Estimating area of inclusions in anisotropic plates from boundary data

We consider the inverse problem of determining the possible presence of an inclusion in a thin plate by boundary measurements. The plate is made by non-homogeneous linearly elastic material belonging to a general class of anisotropy. The inclusion is made by different elastic material. Under some a priori assumptions on the unknown inclusion, we prove constructive upper and lower estimates of the area of the unknown defect in terms of an easily expressed quantity related to work, which is given in terms of measurements of a couple field applied at the boundary and of the induced transversal displacement and its normal derivative taken at the boundary of the plate.


Introduction
In this paper we consider an inverse problem in linear elasticity consisting in the identification of an inclusion in a thin plate by boundary measurements.
Let Ω denote the middle plane of the plate and let h be its constant thickness. The inclusion D is modelled as a plane subdomain compactly contained in Ω.
Suppose we make the following diagnostic test. We take a reference plate, i.e. a plate without inclusion, and we deform it by applying a couple fieldM at its boundary. Let W 0 be the work exerted in deforming the specimen. Now, we repeat the same experiment on a possibly defective plate. The exerted work generally changes and assumes, say, the value W . In this paper we want to find constructive estimates, from above and from below, of the area of the unknown inclusion D in terms of the difference |W − W 0 |.
From the mathematical point of view, see [9], [10] the infinitesimal deformation of the defective plate is governed by the fourth order Neumann boundary value problem div (div ((χ Ω\D P + χ D P)∇ 2 w)) = 0, in Ω, (1.1) (P∇ 2 w)n · n = − M n , on ∂Ω, (1.2) div (P∇ 2 w) · n + ((P∇ 2 w)n · τ ), s = ( M τ ), s , on ∂Ω, (1.3) where w is the transversal displacement of the plate and M τ , M n are the twisting and bending components of the assigned couple field M , respectively. In the above equations χ D denotes the characteristic function of D and n, τ are the unit outer normal and the unit tangent vector to ∂Ω, respectively. The plate tensors P, P are given by P = h 3 12 C, P = h 3 12 C, (1.4) where C is the elasticity tensor describing the response of the material in the reference plate Ω, whereas C denotes the (unknown) corresponding tensor for the inclusion D. The work exerted by the couple field M has the expression W = − ∂Ω M τ,s w + M n w, n . (1.5) When the inclusion D is absent, the equilibrium problem (1.1)-(1.3) becomes div (div (P∇ 2 w 0 )) = 0, in Ω, (1.6) (P∇ 2 w 0 )n · n = − M n , on ∂Ω, (1.7) div (P∇ 2 w 0 ) · n + ((P∇ 2 w 0 )n · τ ), s = ( M τ ), s , on ∂Ω, (1.8) where w 0 is the transversal displacement of the reference plate. The corresponding external work exerted by M is given by Our main result (see Theorem 3.2) states that if, for a given h 1 > 0, the following fatness-condition area ({x ∈ D| dist{x, ∂D} > h 1 }) ≥ 1 2 area(D) (1.10) holds, then where the constants C 1 , C 2 only depend on the a priori data. Estimates (1.11) are established under some suitable ellipticity and regularity assumptions on the plate tensor C and on the jump C − C.
Analogous bounds in plate theory were obtained in [15] and [16] and recently in the context of shallow shells in [8]. The reader is referred to [12], [5], [7] for size estimates of inclusions in the context of the electrical impedance tomography and to [11], [2], [3], [4] for corresponding problems in two and three-dimensional linear elasticity. See also [13] for an application of the size estimates approach in thermography. However, differently from [15] and [16], here we work under very general assumptions on the constitutive properties of the reference plate, which is assumed to be made by nonhomogeneous anisotropic elastic material satisfying the dichotomy condition (3.9a)-(3.9b) only. This choice introduces significant difficulties in obtaining the upper bound for area(D), as we shall discuss shortly.
The first step of the proof of area estimates (1.11) consists in proving that the strain energy of the reference plate stored in the set D is comparable with the difference between the works exerted by the boundary couple fields in deforming the plate with and without the inclusion. More precisely, we have the following double inequality for suitable constants K 1 , K 2 only depending on the a priori data (see Lemma 4.1). The proof of these bounds is based on variational considerations and has been obtained in [15] (Lemma 5.1). The lower bound for area(D) follows from the right hand side of (1.12) and from regularity estimates for solutions to the fourth order elliptic equation (1.6) governing the equilibrium problem in the anisotropic case.
In order to obtain the upper bound for area(D) from the left hand side of (1.12), the next issue is to estimate from below D |∇ 2 w 0 | 2 . This task is rather technical and involves quantitative estimates of unique continuation in the form of three spheres inequalities for the hessian ∇ 2 w 0 of the reference solution w 0 to equation (1.6). It is exactly to this point that the dichotomy condition (3.9a)-(3.9b) on the tensor C is needed. More precisely, it was shown in [18] that if C satisfies the dichotomy condition, then the plate operator of equation (1.6) can be written as the sum of a product of two second order uniformly elliptic operators with regular coefficients and a third order operator with bounded coefficients. Then, Carleman estimates can be developed to derive a three spheres inequality for ∇ 2 w 0 (see Theorem 6.2 of [18]). The reader is referred to the paper [18] for the necessary background.
The paper is organized as follows. Some basic notation is introduced in Section 2. In Section 3 we state the main result, Theorem 3.2, which is proved in Section 4. Section 5 is devoted to the proof of the Lipschitz propagation of smallness property (see Proposition 4.2), which is used in the proof of Theorem 3.2.

Notation
We shall denote by B r (P ) the disc in R 2 of radius r and center P . When representing locally a boundary as a graph, we use the following notation. For every x ∈ R 2 we set x = (x 1 , x 2 ), where x 1 , x 2 ∈ R.
Definition 2.1. (C k,1 regularity) Let Ω be a bounded domain in R 2 . Given k, with k ∈ N, we say that a portion S of ∂Ω is of class C k,1 with constants ρ 0 , M 0 > 0, if, for any P ∈ S, there exists a rigid transformation of coordinates under which we have P = 0 and where ψ is a C k,1 function on (−ρ 0 , ρ 0 ) satisfying ψ(0) = 0, ∇ψ(0) = 0, when k ≥ 1, When k = 0, we also say that S is of Lipschitz class with constants ρ 0 , M 0 .
Remark 2.2. We use the convention to normalize all norms in such a way that their terms are dimensionally homogeneous with their argument and coincide with the standard definition when the dimensional parameter equals one. For instance, given a function u : Ω → R, where ∂Ω satisfies Definition 2.1, we denote Given a bounded domain Ω in R 2 such that ∂Ω is of class C k,1 , with k ≥ 1, we consider as positive the orientation of the boundary induced by the outer unit normal n in the following sense. Given a point P ∈ ∂Ω, let us denote by τ = τ (P ) the unit tangent at the boundary in P obtained by applying to n a counterclockwise rotation of angle π 2 , that is τ = e 3 × n, (2.2) where × denotes the vector product in R 3 and {e 1 , e 2 , e 3 } is the canonical basis in R 3 . Given any connected component C of ∂Ω and fixed a point P 0 ∈ C, let us define as positive the orientation of C associated to an arclength parameterization ϕ(s) = (x 1 (s), x 2 (s)), s ∈ [0, l(C)], such that ϕ(0) = P 0 and ϕ ′ (s) = τ (ϕ(s)). Here l(C) denotes the length of C.
Throughout the paper, we denote by w, i , w, s , and w, n the derivatives of a function w with respect to the x i variable, to the arclength s and to the normal direction n, respectively, and similarly for higher order derivatives.
We denote by M 2 the space of 2 × 2 real valued matrices and by L(X, Y ) the space of bounded linear operators between Banach spaces X and Y .
For every pair of real 2-vectors a and b, we denote by a·b the scalar product of a and b. For every 2 × 2 matrices A, B and for every L ∈ L(M 2 , M 2 ), we use the following notation: where, here and in the sequel, summation over repeated indexes is implied. Moreover we say that L ≤ L, (2.5) if and only if, for every 2 × 2 symmetric matrix A,

The main result
Let us consider a thin plate Ω × [− h 2 , h 2 ] with middle surface represented by a bounded domain Ω in R 2 and having uniform thickness h, h << diam(Ω). We assume that ∂Ω is of class C 1,1 with constants ρ 0 , M 0 and that, for a given positive number M 1 , satisfies We shall assume throughout that the elasticity tensor C of the reference plate is known and has cartesian components C ijkl which satisfy the following symmetry conditions On the elasticity tensor C let us make the following assumptions: i) Ellipticity (strong convexity) There exists a positive constant γ such that for every 2 × 2 symmetric matrix A.
ii) C 1,1 regularity There exists M > 0 such that Condition (3.2) implies that instead of 16 coefficients we actually deal with 6 coefficients and we denote Let S(x) be the following 7 × 7 matrix and On the elasticity tensor C we make the following additional assumption: iii) Dichotomy condition where D(x) is defined by (3.8).
Remark 3.1. Whenever (3.9a) holds we denote We emphasize that, in all the following statements, whenever a constant is said to depend on µ (among other quantities) it is understood that such dependence occurs only when (3.9a) holds.
for some positive constant d 0 .
Concerning the material forming the inclusion, we assume that the corresponding elasticity tensor C = C(x) belongs to L ∞ (Ω, L(M 2 , M 2 )) and has Cartesian components which satisfy the symmetry conditions C ijkl (x) = C klij (x) = C lkij (x), i, j, k, l = 1, 2, a.e. in Ω. (3.12) Moreover, we assume the following jump conditions on C: either there exist η 0 > 0 and η 1 > 1 such that in Ω, (3.13) or there exist η 0 > 0 and 0 < η 1 < 1 such that in Ω. (3.14) Let us assume that the body forces inside the plate are absent and that a couple fieldM is acting on the boundary of Ω. We shall assume: where Γ is an open subarc of ∂Ω, such that for some positive constant δ 0 . Moreover, we obviously assume the compatibility conditions on the boundary couple field M ∂Ω M α = 0, α = 1, 2, (3.18) and that, for a given constant F > 0, Let us notice that, following a standard convention in the theory of plates, we represent the boundary couple field M in cartesian coordinates as on ∂Ω.
We are now in position to state the main result of this paper.

Theorem 3.2.
Let Ω be a bounded domain in R 2 , such that ∂Ω is of class C 2,1 with constants ρ 0 , M 0 and satisfying (3.1). Let D be a measurable subset of Ω satisfying (3.11) and If, conversely, (3.14) holds, then we have only depend on the same quantities and also on δ 0 , h 1 and F .
If, instead, (3.14) holds, then we have The proof of the above lemma is given in [15], Lemma 5.1.
Proof of Theorem 3.2. By the hypotheses made on P, the inequality (4.1) is satisfied with ξ 0 = γ h 3 12 , ξ 1 = h 3 6 M, so that Lemma 4.1 can be applied. By standard interior regularity estimates (see, for instance, Theorem 8.3 in [15]) and by the Sobolev embedding theorem, we have with C only depending on γ, h, M and d 0 . From (4.5), Poincaré inequality, (4.1), (3.24), we have where the constant C only depends on γ, h, M, d 0 , M 0 and M 1 . The lower bound for |D| in (3.26), (3.27) follows from the right hand side of (4.2), (4.3) and from (4.6).
where s is as in Proposition 4.2. Let us cover D h 1 ρ 0 with internally non overlapping closed squares Q l of side ǫρ 0 , for l = 1, ..., L. By the choice of ǫ the squares Q l are contained in D. Letl be such that Letx be the center of Ql. From (4.1), (4.7), estimate (4.4) with ρ = ǫ 2 ρ 0 , from (3.24) and by our hypothesis (3.25) we have
A proof of the above proposition can be easily obtained by Theorem 6.5 in [18].
In order to prove Proposition 4.2, we need the estimate stated in the following Lemma (for the proof see [15], Lemma 7.1).

Lemma 5.2.
Let Ω be a bounded domain in R 2 , such that ∂Ω is of class C 2,1 with constants ρ 0 , M 0 . Let the fourth order tensor P be defined by (1.

(5.25)
By an iterated application of the three spheres inequality (5.2) over the discs of center x i and radii r k(ρ) , 3r k(ρ) , 4r k(ρ) , we obtain where C > 1 only depends on γ, M and µ.