Detecting an Inclusion in an Elastic Body by Boundary Measurements

SIAM REVIEW c© 2004 Society for Industrial and Applied MathematicsVol. 46, No. 3, pp. 477–498

Detecting an Inclusionin an Elastic Body byBoundary Measurements∗

Giovanni Alessandrini†

Antonino Morassi‡

Edi Rosset†

Abstract. We consider the problem of determining, within an elastic isotropic body Ω, the possiblepresence of an inclusion D made of different elastic material from boundary measurementsof traction and displacement. We prove that the volume (size) of D can be estimated,from above and below, by an easily expressed quantity related to work depending only onthe boundary traction and displacement.

Key words. inverse boundary problem, elasticity, size estimates, strong unique continuation

AMS subject classifications. Primary, 35R30; Secondary, 35R25, 73C02

DOI. 10.1137/S0036144504442098

1. Introduction. In this paper we address the following problem of nondestruc-tive testing: To determine, within an elastic body Ω, the possible presence of an inclu-sion D made of a different elastic material (i.e., harder or softer) from measurementsof traction and displacement taken at the exterior boundary of Ω.

In mathematical terms we are dealing with an inverse problem associated with aboundary value problem for a system of partial differential equations.

The direct boundary value problem under consideration is composed of the systemof linearized elasticity,

(1.1) div ((χΩ\DC+ χDC)∇u) = 0 in Ω,

coupled with the Neumann-type boundary condition

(1.2) (C∇u)ν = ϕ on ∂Ω,

where the given vector field ϕ represents the boundary traction forces.Here D is a set strictly contained in Ω; C and C denote the elasticity tensor fields

in Ω \D and in D, respectively; ν is the unit exterior normal to ∂Ω; and χE denotesthe characteristic function of a set E.

∗Published electronically July 30, 2004. This paper originally appeared in SIAM Journal onMathematical Analysis, Volume 33, Number 6, 2002, pages 1247–1268. The current version has beenrevised and updated, and was supported in part by MIUR grants MM01111258, 2002013279, and2003082352 and by IAC-CNR.

http://www.siam.org/journals/sirev/46-3/44209.html†Dipartimento di Matematica e Informatica, Universita degli Studi di Trieste, Trieste, Italy

([email protected], [email protected]).‡Dipartimento di Georisorse e Territorio, Universita degli Studi di Udine, Udine, Italy (antonino.

[email protected]).

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478 G. ALESSANDRINI, A. MORASSI, AND E. ROSSET

In this direct problem, D is assumed to be known, and the unknown is the dis-placement field u in Ω.

In the inverse problem we are interested in, on the other hand, we consider D tobe unknown and want to determine it by measuring the displacement at the boundary,

(1.3) u = g on ∂Ω,

for a given, nontrivial choice of the boundary traction ϕ.This appears to be an extremely difficult problem. A similar problem in elec-

trical impedance tomography (for which the direct problem involves a single scalarelliptic partial differential equation, rather than a system) has received a great dealof attention in recent years. In this case, the direct problem is governed by the scalarNeumann boundary value problem

(1.4) div ((χΩ\D + 2χD)∇u) = 0 in Ω,

(1.5) ∇u · ν = ϕ on ∂Ω,

which models electrical conduction in a body Ω of uniform conductivity σ ≡ 1 andwhich contains an inclusion whose conductivity is σ ≡ 2, u being here the voltagepotential and ϕ the prescribed boundary current density. Then the inverse problemconsists of determining D when, for a given nontrivial ϕ, the boundary voltage u|∂Ω =g is measured. The uniqueness issue remains a challenging open problem, althoughin the last two decades numerous and interesting partial results have been obtained.See, for instance, Friedman [Fr87], Friedman and Gustafsson [FrG87], Friedman andIsakov [FrI89], Alessandrini, Isakov, and Powell [AIP95], and Alessandrini and Isakov[AI96]; see also [Is98], [Al99] for more extended accounts on the results and morecomplete reference lists. Let us also recall the uniqueness result of Isakov [Is88] whenthe boundary measurement u|∂Ω is available for all possible current densities ϕ, andalso that, still in this more favorable setting, it has been shown by examples thatthe problem remains severely ill-posed ; see Di Cristo and Rondi [DCR03]. See alsoAlessandrini and Di Cristo [ADC04] for a positive result in this direction.

Here, following the line of research initiated in Alessandrini and Rosset [AR98],Kang, Seo, and Sheen [KSS97], Alessandrini, Rosset, and Seo [ARS00], and Alessan-drini and Rosset [AR04] in the electrostatic setting, and in Ikehata [Ik98] for elasticity,we pose a relatively modest but realistic goal: Can we estimate the size (i.e., volume)of the unknown inclusion D from one set of boundary measurements of traction anddisplacement?

In the present paper we restrict our attention to the Lame system of linearizedelasticity, corresponding to the system (1.1) when the material is isotropic, and weconsider the case when the elasticity tensor C within D is finite and nondegenerate.Let us observe, however, that also the extreme cases, when D is totally rigid orrepresents a cavity, are of great interest, and size estimates for such cases have indeedbeen obtained in [MR02]. We refer also to [AMR03] for an overview of all such results,in both the electrical and elastic settings.

In order to illustrate our main results it is convenient to consider the solution u0to the Neumann problem (1.1)–(1.2) when D is the empty set, that is, the solutionu0 to the boundary value problem

(1.6) div (C∇u0) = 0 in Ω,

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DETECTING AN INCLUSION IN AN ELASTIC BODY 479

(1.7) (C∇u0)ν = ϕ on ∂Ω.

Also we introduce the quantities

(1.8) W =∫∂Ωg · ϕ,

(1.9) W0 =∫∂Ωg0 · ϕ,

where g0 = u0|∂Ω . Note that such numbers W , W0 represent the work exerted by thesurface forces ϕ when the inclusion D is, respectively, present or absent.

In Theorem 2.3 we find that for a suitable p > 1, we have

(1.10) C1

∣∣∣∣W −W0

W0

∣∣∣∣ |D| C2

∣∣∣∣W −W0

W0

∣∣∣∣ 1p

.

(See section 2 below for the precise statement.)We believe that these estimates should be useful in practice as a decision tool

in quality control tests. Namely, one can fix experimentally a threshold parameterT > 0 in such a way as to say that D is absent or negligible if |W−W0

W0| ≤ T , whereas

D is significant if |W−W0W0

| ≥ T .The main underlying idea in these estimates is that the integral

(1.11)∫D

|∇u0|2 is comparable to |W −W0|,

where ∇u0 = 12 (∇u0 + (∇u0)T ) is the strain tensor field; see Ikehata [Ik98] and also

Lemma 5.2 below.The next point is to control the above integral in terms of the measure (volume)

of D. On the one hand, this task involves upper bounds on |∇u0|2, which are standardin the regularity theory of elliptic systems like (1.1). On the other hand, it involveslocal lower bounds on |∇u0|2. Rather than regularity theory, this task is more strictlyrelated to the issue of unique continuation, namely, the study of the character ofzeros (i.e., order of vanishing and size of the zero sets) of nontrivial solutions tosystem (1.1). Unique continuation is very well studied and understood for the caseof linear elliptic equations. (See, for instance, Aronszajn, Krzywicki, and Szarski[AKS62], Garofalo and Lin [GL86], [GL87], and Koch and Tataru [KT01].) However,until recently, only results of weak unique continuation for the elasticity system wereknown; see Weck [W69], [W01], Leis [L86], Dehman and Robbiano [DR93], Ang et al.[AITY98], and Eller et al. [EINT02]. In this paper we apply some of the estimatesof unique continuation found in [AM01] (three spheres inequalities [AM01, (5.2)] andespecially doubling inequalities yielding strong unique continuation [AM01, (5.5)]),and we further elaborate on this topic. We also recall [LW03] for recent new resultsof strong unique continuation for the two-dimensional Lame system. The main resultin this direction here are new doubling inequalities for the reference solution u0 (seeTheorem 3.9) and for its symmetrized gradient ∇u0 (see Corollary 3.10). Such aninequality allows us to prove for |∇u0|2 the property that it is a Muckenhoupt weight(Coifman and Fefferman [CF74], Garcia-Cuerva and Rubio de Francia [GCRDF85]).This is a property of homogeneity in the average at all scales, which was first provedfor solutions of scalar elliptic equations by Garofalo and Lin [GL86].

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Let us comment now on the constants C1, C2, p appearing in (1.10). It is in factclear that their accurate evaluation would be essential for concrete applications. Theanalytical procedure by which they are found in Theorem 2.3 is indeed constructive,but, in practice, it may be difficult (and perhaps too pessimistic) to compute themby this theoretical approach. We believe that, alternatively, they should be evalu-ated by numerical simulation and then used in real-life situations. This approach iscurrently under development by the present authors and Bilotta, Formica, and Turco[ABFMRT].

From another point of view, however, the theoretical evaluation of such constantsmay provide useful hints for the setup of experiments. In fact, it turns out that acrucial parameter in the evaluation of the constants C2 and p in the right-hand sideof (1.10) is given by the number

F [ϕ] =‖ϕ‖L2(∂Ω)

‖ϕ‖H−1/2(∂Ω),

which we call the frequency of the boundary traction ϕ. In fact F [ϕ] is a measureof the oscillation character of ϕ, and it turns out that C2, p are increasing functionsof F [ϕ]. Thus, it is advisable, when possible, to choose ϕ in such a way that F [ϕ]is as small as possible. As will be seen in Proposition 4.4, the number p controls, inan average sense, the maximal vanishing rate of |∇u0|, and, in turn, p is controlledby F . It is apparent from (1.11) that the detectability of D (and the evaluation ofits size) deteriorates when D clusters near the zero set of |∇u0|. See [ARS00] fora more detailed discussion and examples in the scalar case. Therefore, it would beconvenient to choose ϕ in such a way that |∇u0| never vanishes, or at least has a lowvanishing rate. An example of this sort was discussed in Ikehata [Ik98], in the casewhen the materials within and outside D are uniform (that is, C, C are known to beconstant tensors) and ϕ is chosen to be ϕ = (CM)ν, where M is a given constantmatrix such that M = 1

2 (M + MT ) = 0. In this case we have that u0 is linear,|∇u0| = |M | = const = 0, and it turns out that p = 1. Unfortunately, besides thisvery special case, it seems quite difficult to give a general procedure to decide whichare the boundary tractions ϕ for which |∇u0| never vanishes. This is why we foundit necessary to follow the present approach.

The plan of the paper is as follows. In section 2 we introduce some notationand state our main result (Theorem 2.3). Section 3 is devoted to the derivationof quantitative estimates of unique continuation for solutions to the Lame system,following ideas introduced in Alessandrini and Morassi [AM01]. In section 4 we firstderive an interior average lower bound on |∇u0|2 on small balls contained inside Ω (seeProposition 4.1). Moreover, we rephrase the doubling inequalities obtained in section3 in terms of the boundary data (see Proposition 4.3), and we show that |∇u0|2 isa Muckenhoupt weight (see Proposition 4.4). Finally, section 5 contains the proof ofthe main theorem.

2. The Main Result. Let us introduce some notation which will be useful inwhat follows. We restrict our attention to the dimensions n = 2, 3, which are thosephysically relevant for elasticity.

Given a bounded domain Ω ⊂ Rn, n = 2, 3, for any h > 0 we shall denote

(2.1) Ωh = x ∈ Ω | dist(x, ∂Ω) > h.

When locally representing a boundary as a graph, it will be convenient to use the

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following notation. For every x ∈ Rn we shall set x = (x′, xn), where x′ ∈ Rn−1,xn ∈ R.

Definition 2.1. Given a bounded domain Ω ⊂ Rn, we shall say that ∂Ω is

of class C1,1 with constants r0,M0 > 0 if, for any x0 ∈ ∂Ω, there exists a rigidtransformation of coordinates under which we have x0 = 0 and

Ω ∩Br0(0) = x ∈ Br0(0) | xn > ϕ(x′),

where ϕ is a C1,1 function on Br0(0) ⊂ Rn−1 satisfying

ϕ(0) = |∇ϕ(0)| = 0

and

‖ϕ‖C1,1(Br0 (0)) ≤M0r0.

Notice that this quantitative formulation of the smoothness of the boundary alsoinvolves the introduction of the dimensional parameter r0, which gives us the scale atwhich the boundary is representable as a graph.

Remark 2.1. We have chosen to normalize all norms in such a way that theirterms are dimensionally homogeneous and coincide with the standard definition whenthe dimensional parameter equals one. For instance, the norm appearing above ismeant as follows:

(2.2) ‖φ‖C1,1(Br0 (0)) = ‖φ‖L∞(Br0 (0)) + r0‖∇φ‖L∞(Br0 (0)) + r02‖∇2φ‖L∞(Br0 (0)).

Similarly, given a function f : Ω → R, where ∂Ω satisfies Definition 2.1, we shalldenote

(2.3) ‖f‖C1,1(Ω) = ‖f‖L∞(Ω) + r0‖∇f‖L∞(Ω) + r02‖∇2f‖L∞(Ω).

Notice also that when Ω = BR(0), Ω then satisfies Definition 2.1 with r0 = R.We consider weak solutions u ∈ H1(Ω,Rn) to the displacement equation of equi-

librium when body forces are absent:

(2.4) div (C(x)(∇u(x))) = 0 in Ω;

see Gurtin [Gur72].We shall assume throughout that the elasticity tensor field C = C(x) of the mate-

rials under consideration have components Cijkl which satisfy the following conditions:

(2.5) Cijkl ∈ L∞(Ω,R) ∀i, j, k, l = 1, . . . , n,

(2.6) Cijkl = Cklij = Cklji ∀i, j, k, l = 1, . . . , n a.e. in Ω.

We recall that the symmetry conditions (2.6) are equivalent to

(2.7) CA = CA,

(2.8) CA is symmetric,

(2.9) CA ·B = CB ·A

for every pair of n× n matrices A,B.

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Here, and in the remainder of the paper, the following notation is used:

(2.10) (CA)ij =n∑

k,l=1

CijklAkl,

(2.11) A ·B =n∑

i,j=1

AijBij ,

(2.12) |A| = (A ·A) 12 ,

(2.13) A =12(A+AT )

for every pair of n× n matrices A,B.We shall also use the following conventions for inequalities. Given C, C satisfying

(2.5), (2.6) we shall say that

(2.14) C ≤ C

if and only if

(2.15) CA ·A ≤ CA ·A

for every symmetric n× n matrix A.We shall say that C is strongly convex in Ω if there exists a positive constant ξ0

such that

(2.16) C(x)A ·A ≥ ξ0|A|2 for a.e. x ∈ Ω,

for any symmetric n× n matrix A.C is said to be strongly elliptic in Ω if there exists a positive constant κ0 such

that

(2.17) C(x)A ·A ≥ κ0|A|2 for a.e. x ∈ Ω,

for any matrix A of the form Aij = aibj , where a and b are n-vectors. It is well knownthat if C is strongly convex, then it is also strongly elliptic.

When the elastic material is isotropic, then the elasticity tensor C takes the form

(2.18) Cijkl(x) = λ(x)δijδkl + µ(x)(δkiδlj + δliδkj),

where λ = λ(x) and µ = µ(x) ∈ L∞(Ω,R) are the Lame moduli. Hence, in this case,denoting by In the n× n identity matrix, we have

(2.19) C(x)A = λ(x)(A · In)In + 2µ(x)A,

and the displacement equation of equilibrium (2.4) becomes the Lame system

(2.20) div (2µ∇u) +∇(λdivu) = 0 in Ω.

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In the isotropic case, the strong convexity condition takes the form

(2.21) µ(x) ≥ α0, 2µ(x) + nλ(x) ≥ γ0 for a.e. x ∈ Ω

and the strong ellipticity condition is expressed by

(2.22) µ(x) ≥ α0, 2µ(x) + λ(x) ≥ β0 for a.e. x ∈ Ω,

where α0, β0, γ0 are positive constants.Let Ω be a bounded domain whose boundary is of class C1,1 with given constants

r0,M0 > 0. Let D be a measurable, possibly disconnected, subset of Ω such that,given d0 > 0,

(2.23) dist(D, ∂Ω) ≥ d0.

Given elasticity tensors C, C satisfying (2.5), (2.6) we shall consider traction problemsin Ω when the elasticity tensor is either χΩ\DC+ χDC or C.

We shall prescribe a boundary traction field ϕ ∈ L2(∂Ω,Rn) satisfying the com-patibility conditions

(2.24)∫∂Ωϕ · r = 0

for every infinitesimal rigid displacement r, that is, r(x) = c +Wx, where c is anyconstant n-vector andW is any constant skew n×n matrix. Namely, we shall considerweak solutions u, u0 ∈ H1(Ω,Rn) of the following problems:

(2.25) div ((χΩ\DC+ χDC)∇u) = 0 in Ω,

(2.26) (C∇u)ν = ϕ on ∂Ω;

(2.27) div (C∇u0) = 0 in Ω,

(2.28) (C∇u0)ν = ϕ on ∂Ω.

Regarding existence, we recall that, provided the compatibility condition (2.24) issatisfied, a solution of the traction problem exists as long as the involved elasticitytensor either satisfies the strong convexity condition or is continuous and satisfies thestrong ellipticity condition; see, for instance, Valent [V88, section III].

With respect to uniqueness we recall that it is well known that solutions u, u0 tothe above problems are uniquely determined up to an infinitesimal rigid displacement.In order to uniquely identify such solutions, we shall assume from now on that bothu and u0 satisfy the following normalization conditions:

(2.29)∫

Ωu = 0,

∫Ω(∇u− (∇u)T ) = 0.

We set g, g0 ∈ H1/2(∂Ω,Rn) to be the traces of u, u0, respectively, on ∂Ω.Now we are in position to state our main result on the estimates for the size of

the inclusion.

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We shall use the following assumptions on the elasticity tensors C, C:(i) C satisfies the isotropy condition (2.18) and the strong convexity (2.21);(ii) (bounds on the jump and uniform strong convexity for C)

either there exist η > 0 and δ > 1 such that

(2.30) ηC ≤ C− C ≤ (δ − 1)C a.e. in Ω

or there exist η > 0 and 0 < δ < 1 such that

(2.31) −(1− δ)C ≤ C− C ≤ −ηC a.e. in Ω;

(iii) (C1,1 regularity for C)there exists M > 0 such that

(2.32) ‖µ‖C1,1(Ω) + ‖λ‖C1,1(Ω) ≤M.

Remark 2.2. It is worth noting that very mild assumptions are made on theunknown inclusion; namely, the inclusion D may consist of an anisotropic materialwhich is either harder (case (2.30)) or softer (case (2.31)) than the surrounding mate-rial in Ω, and no additional regularity assumption is required on the elasticity tensorinside D.

Theorem 2.3. Let Ω be a bounded domain in Rn such that ∂Ω is of class C1,1

with constants r0, M0. Let D be any measurable subset of Ω satisfying (2.23). Let C,C satisfy (i), (ii), (iii). If (2.30) holds, then we have

(2.33)1

δ − 1C+

1

∫∂Ω(g0 − g) · ϕ∫∂Ω g0 · ϕ

≤ |D| ≤(δ

η

) 1p

C+2

(∫∂Ω(g0 − g) · ϕ∫∂Ω g0 · ϕ

) 1p

.

If, conversely, (2.31) holds, then we have

(2.34)δ

1− δC−1

∫∂Ω(g − g0) · ϕ∫∂Ω g0 · ϕ

≤ |D| ≤(1η

) 1p

C−2

(∫∂Ω(g − g0) · ϕ∫∂Ω g0 · ϕ

) 1p

,

where C+1 , C−1 depend only on the geometrical parameters |Ω|, r0, M0 (see Definition

2.1), d0 (see (2.23)) and on the bounds on the Lame moduli α0, γ0 (see (2.21)), M(see (2.32)), whereas p > 1, C+

2 , C−2 depend only on |Ω|, r0, M0, d0, α0, γ0, M , and‖ϕ‖L2(∂Ω)/‖ϕ‖H−1/2(∂Ω).

Remark 2.4. Other choices of the norms appearing in the quotient

F [ϕ] =‖ϕ‖L2(∂Ω)

‖ϕ‖H−1/2(∂Ω)

could serve the same purpose. For instance in [AMR03] it was replaced with thequotient

‖ϕ‖H−1/2(∂Ω)

‖ϕ‖H−1(∂Ω).

This other choice has the advantage of being finite for every nontrivial ϕ ∈ H−1/2(∂Ω),which is the natural space for the variational formulation of the boundary value prob-lems (2.25)–(2.26), (2.27)–(2.28).

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3. Quantitative Estimates of Unique Continuation. In this section we shallprove quantitative estimates of unique continuation in the form of three spheres in-equalities and doubling inequalities for solutions u ∈ H1(Ω,Rn) to the Lame systemof linearized elasticity (2.20) in a bounded domain Ω satisfying Definition 2.1 withconstants r0, M0. Throughout this section the Lame moduli µ = µ(x), λ = λ(x) areassumed to satisfy the strong ellipticity condition (2.22) and the regularity assump-tion (2.32). Following ideas introduced in [AM01], the first step consists of reducingthe Lame system (2.20) to a weakly coupled elliptic system with Laplacian principalpart. We denote by Mm×n the space of m× n real-valued matrices.

Proposition 3.1. Under the above assumptions, there exist B ∈ L∞(Ω,L(M(n+1)×n,Rn+1)) and V ∈ L∞(Ω, L(Rn+1,Rn+1)) such that, for every weak so-lution u ∈ H1(Ω,Rn) to (2.20), the Rn+1-valued function U given by

(3.1) U =(

u

r0 div u

)belongs to W 2,p

loc (Ω,Rn+1) for every p <∞ and satisfies

(3.2) −∆U +B(∇U) + V (U) = 0 in Ω.

Moreover

(3.3) r0‖B‖L∞(Ω,L(M(n+1)×n,Rn+1)) + r20‖V ‖L∞(Ω,L(Rn+1,Rn+1)) CM,

where C > 0 depends only on α0 and β0.Proof. The proof is essentially contained in [AM01, Theorem 2.1]. Here the

statement is slightly modified in order to encompass the scaling invariance of thenorms introduced in the present paper.

Three spheres inequalities and doubling inequalities for solutions u to systems ofthe form (3.2), under the assumption (3.3), were derived in [AM01, Theorems 3.1 and4.1]. Next, one can obtain analogous estimates for solutions u to the Lame system(2.20) via the reduction described in Proposition 3.1.

Proposition 3.2 (see [AM01, Theorem 5.1]). Let Ω = BR = x ∈ Rn | |x| < R.Under the above assumptions, there exists θ, 0 < θ 1, depending only on α0, β0,M , such that for every weak solution u ∈ H1(BR,Rn) to (2.20) and for every r1, r2,r3, 0 < r1 < r2 < r3 θR, we have

(3.4)∫Br2

|u|2 C(∫

Br1

|u|2)δ (∫

Br3

|u|2)1−δ

,

where C > 0, 0 < δ < 1 depend only on α0, β0,M, r1r3 , andr2r3.

Proof. The proof can be found in [AM01]. We notice that here, in view of ourscaling on the norms (see Remark 2.1), the constant C does not explicitly depend onR.

In view of the applications in section 5, we need the analogous result for ∇u.Corollary 3.3. Under the same hypotheses of Proposition 3.2, for every weak

solution u ∈ H1(BR,Rn) to (2.20) and for every r1, r2, r3, 0 < r1 < r2 < r3 θR,we have

(3.5)∫Br2

|∇u|2 C(∫

Br1

|∇u|2)δ (∫

Br3

|∇u|2)1−δ

,

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where θ, 0 < θ 1, is the same as in Proposition 3.2 and C > 0, 0 < δ < 1 dependonly on α0, β0, M , r1r3 , and

r2r3.

In order to prove Corollary 3.3 it is convenient to recall the following two inequal-ities.

Lemma 3.4 (Caccioppoli-type inequality). If C satisfies (2.18), (2.22), and(2.32), then for every solution u ∈ H1(BR,Rn) to (2.4) and for every r, 0 < r < R,we have

(3.6)∫Br

|∇u|2 C

(R− r)2∫BR

|u|2,

where C > 0 depends only on α0, β0,M.Proof. The proof follows by a standard cut-off argument from Garding’s inequality

[V88].Given u ∈ H1(BR,Rn) and r, 0 < r < R, set

(3.7) ur =1|Br|

∫Br

u,

(3.8) Wr =1

2|Br|

∫Br

(∇u− (∇u)T ).

Lemma 3.5 (Korn inequality). There exists an absolute constant C > 0 suchthat for every u ∈ H1(BR,Rn) and every r, 0 < r < R, we have

(3.9)∫BR

|∇u−Wr|2 +1R2 |u− ur −Wrx|

2 C(R

r

)4n−2 ∫BR

|∇u|2.

Remark 3.6. When r = R this is the well-known second Korn inequality, whichis known to hold in every sufficiently regular domain Ω (see [Fi72], [T99]). Here weintroduce a minor variant, in which the averages of u and of the skew part of ∇u aretaken on the smaller ball Br. We refer the interested reader to [AMR02] for a proof.

Proof of Corollary 3.3. The function v defined in BR by

(3.10) v = u− ur1 −Wr1x

satisfies (2.20) and 1|Br1 |

∫Br1

v = 0, 12|Br1 |

∫Br1∇v− (∇v)T = 0. By applying to v the

Caccioppoli-type inequality (3.6) and the three spheres inequality (3.4) and using theKorn inequality (3.9) twice, we have

(3.11)∫Br2

|∇u|2 =∫Br2

|∇v|2 C1

(r3 − r2)2∫B r2+r3

2

|v|2

C2

(r3 − r2)2

(∫Br1

|v|2)δ (∫

Br3

|v|2)1−δ

C3

(∫Br1

|∇u|2)δ (∫

Br3

|∇u|2)1−δ

,

where C1, C2, C3 are constants depending only on α0, β0,M, r1r3 , andr2r3.

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In order to obtain the doubling inequality for solutions to the Lame system (2.20),we need to state a slightly modified version of the doubling inequality for solutions Uto (3.2) contained in [AM01, Theorem 4.1].

Proposition 3.7. Let Ω = BR and let R‖B‖∞ + R2‖V ‖∞ E. There existsθ∗, 0 < θ∗ 1, depending only on E, such that for every nonzero solution U ∈H1(BR,Rn+1) to (3.2) we have

(3.12)∫B2r

|U |2 C∫Br

|U |2 for every r, 0 < r θ∗R

2,

where C depends only on E and N0(θ∗R), where

(3.13) N0(r) = N0(U ; r) =r∫Br|∇U |2∫

∂Br|U |2 , 0 < r R.

Moreover C is increasing with N0(θ∗R).Proof. By Theorem 4.1 in [AM01] and by a rescaling argument, it easily follows

that (3.12) holds, with C depending only on E and on N(θ∗R), where

(3.14) N(r) = N(U ; r) =r∫Br|∇U |2 +B(∇U) · U + V (U) · U∫

∂Br|U |2 , 0 < r R,

the dependence on this last variable being monotonically increasing. Hence, we haveto show that N(r) can be bounded from above in terms of N0(r). It is convenient torecall the following notation introduced in [AM01]:

(3.15) G(r) =∫Br

|U |2,

(3.16) H(r) =∫∂Br

|U |2,

(3.17) I(r) =∫Br

|∇U |2 +B(∇U) · U + V (U) · U,

(3.18) D(r) =∫Br

|∇U |2

for 0 < r R. We easily have

(3.19) I(r) D(r) +E

R(D(r)G(r))

12 +

E

R2G(r) C(D(r) +

G(r)R2

),

with C depending only on E. Moreover, from Lemma 3.3 in [AM01] we have

(3.20) G(r) rH(r) for r θ∗R.

Hence, for r θ∗R we have

(3.21) N(r) =rI(r)H(r)

C(N0(r) +

r2

R2

) C(N0(r) + 1),

where C depends only on E.

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Remark 3.8. Let us notice that analogously to (3.21) we also have

(3.22) N0(r) C(N(r) + 1) for r θ∗R,

where C depends only on E.Theorem 3.9. Let Ω = BR. There exists θ∗, 0 < θ∗ 1, depending only on

α0, β0,M , such that for every nonzero weak solution u ∈ H1(BR,Rn) to (2.20) wehave

(3.23)∫B2r

|u|2 K∫Br

|u|2 for every r, 0 < r θ∗R

2,

where K > 0 depends only on α0, β0,M and on N0(θ∗R), where

(3.24) N0(r) =r2∫Br|∇u|2 +R2|∇(divu)|2∫Br|u|2 +R2|divu|2 , 0 < r R.

Moreover K is increasing with N0(θ∗R).Proof. By applying (3.12) of Proposition 3.7 to the solution U to (3.2) given by

the position (3.1), and recalling (3.3), we have

(3.25)∫B2r

|u|2 +R2|divu|2 C∫Br

|u|2 +R2|divu|2 for every r, 0 < r θ∗R

2,

where C depends only on α0, β0,M and on N0(θ∗R), with N0(r) given by (3.13),the dependence on this last variable being monotonically increasing. By an iteratedapplication of (3.25) and by the Caccioppoli-type inequality (3.6), we have

(3.26)∫B2r

|u|2 C(1 +

R2

r2

)∫Br

|u|2 for every r, 0 < r θ∗R

2.

Let ρ be such that 0 < ρ 1, and let

uρ(x) = u(ρx) in BRρ.

We have that uρ is a solution in BRρto the Lame system (2.20) with Lame moduli

satisfying uniformly the bounds (2.22), (2.32). Therefore, by (3.26) we have that

(3.27)∫B2r

|uρ|2 Cρ(1 +

R2

ρ2r2

)∫Br

|uρ|2 for every r, 0 < r θ∗R

2ρ,

where Cρ depends only on α0, β0,M , and N0(Uρ; θ∗Rρ ), where Uρ is given by (3.1)

when u, r0 are replaced with uρ, Rρ , respectively. Here, again, Cρ is increasing withN0(Uρ; θ

∗Rρ ). We have for any r, 0 < r θ∗R

2ρ ,

(3.28) N0(Uρ; r) =r∫Br|∇uρ|2 + R2

ρ2 |∇divuρ|2∫∂Br|uρ|2 + R2

ρ2 |divuρ|2

=r∫Brρ2|∇u(ρx)|2 + R2

ρ2 · ρ4|∇divu(ρx)|2∫∂Br|u(ρx)|2 + R2

ρ2 · ρ2|divu(ρx)|2

=ρr∫Bρr|∇u|2 +R2|∇divu|2∫

∂Bρr|u|2 +R2|divu|2 = N0(U ; ρr).

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Hence, in particular,

(3.29) N0

(Uρ;

θ∗R

ρ

)= N0(U ; θ∗R).

Consequently, the quantity Cρ appearing in (3.27) is uniformly bounded from abovewith respect to ρ ∈ (0, 1]. Taking r = θ∗R

2ρ in (3.27) and setting s = rρ, we obtain

(3.30)∫B2s

|u|2 K∫Bs

|u|2 for every s, 0 < s θ∗R

2,

where K depends only on α0, β0,M and on N0(U ; θ∗R), the dependence on this lastvariable being monotonically increasing. Recalling (3.13), (3.20), (3.24), we have thatN0(U ; θ∗R) N0(θ∗R).

From Theorem 3.9, by using arguments analogous to those employed in the proofof Corollary 3.3, the following doubling inequality for ∇u follows.

Corollary 3.10. Let Ω = BR. There exists θ∗, 0 < θ∗ 1, depending onlyon α0, β0,M , such that for every nonzero weak solution u ∈ H1(BR,Rn) to (2.20) wehave

(3.31)∫B2r

|∇u|2 Kr∫Br

|∇u|2 for every r, 0 < r θ∗R

4,

where Kr > 0 depends only on α0, β0,M , and N0(v; θ∗R) given by (3.24), wherev = u− ur −Wrx, with ur and Wr defined by (3.7) and (3.8), respectively. MoreoverKr is increasing with N0(v; θ∗R).

4. Estimates in Terms of the Boundary Data. In this section we shall considerthe traction problem (2.27), (2.28) for a given ϕ ∈ L2(∂Ω,Rn) satisfying (2.24).For simplicity of notation we shall denote by u the solution (instead of u0). Thenormalization (2.29) is understood throughout.

Regarding the elasticity tensor C we assume the isotropy condition (2.18), thestrong ellipticity (2.22), and the C1,1 regularity (2.32).

Proposition 4.1 (Lipschitz propagation of smallness). For every ρ > 0 and forevery x ∈ Ω 4ρ

θ

, we have

(4.1)∫Bρ(x)

|∇u|2 Cρ∫

Ω|∇u|2,

where θ, 0 < θ < 1, depends only on α0, β0,M and Cρ depends only on α0, β0, M ,|Ω|, r0, M0, ‖ϕ‖L2(∂Ω)/‖ϕ‖H−1/2(∂Ω), and ρ.

We adapt arguments from [ARS00, Theorem 2.2]. We start with the followingauxiliary lemma.

Lemma 4.2.

(4.2)∫

Ω\Ω 5ρθ

|∇u|2 Cρ1/n‖ϕ‖2L2(∂Ω),

where C depends only on α0, β0,M, r0,M0, |Ω|.Proof of Lemma 4.2. By Holder’s inequality

(4.3) ‖∇u‖2L2(Ω\Ω 5ρθ

) |Ω \ Ω 5ρθ

|1/n‖∇u‖2L2n/(n−1)(Ω\Ω 5ρθ

),

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and by the Sobolev inequality (see, for instance, [Ad75])

(4.4) ‖∇u‖2L2n/(n−1)(Ω) C‖∇u‖2H1/2(Ω),

we have

(4.5) ‖∇u‖2L2(Ω\Ω 5ρθ

) C|Ω \ Ω 5ρθ

|1/n‖u‖2H3/2(Ω),

where C depends only on r0, M0, |Ω|.Moreover, we have

(4.6) ‖u‖H3/2(Ω) C‖ϕ‖L2(∂Ω),

where C depends only on α0, β0,M, r0,M0, |Ω|. Inequality (4.6) follows, by interpo-lation (see [LM72]), from the global estimates for the Neumann problem

(4.7) ‖u‖H1(Ω) C1‖ϕ‖H−1/2(∂Ω),

(4.8) ‖u‖H2(Ω) C2‖ϕ‖H1/2(∂Ω),

where C1 and C2 depend only on α0, β0, M , r0, M0, |Ω| (see [ADN64]).Moreover,

(4.9) |Ω \ Ω 5ρθ

| Cρ,

where C depends only on r0, M0, |Ω|. (See (A.3) in [AR98] for details.) From (4.5),(4.6), and (4.9) the thesis follows.

Proof of Proposition 4.1. Let us fix ρ0, depending only on r0,M0, such that Ω 4ρθ

isconnected for every ρ ρ0. Without loss of generality, we may assume ρ ρ0 for thisproof. Given any y ∈ Ω 4ρ

θ

, let γ be an arc in Ω 4ρθ

joining x and y. Let us define xi,i = 1, . . . , L, as follows: x1 = x, xi+1 = γ(ti), where ti = maxt | |γ(t)− xi| = 2ρ if|xi − y| > 2ρ; otherwise let i = L and stop the process. Then, by construction, theballs Bρ(xi) are pairwise disjoint, |xi+1 − xi| = 2ρ for i = 1, . . . , L− 1, |xL − y| 2ρ.

Since xi ∈ Ω 4ρθ

, we may apply (3.5) for x = xi, r1 = ρ, r2 = 3ρ, r3 = 4ρ, fori = 1, . . . , L− 1, obtaining

(4.10)‖∇u‖L2(Bρ(xi+1))

‖∇u‖L2(Ω) C

(‖∇u‖L2(Bρ(xi))

‖∇u‖L2(Ω)

)δ,

where C > 0 and δ, 0 < δ < 1, depend only on α0, β0, and M . By induction we have

(4.11)‖∇u‖L2(Bρ(y))

‖∇u‖L2(Ω) C1/(1−δ)

(‖∇u‖L2(Bρ(x))

‖∇u‖L2(Ω)

)δL.

Let us note that L |Ω|ωnρn

.Let us cover Ω 5ρ

θ

with internally nonoverlapping closed cubes of side l = 2ρ/√nθ.

Clearly, any such cube is contained in a ball of radius ρ and center in Ω 4ρθ

and thenumber of such cubes is controlled by N = |Ω|nn/2θn

2nρn . Therefore, from (4.11) we have

(4.12)‖∇u‖L2(Ω 5ρ

θ

)

‖∇u‖L2(Ω) C

ρn/2

(‖∇u‖L2(Bρ(x))

‖∇u‖L2(Ω)

)δL,

where C depends only on α0, β0,M, |Ω|.

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Now, let us estimate from below the left-hand side of (4.12) by means of a positiveconstant. Let us set

(4.13)‖∇u‖2L2(Ω 5ρ

θ

)

‖∇u‖2L2(Ω)

= 1−

∫Ω\Ω 5ρ

θ

|∇u|2∫Ω |∇u|2

.

By a trace inequality (see, for instance, [LM72]) and by the Korn inequality (3.9), wehave

(4.14) ‖ϕ‖H−1/2(∂Ω) C‖∇u‖L2(Ω),

where C depends only on α0, β0, r0,M0, |Ω|. Hence, by (4.2) and (4.14), we have thatthere exists ρ > 0, depending only on α0, β0,M, r0,M0, |Ω|, and ‖ϕ‖L2(∂Ω)/‖ϕ‖H−1/2(∂Ω),such that

(4.15)‖∇u‖2L2(Ω 5ρ

θ

)

‖∇u‖2L2(Ω)

12

for every ρ, 0 < ρ ρ.Finally, from (4.12) and (4.15) the thesis follows when 0 < ρ ρ; for larger values

of ρ, inequality (4.1) is trivial.Proposition 4.3. There exists θ∗, 0 < θ∗ 1, depending only on α0, β0,M ,

such that for every r > 0 and every x0 ∈ Ωr we have

(4.16)∫B2r(x0)

|u|2 K∫Br(x0)

|u|2 for every r, 0 < r θ∗r

2,

(4.17)∫B2r(x0)

|∇u|2 K∫Br(x0)

|∇u|2 for every r, 0 < r θ∗r

4,

where K > 0 depends only on α0, β0,M , r0, M0, |Ω|, r, and ‖ϕ‖L2(∂Ω)/‖ϕ‖H−1/2(∂Ω),and θ∗ is the quantity introduced in Corollary 3.10.

Proof. The proofs of (4.16) and (4.17) are similar. Let us prove (4.17), whichtakes a little bit more work. Given x0 ∈ Ωr and r, 0 < r < θ∗r

4 , we may applyCorollary 3.10 with R = r, obtaining (4.17) with K depending only on α0, β0,M , and

(4.18) N0(v; θ∗r) =(θ∗r)2

∫Bθ∗r(x0) |∇v|

2 + r2|∇(div v)|2∫Bθ∗r(x0) |v|2 + r2|div v|2

,

the dependence on this last variable being monotonically increasing and where v isdefined in Br(x0) by

(4.19) v = u− c−W (x− x0),

with

(4.20) c =1

|Br(x0)|

∫Br(x0)

u, W =1

2|Br(x0)|

∫Br(x0)

∇u− (∇u)T .

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We have that ∇v = ∇u −W , ∇v = ∇u, and div v = divu. Moreover, by interiorregularity estimates (see [ADN64]), we have

(4.21) |W | 1|Br(x0)|

∫Br(x0)

|∇u| ‖∇u‖L∞(Br(x0))

C‖u‖H1(Ω) C‖ϕ‖H−1/2(∂Ω),

where C depends only on α0, β0, M , r0, and r. Hence

(4.22)∫Bθ∗r(x0)

|∇v|2 2∫Bθ∗r(x0)

|∇u|2 + |W |2 C‖ϕ‖2H−1/2(∂Ω),

where C depends only on α0, β0, M , r0, and r. By the Caccioppoli-type inequality(3.6) we have

(4.23)∫Bθ∗r(x0)

|v|2 C∫B θ∗r

2(x0)|∇v|2 C

∫B θ∗r

2(x0)|∇u|2,

where C depends only on α0, β0, M , r. If θ∗

2 θ4 , we may apply Proposition 4.1 with

ρ = θ∗r2 , whereas if θ

∗

2 θ4 , we may apply Proposition 4.1 with ρ = θr

4 . In both casesby trace theorems and the standard Korn inequality we obtain

(4.24)∫B θ∗r

2(x0)|∇u|2 C

∫Ω|∇u|2 C‖ϕ‖2H−1/2(∂Ω),

where C depends only on α0, β0, M , r0, M0, |Ω|, r, and ‖ϕ‖L2(∂Ω)/‖ϕ‖H−1/2(∂Ω). By(4.22)–(4.24) and interior regularity estimates (see [ADN64]) we have

(4.25) N0(v; θ∗r) C,

where C depends only on α0, β0, M , r0 M0, |Ω|, r, and ‖ϕ‖L2(∂Ω)/‖ϕ‖H−1/2(∂Ω).Hence the thesis follows.

Proposition 4.4 (Ap property). For every r > 0 there exist B > 0 and p > 1such that for every x0 ∈ Ωr we have

(4.26)

(1

|Br(x0)|

∫Br(x0)

|∇u|2)(

1|Br(x0)|

∫Br(x0)

|∇u|−2/(p−1)

)p−1

B

for every r, 0 < r θ∗r

4,

where θ∗ is the quantity introduced in Corollary 3.10 and where B, p depend only onα0, β0, M , r0, M0, |Ω|, r, and ‖ϕ‖L2(∂Ω)/‖ϕ‖H−1/2(∂Ω).

Proof. In view of the results in [CF74] it is enough to prove a reverse Holder’sinequality for |∇u|2. Let v = u − c −W (x − x0), with c = 1

|B2r(x0)|∫B2r(x0) u, W =

12|B2r(x0)|

∫B2r(x0)∇u − (∇u)T . By interior regularity estimates, the Korn inequality

(3.9), and Proposition 4.3 we have

(4.27) ‖∇u‖L∞(Br(x0)) = ‖∇v‖L∞(Br(x0)) C

r(n+2)/2 ‖v‖H1(B2r(x0))

C

rn/2‖∇u‖L2(B2r(x0))

C

rn/2‖∇u‖L2(Br(x0)),

where C depends only on α0, β0,M , r0,M0, |Ω|, r, and ‖ϕ‖L2(∂Ω)/‖ϕ‖H−1/2(∂Ω).

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5. Proof of Theorem 2.3. We base the proof of our main theorem on two aux-iliary lemmas.

Lemma 5.1. Let the elasticity tensor fields C(x) and C(x) satisfy (2.5), (2.6) inΩ. Suppose that weak solutions u, u0 ∈ H1(Ω,Rn) to the traction problems (2.25)–(2.26), (2.27)–(2.28) exist. The following identities hold:

(5.1)∫

Ω(χΩ\DC+ χDC)∇(u− u0) · ∇(u− u0)

−∫D

(C− C)∇u0 · ∇u0 =∫∂Ω

(g − g0) · ϕ,

(5.2)∫

ΩC∇(u− u0) · ∇(u− u0) +

∫D

(C− C)∇u · ∇u =∫∂Ω

(g0 − g) · ϕ,

(5.3)∫D

(C− C)∇u · ∇u0 =∫∂Ω

(g0 − g) · ϕ,

where g, g0 ∈ H1/2(∂Ω,Rn) are the traces of u, u0, respectively, on ∂Ω.Proof of Lemma 5.1. Let us denote H = (C−C) in Ω. From the weak formulation

of the problem (2.25)–(2.26) with D = D1 and D = D2 we get the identity

(5.4)∫

Ω(C+ χD1H)∇u1 · ∇w

=∫

Ω(C+ χD2H)∇u2 · ∇w for every w ∈ H1(Ω,Rn),

where ui is the solution to (2.25)–(2.26) with D = Di, i = 1, 2, respectively. Sub-tracting the quantity

∫Ω(C+ χD1H)∇u2 · ∇w from both sides of (5.4) we have

(5.5)∫

Ω(C+ χD1H)∇(u1 − u2) · ∇w

=∫

Ω(χD2 − χD1)H∇u2 · ∇w for every w ∈ H1(Ω,Rn).

Choosing w = u1 in (5.5) we get

(5.6)∫

Ω(C+ χD1H)∇(u1 − u2) · ∇u1 =

∫Ω(χD2 − χD1)H∇u2 · ∇u1.

By using the weak formulation of the traction problems for u1 and u2, the left-handside of (5.6) can be rewritten as

(5.7)∫

Ω(C+ χD1H)∇(u1 − u2) · ∇u1 =

∫∂Ω

(g1 − g2) · ϕ,

where gi = ui |∂Ω, i = 1, 2, and therefore (5.6) becomes

(5.8)∫∂Ω

(g1 − g2) · ϕ =∫

Ω(χD2 − χD1)H∇u2 · ∇u1.

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Choosing w = u1 − u2 in (5.5) and using (5.8) we get

(5.9)∫

Ω(C+ χD1H)∇(u1 − u2) · ∇(u1 − u2)

=∫

Ω(χD1 − χD2)H∇u2 · ∇u2 +

∫∂Ω

(g1 − g2) · ϕ,

and finally we obtain the fundamental identity

(5.10)∫

Ω(C+ χD1H)∇(u1 − u2) · ∇(u1 − u2) +

∫D2\D1

H∇u2 · ∇u2

=∫∂Ω

(g1 − g2) · ϕ+∫D1\D2

H∇u2 · ∇u2,

which is the analogue to the identity found in Kang, Seo, and Sheen [KSS97] for theinverse conductivity problem.

By choosing D1 = D and D2 = ∅ we obtain the first identity (5.1) of the lemma.The second identity (5.2) follows from (5.10) for D1 = ∅ and D2 = D.

To get the third identity (5.3), we choose w = u0 and w = u in the weak formu-lation of the traction problem (2.25)–(2.26) for D = D1 and D = ∅, respectively:

(5.11)∫

Ω(C+ χDH)∇u · ∇u0 =

∫∂Ωg0 · ϕ,

(5.12)∫

ΩC∇u0 · ∇u =

∫∂Ωg · ϕ.

Subtracting (5.12) from (5.11) we obtain identity (5.3).Lemma 5.2. Let C(x) and C(x) satisfy (2.5), (2.6) in Ω. Let ξ0, ξ1, 0 < ξ0 < ξ1,

be such that

(5.13) ξ0|A|2 ≤ C(x)A ·A ≤ ξ1|A|2 for a.e. x ∈ Ω,

for any symmetric n × n matrix A, and let the jump (C(x) − C(x)) satisfy either(2.30) or (2.31). Let u, u0 ∈ H1(Ω,Rn) be the weak solutions to the traction problems(2.25)–(2.26), (2.27)–(2.28), respectively.

If (2.30) holds, then we have

(5.14)ηξ0δ

∫D

|∇u0|2 ≤∫∂Ω

(g0 − g) · ϕ ≤ (δ − 1)ξ1∫D

|∇u0|2.

If instead (2.31) holds, then we have

(5.15) ηξ0

∫D

|∇u0|2 ≤∫∂Ω

(g − g0) · ϕ ≤1− δδ

ξ1

∫D

|∇u0|2.

Proof of Lemma 5.2. Suppose that (2.30) holds. Then from identity (5.1) we have

(5.16)∫∂Ω

(g0 − g) · ϕ ≤∫D

H∇u0 · ∇u0,

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where H = (C − C) in Ω. The inequality below follows by the symmetry properties(2.7), (2.8), (2.9) and the positivity condition (2.30):

(5.17)∫D

H∇u0 · ∇u0 ≤ (1 + ε)∫D

H∇(u− u0) · ∇(u− u0)

+(1 +

1ε

)∫D

H∇u · ∇u for every ε > 0.

Then from (2.30) we have

(5.18)∫D

H∇u0 · ∇u0

≤ (1 + ε)(δ − 1)[∫D

C∇(u− u0) · ∇(u− u0) +1

ε(δ − 1)

∫D

H∇u · ∇u].

Choosing ε = 1δ−1 in (5.18) and employing identity (5.2) we get

(5.19)∫D

H∇u0 · ∇u0 ≤ δ∫∂Ω

(g0 − g) · ϕ.

The double inequality (5.14) follows from (5.16), (5.19) and (5.13), (2.30).In the case where (2.31) holds, from (5.1) we have

(5.20)∫∂Ω

(g − g0) · ϕ ≥∫D

−H∇u0 · ∇u0.

From (5.3) we obtain∫∂Ω(g − g0) · ϕ =

∫D−H∇u · ∇u0, and then, reasoning as in

(5.17), we find

(5.21)∫∂Ω

(g − g0) · ϕ ≤ε

2

∫D

−H∇u · ∇u

+12ε

∫D

−H∇u0 · ∇u0 for every ε > 0.

By using (5.2), (2.31), and (5.1) we have

(5.22)∫D

−H∇u · ∇u =∫∂Ω

(g − g0) · ϕ+∫

ΩC∇(u− u0) · ∇(u− u0)

≤∫∂Ω

(g − g0) · ϕ+1δ

∫Ω(C+H)∇(u− u0) · ∇(u− u0)

=∫∂Ω

(g − g0) · ϕ+1δ

[∫∂Ω

(g − g0) · ϕ+∫

ΩH∇u0 · ∇u0

]=δ + 1δ

∫∂Ω

(g − g0) · ϕ+1δ

∫D

H∇u0 · ∇u0.

Inserting inequality (5.22) into (5.21), we obtain

(5.23)∫∂Ω

(g − g0) · ϕ ≤ α(ε)∫D

−H∇u0 · ∇u0,

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where α(ε) = δ−ε2ε(2δ−εδ−ε) . The minimum of α(ε) occurs when ε = δ, and in this case

we have

(5.24)∫∂Ω

(g − g0) · ϕ ≤1δ

∫D

−H∇u0 · ∇u0.

The double inequality (5.15) follows from (5.20), (5.24) and (5.13), (2.31).Proof of Theorem 2.3. By (2.21), the inequality (5.13) holds, with ξ0 =

min2α0, γ0, ξ1 = (n+ 2)M , so that Lemma 5.2 may be applied.By standard regularity estimates for elliptic systems (see Agmon, Douglis, and

Nirenberg [ADN64]), by the Korn inequality, by (5.13), and by the weak formulationof the Neumann problem (2.27)–(2.28), we have

(5.25) ‖∇u0‖L∞(D) ≤ C‖u0‖H1(Ω) ≤ C‖∇u0‖L2(Ω) ≤ C(∫

∂Ωg0 · ϕ

) 12

,

where the constant C depends only on α0, γ0, M , d0, and |Ω|.The lower bound for |D| in (2.33), (2.34) follows from the right-hand sides of

(5.14), (5.15) and from (5.25).Let r = d0

2 and ε = min θ∗d04√n, θd0

4 , where θ∗ is as in Proposition 4.4. Let uscover D with internally nonoverlapping closed cubes Qj , j = 1, . . . , J , with side ε. ByHolder’s inequality we have

(5.26) |D| ≤(∫

⋃Jj=1Qj

|∇u0|− 2p−1

) p−1p (∫

D

|∇u0|2) 1p

,

where p is chosen as in Proposition 4.4. By applying Proposition 4.4 to the balls Bjcircumscribing each Qj , j = 1, . . . , J , we have

(5.27)

(∫⋃Jj=1Qj

|∇u0|− 2p−1

) p−1p

=

εn J∑j=1

1|Qj |

∫Qj

|∇u0|− 2p−1

p−1p

≤

εn J∑j=1

B[C(n)]p

1|Qj |

∫Qj|∇u0|

2

1p−1

p−1p

≤ (Jεn)p−1p B

1pC(n)

minj(

1|Qj |

∫Qj|∇u0|

2) 1p

,

where C(n) = ωn(√n

2 )n and B is as in Proposition 4.4. Now Jεn =∑Jj=1 |Qj | ≤ |Ω|

and hence, from (5.26), we have

(5.28) |D| ≤ |Ω|p−1p B

1pC(n)

εn∫D|∇u0|

2

minj∫Qj|∇u0|

2

1p

.

By Proposition 4.1, (5.28), (5.13), and the weak formulation of (2.27)–(2.28), we have

(5.29)∫D

|∇u0|2≥(K

∫∂Ωg0 · ϕ

)|D|p,

where K depends only on α0, β0, d0, |Ω|, r0, M0, M , h1, ‖ϕ‖L2(∂Ω)/‖ϕ‖H−1/2(∂Ω).The right-hand sides of (2.33), (2.34) follow from the left-hand sides of (5.14), (5.15),and (5.29).

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Acknowledgment. The authors wish to thank Mikhail Belishev, who kindlypointed out the reference [L86].

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Detecting an Inclusion in an Elastic Body by Boundary Measurements

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