AMERICAN MATHEMATICAL SOCIETY Volume 358, Number 7, Pages 2837–2853 S 0002-9947(06)03938-9
Article electronically published on February 6, 2006
UNIQUE CONTINUATION FOR THE TWO-DIMENSIONAL ANISOTROPIC ELASTICITY SYSTEM AND ITS
APPLICATIONS TO INVERSE PROBLEMS
GEN NAKAMURA AND JENN-NAN WANG
Abstract. Under some generic assumptions we prove the unique continu-ation property for the two-dimensional inhomogeneous anisotropic elasticity system. Having established the unique continuation property, we then inves-tigate the inverse problem of reconstructing the inclusion or cavity embedded in a plane elastic body with inhomogeneous anisotropic medium by infinitely many localized boundary measurements.
1. Introduction
Assume thatB is an anisotropic elastic body and the reference configuration of B is Ω, a bounded open connected set in R2. Let C(x) = (C
ijkl(x)) be the elastic
tensor. Here and below, all Latin indices are set to be from 1 to 2. We assume that the elastic tensor C satisfies the full symmetric properties
(1.1) Cijkl= Cjikl= Cklij ∀ i, j, k, l.
The displacement equation of equilibrium when there is no body force is given by
(1.2) LCu =∇ · (C(x)∇u) = 0 in Ω,
where (∇u)kl = ∂luk and (∇ · G)i =
j∂jgij for any matrix function G = (gij).
Also, we have used the convention notation (CH)ij=
kl
Cijklhkl,
where H = (hij) is a 2× 2 matrix. Here u =t[u1, u2] is the displacement vector. In this paper we are concerned with the unique continuation property (UCP) for (1.2). Precisely, we want to know whether u vanishes identically in Ω whenever it vanishes in some nonempty open subset of Ω.
The unique continuation property for differential equations has a long history. There exists a vast number of literature in this field, especially for scalar differential equations. We will not try to give a full account of its development here. Instead, we only mention some related results on the elasticity system. When the medium is isotropic, the UCP has been established in [1], [4] and [21]. Moreover, a strong
Received by the editors January 5, 2004.
2000 Mathematics Subject Classification. Primary 35B60, 74B05; Secondary 74G75.
Key words and phrases. Unique continuation, anisotropic elasticity system, inverse problems.
The first author was partially supported by Grant-in-Aid for Scientific Research (B)(2) (No.14340038) of the Japan Society for the Promotion of Science.
The second author was partially supported by the National Science Council of Taiwan.
c
2006 American Mathematical Society
Reverts to public domain 28 years from publication
unique continuation property for the isotropic system was recently proven in [2] and in [15]. Unlike the isotropic case, the unique continuation property for the inhomogeneous anisotropic elasticity has not been fully explored. In this direction, the authors of this paper have proved the UCP for an elasticity system with residual stress [18]. It is known that this system is no longer isotropic due to the existence of residual stress. A strong unique continuation for the elasticity system with residual stress was recently proved in [14].
In this paper we want to investigate the UCP for (1.2) when the elastic tensor is anisotropic with finite smoothness. To the authors’ knowledge, this problem has not been studied before. For our UCP result in this paper, we assume that the elastic tensor is merely a locally Lipschitz function in Ω. One of the key ideas in our proof is that under suitable conditions the original elasticity system (1.2) can be transformed into a first order elliptic system locally (see (2.11)). Therefore, proving the UCP for (1.2) is now reduced to proving the UCP for (2.11). There were several results on the UCP for first order elliptic systems; see, for example, [3], [5] and [8]. However, in those papers, the matrix function N in (2.11) is assumed to be either (locally) diagonalizable ([3], [5]) by a C1 invertible matrix or normal ([8]). In this work we are not assuming that N is normal. Instead, we will suppose that near every point x0 ∈ Ω the quadratic pencil Λ11p2+ Λ12p + Λ22 (see (2.12)) has at least one eigenvalue θ(x) with associated eigenvector z(x), and both are Lipschitz, such that the matrix function [z, ¯z] is nonsingular. Under this assumption, we can show that the diagonal blocks of N are Lipschitz diagonalizable. From now on, we say that a matrix function is Lipschitz diagonalizable if it can be similarly transformed to a diagonal matrix with Lipschitz entries by a Lipschitz invertible matrix. Nevertheless, it should be emphasized that the whole matrix N is not necessarily Lipschitz diagonalizable. There are two obstructions preventing it from having this property. On one hand, the two diagonal blocks may have common eigenvalues. On the other hand, the lower left block of N , i.e. the (2, 1) block of N , is only L∞. In view of these two points, it is, in general, not possible to find a Lipschitz invertible matrix to diagonalize N . In other words, we have to treat a lower triangular matrix function. Our proof of the UCP for (1.2) via (2.11) relies on some delicate Carleman estimates. To deal with the lower triangular matrix function N , we will borrow some ideas from [21] (or [18]). The proof of the UCP is given in Section 2.
Also in Section 2, we give an example to show that the assumption on one eigenpair of the quadratic pencil Λ11p2+ Λ12p + Λ22 is not too restrictive. This condition is in fact generic. Moreover, it is easy to verify that this assumption holds true when the elastic tensor is isotropic. Therefore, as a by-product of our UCP result, we prove that the UCP holds for the two-dimensional isotropic elasticity system with locally Lipschitz Lam´e coefficients, which is an improvement of previous UCP results for the same system in two dimensions. We remark that for the UCP result obtained in [1] or [21], the Lam´e coefficients are required to be at least C2, while for the strong UCP proved in [2], one needs C1,1 coefficients.
Our study of the UCP for the inhomogeneous two-dimensional anisotropic elas-ticity system (1.2) is motivated by its application to inverse problems. Having proved the UCP for (1.2), in Section 3 we will establish reconstruction algorithms for identifying the inclusion or cavity embedded inside a plane anisotropic elastic body by infinitely many localized boundary measurements with the help of Runge’s
approximation property. For the inclusion identification in the conductive medium, we refer to the pioneer work by Isakov [13]. It is well known that Runge’s ap-proximation property is an easy consequence of the UCP. Here the reconstruction algorithms of determining the inclusion or cavity are based the probe method de-veloped by Ikehata in [9] (see [10] and references therein for related results). It should be noted that Runge’s approximation theorem for the anisotropic elasticity was proved in [11] and [12]. However, the elasticity tensor there was assumed to be either homogeneous or real-analytic. The UCP is an obvious fact in these two situations. For other interesting inverse boundary value problems, we refer readers to a nice survey article [20].
2. Unique continuation property
This section is devoted to the proof of the UCP for (1.2) when the elastic tensor C(x) is locally Lipschitz. We will first transform the system (1.2) into a first order elliptic system with appropriate matrix coefficients. Next we will derive some useful Carleman estimates. The proof of the UCP for (1.2) is carried out with the help of those Carleman estimates. Throughout this section, in addition to the symmetric properties (1.1), we assume that the elastic tensor satisfies the strong ellipticity condition, i.e. there exists δ > 0 so that for any vectors a =t[a1, a2] and b =t[b1, b2] we have that
(2.1) Cijkl(x)aibjakbl≥ δ|a|2|b|2
for all x∈ Ω.
2.1. First order elliptic system. Our goal here is to transform (1.2) into a first order elliptic system. To this end, we would like to express (1.2) in a more detailed form, namely,
(2.2) LCu = Λ11∂12u + Λ12∂1∂2u + Λ22∂22u + R(u) = 0 a.e. in Ω, where
Λ11= (Ci1k1), Λ22= (Ci2k2), Λ12= Φ +tΦ with Φ = (Ci2k1)
and
R(u) =
jkl
(∂jCijkl)∂luk.
Since C(x) is locally Lipschitz, it is well known that ∂jCijkl(x) is locally bounded
for all i, j, k, l. In other words, R is a first order differential operator with locally bounded coefficients. From the strong ellipticity condition (2.1), we can see that Λ11 and Λ22 are invertible (in fact, positive definite). Let x0∈ Ω and ωx0 ⊂ Ω be a small open neighborhood of x0. We now define W =t[w1, w2] =t[u, ∂1u + T ∂2u] for x∈ ωx0, where T = T (x) is a Lipschitz invertible matrix (in ωx0) which will be chosen later. Thus, we can compute that
(2.3)
∂1w1= ∂1u = (∂1u + T ∂2u)− T ∂2u =−T ∂2w1+ w2, ∂1w2= ∂12u + T ∂1∂2u + ∂1T ∂2w1.
Here and below, to simplify expressions, all equations are interpreted in the sense of a.e. in ωx0 unless otherwise indicated. It follows from (2.2) that
Using the definition of w2, we immediately see that ∂2w2= ∂1∂2u + T ∂22u + ∂2T ∂2w1, which implies
(2.5) ∂1∂2u =−T ∂22u− ∂2T ∂2w1+ ∂2w2. Plugging (2.5) into (2.4) yields
(2.6) ∂21u =−Λ−111Λ12(−T ∂22u− ∂2T ∂2w1+ ∂2w2)− Λ−111Λ22∂ 2
2u− Λ−111R(u). Now substituting (2.5) and (2.6) into the second equation of (2.3) we have that (2.7) ∂1w2 ={−Λ−111Λ12(−T ∂22u− ∂2T ∂2w1+ ∂2w2)− Λ−111Λ22∂22u− Λ−111R(u)} +T (−T ∂2 2u− ∂2T ∂2w1+ ∂2w2) + ∂1T ∂2w1 =−(T2− Λ−1 11Λ12T + Λ−111Λ22)∂22u + (Λ−111Λ12∂2T− T ∂2T + ∂1T )∂2w1 +(T − Λ−111Λ12)∂2w2− Λ−111R(u).
By the relations ∂1u = ∂1w1 and ∂2u = −T−1∂1w1+ T−1w2, we can see that R is a first order linear operator containing only x1 derivative acting on w1. More precisely, we can derive
(2.8)
R(u) = ∂1Λ11∂1u + ∂1tΦ∂2u + ∂2Φ∂1u + ∂2Λ22∂2u
= (∂1Λ11+ ∂2Φ)∂1w1+ (∂2Λ22+ ∂1tΦ)(−T−1∂1w1+ T−1w2)
= (∂1Λ11+ ∂2Φ− T−1∂2Λ22− T−1∂1tΦ)∂1w1+ (∂2Λ22+ ∂1tΦ)T−1w2. Replacing R(u) in (2.7) by (2.8) and choosing T such that
(2.9) T2− Λ−111Λ12T + Λ−111Λ22= 0 ∀ x ∈ ωx0 or equivalently
Λ11(−T )2+ Λ12(−T ) + Λ22= 0 ∀ x ∈ ωx0, we get from (2.3) that
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ∂1w1+ T ∂2w1− w2= 0, Λ−111(∂1Λ11+ ∂2Φ− T−1∂2Λ22− T−1∂1tΦ)∂1w1 +∂1w2− (Λ−111Λ12∂2T− T ∂2T + ∂1T )∂2w1 +(Λ−111Λ12− T )∂2w2+ Λ−111(∂2Λ22+ ∂1 tΦ)T−1w 2= 0. Equivalently, in the matrix form, we have that
(2.10) A∂1W + B∂2W + F W = 0, where A = I 0 Λ−111(∂1Λ11+ ∂2Φ− T−1∂2Λ22− T−1∂1tΦ) I , B = T 0 −(Λ−111Λ12∂2T− T ∂2T + ∂1T ) Λ−111Λ12− T , and F = 0 −I 0 Λ−111(∂2Λ22+ ∂1tΦ)T−1 .
The matrix A is obviously invertible. We at once deduce the following first elliptic system from (2.10):
with N = A−1B = T 0 n Λ−111Λ12− T and M = A−1F, where n =−Λ−111(∂1Λ11+ ∂2Φ− T−1∂2Λ22− T−1∂1tΦ)T− (Λ−111Λ12∂2T− T ∂2T + ∂1T ). It should be noted that n(x) is in L∞(ωx0) and so is M .
Now we are at a position to discuss the solvability of (2.9) and the Lipschitz diagonalizabilities of T and Λ−111Λ12− T .
Proposition 2.1. Let ωx0 ⊂ Ω be a sufficiently small open neighborhood of x0and
let (θ(x), z(x)) be a local eigenpair of the quadratic pencil
(2.12) Λ11p2+ Λ12p + Λ22,
i.e.
(Λ11θ2+ Λ12θ + Λ22)z = 0 ∀ x ∈ ωx0.
Assume that θ(x), z(x) are Lipschitz and the matrix function [z, ¯z] is nonsingular for all x∈ ωx0. Then by choosing
(2.13) −T = [z, ¯z]diag(θ, ¯θ)[z, ¯z]−1
we have that T (x) satisfies (2.9) and is Lipschitz diagonalizable. Moreover, by possibly shrinking the neighborhood ωx0, there exist a Lipschitz invertible matrix
Q(x) and a Lipschitz diagonal matrix diag(ρ, ¯ρ) such that
Q(x)−1(Λ−111Λ12− T )Q(x) = diag(ρ(x), ¯ρ(x)) ∀ x ∈ ωx0.
Proof. By the strong ellipticity condition (2.1) and the fact that Λ11, Λ12, Λ22 are real, we can see that the eigenvalues of the quadratic pencil Λ11p2+ Λ12p + Λ22 are all genuine complex values (with nonvanishing imaginary parts) and appear in conjugate pairs. Therefore, if (θ, z) is an eigenpair of the quadratic pencil, then so is (¯θ, ¯z), i.e.
(Λ11θ¯2+ Λ12θ + Λ¯ 22)¯z = 0.
Clearly ¯θ(x) and ¯z(x) are all Lipschitz. Let T be defined by (2.13); then we have that
(Λ11(−T )2+Λ12(−T )+Λ22)[z, ¯z] = [(Λ11θ2+Λ12θ+Λ22)z, (Λ11θ¯2+Λ12θ+Λ¯ 22)¯z] = 0, which implies (2.9).
Having found T satisfying (2.9), we observe that
(2.14)
det(T − Λ−111Λ12− pI) det(−T − pI) = det(T− Λ−111Λ12− pI)(−T − pI)
= detp2I + Λ−111Λ12p + Λ−111Λ12T − T2 = detp2I + Λ−1 11Λ12p + Λ−111Λ22 = det(Λ−111) det(Λ11p2+ Λ12p + Λ22).
It is easily seen from (2.14) that if {θ, ¯θ, −ρ, −¯ρ} are eigenvalues of the quadratic pencil Λ11p2+ Λ12p + Λ22, then {ρ, ¯ρ} are eigenvalues of Λ−111Λ12− T . Note that ρ(x) is a genuine complex function and, hence, ρ(x) and ¯ρ(x) are distinct. In other words, ρ(x) and ¯ρ(x) are simple eigenvalues of the matrix Λ−111Λ12− T . Note that the matrix Λ−111Λ12− T is not continuously differentiable. So we cannot use the classical implicit function theorem to conclude that ρ(x) is Lipschitz. However,
since ρ(x) and ¯ρ(x) are simple roots of a polynomial with Lipschitz coefficients, we can apply a generalized version of the implicit function theorem proved by Hildebrandt and Graves [7] (or see [22, Theorem 4.B]) to get that ρ(x) and ¯ρ(x) are indeed Lipschitz in a possibly smaller ωx0. Now let ˜q1be a constant vector so that [(Λ−111Λ12− T )(x0)− ¯ρ(x0)I]˜q1= 0; then q1(x) = [(Λ−111Λ12− T )(x) − ¯ρ(x)I]˜q1= 0 and [(Λ−111Λ12−T )(x)−ρ(x)I]q1(x) = 0 for all x∈ ωx0 (possibly smaller). Note that
q1(x) is as smooth as Λ−111Λ12− T and ¯ρ. Repeating the same argument for ¯ρ, we can construct q2(x), which is an eigenvector associated with ¯ρ(x). Since ρ(x) and ¯
ρ(x) are distinct in ωx0, Q(x) = [q1, q2](x) must be nonsingular there (see similar
arguments in [8]).
Now we provide an example showing that the assumptions given in Proposi-tion 2.1 are in fact generic.
Example 1. Consider the two-dimensional orthotropic medium where the
associ-ated matrices Λ11, Λ12, and Λ22 are defined as Λ11= C11 0 0 C66 , Λ12= 0 C12+ C66 C12+ C66 0 , and Λ22= C66 0 0 C22 .
Note that C11, C22, C66are all positive. The related quadratic pencil is now written
as
C11p2+ C66 (C12+ C66)p (C12+ C66)p C66p2+ C22
.
We can show that if all C’s are locally Lipschitz functions and satisfy the relation (2.15) (C11C22− C122 − 2C12C66)2− 4C11C22C662 =: ∆ > 0 and C12+ C66= 0 ∀ x ∈ Ω, or (2.16) ∆ < 0, C12+ C66= 0, and C11C22− C66= 0 ∀ x ∈ Ω,
then the assumptions in Proposition 2.1 hold true. To verify this, we first observe that if ∆= 0, then det C11p2+ C66 (C12+ C66)p (C12+ C66)p C66p2+ C22 = C11C66p4+ [(C11C22+ C662 )− (C12+ C66)2]p2+ C22C66= 0 gives rise to four distinct roots, which are locally Lipschitz. If ∆ > 0, then all roots are purely imaginary (p2is real-valued). Now let θ(x) be any root (eigenvalue) and choose the associated eigenvector
z(x) = 1 −(C11θ2+C66) (C12+C66)θ .
Then it is easy to see that [z, ¯z] is nonsingular because the second component of z has a nonvanishing imaginary part. Thus, we have verified condition (2.15).
Now we look at condition (2.16). Let θ(x) and z(x) be chosen as above. In this case, θ2is a complex-valued function (with nonvanishing real and imaginary parts) and so is θ. To see that [z, ¯z] is nonsingular, it suffices to show that (C11θ2+C66)
a nonvanishing imaginary part. We prove this by contradictory arguments. Assume that (C11θ2+C66)
(C12+C66)θ becomes real-valued at some point x0. Then we must have that (C11θ2+ C66) (C12+ C66)θ =(C11 ¯ θ2+ C 66) (C12+ C66)¯θ at x0 which leads to C11|θ|2= C66 at x0. Note that|θ|2= C22
C11. Therefore, we have that
√
C11C22= C66 at x0, which is a
contradiction.
In next example we consider the isotropic case. It turns out that the assumptions in Proposition 2.1 hold without further restriction on the Lam´e coefficients other than the usual strong ellipticity condition.
Example 2. Let the medium be isotropic; then we have that
C11= C22= λ + 2µ, C12= λ, C66= µ
in Example 1, where λ(x) and µ(x) are locally Lipschitz functions in Ω. Assume that µ(x) > 0 and λ(x)+2µ(x) > 0 for all x∈ Ω. Then the eigenvalues of the related quadratic pencil are±i (repeated) with associated eigenvectorst[1,∓i]. Therefore, if we choose θ = i and z =t[1,−i], then the assumption of Proposition 2.1 is clearly satisfied. Furthermore, we can compute that
Λ−111Λ12− T = 0 λ+2µ−µ λ+2µ µ 0 . So when we take Q = 1 1 −λ+2µ µ i λ+2µ µ i , we get that Q−1(Λ−111Λ22− T )Q = diag(i, −i). Now, in view of Proposition 2.1, we define a Lipschitz invertible matrix
P (x) =
[z, ¯z] 0
0 Q
and set W = P V in ωx0. Then from (2.11) we obtain that (2.17) ∂1V + N ∂2V + M V = 0 (a.e. in ωx0), where N = diag(−µ, −¯µ) 0 n diag(ρ, ¯ρ) and M = P−1∂1P + P−1N ∂2P + P−1M P ∈ L∞(ωx0).
As we have mentioned in the Introduction, N is not necessarily Lipschitz diagonal-izable since−µ(x) and ρ(x) may coincide at some points in ωx0, and n(x) is merely essentially bounded. Now, to prove the UCP for (1.2), it suffices to prove that for (2.17). To do this, we will derive some Carleman estimates, which will be carried out in the following section.
2.2. Carleman estimates. Let Lλv = ∂1v +λ(x)∂2v be a first order scalar elliptic operator, where λ(x) = α(x) + iβ(x) is a locally Lipschitz function and β(x) never vanishes inR2. Given 0 < r0< 1, define U ={v ∈ C0∞(R2\ {0}) : supp(v) ⊂ Br0}, where Br0 is the disk centered at the origin with radius r0. From now on, we use c or ˜c to denote general positive constants whose values may vary from line to line.
Theorem 2.2. There exist a constant c > 0 and a sufficiently large number s0> 0 such that for all v∈ U and s ≥ s0 we have that
(2.18) s2
r−σ−s−2φ2s|v|2dx≤ c
r−σφ2s|Lλv|2dx,
where r =|x|, φs= exp(r−s) and σ = σs= σ0+ σ1s with σ0, σ1∈ R.
Proof. The proof of the theorem is motivated by Hile and Protter’s paper [8] in which they considered the system of equations without the parameter σ. As in [8], we will work on a slightly different operator instead of Lλ. For simplicity, we
denote L := Lλ. Define the scalar function
η(x) =1 2( ∂1β β + i∂2β + α ∂2β β ) and consider the operator
Lv = Lv + ηv = ∂1v + λ∂2v + ηv. It is easily seen that there exists a constant c > 0 such that
(2.19) |Lv| ≤ |Lv| + c|v|.
Now we set ψ(r) = r−s−σ
2log r and denote w = e
ψv. Then we can find that
(2.20)
e2ψ|Lv|2dx =
|∂1w + λ∂2w− ∂1ψw− λ∂2ψw + ηw|2dx. Following the notations in [8] we set
a1= ∂1w, a2= α∂2w, a3= iβ∂2w, a4=−(∂1ψw + α∂2ψw), a5=−iβ∂2ψw, a6= 12∂β1βw, a7= 12i∂2βw, a8= 12(α∂2ββw). Now from (2.20) we can write
(2.21) e2ψ|˜Lv|2dx =|8 j=1aj| 2dx ={|a1+ a2+ a5+ a6+ a8|2+ 2Re
(a1+ a2+ a5+ a6+ a8)· (a3+ a4+ a7) +|a3+ a4+ a7|2}dx
≥ 2Re(a1+ a2+ a5+ a6+ a8)· (a3+ a4+ a7)dx.
It is readily seen that Re(a2¯a3) = Re(a5¯a4) = Re(a6¯a7) = Re(a8¯a7) = 0 and 2Re(a2¯a7+ a8¯a3) = Re(−iα∂2β( ¯w∂2w + w∂2w)) = 0.¯
We at once deduce that 2Re a2¯a3dx = 2Re a5¯a4dx = 2Re a6¯a7dx = 2Re a8¯a7dx = 2Re (a2¯a7+ a8¯a3)dx = 0.
Next, note that
Re(∂2(iβw∂1w) + ∂¯ 1(−iβw∂2w))¯ = Re(2iβ∂2w∂1w + i∂¯ 2βw∂1w¯− i∂1βw∂2w)¯ = 2Re(a1a¯3+ a1¯a7+ a6a¯3).
Therefore, performing the integration by parts leads to 2Re
(a1¯a3+ a1¯a7+ a6¯a3)dx = 0.
Thus, there are five terms left on the right side of (2.21) needed to be estimated, namely,
(2.22) 2Re
(a1¯a4+ a2¯a4+ a5(¯a3+ ¯a7) + a6¯a4+ a8¯a4)dx.
We begin with the first term of (2.22). Using the integration by parts, we get that (2.23) 2Re a1¯a4dx =−2Re ∂1w(∂1ψ ¯w + α∂2ψ ¯w)dx =−(∂1ψ∂1|w|2+ α∂2ψ∂1|w|2)dx ={(∂12ψ + α∂1∂2ψ)|w|2+ ∂1α∂2ψ|w|2}dx ≥(∂2 1ψ + α∂1∂2ψ)|w|2dx− c |∂2ψ||w|2dx. Similarly, we can find that
(2.24) 2Rea2¯a4dx =−2Re α∂2w(∂1ψ ¯w + α∂2ψ ¯w)dx =−(α∂1ψ∂2|w|2+ α2∂2ψ∂2|w|2)dx ={(α∂1∂2ψ + α2∂22ψ)|w|2+ (∂2α∂1ψ + ∂2α2∂2ψ)|w|2}dx ≥(α∂1∂2ψ + α2∂22ψ)|w|2dx− c (|∂2ψ| + |∂1ψ|)|w|2dx and (2.25) 2Rea5¯a3dx =−2Re β2∂2ψw∂2wdx¯ =−β2∂2ψ∂2|w|2dx =β2∂22ψ|w|2dx + ∂2β2∂2ψ|w|2dx ≥β2∂22ψ|w|2dx− c |∂2ψ||w|2dx. Furthermore, we can estimate
(2.26) ⎧ ⎪ ⎨ ⎪ ⎩ 2Re a5¯a7dx≥ −c |∂2ψ||w|2dx, 2Re a6¯a4dx≥ −c (|∂1ψ| + |∂2ψ|)|w|2dx, 2Re a8¯a4≥ −c (|∂1ψ| + |∂2ψ|)|w|2dx. Combining (2.23), (2.24), (2.25) and (2.26) yields
(2.27) 2Re (a1¯a4+ a2¯a4+ a5(¯a3+ ¯a7) + a6¯a4+ a8¯a4)dx ≥(∂2 1ψ + 2α∂1∂2ψ + (α2+ β2)∂22ψ)|w|2dx− c (|∂1ψ| + |∂2ψ|)|w|2dx. Through direct computations, we obtain that
∂jψ =− sr−s−2xj− (σ/2)r−2xj,
∂j2ψ =s(s + 2)r−s−4x2j− sr−s−2+ σr−4x2j− (σ/2)r−2, j = 1, 2, and
∂1∂2ψ = s(s + 2)r−s−4x1x2+ σr−4x1x2. Therefore, by taking s sufficiently large, we can get from (2.27) that (2.28) 2Re (a1¯a4+ a2¯a4+ a5(¯a3+ ¯a7) + a6¯a4+ a8¯a4)dx ≥ s2r−s−4(x2 1+ 2αx1x2+ (α2+ β2)x22)|w|2dx− cs r−s−2|w|2dx.
Note that
(2.29) x21+ 2αx1x2+ (α2+ β2)x22=|x1+ λx2|2. By easy computations, we have the following identity:
(1 +|λ|2)|x1+ λx2|2= β2(x21+ x 2 2) + (x1+ αx2) 2+ (αx1+ α2x 2+ β2x22) 2, which leads to (2.30) |x1+ λx2|2≥ β2r2 1 + sup Br0 |λ|2 ∀ x ∈ Br0.
In view of the assumption on the imaginary part of λ, we immediately get that (2.31) |β(x)| ≥ δ > 0 ∀x ∈ Br0.
Substituting (2.29), (2.30), (2.31) into (2.28) and taking s large enough, we conclude that
2Re
(a1¯a4+ a2¯a4+ a5(¯a3+ ¯a7) + a6¯a4+ a8¯a4)dx≥ cs2 r−s−2|w|2dx and, hence, e2ψ|Lv|2dx = r−σφ2s|Lv|2dx≥ cs2 r−s−2r2ψ|v|2dx = cs2 r−σ−s−2φ2s|v|2dx.
Finally, in view of (2.19), we obtain the estimate (2.18). To handle the possible nonzero off-diagonal block in N , we need another Carle-man estimate.
Theorem 2.3. There exist a constant c > 0 and a sufficiently large number s0> 0 such that for all v∈ U and s ≥ s0 we have the following estimate:
(2.32)
r−σφ2s|∂2v|2dx≤ c
r−σφ2s(|Lλv|2+ s2r−2s−2|v|2)dx.
Proof. To begin with, we set v = vR+ ivI and Lλ= L. Now we can compute
(2.33) r−σφ2 s|Lv|2dx = r−σφ2 s{|∂1v|2+|λ|2|∂2v|2+ 2Re(∂1vλ∂2v)}dx = r−σφ2 s(|∂1v|2+ (α2+ β2)|∂2v|2)dx + 2 r−σφ2 sα(∂1vR∂2vR+ ∂1vI∂2vI)dx −2r−σφ2 sβ(∂1vR∂2vI− ∂1vI∂2vR)dx.
Using the inequality
(2.34) |ab| ≤ (εa2+ ε−1b2)/2 for any ε > 0, we have that (2.35) | r−σφ2sα(∂1vR∂2vR+ ∂1vI∂2vI)dx| ≤1 2 r−σφ2s{(∂1vR)2+ (∂1vI)2}dx +12 r−σφ2sα2{(∂2vR)2+ (∂2vI)2}dx =ε1 2 r−σφ2 s|∂1v|2dx +12ε−11 r−σφ2 sα2|∂2v|2dx,
where ε1 > 0 will be chosen later. On the other hand, from the integration by parts, one can easily derive that
(2.36) |r−σφ2 sβ(∂1vR∂2vI− ∂1vI∂2vR)dx| =| −∂1(r−σφ2 sβ)∂2vIvRdx + ∂2(r−σφ2 sβ)∂1vIvRdx| ≤ |{(−σr−σ−2− 2sr−σ−s−2)φ2 sx1β∂2vIvR+ r−σφ2s∂1β∂2vIvR}dx| +|{(−σr−σ−2− 2sr−σ−s−2)φ2sx2β∂1vIvR+ r−σφ2s∂2β∂1vIvR}dx| ≤ |(−σ)r−σ−2φ2sx1β∂2vIvRdx| + | (−2s)r−σ−s−2φ2sx1β∂2vIvRdx| +|(−σ)r−σ−2φ2sx2β∂1vIvRdx| + | (−2s)r−σ−s−2φ2sx2β∂1vIvRdx| +|r−σφ2 s∂1β∂2vIvRdx| + | r−σφ2 s∂2β∂1vIvRdx| ≤ 1 2ε2 r−σφ2 sβ2(∂2vI)2dx + 12|σ|2ε−12 r−σ−4φ2 sx21v2Rdx +12ε2 r−σφ2sβ2(∂2vI)2dx + 2s2ε−12 r−σ−2s−4φ2sx21v2Rdx +12ε2 r−σφ2 s(∂1vI)2dx + 12|σ|2ε−12 r−σ−4φ2 sx22β2vR2dx +12ε2 r−σφ2s(∂1vI)2dx + 2s2ε−12 r−σ−2s−4φ2sx22β2v2Rdx +12ε2 r−σφ2sβ2(∂2vI)2dx + 12ε−12 r−σφ2s(∂1β β ) 2v2 Rdx +12ε2 r−σφ2 s(∂1vI)2dx + 12ε−12 r−σφ2 s(∂1β)2v2Rdx ≤ 3ε2 2 r−σφ2s{(∂1vI)2+ β2(∂2vI)2}dx + cs2ε−12 r−σ−2s−2φ2sv2Rdx,
where ε2> 0 will be determined below. In deriving (2.36), we once again used the inequality (2.34). Combining (2.33), (2.35), (2.36) and choosing 3ε2= 1− ε1 with 0 < ε1< 1, we obtain that (2.37) r−σφ2s|Lv|2dx ≥ (1 − ε1− 3ε2) r−σφ2 s|∂1v|2dx + r−σφ2 s(α2+ β2− ε−11 α2− 3ε2β2)|∂2v|2dx −2cε−1 2 s 2 r−σ−2s−2φ2 s|v|2dx ≥ r−σφ2 s((1− ε−11 )α2+ ε1β2)|∂2v|2dx− 6c(1 − ε1)−1s2 r−σ−2s−2φ2 s|v|2dx.
Since|β(x)| ≥ δ > 0 for all x ∈ Br0, we can find an ε1 sufficiently close to 1 such that
(2.38) (1− ε−11 )α2+ ε1β2≥ δ > 0 ∀ x ∈ Br0.
Now the Carleman estimate (2.32) follows from (2.37) and (2.38). 2.3. Proof of the UCP. Here we will prove a weaker version of UCP for (1.2). Namely, we would like to show that a solution u of (1.2) that vanishes at one point in Ω of any exponential order must vanish in Ω provided that the assumption of Proposition 2.1 holds near every point in Ω. It is clear that the UCP follows from this version of continuation property. We say that u vanishes at x0 of any exponential order if
lim
|x−x0|→0
exp(|x − x0|−s)u(x) = 0 ∀ s > 0.
Theorem 2.4. Assume that the assumptions of Proposition 2.1 are satisfied near
every point in Ω. Let u vanish at some x0∈ Ω of any exponential order; then u ≡ 0 in Ω.
Proof. Without loss of generality, we assume x0 = 0. Let the assumptions of Proposition 2.1 hold in ω0 ⊂ Ω, where ω0 is a small neighborhood of 0. Then we can convert (1.2) into (2.17) in ω0. Define r0 = min{1/2, dist(0, ∂ω0)}. Now let χ ∈ C0∞(R2) be a cut-off function satisfying 0 ≤ χ ≤ 1, χ|
Br0/2 = 1 and
supp (χ)⊂ Br0. Define V =
t(˜v
1, ˜v2, ˜v3, ˜v4) = χV . Then from (2.17), we get that (2.39) ∂1V + N ∂2V = −M V + G,
where G =t(g1, g2, g3, g4)∈ L2(Ω) is supported in Br0\ Br0/2. Denote λ1= ¯λ2= −θ and λ3= ¯λ4= ρ. We set H = γr−sφ2 s(|Lλ1˜v1| 2+|L λ2˜v2| 2)dx +φ2 s(|Lλ3˜v3| 2+|L λ4˜v4| 2)dx, K = γr−sφ2 s(|˜v1|2+|˜v2|2)dx + φ2 s(|˜v3|2+|˜v4|2)dx, J = γr−sφ2 s(|g1|2+|g2|2)dx + φ2 s(|g3|2+|g4|2)dx, where γ > 0 will be chosen later. It follows from (2.39) that
(2.40) H ≤ c(K + J +
φ2s(|∂2˜v1|2+|∂2v˜2|2)dx).
By the standard approximation argument, we can see that ˜vj satisfies estimates
(2.18) and (2.32) for all 1≤ j ≤ 4. Now replacing the last term of (2.40) by the estimate (2.32) for Lλ1 and Lλ2 with σ = 0, we have that
(2.41) H≤ c(K+J+ φ2s(|Lλ1˜v1| 2+|L λ2˜v2| 2)dx+s2 r−2s−2φ2s(|˜v1|2+|˜v2|2)dx). Substituting the estimate (2.18) for Lλ1 and Lλ2 with σ =−s into the last term of (2.41) leads to (2.42) H ≤ c(K + J + r−sφ2s(|Lλ1˜v1| 2+|L λ2˜v2| 2)dx).
In deriving (2.42), we also use the obvious inequality r−s > 1 for any 0 < r < 1 and s > 0. Taking γ 1, we can absorb the last term of (2.42) and get
(2.43) H ≤ c(K + J).
From now on we fix the parameter γ.
Next repeatedly using σ = s in (2.18) for Lλ1, Lλ2 and σ = 0 in (2.18) for Lλ3,
Lλ4, we find that (2.44) s2r−2s−2φ2 s(|˜v1|2+|˜v2|2)dx + s2 r−s−2φ2 s(|˜v3|2+|˜v4|2)dx ≤ c r−sφ2 s(|Lλ1˜v1| 2+|L λ2˜v2| 2)dx + φ2 s(|Lλ3˜v3| 2+|L λ4˜v4| 2)dx ≤ cH ≤ c(K + J).
In view of the inequalities 1 < r−s < r−s−2 < r−2s−2 for 0 < r < 1, by taking s 1, we get from (2.44) that
s2r−s−2φ2s|V|2dx ≤ s2 r−2s−2φ2 s(|˜v1|2+|˜v2|2)dx + s2 r−s−2φ2s(|˜v3|2+|˜v4|2)dx ≤ c r−sφ2s(|˜v1|2+|˜v2|2)dx + φ2s(|˜v3|2+|˜v4|2)dx +c r−sφ2s(|g1|2+|g2|2)dx + φ2s(|g3|2+|g4|2)dx ≤ c r−s−2φ2s|V|2dx + cr−s−2φ2s|G|2dx ≤ ˜c r−s−2φ2s|G|2dx, which leads to (2.45) s2 Br0/2 r−s−2φ2s|V|2dx≤ c Br0\Br0/2 r−s−2φ2s|G|2dx.
Since r−s−2φ2s is a strictly decreasing function, (2.45) implies that
s2 Br0/2 V dx≤ c Br0\Br0/2 |G|2dx <∞
and thus V = V = 0, i.e. u = 0, on Br0/2 if we choose s sufficiently large. Now the standard arguments imply that the set {x ∈ Ω : u = 0} is open and closed in Ω.
Since Ω is connected, we must have u≡ 0 in Ω.
In view of Example 2, we know that the assumption of Proposition 2.1 holds automatically for the isotropic elastic tensor. So from Theorem 2.4 we immedi-ately conclude that the UCP is valid for the isotropic elasticity system with locally Lipschitz Lam´e coefficients.
Corollary 2.5. Let C(x) be isotropic, i.e.
Cijkl(x) = λ(x)δijδkl+ µ(x)(δikδjl+ δilδjk).
Assume that λ, µ are locally Lipschitz in Ω and λ + 2µ > 0, µ > 0 for all x∈ Ω. Then (1.2) possesses the UCP.
3. Applications to inverse problems
As mentioned in the Introduction, we would like to investigate some applications of the UCP proved in the previous section to the object identification problem. Another application of the UCP to the limiting absorption principle for the same system was considered by the authors in [19]. To study our inverse problems here, the Runge approximation property with Dirichlet constraints for (1.2) is a key ingredient. It has been known that this Runge property is an easy consequence of the UCP. Its proof can be found, for example, in [11] or [12].
Theorem 3.1. Let O and Ω be two open bounded domains with Lipschitz
bound-aries such that ¯O ⊂ Ω. Assume that all components of C are uniformly Lipschitz functions in Ω. Moreover, suppose that the quadratic pencil Λ11p2+ Λ12p + Λ22 has at least one eigenpair (θ(x), z(x)) which is uniformly Lipschitz in Ω and [z, ¯z] is nonsingular for all x∈ ¯Ω. Denote Γ0a subset of the boundary ∂Ω. Let u∈ H1(O) satisfy
LCu = 0 in O.
Then for any compact subset K ⊂ O such that Ω \ K is connected and any ε > 0, there exists w∈ H1(Ω) satisfying
LCw = 0 in Ω
with supp (w|∂Ω)⊂ Γ0 such that
w − u H1(K)< ε.
Remark. Under the assumptions of Theorem 3.1, we can extend C, denoted by ˆC, to a slightly larger domain ˆΩ such that the same assumptions are satisfied for ˆC in
ˆ
Ω. Therefore, the UCP holds for (1.2) with ˆC in ˆΩ.
In this section we consider the inverse problem of identifying inclusions or cavities embedded in an anisotropic elastic plane region by boundary measurements. To begin, assume that D is an open subset of Ω with Lipschitz boundary such that Ω\ ¯D is connected. The domain D stands for the region of the inclusion or cavity embedded in Ω. In the degenerate case where the domain D represents the crack, the inverse problem of identifying D by near-field measurements was consider in [16] and [17]. To simply our presentation, we will not discuss this matter here.
Let the reference elastic tensor C(x) satisfy the conditions given in Theorem 3.1. Here we require that the elastic tensor C(x) satisfies the strong convexity condition, namely, there exists κ > 0 such that for any symmetric matrix E we have
(3.1) C(x)E· E ≥ κ
i,j
EijEij = κ|E|2 ∀ x ∈ Ω.
It is obvious that (3.1) implies the strong ellipticity condition (2.1). Next we as-sume that ˜C(x) is some fourth-rank symmetric tensor with L∞ components such that C + χDC satisfies the strong convexity condition (3.1), where χ˜ Ddenotes the
characteristic function of D. Then it is easy to show that there exists a unique solution u∈ H1(Ω) to
∇ · ((C + χDC)˜ ∇u) = 0 in Ω,
u = f on ∂Ω
for any f ∈ H1/2(∂Ω). So we can define the Dirichlet-to-Neumann (displacement-to-traction) map Λinc: H1/2(∂Ω)→ H−1/2(∂Ω) by
Λinc(f ) = (C∇u)ν|∂Ω,
where ν stands for the unit outer normal of ∂Ω (or ∂D). Equivalently, Λinc can be
defined by the formula
Λinc(f ), g =
Ω
(C + χDC)˜ ∇u · ∇vdx,
where v∈ H1(Ω) with v|
∂Ω= g. The region D is called an inclusion if the medium
in D is different from the reference medium. To describe it more precisely, we assume that ˜C satisfies the following jump condition:
(3.2)
∀ x ∈ ∂D, ∃ Cx> 0, ∃ δx> 0 such that ˜C(y)E·E ≥ Cx|E|2or ˜C(y)E·E ≤ −Cx|E|2
for almost all y∈ Bδx(x)∩ D and any symmetric matrix E. We are interested in
the following inverse problem
IP.I Reconstruct the inclusion D from the knowledge of Λinc(f )|Γ0for infinitely many f ∈ H1/2(∂Ω) with supp (f )⊂ Γ
0, where Γ0 is a nonempty subset of ∂Ω. Likewise, in the extreme case, if the tensor ˜C becomes−C, then the domain D corresponds to a cavity. In the same way, we can prove that there exists a unique solution u∈ H1(Ω\ ¯D) to the following boundary value problem:
∇ · (C∇u) = 0 in Ω \ ¯D,
(C∇u)ν = 0 on ∂D, (C∇u)ν = g on ∂Ω
for any g ∈ H1/2(∂Ω). Therefore, we can define the Dirichlet-to-Neumann map Λcav: H1/2(∂Ω)→ H−1/2(∂Ω) by
Λcav(g) = (C∇u)ν|∂Ω.
Similarly, we will consider the inverse problem
IP.C Reconstruct the cavity D from the knowledge of Λcav(g)|Γ0 for infinitely many g with supp (g)⊂ Γ0.
Note that uniqueness theorems of determining the inclusion or cavity embedded in an elastic body have been established in [11] and [12], where the reference medium is assumed to be either inhomogeneous isotropic or anisotropic with homogeneous
or analytic elasticity tensors. Besides, a reconstruction algorithm for recovering the cavity is given in [12]. A similar algorithm can be developed for the inclusion case. Having the Runge approximation property Theorem 3.1 at hand, we can now apply the methods in [11] and [12] to solve IP.I and IP.C. For IP.I, we prove that (see [11])
Theorem 3.2 (Identification of inclusion). Let (D1, ˜C1) and (D2, ˜C2) be two sets of inclusion data satisfying all conditions stated in this section. If
Λinc1(f ) = Λinc2(f ) on Γ0
for all f∈ H1/2(∂Ω) with supp (f )⊂ Γ0, then D1= D2.
The proof of Theorem 3.2 is based on integral inequalities (3.3) D {C−1−(C + ˜C)−1}C∇w ·C∇wdx ≤ (Λ inc−Λ∅)f, f ≤ D ˜ C∇w ·∇wdx, where w∈ H1(Ω) solves (3.4) ∇ · (C∇w) = 0 in Ω, w|∂Ω= f
and Λ∅ is the Dirichlet-to-Neumann map when D is absent, namely, Λ∅ is the map associated with (3.4). Here C−1 (also (C + ˜C)−1) is called the compliance tensor (see, e.g., [6]). Note that we do not assume ˜C1 = ˜C2 in Theorem 3.2. Also, the regularity of the medium inside of the inclusions is only assumed to be essentially bounded. Theorem 3.2 provides the uniqueness result in determining the inclusion embedded in an inhomogeneous anisotropic elastic plane region by the localized Dirichlet to Neumann map. For the sake of completeness, we want to briefly describe a reconstruction algorithm for identifying the inclusion. Let y∈ Ω and G0(·; y) be the fundamental solution for the operator ∇ · C(y)∇. One can find e(·; y) such that
∇ · (C(x)∇e(·; y)) = 0 in Ω \ {y} and
(e(·; y) − G0(· − y; y)b)y∈Ω is bounded in H1(Ω),
where b is a nonzero constant vector. The proof for the existence of e(·; y) is elementary and therefore we leave it to the reader. It has been proved in [11] that if y∈ ∂D, then
(3.5)
D∩Br(y)
|Sym∇{G0(x− y; y)b}|2dx =∞
for any ball Br(y) centered at y with radius r and nonzero vector b, where Sym(·)
denotes the symmetric part of a matrix.
A continuous map c : [0, 1]→ ¯Ω is called a needle if it satisfies (i) c(0), c(1) ∈ ∂Ω; (ii) c(t) ∈ Ω for 0 < t < 1. In view of Theorem 3.1, we can see that for each needle and t∈ (0, 1), there exists a sequence {fj} = {fj(·; c(t))} in H1/2(∂Ω) with
supp (fj)⊂ Γ0such that the solution wjof (3.4) with f = fjsatisfies wj→ e(·; c(t))
in H1
loc(Ω\ {c(t) : 0 < t ≤ t}) as j → ∞. We call {fj} a fundamental sequence
with respect to Γ0. For each needle c, define
It should be noted that 0 < t(c)≤ 1, and if t(c) = 1, then c never touches ∂D. On the other hand, if t(c) < 1, then c touches ∂D at t = t(c) at the first time. Since Ω\ ¯D is connected, we have that
(3.6) ∂D ={c(t(c)) : c is a needle and t(c) < 1}. Denote
Iinc(t, c) = lim
j→∞(Λinc− Λ∅)fj(·; c(t)), fj(·; c(t))
and
Tinc(c) ={0 < s < 1 : Iinc exists for all 0 < t < s and sup
0<t<s|I
inc(t, c)| < ∞}.
Using (3.2), (3.3), (3.5) and pursuing the arguments in [11], we can show that Tinc(c) = (0, t(c)) and therefore t(c) = supTinc(c) (see similar arguments in [12]).
In summary, we have a reconstruction algorithm for determining the inclusion as follows.
Reconstruction algorithm for IP.I.
(i) For each needle c and each t∈(0, 1), find the fundamental sequence {fj(·; c(t))}
with respect to Γ0.
(ii) ComputeTinc(c) and set t(c) = supTinc(c).
(iii) Use the formula (3.6) to reconstruct ∂D. Now for IP.C, we show that (see [12])
Theorem 3.3 (Identification of cavity). Assume that D1 and D2 are two cavities and Ω\ ¯D1 and Ω\ ¯D2 are connected. Let
Λcav1(f ) = Λcav2(f ) on Γ0
for all f∈ H1/2(∂Ω) with supp (f )⊂ Γ0; then D1= D2.
As for reconstructing the cavity, we follow the lines of the above algorithm and define
Icav(t, c) = lim
j→∞(Λ∅− Λcav)fj(·; c(t)), fj(·; c(t))
and
Tcav(c) ={0 < s < 1 : Icav exists for all 0 < t < s and sup
0<t<sI
cav(t, c) <∞}.
Note that (Λ∅− Λcav)f, f ≥ 0 for all f ∈ H1/2(∂Ω). Now using (3.5) and the
inequalities 1 M D |∇e(x; c(t))|2dx≤ I cav(t, c)≤ M D |∇e(x; c(t))|2dx
for some constant M > 0, one can prove that Tcav(c) = (0, t(c)) and thus t(c) =
supTcav(c) (see the arguments in [12]). So a reconstruction algorithm for identifying
the cavity is described as follows.
Reconstruction algorithm for IP.C.
(i) For each needle c and each t∈(0, 1), find the fundamental sequence {fj(·; c(t))}
with respect to Γ0.
(ii) ComputeTcav(c) and set t(c) = supTcav(c).
References
[1] D.D. Ang, M. Ikehata, D.D. Trong and M. Yamamoto, Unique continuation for a
station-ary isotropic Lam´e system with varaiable coefficients, Comm. in PDE, 23 (1998), 371-385.
MR1608540 (98j:35049)
[2] G. Alessandrini and A. Morassi, Strong unique continuation for the Lam´e system of elasticity,
Comm. in PDE., 26 (2001), 1787-1810. MR1865945 (2003j:35030)
[3] T. Carleman, Sur un probl`em d’unicit´e pour les systemes d’´equations aux d´eriv´ees partielles `
a deux varaibles ind´ependentes, Ark. Mat., Astr. Fys., 26B (1939), 1-9.
[4] B. Dehman and L. Rabbiano, La propri´et´e du prolongement unique pour un syst`eme el-liptique: le syst`eme de Lam´e, J. Math. Pures Appl., 72 (1993), 475-492. MR1239100 (94h:35051)
[5] A. Douglis, Uniqueness in Cauchy problem for elliptic systems of equations, Comm. Pure Appl. Math., 6 (1953), 291-298. MR0064278 (16:257d)
[6] M. Gurtin, The Linear Theory of Elasticity, Mechanics of Solids, Vol. II (Thruesdell C., ed.), Springer-Verlag, Berlin, 1984.
[7] T. Hildebrandt and L. Graves, Implicit functions and their differentials in general analysis, Trans. Amer. Math. Soc., 29 (1927), 127-153. MR1501380
[8] G. Hile and M. Protter, Unique continuation and the Cauchy problem for first order systems
of partial differential equations, Comm. in PDE, 1 (1976), 437-465. MR0481439 (58:1555)
[9] M. Ikehata, Reconstruction of the shape of the inclusion by boundary measurements, Comm. in PDE, 23 (1998), 1459-1474. MR1642619 (99f:35222)
[10] M. Ikehata, The probe method and its applications, Inverse problems and related topics (Kobe, 1998), 57-68, Chapman and Hall/CRC Res. Notes Math., 419, Chapman & Hall/CRC, Boca Raton, FL, 2000. MR1761338 (2001c:35245)
[11] M. Ikehata, G. Nakamura, and K. Tanuma, Identification of the shape of the inclusion in the
anisotropic elastic body, Appl. Anal., 72 (1999), 17-26. MR1775432 (2001d:74026)
[12] M. Ikehata and G. Nakamura, Reconstruction of cavity from boundary measurements, preprint.
[13] V. Isakov, On uniqueness of recovery of a discontinuous conductivity coefficients, Comm. Pure Appl. Math, 41 (1988), 865-877. MR0951742 (90f:35205)
[14] C.-L. Lin, Strong unique continuation for an elasticity system with residual stress, Indiana Univ. Math. J., 53 (2004), 557-582. MR2060045 (2005f:35033)
[15] C.-L. Lin and J.-N. Wang, Strong unique continuation for the Lam´e system with Lipschitz coefficients, Math. Annalen, to appear.
[16] G. Nakamura, G. Uhlmann, and J.-N. Wang, Reconstruction of cracks in an anisotropic
elastic medium, J. Math. Pures Appl., 82 (2003), 1251-1276. MR2020922 (2005b:35299)
[17] G. Nakamura, G. Uhlmann, and J.-N. Wang, Reconstruction of cracks in an inhomogeneous
anisotropic medium using point sources, Advances in Appl. Math., to appear.
[18] G. Nakamura and J.-N. Wang, Unique continuation for an elasticity system with
resid-ual stress and its applications, SIAM J. Math. Anal., 35 (2003), 304-317. MR2001103 (2004k:35037)
[19] G. Nakamura and J.-N. Wang, The limiting absorption principle for the two-dimensional
inhomogeneous anisotropic elasticity system, Trans. Amer. Math. Soc., 358 (2006), 147-165.
[20] G. Uhlmann, Developments in inverse problems since Calderon’s foundational paper, Har-monic analysis and partial differential equations (Chicago, IL, 1996), 295-345, Chicago Lec-tures in Math., Univ. Chicago Press, Chicago, IL, 1999. MR1743870 (2000m:35181) [21] N. Weck, Unique continuation for systems with Lam´e principal part, Math. Methods Appl.
Sci., 24 (2001), 595–605. MR1834916 (2002f:35048)
[22] E. Zeidler, Nonlinear Function Analysis and its Applications, I. Fixed-Point Theorems, Springer-Verlag, New York, 1986. MR0816732 (87f:47083)
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
E-mail address: [email protected]
Department of Mathematics, National Taiwan University, Taipei 106, Taiwan
