented by the Dirichlet to Neumann map. As a matter of fact, since we will restrict our measurements to boundary data that have support on some subset of the boundary, we will make use of a local Dirichlet to Neumann map.
We define the local Dirichlet to Neumann linear map Λ Σ C as follows: Note that for Σ = ∂Ω we get the usual Dirichlet to Neumann map. For this reason we will set Λ C := Λ ∂Ω C .
The map Λ Σ C can be identified with the bilinear form on H 1/2 for all ψ, φ ∈ H 1/2 co (Σ) and where u solves (6) and v is any H 1 (Ω) function such that v = φ on ∂Ω.
We shall denote by · ⋆ the norm in L H We finally recall an extension to systems of Alessandrini's identity [Al1], [Is]. Let u 1 and u 2 be the solutions to div(C k ∇u k ) = 0 in Ω for k = 1, 2 respectively and with C k , k = 1, 2, satisfying (3). Then where Λ C 1 , Λ C 2 denote the Dirichlet to Neumann map corresponding to C 1 , C 2 respectively.

Main assumptions and statement of the main result
Let A, L, α 0 , β 0 , N be given positive numbers such that N ∈ N, α 0 ∈ (0, 1), β 0 ∈ (0, 2) and L ≥ 1. We shall refer to them as the a priori data. Let r 0 be a positive number. Our main assumptions are: and we assume that Ω = ∪ N j=1 D j , where D j , j = 1, . . . , N are connected and pairwise non overlapping open domains of Lipschitz class with constants r 0 , L.
We also assume that there exists one region, say D 1 such that ∂D 1 ∩ ∂Ω contains an open flat portion Σ and that for every j ∈ {2, . . . , N } there exist j 1 , . . . , j M ∈ {1, . . . , N } such that and, for every k = 2, . . . , M ∂D j k−1 ∩ ∂D j k contains a flat portion Σ k such that Furthermore, for k = 1, . . . , M , we assume there exists P k ∈ Σ k and a rigid transformation of coordinates such that P k = 0 and where we set Σ 1 := Σ.
For simplicity we will call D j1 , . . . , D jM a chain of domains connecting D 1 to D j . For any k ∈ {1, ...M } we will denote by n k the exterior unit vector to ∂D k in P k .
(A2) We assume the tensor C to be isotropic piecewise constant of the form where C j = λ j I 3 ⊗ I 3 + 2µ j I Sym with constant Lamé coefficients λ i and µ i that satisfy For j = 1, . . . , N , we denote the Poisson's ratio by ν j = λj 2(λj +µj ) . Note that each ν j satisfies (5).
In the sequel we will introduce a number of constants that we will always denote by C. The values of these constants might differ from one line to the other.
Theorem 2.4. Let Ω and Σ satisfy assumption (A1). Then there exists a positive constant C depending on L, A, N, α 0 , β 0 only such that, for any C k , k = 1, 2 satisfying assumption (A2), we have A better evaluation of constant C is given in Remark 4.7.
In order to prove Theorem 2.4 we will first state it in terms of the forward map that maps Lamé parameters to the corresponding Dirichlet to Neumann map. Then, we will apply to the forward map the following general result: Proposition 2.5. Let M 1 and M 2 be positive numbers and d ∈ N. Let A and K be an open subset and a compact subset of R d respectively. Assume that K ⊂ A, Let B be a Banach space and let F : A → B be such that: then we have This proposition holds also in infinite dimensional spaces. For a proof of Proposition 2.5 in finite dimensional space we refer to [BaV,Prop.5 ].
Let us now introduce the forward map corresponding to our problem. In order to represent the set of Lamé parameters, we will use the following notation: For each vector L ∈ A we can define a piecewise constant isotropic elastic tensor C L (as in (9)) with Lamé parameters λ j and µ j for j = 1, . . . , N . In this case, C L ∞ is equal to the norm in R 2N given by The (nonlinear) forward map can be defined as follows: Definition 2.6. Let Ω and Σ satisfy assumption (A1).
We can identify F withF : A → B such thatF (L) =Λ Σ CL (defined in (7)), where B is the Banach space of bilinear form on H Let ψ and φ ∈ H 1/2 co (Σ) and let u L be the solution to In the sequel, we will write F and Λ Σ CL instead ofF andΛ Σ CL .
With the above notation, Theorem 2.4, can be stated as follows: Theorem 2.7. Let Ω and Σ satisfy assumption (A1) and let K ⊂ A be the compact subset Then, there exists a positive constant C, depending on L, A, N, α 0 , β 0 only such that Notice that Theorem 2.7 means that F is invertible on K and its inverse is Lipschitz continuous.
In Section 4 we will show that the forward map of definition 2.6 satisfies all the assumptions of Proposition 2.5 for A and K defined as in (12) and (13) respectively and B is the space of bilinear form on H 1/2 co (Σ) × H 1/2 co (Σ). Then, Theorem 2.7 is a consequence of Proposition 2.5.

Further notation and definitions
In order to prove the main theorem we need to introduce some further notation and definitions.

Construction of an augmented domain and extension of C.
First we extend the domain Ω to a new domain Ω 0 such that ∂Ω 0 is of Lipschitz class and B r0/C (P 1 ) ∩ Σ ⊂ Ω 0 , for some suitable constant C ≥ 1 depending only on L. We proceed as in [Al-Ro-R-Ve, Sect. 6]. Set and define, for every It is straightforward to verify that i) Ω 0 has Lipschitz boundary with constants r0 3 , 3L. ii) Let C be an isotropic tensor that satisfies assumption (A2). We still denote by C its extension to Ω 0 such that C |D0 A = 2Â for every 3 × 3 matrix A. This extended tensor is still an isotropic elasticity tensor of the form where each C j for j = 0, . . . , N has Lamé parameters satisfying (10).

Construction of a walkway
Let us fix j ∈ {1, ...N } and let D j1 , . . . , D jM be a chain of domains connecting where ρ 1 is as in (14).
Let us introduce the following sets: i) Q (k) , k = 1, . . . , M , is the cylinder centered at P k such that by a rigid transformation of coordinates under which P k = 0 and Σ k belongs to the plane {(x ′ , 0)}, is given by It is straightforward to verify that K h is connected and of Lipschitz class for every h ∈ (0, h 0 ) and that (in a suitable coordinate system)

Fundamental solution in the biphase laminate
In our proof of Lipschitz stability estimates, as in the approach used by Alessandrini and Vessella for the conductivity equation [AV], a crucial role is played by singular solutions for the Lamé system. As a matter of fact we are not only interested in the existence of such singular solutions, but also in their asymptotic behavior close to the interfaces. In the scalar case, this tool is granted by the existence of Green functions and by explicit expressions for solutions in the presence of an interface.
Unfortunately, the existence of the Green matrix cannot be inferred for elliptic systems with bounded coefficients in dimension 3 or higher.
Moreover, whereas for the scalar (even complex valued) conductivity equation is fairly easy to get a solution of the equation close to a flat interface by using the fundamental solution for the Laplace equation and suitable reflection arguments, this seems not to be possible for the Lamé system.
In order to construct singular solutions, we make use of special fundamental solutions constructed by Rongved in [R] for isotropic biphase laminates.
Consider the isotropic tensor where C + and C − are constant isotropic elastic tensors given by with λ and µ and λ ′ and µ ′ satisfy (10). Denote Poisson's parameters by ν and ν ′ . In [R] an explicit formula for a fundamental solution Γ : is given. Here δ y is the Dirac distribution concentrated at y. This explicit formula is quite involved and some alternative formulations in more convenient tensor form have been proposed, for example in [MeRe].
For the purpose of the present work we need to point out some properties of biphase fundamental solution. First of all, it is a fundamental solution, in the sense that Γ(x, y), is continuous in and, for any r > 0, where C depends on α 0 , β 0 only. We will also need to use the explicit representation of the some components of biphase fundamental solution. In particular, we will make use of explicit expression of the third column of Γ. For x = (x 1 , x 2 , x 3 ) with x 3 > 0 and y = (0, 0, c) with c > 0, from [R] we have, where It is worth noticing that, for C + = C − , the matrix Γ coincides with the free space fundamental matrix for constant isotropic elasticity tensor.

Singular solutions
With the aid of the biphase fundamental solution Γ, let us construct singular solutions with singularities in some subset of the domain. Since Γ is defined only in the case of a flat interface, we will need to keep away from interfaces that are not flat.
For this reason we need to set the following notation: let F be the union of the flat parts of ∪ N j=0 ∂D j . By flat parts we intend that they can be represented as the graphs of a constant function in at least a ball of radius r0 3 (as Σ k in assumption (A1)). Let D = ∪ N j=0 ∂D j \ F . The set D contains the non flat parts of the interfaces. Let C = N j=0 C j χ Dj with tensors C j satisfying (A2) for all j. Let y ∈ Ω 0 \D and let r = min(r 0 /4, dist(y, D ∪ ∂Ω 0 )). Then, in the sphere B r (y) either C is constant, C = C j or, by a suitable choice of the coordinate system, C = C j + (C j+1 − C j )χ {x3>a} for some j = 0, 1, · · · , N and some a with |a| < r. Let and consider the biphase fundamental solution to div(C y ∇Γ(·, y)) = δ y I 3 in R 3 .

Two useful properties of the biphase fundamental solution.
Let C b , C b be given by where C + , C − and C − are constant and strongly convex isotropic tensors whose Lamé coefficients satisfy (4) Let Γ C b and Γ C b be the biphase fundamental solutions relative to operators div C b ∇· and div C b ∇· , respectively.
Proposition 3.3. Let h, k be real numbers and let H be the fourth order tensor For every l, m ∈ R 3 and every y, z ∈ R 3 + , y = z, we have Proof. Let us fix l, m ∈ R 3 and y, z ∈ R 3 + , y = z. Let t 0 be a positive number such that for every t ∈ (−t 0 , t 0 ) the tensor C b + tH is strongly convex.
Since H(x) = 0 for every x ∈ R 3 + we have trivially Hence, for every t ∈ (−t 0 , t 0 ) \ {0}, By (33) and by Proposition 3.2 we get Now, by straightforward calculation on the biphase fundamental solution given in [R], we have, for every and, for every t ∈ (−t 0 , t 0 ), where C depends on α 0 , β 0 , y, z, l and m only. By (35), (36), and applying the dominated convergence theorem, we get Finally, by (37) and (34) the thesis follows.

Some estimates for solutions to the Lamé system
In this section we collect some properties concerning solutions to the linearized elasticity system with piecewise constant elasticity tensor that will be crucial to proe our main result. First we state a regularity result for solutions of elliptic systems in composite materials from [LN] and [CKVC]. Afterwords we use this result in order to obtain Proposition 3.5, which is then used to prove a quantitative estimate of unique continuation for solutions of systems satisfying assumptions (A1) and (A2).
Proposition 3.4. Let C + and C − be two isotropic elasticity tensors with constant Lamé coeffcients λ, µ and λ ′ , µ ′ respectively, satisfying assumption (A2), let R > 0 and Then, for every multiindex β ′ , D β ′ x′ v ∈ C 0 (B R ) and v ∈ C ∞ (B ± R ). Moreover for any δ > 0 and k ≥ 0 Beside this regularity result, the principal ingredients for proving estimates of unique continuation are the three sphere inequality, some stability estimates for the Cauchy problem and a smallness propagation estimate in a cone. All these results holds for constant elasticity tensors and are precisely described in the Appendix.
The flatness assumption on interfaces allows us to get a better estimate for the Lipschitz constant. The reason is the fact that solutions to the system with piecewise constant elasticity tensor have analytic extension beyond the flat interface. This property is stated in the following Proposition.
Here we use the notation of section 3.2.1 and assumptions (A1) and (A2).
Proposition 3.5. Let v ∈ H 1 loc (K) be a solution to div C ∇v = 0 in K.
Let us fix k ∈ {0, . . . , M }. Then there exist two positive constants C 1 and C, depending only on α 0 , β 0 and L, such that v |D k can be extended by a function v in the set and Proof of Proposition 3.5 It is not restrictive to assume that P k = 0, Σ k belongs to the plane {x 3 = 0} and B ). Let us recall the following Caccioppoli inequality ( [BBFM,pag.20]): let u be a solution to (40) then, for every for 0 < ρ 2 < ρ 1 < R where C depends on α 0 and β 0 only. Denote by We now prove that φ and ψ are analytic in Σ k ∩ B R and we estimate the Starting from (43) and using the same iterative procedure followed to prove inequality (39) in [BF] we obtain for every N 0 ∈ N, where C depends on α 0 and β 0 only. On the other side, by Proposition 3.4 we have Hence, proceeding as in [BF,(41)] and applying Proposition 3.4 to D β ′ x ′ v, we get and where C depends on α 0 and β 0 only. By (45) and (46) and by the Cauchy-Kowalevski theorem we have that the solutionṽ to the Cauchy problem is analytic in the neighborhood Ξ C1 k+1 of Σ k ∩B R . Therefore, taking into account (44),ṽ is the analytic extension of v + in Ξ C1 k+1 and estimate (42) follows.
Finally we state a quantitative estimate of unique continuation.
The proof of the above Proposition is given in the Appendix.

Proof of the main result
This section contains the proof of the main result that consists in showing that the forward map introduced in definition 2.6 satisfies all the assumptions of Proposition 2.5.

Differentiability of F
Proposition 4.1. The map is Lipschitz continuous with Lipschitz constant C F ′ depending on A, L, α 0 , β 0 only.
Proof. Fix L ∈ A and let H ∈ R 2N such that H ∞ is sufficiently small. By (8) we have Hence, by setting, hence Ω C L ∇w : ∇w = Ω H ∇u L+H : ∇w.

By Lax-Milgram Theorem and Korn inequality we get
where C depends on A, L, α 0 , β 0 only. By inserting (51) into (49) we get where C on depends on A, L, α 0 , β 0 only, that yields (48). Let us now prove the Lipschitz continuity of F ′ . Let L 1 , L 2 ∈ A and set By reasoning as we did to derive (52) we obtain , where C F ′ depends only on A, L, α 0 , and β 0 .

Injectivity of F |K and uniform continuity of (F |K ) −1
In the present subsection we will prove Theorem 4.2 whose statement is given below. Let where δ ∈ (0, 1) is as in Proposition 3.6. The function σ is strictly increasing, concave, and lim t→0 σ(t) = 0. We have Theorem 4.2. For every L 1 , L 2 ∈ K the following inequality holds true where σ N (·) is the composition of the function σ(·) defined in (53) with itself N times and C * is a constant depending on A, L, α 0 , β 0 , N only.
Remark 4.3. Observe that Theorem 4.2 provides the injectivity of F |K and an estimate of the modulus of continuity of (F |K ) −1 .
In order to prove Theorem 4.2 we need to prove first some preliminary results. Let j ∈ {1, . . . , N } be such that and let D j1 , . . . , D jM be a chain of domains connecting D 1 to D j . For the sake of brevity set D k = D j k . Consider K, K 0 and K h defined as in Section 3. Let W k = Int(∪ k j=0 D j ), U k = Ω 0 \W k , for k = 1, . . . , M − 1. The tensors C L 1 and C L 2 are extended as in (15) in all of Ω 0 . To simplify the notation we will set C := C L 1 andC := C L 2 . Finally letK k = K h ∩ W k and for y, z ∈K k define the matrix-valued function S k (y, z) := U k (C −C)(·) ∇G(·, y) : ∇Ḡ(·, z), whose entries are given by S (p,q) k (y, z) := U k (C −C)(·) ∇G (p) (·, y) : ∇Ḡ (q) (·, z) p, q = 1, 2, 3 and where G (p) (·, y) andḠ (q) (·, z) denote respectively the p-th column and the q-th columnn of the singular solutions of Proposition 3.1 corresponding to the tensors C andC respectively. From (25) we have that where the constant C depends on the a priori parameters only and d(y) = d(y, U k ), d(z) = d(z, U k ). For any fixed q = 1, 2, 3 let us denote by S (·,q) k (·, z) the vector valued function whose elements are S (p,q) k (y, z), p = 1, 2, 3; analogously we define S (p,·) k (y, ·), for any fixed p = 1, 2, 3. First we prove Proposition 4.4. For all y, z ∈K k we have that S (·,q) k (·, z) and S (p,·) k (y, ·) belong to H 1 loc (K k ) and for any fixed q ∈ {1, 2, 3} and for any fixed p ∈ {1, 2, 3} Proof. For seek of simplicity, in the proof we will omit the index k. Let us fix q ∈ {1, 2, 3} and let us first show that the vector valued function S (·,q) (·, z) ∈ H 1 loc (K) for fixed z ∈K. Let φ ∈ C ∞ 0 (B r (y 0 )) where y 0 ∈K and B r (y 0 ) ⊂K. Consider, for fixed p, q ∈ {1, 2, 3} Observe now that by (25) and by the fact that B r (y 0 ) ⊂K we have Hence, by Schwarz inequality, we have that, for fixed z ∈K, so that we can interchange the order of integration and get and using the symmetry of G almost everywhere in U, (26), we get Now recalling that G(y, x) = w(y, x) + Γ(y, x), by the properties of Γ and by the boundary value problem satisfied by ∂ xj w(y, x) for any j = 1, 2, 3 it is straightforward to see that ∇ y ∂ xj w(y, x) ∈ L 2 (U × B r (y 0 )) and hence Now, arguing as in the first part of the proof and considering now a vectorvalued test function Φ ∈ C ∞ 0 (B r (y 0 )) and by (26) we have for all p = 1, 2, 3 we finally have for all Φ ∈ C ∞ 0 (B r (y 0 )) and since y 0 is arbitrary (55) follows. Analogously we get (56).
Proof. Fix z ∈ K 0 and consider the function v(y) := S (·,q) k (y, z), for fixed q. By Proposition 4.4 we know that v is solution to div(C ∇v(·)) = 0 inK k .

Proof of Theorem 4.2 Observe first that
Then from identity (8), we derive that for every y, z ∈ K 0 and for |l| = |m| = 1.
For seek of simplicity in what follows we will omit the indices k and k + 1 and write µ = µ k , µ ′ = µ k+1 and, in a similar way, define λ, λ ′ , ν, ν ′ . We will bar corresponding Lamé coefficients forC.
be the matrix valued function whose elements are given by Moreover, for any fixed q = 1, 2, 3, let us denote by T (·,q) i (·, z) the vector valued function whose elements are T (p,q) i (·, z), p = 1, 2, 3; analogously we define T (p,·) i (y, ·), for any fixed p = 1, 2, 3. Let us fix l, m unit vectors of R 3 . By (78) we have, for every y, z ∈ K 0 , |T i (y, z) l · m| ≤ Ω H 0 ∇G(·, y) l : ∇G(·, z) m where C depends on α 0 , β 0 , L and A only. Arguing similarly to the proof of Proposition 4.5, we have that there exist C 0 , C 1 such that for every r ∈ (0, r 0 /C 0 ) the following inequality holds true (recall r 1 = ρ1 4L where ρ 1 is defined in (14)) where ς is defined as in (62). Now we have trivially On the other side we have where C depend on A, L, α 0 , β 0 and M only. By the above inequality, (80) and (79) we have where C depend on A, L, α 0 , β 0 and M only. Denote by H the tensor given by From Proposition 3.1 we have, for every c ∈ (1/4, 1/2) and choosing l = m = e 3 , where C depend on A, L, α 0 , β 0 and M only. Now, performing the change of variables x = rξ in the integral on the right hand side of (81) we obtain Therefore, for every ̺ ∈ (0, 1/C 1 ), we have where C depend on A, L, α 0 , β 0 and M only. Now, if ω i (q 0 ) < e −1 /2 then we choose ̺ = 1 2C1 |log ω i (q 0 )| − 1 4δ , otherwise if ω i (q 0 ) ≥ e −1 /2 then we estimate from above the right hand side of (82) trivially. Hence, by Proposition 3.3 we have where σ is defined by (53) and C depends on A, L, α 0 , β 0 and M only. By explicit calculation from (21), denoting by and by for F 1 and F 2 as in (22) and (23), we get Therefore, from (83) and (5) we find easily The first condition of (84) gives hence, by recalling (5) and (10), we have |k 0,i+1 | ≤ Cσ(ω i (q 0 )).
Finally, by iteration we get and the thesis follows.
Remark 4.7. Observe that the above proposition implies that the Frechét derivative F ′ (L) is injective for every L ∈ K and hence point (v) of Proposition 2.5 is completely proved. Therefore we have α0 , σ 2 (·) = C * σ N (·), q 0 = σ N −1 (1/C ⋆ ), δ 1 = 1 2 min{δ 0 , M 2 } and δ 0 = q0 2C F ′ and we recall that C * is the constant that occurs in Theorem 4.2, C F ′ is the Lipschitz constant of F ′ introduced in Proposition 4.1, and C * has been introduced in Proposition 4.6.

Appendix
For the convenience of the reader, we recall here some quantitative estimates of unique continuation. Although such estimates have been proved in the general case where the elasticity tensor is of class C 1,1 , here we give the statements in the special case we are interested in, namely we assume that where λ and µ are real numbers satisfying (4). The following theorem is an immediate consequence of [Al-M, Theorem 5.1] and standard estimate of smallness propagation [Al-Ro-R-Ve, proof of Theorem 1.10] Theorem 5.1 (Three sphere inequality). Let u be a solution to the Lamé system div C ∇u = 0 in Br, for some positive numberr. Then, for every r 1 , r 2 , r 3 , such that 0 < r 1 ≤ r 2 < r 3 ≤r, we have where C and θ 0 , 0 < θ 0 < 1, only depend on α 0 , β 0 , r2 r3 and, increasingly, on r1 r3 .
The following theorem has been proved in [M-R].
Proposition 5.3. Let C be as in (86) and let u be a solution to the Lamé system div C ∇u = 0 in C ρ (γ 3 ).
Iterating the last inequality and taking into account the first inequality in (91) we get , for every k ≥ 2. Now, we choose k = k 0 in the above inequality and notice that where C depends on α 0 , β 0 , γ 1 , γ 2 and γ 3 only. Finally, by (90) and by the estimate (39), where C depends on α 0 and β 0 only and (92) follows.
We finally end this appendix by proving our main result on quantitative estimate of unique continuation for solutions of Lamé system with piecewise constant coefficients. In this proof the elasticity tensor C is of the form (15).
Proof of Proposition 3.6 Denote by C 2 = max 6, 4LC L , 4C 1 , 2r0 h0 and by ρ 2 = r0 C2 , where C L and C 1 are defined in (14) and in Proposition 3.5 respectively and h 0 is defined in (16). Notice that C 2 does not depend on r 0 and that ρ 2 ≤ ρ1 16 . It is not restrictive to assume n 1 = e 3 . Let us denote x 0 = P 1 + n 1 3 16 ρ 1 . We have by (18) Moreover, by Proposition 3.5 we have that the function v |B ′ ρ 2 (P1)×(0, ρ 1 4 ) can be extended analytically to a function v 0 on B ′ ρ2 (P 1 ) × − r0 4C1 , ρ1 where C depends on A, L, α 0 , β 0 and N only. Let us construct a chain of spheres of radius ρ 2 /4 such that the first is B ρ2/4 (x 0 ), all the spheres are externally tangent and the last one is centered at P 1 + ρ2 2 n 1 . We choose such a chain so that the spheres of radius ρ 2 concentric with those of the chain are contained in B ′ ρ2 (P 1 ) × − r0 4C1 , ρ1 4 . The number of spheres of the chain is certainly smaller than a constant m 1 depending on L, α 0 and β 0 only.
By an iterated application of three sphere inequality (87) with r 1 = ρ2 4 , r 2 = K h0/2 ∩ D k+1 ∪ (Ξ C1 k+1 ∩D k ) where h 0 , K h0/2 and Ξ C1 k+1 are defined by (16), (17) and (41) respectively. Let us denote x k = P k − ρ k 32 n k . By the induction hypothesis we have trivially v k L 2 (B ρ k+1 /64 (x k )) ≤ Cε where C depends on A, L, α 0 , β 0 and (increasingly) k only. Let us construct a chain of spheres of radius ρ k+1 /4 · 64 such that the first is B ρ k+1 /4·64 (x k ), all the spheres are externally tangent and the last one is centered at P k+1 + ρ k+1 4·64 n k+1 . We choose such a chain so that the spheres of radius ρ k+1 /64 concentric with those of the chain are contained in K h0/2 ∩ D k . The number of spheres of the chain is certainly smaller than a constant m 2 .