Optimal identification of cavities in the Generalized Plane Stress problem in linear elasticity

For the Generalized Plane Stress (GPS) problem in linear elasticity, we obtain an optimal stability estimate of logarithmic type for the inverse problem of determining smooth cavities inside a thin isotropic cylinder {}from a single boundary measurement of traction and displacement. The result is obtained by reformulating the GPS problem as a Kirchhoff-Love plate-like problem in terms of the Airy's function, and by using the strong unique continuation at the boundary for a Kirchhoff-Love plate operator under homogeneous Dirichlet conditions, which has been recently obtained in \cite{l:arv}.


Introduction
In this paper we consider the inverse problem of detecting cavities inside a thin isotropic elastic plate Ω × − h 2 , h 2 , where the middle plane Ω is a bounded domain in R 2 and h is the constant thickness, subject to a single experiment consisting in applying in-plane boundary loads and measuring the induced displacement at the boundary. Practical applications concern the use of non-destructive techniques for the identification of possible defects, such as cavities, inside the plate.
(1. Here, N = N 1 e 1 + N 2 e 2 is the in-plane load field applied to ∂Ω satisfying the compatibility condition (1.6) ∂Ω N · r = 0, for every r ∈ R 2 , where R 2 is the linear space of infinitesimal two-dimensional rigid displacements. Here, E = E(x) and ν = ν(x) are the Young's modulus and the Poisson's coefficient of the material, respectively. Under suitable strong convexity assumptions on the elastic tensor of the material (see Section 3 for details), and assuming N ∈ H − 1 2 (∂Ω, R 2 ), problem (1.1)-(1.6) admits a unique solution a ∈ H 1 (Ω \ D, R 2 ) satisfying the normalization conditions (1.7) Ω\D a = 0, Ω\D (∇a − ∇ T a) = 0, and such that a H 1 (Ω\D) ≤ C N H − 1 2 (∂Ω,R 2 ) . In this work we face the inverse problem of determining the cavity D from a single pair of Cauchy data {a, N} given on ∂Ω. More precisely, we are interested to obtain quantitative stability estimates, which are useful to control the effect that possible errors on the measurements have on the results of reconstruction procedures. The arbitrariness of the normalization conditions (1.7), which are related to the non-uniqueness of the solution to the direct problem (1.1)-(1.6), leads to the following formulation of the stability issue: given two solutions a (i) ∈ H 1 (Ω, R 2 ), i = 1, 2, to the direct problem (1.1)-(1.6) with D = D i , satisfying, for some ε > 0, to control the Hausdorff distance d H (D 1 , D 2 ) in terms of ε when ε goes to zero, where Σ is an open subset of ∂Ω.
Assuming D ∈ C 6,α , 0 < α ≤ 1, we prove where C > 0 and η > 0 are constants only depending on the a priori data. We refer to Theorem 3.1 for a precise statement. Let us notice that, in view of the counterexamples obtained in the simpler context of electrical conductivity (see, for instance, [Al], [Ma], [DiC-R]), we can infer the optimality of the stability estimate (1.9). The general scheme of our proof is inspired to the seminal paper [Al-Be-Ro-Ve], which established the first optimal logarithmic estimate for the determination of unknown boundaries in electrostatics. The key tool in [Al-Be-Ro-Ve] was, among others, the polynomial vanishing rate for solutions to the second order elliptic equation of electrostatics, satisfying either homogeneous Dirichlet or homogeneous Neumann boundary conditions, ensured by a doubling inequality at the boundary established in [A-E]. Aiming at obtaining a strong unique continuation property at the boundary (SUCB) for solutions to the GPS elliptic system, in this paper we have exploited the two dimensional character of the problem (1.1)-(1.6) by using the classical Airy's transformation, which (locally) reduces the GPS system with homogeneous Neumann boundary conditions to a scalar fourth order Kirchhoff-Love plate's equation under homogeneous Dirichlet boundary conditions. This reformulation allows us to use the finite vanishing rate at the boundary for homogeneous Dirichlet boundary conditions recently obtained in [A-R-V] in the form of a three spheres inequality at the boundary with optimal exponent, and in [M-R-V3] in the form of a doubling inequality at the boundary.
It is worth noticing that the present approach, here applied to the GPS problem, allows also to cover the analogous inverse problem of detecting cavities in a two-dimensional elastic body made by inhomogeneous Lamé material, thus improving the log − log stability result previously obtained in [M-R]. An optimal log-type estimate in dimension three remains a challenging open problem. Let us mention that the Airy's transformation has been used in [L-U-W] to prove global identifiability of the viscosity in an incompressible fluid governed by the Stokes and the Navier-Stokes equations in the plane by using boundary measurements.
The paper is organized as follows. Notation is presented in Section 2. Section 3 contains the formulation of the inverse problem and the statement of our stability result. The Airy's transformation is illustrated in Section 4. The proof of the main result, given in Section 5, is based on a series of auxiliary propositions concerning Lipschitz propagation of smallness (Proposition 5.1), finite vanishing rate in the interior (Proposition 5.2), finite vanishing rate at the boundary (Proposition 5.3), stability estimate from Cauchy data (Proposition 5.4). Finally, for the sake of completeness, in Section 6 we recall a derivation of the GPS problem from the corresponding three dimensional elasticity problem for a thin plate subject to in-plane boundary loads.

Notation
Let P = (x 1 (P ), x 2 (P )) be a point of R 2 . We shall denote by B r (P ) the disk in R 2 of radius r and center P and by R a,b (P ) the rectangle of center P and sides parallel to the coordinate axes, of length 2a and 2b, namely Definition 2.1. (C k,α regularity) Let Ω be a bounded domain in R 2 . Given k, α, with k ∈ N, 0 < α ≤ 1, we say that a portion S of ∂Ω is of class C k,α with constants r 0 , M 0 > 0, if, for any P ∈ S, there exists a rigid transformation of coordinates under which we have P = 0 and We use the convention to normalize all norms in such a way that their terms are dimensionally homogeneous and coincide with the standard definition when the dimensional parameter equals one. For instance, , and so on for boundary and trace norms. Given a bounded domain Ω in R 2 such that ∂Ω is of class C k,α , 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 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 ψ(s) = (x 1 (s), x 2 (s)), s ∈ [0, l(C)], such that ψ(0) = P 0 and ψ ′ (s) = τ (ψ(s)). Here l(C) denotes the length of C.
Throughout the paper, we denote by w, α , α = 1, 2, w, s , and w, n the derivatives of a function w with respect to the x α variable, to the arclength s and to the normal direction n, respectively, and similarly for higher order derivatives.
We denote by M n the space of n × n real valued matrices and by L(X, Y ) the space of bounded linear operators between Banach spaces X and Y .
Given A, B ∈ M n and K ∈ L(M n , M n ), we use the following notation: We denote by I n the n × n identity matrix, and by tr(A) the trace of A. When n = 2, we replace the Latin indexes with Greek ones. The linear space of the infinitesimal rigid displacements, for n = 2, 3, is defined as 3 Inverse problem and main result i) A priori information on the geometry.
Let Ω be a bounded domain in R 2 and let us assume that the cavity D is an open subset compactly contained in Ω, such that Moreover, let us assume that, given positive numbers (3.4) ∂Ω is of class C 1,α with constants r 0 , M 0 , (3.5) ∂D is of class C 6,α with constants r 0 , M 0 , with α such that 0 < α ≤ 1. Let us denote by Σ the open portion of ∂Ω where measurements are taken. We assume that there exists P 0 ∈ Σ such that and (3.7) Σ is of class C 2,α with constants r 0 , M 0 .
Let us notice that, without loss of generality, we have chosen M 0 ≥ 1 2 to ensure that B r 0 (P ) ⊂ R r 0 ,2M 0 r 0 (P ) for every P ∈ ∂Ω.
ii) A priori information on the Neumann boundary data.
We assume that and that, for a given constant F > 0, iii) A priori information on the elasticity tensor.
The constitutive equation (1.4) can be written as where the elasticity tensor C = (C αβγδ ) is defined as for every 2 × 2 matrix A, where the Young's modulus E and the Poisson's coefficient ν are given in terms of the Lamé moduli as follows (3.14) .
On the Lamé coefficients µ = µ(x), λ = λ(x), µ : Ω → R, λ : Ω → R, we assume for positive constants α 0 and γ 0 . The above assumptions ensure that C satisfies the minor and major symmetries (3.16) C αβγδ = C βαγδ = C αβδγ , C αβγδ = C γδαβ , for every α, β, γ, δ = 1, 2, in Ω, and that it is strongly convex in Ω, precisely for some positive constant Λ 0 . We note that the equilibrium problem (1.1)-(1.5) can be written in compact form as The weak formulation of (3.19)-(3.21) consists in finding a = a( Under our assumptions, there exists a unique solution to (3.22) up to addition of a rigid displacement. In order to select a single solution, we shall assume the normalization conditions which imply the following stability estimate for the direct problem (3.19)-(3.21) In what follows, we shall refer to the set of constants h, α 0 , γ 0 , Λ 0 , α, M 0 , M 1 and F as the a priori data.
Remark 3.2. Let us notice that, as it will be clear from the proof, the above stability result holds true also when the domain Ω contains a finite number of connected cavities D (j) , j = 1, . . . , J, such that ∂D (j) ∈ C 6,α with constants r 0 , M 0 , and dist(∂D (j) , ∂D (k) ) ≥ r 0 , for j = k.

Airy's transformation
It is known that the boundary value problem in plane linear elasticity can be formulated in terms of an equivalent Kirchhoff-Love plate-like problem involving a scalar-valued function called Airy's function. Although this argument is well established, see, for instance, [G] and [Fic], for reader convenience in what follows we recall the essential points of the analysis. For the sake of completeness, we consider a mixed boundary value problem, in order to describe the transformation of both Dirichlet and Neumann boundary conditions. Let a = a 1 e 1 + a 2 e 2 , a ∈ H 1 (U, R 2 ), be the solution to the GPS problem where N ∈ H −1/2 (∂ t U, R 2 ) and a ∈ H 1/2 (∂ u U, R 2 ) are given Neumann and Dirichlet data, respectively. Here, The equilibrium equations (4.1), and the simply connectness of U, ensure the existence of a single-valued function where the matrix e αγ is defined as follows: e 11 = e 22 = 0, e 12 = 1, e 21 = −1; see [Ai]. We recall that, by construction, the function ϕ and its first partial derivatives ϕ ,1 , ϕ ,2 are uniquely determined up to an additive arbitrary constant.
We now consider the transformation of the Neumann boundary condition (4.2) on ∂ t U. By (4.6), the condition on ∂ t U can be written as (4.16) e αγ e βδ ϕ ,γδ n β = N α , that is, recalling that τ δ = e βδ n β on ∂U, where s is an arc length parametrization on ∂U. By integrating the above equations with respect to s, from P 0 ∈ ∂ t U to P ∈ ∂ t U, with s(P 0 ) = 0 and s(P ) = s, the gradient of ϕ on ∂ t U can be determined up to an additive constant vector c = c 1 e 1 + c 2 e 2 , namely where g(s) = g 1 (s)e 1 + g 2 (s)e 2 , g 1 (s) = − s 0 N 2 (ξ)dξ, g 2 (s) = s 0 N 1 (ξ)dξ. It follows that the normal derivative of ϕ on ∂ t U is prescribed in terms of the Neumann data N , that is, whereas, integrating once more (4.18) from P 0 to P , we have where C = ϕ(0) =constant, and G(s) = s 0 (c + g(ξ)) · τ (ξ)dξ. We notice that it is always possible to select the two arbitrary constants occurring in the construction of ∇ϕ such that c 1 = c 2 = 0 (see, for example, [S] for details). In particular, if the Neumann data N vanishes on ∂ t U, then we can also choose the third constant C = 0, so that ϕ(s) = 0 on ∂ t U. In this case, the homogeneous Neumann boundary conditions for the GPS problem are transformed into the homogeneous Dirichlet boundary conditions for the Airy's function: The determination of the boundary conditions satisfied by ϕ on ∂ u U is less obvious, since the corresponding boundary conditions in the original twodimensional elasticity problem are not explicitly expressed in terms of the Airy's function or its derivatives. In dealing with this boundary condition, we need to assume C 1,1 -regularity for ∂U. We adopt a variational-like approach.
Without loss of generality, we can assume ∂ u U = ∂U.
Let ϕ, ϕ : U → R, be a C ∞ -test function, and define the associated Airy stress field Multiplying (4.23) by the displacement field a = a 1 e 1 + a 2 e 2 solution to (4.1)-(4.5), and integrating by parts, we obtain We first work on the integral of the left hand side of (4.24). After two integrations by parts, we obtain We elaborate the second integral I on the right hand side of the above equation in terms of the local coordinates. Recalling that τ α = e βα n β on ∂U and ϕ ,α = n α ϕ ,n + τ α ϕ ,s on ∂U, α, β = 1, 2, we have where, to simplify the notation, we have introduced on ∂U the two functions (4.27) K nn = K γδ n δ n γ , K nτ = K γδ n δ τ γ (= K τ n ).
Therefore, the left hand side of (4.24) takes the form We next elaborate the integral appearing on the right hand side of (4.24). Let us introduce the boundary displacement functions associated to the Dirichlet data a: (4.30) U γ = e αγ a α , on ∂U.
Passing to local coordinates, after an integration by parts, we have Expressing again ∇ ϕ in terms of local coordinates, and integrating by parts, by the regularity of ∂U we obtain (4.32) Finally, by rewriting (4.24) using (4.29) and (4.32), the strain functions K γδ satisfy the condition (4.33) for every ϕ ∈ C ∞ (U). By the arbitrariness of the test function ϕ, and of the traces of ϕ and ϕ ,n on ∂U, we determine the conditions satisfied by K γδ , namely, the field equation (4.34) K γδ,γδ = 0, in U, which coincides with (4.14), and the two boundary conditions (4.35) K nn = − U γ,s n γ , on ∂U, (4.36) K γδ,δ n γ + K τ n,s = −(τ γ U γ,s ) ,s , on ∂U.
The above equations (4.34) and (4.35), (4.36) are known as compatibility field equation and compatibility boundary conditions for the strain functions K γδ , respectively. In conclusion, under the assumption N = 0 on ∂ t U, the twodimensional elasticity problem (4.1)-(4.5) can be formulated in terms of the Airy's function as follows: There is an important analogy connected with the above boundary value problem. Equations (4.37)-(4.42) describe the conditions satisfied by the transversal displacement ϕ = ϕ(x 1 , x 2 ) of the middle surface U of a Kirchhoff-Love thin elastic plate made by isotropic material. The plate is clamped on ∂ t U, and subject to a couple field M = M τ n + M n τ assigned on ∂ u U, with M n = − U γ,s n γ and M τ = τ γ U γ,s , see, for example, [M-R-V1]. Within this analogy, the strain functions K αβ = K αβ (x 1 , x 2 ) play the role of the bending moments (for α = β) and the twisting moments (for α = β) of the plate at (x 1 , x 2 ) ∈ Ω (per unit length), and the bending stiffness of the plate is equal to (Eh) −1 . Let us observe that the geometry of the inverse problem here considered, that is U = Ω \ D does not ensure the existence of a globally defined Airy's function, since the hypotheses of simple connectedness is missing. For this reason, in the following Section 5 we shall make use of local Airy's functions, defined either in interior discs (see the proof of Proposition 5.2) or in neighbourhoods of the boundary of the cavity (see the proof of Proposition 5.3).

Proof of the main result
where τ 0 ≥ 1 only depends on α 0 , γ 0 , Λ 0 , M 0 , M 1 , r 0 r and F . Proof. We can introduce in Br(x) a locally defined Airy's function ϕ associated to the solution a. The proof follows by adapting the arguments in the proof of the analogous Proposition 3.5 in [M-R-V2] which applies to Kirchhoff-Love plate equation. The main difference consists in estimating the L 2 norms of ϕ and |∇ϕ| appearing in (3.21) of [M-R-V2] in terms of the L 2 norm of |∇ 2 ϕ| and using (4.43), the stability estimate (3.24) and Proposition 5.1.
Let us notice that, under our assumptions, the fourth order tensor L satisfies the strong convexity condition for every 2 × 2 symmetric matrix A. We also notice that the coefficients of the terms involving second and third-order derivatives of ϕ in (5.5) are of class C 2 and C 3 in R r 0 ,2M 0 r 0 (x) ∩ Ω \ D, respectively, with corresponding C 2 and C 3 -norm bounded by a constant only depending on h, α 0 , γ 0 and Λ 0 . Therefore, we can apply the results obtained in [A-R-V]. Precisely, by Corollary 2.3 in [A-R-V], there exist c < 1, only depending on M 0 and α, and C > 1, only depending on α 0 , γ 0 , Λ 0 , M 0 and α, such that, for every r 1 < r 2 < cr 0 , we have (5.8) where B > 1 is given by Br 2 (x)∩(Ω\D) ϕ 2 . Let us choose r 2 = c 0 r 0 , with c 0 = c 2 . We need to estimate the quantity B. By (5.10)-(5.11), using the stability estimate of the direct problem (3.24) and Proposition 5.1, we can estimate B ≤ C, with C only depending on α 0 , γ 0 , Λ 0 , M 0 , α, M 1 and F . By using again Poincaré inequality, Lemma 4.7 in [A-R-V] and (4.43), we obtain the thesis.
From now on, we shall denote by G the connected component of Ω \ (D 1 ∪ D 2 ) such that Σ ⊂ ∂G.
Proof. The proof can be easily obtained by adapting the proof of the analogous estimates contained in Proposition 3.5 and Proposition 3.6 in [M-R].
The only difference consists in replacing the auxiliary function w = a (1) −a (2) with w = a (1) − a (2) − r, where r ∈ R 2 is the minimizer of problem (3.25), and noticing that ∇r = 0.
Proof of Theorem 3.1. It is convenient to introduce the following auxiliary distances: Let η > 0 such that Step 1. Let us assume η ≤ r 2 0 N 2 H −1/2 (∂Ω,R 2 ) . We have where τ has been introduced in Proposition 5.3 and C is a positive constant only depending on the a priori data.
Proof. Without loss of generality, let x 0 ∈ ∂D 1 such that and then, by (5.18), Let us distinguish two cases. First, let where c 0 is the positive constant appearing in Proposition 5.3. By applying this proposition, we have where C > 0 is a positive constant only depending on α 0 , γ 0 , Λ 0 , α, M 0 , M 1 and F . By Proposition 5.1, we have where C > 0 is a positive constant only depending on α 0 , γ 0 , Λ 0 , α, M 0 , M 1 , F , from which we can estimate d m , obtaining (5.19).
Since d > 0, we have y 0 ∈ D 2 \ D 1 . Let There are three cases to consider: Here the number d 0 , 0 < d 0 < r 0 , is such that d 0 r 0 only depends on M 0 , and it is the same constant appearing in Proposition 5.4. In particular, Proposition 3.6 in [Al-Be-Ro-Ve] shows that there exists an absolute constant where C > 0 only depends on α 0 , γ 0 , Λ 0 , M 0 , M 1 and F . For η small enough, where C > 0 only depends on α 0 , γ 0 , Λ 0 , M 0 , M 1 and F . Collecting the three cases, the thesis follows.
Step 3. We have Proof. The proof is based on purely geometrical arguments, we refer to [M-R-V2, Proof of Theorem 3.1, Step 3].

Generalized Plane Stress problem
In this section we derive the Generalized Plane Stress (GPS) problem for the statical equilibrium of a thin elastic plate under in-plane boundary loads. Our analysis follows the classical approach of the theory of structures, according to the original idea introduced by Filon [Fil]. Alternative, more formal derivations have been proposed to justify the GPS problem. The interested reader can refer, among others, to the contributions [C-D], [A-B-P] and [P]. Let U be a bounded domain in R 2 , and consider the cylinder C = U × − h 2 , h 2 with middle plane U ×{x 3 = 0} (which we will simply denote by U in what follows) and thickness h. Here, {O, x 1 , x 2 , x 3 } is a Cartesian coordinate system, with origin O belonging to the plane x 3 = 0 and axis x 3 orthogonal to U. Such cylinder is called plate if h is small with respect to the linear dimensions of U, e.g., h << diam(U).