UNIQUE CONTINUATION FOR THE ELASTICITY SYSTEM WITH RESIDUAL STRESS AND ITS APPLICATIONS

GEN NAKAMURA AND JENN-NAN WANG

Abstract. In this paper we prove the unique continuation property for the elasticity system with small residual stress. The elasticity system becomes anisotropic due to the existence of residual stress. The main technique in the proof is Carleman estimates. Having proved the unique continuation property, we will study the inverse problem of identifying the inclusion or cavity.

1. Introduction

Let B be an isotropic elastic body with residual stress and the reference configuration of B be Ω, a bounded open set in ℝ^{n} with smooth boundary. The residual stress is modeled by a symmetric, smooth, second-rank tensor T (x) = (t_{ij}(x))_{1≤i,j≤n} satisfying

∂_{x_{j}}t_{ij} = 0 in Ω, 1 ≤ i ≤ n (1.1)

and

t_{ij}ν_{j} = 0 on ∂Ω, 1 ≤ i ≤ n, (1.2)

where ν = (ν_{1}, ⋯, ν_{n}) is the unit outer normal to ∂Ω. Hereafter, we adopt the summation convention. Let u : Ω → ℝ^{n} be the displacement vector, then the first Piola-Kirchhoff stress is written as

σ = T + (∇u)T + λ(trϵ)I + 2μϵ + β_{1}(trϵ)(trT)I + β_{2}(trT)ϵ
+β_{3}((trϵ)T + tr(ϵT)I) + β_{4}(ϵT + T ϵ),

where λ, μ are the Lamé moduli and β_{1} ⋯ β_{4} are material parameters and

ϵ = (ϵ_{ij}) = \frac{1}{2}(∇u + (∇u)^{t})

is the strain tensor [16]. Moreover, we assume that the Lamé moduli satisfy the strong ellipticity condition

μ(x) > δ > 0, λ(x) + 2μ(x) > δ > 0 ∀x ∈ Ω (1.3)

and

β_{3} = β_{4} = 0,

i.e.,

σ = T + (∇u)T + λ̃(trϵ)I + 2μ̃ϵ,

JW is currently a postdoc at MSRI. Research at MSRI is supported in part by NSF grant DMS-9701755. GN is partially supported by Grant in Aid for Scientific Research (C) (No.12640153) of Japan Society for the Promotion of Science.

where

λ̃ = λ + β_{1}(trT), μ̃ = μ + \frac{1}{2}β_{2}(trT).

The stationary elasticity system is expressed as

(Lu)_{i} = (∇ · σ)_{i} + ω^{2}ρ(x)u_{i} = ∂_{j}σ_{ij} + ω^{2}ρ(x)u_{i} = 0 in Ω, 1 ≤ i ≤ n, ω ∈ ℝ, (1.4)

where ρ(x) > 0 is the density of the medium. In another setting, if we define the elasticity tensor ℂ with components

C_{ijkl} = λ̃δ_{ij}δ_{kl} + (μ̃δ_{jl} + t_{jl})δ_{ik} + μ̃δ_{il}δ_{jk} (1.5)

and denote

(ℂE)_{ij} = C_{ijkl}E_{kl} for any matrix E,

then (1.4) is equivalent to

(Lu)_{i} = (∇ · ℂ∇u)_{i} + ω^{2}ρu_{i} = ∂_{j}(C_{ijkl}∂_{l}u_{k}) + ω^{2}ρu_{i} = 0 in Ω, 1 ≤ i ≤ n.

It is clear to see that (1.4) is an anisotropic elasticity system. In this paper, we will investigate the (weak) unique continuation property (UCP) for the system (1.4), i.e. if u ∈ H_{loc}^{2}(Ω) is a solution to (1.4) in Ω and vanishes in a non-empty open subset of Ω, then u vanishes identically in Ω.

The unique continuation property for differential equations has a long history. Many deep results about scalar elliptic equations or elliptic systems have been established. We refer the reader to [3] and references therein for details. Recently, few attempts have been made at studying the unique continuation property for systems of equations in mathematical physics such as the Dirac equations and the Maxwell equations [4], [13], [17], [19], [20]. Here we mention two interesting articles [20] and [17] in which Vogelsang and Okaji, respectively, proved the strong unique continuation property for the Maxwell system with anisotropic coefficients. In this paper we pay our attention to the elasticity system. Several results of weak continuation property for the inhomogeneous isotropic elasticity have been obtained in [1], [5], (stationary) and [6], [12] (non-stationary). Moreover, a strong unique continuation property were recently proven by Alessandrini and Morassi [2]. Unlike the isotropic case, the unique continuation property for the inhomogeneous anisotropic elasticity has not been fully explored.

Our study of the unique continuation property for the inhomogeneous anisotropic elasticity is motivated by its application to inverse problems. It was first recognized by Lax [15] that the Runge approximation property is a consequence of the weak unique continuation property. The Runge approximation property is shown to be a useful technique in dealing with some inverse problems, especially the inverse problem of recovering inclusions or cavities (see [11], [7], [8], [9], [10], [14] and references therein). It should be noted that the Runge approximation property with constraint for the anisotropic elasticity were proved in [9] and [10]. However, the elasticity tensor there is assumed to be either homogeneous or real-analytic. The weak unique continuation property is an obvious fact in these two situations.

To prove the unique continuation property for the general inhomogeneous anisotropic elasticity is very challenging and difficult. Here we want to consider the system (1.4) which has the simplest form of anisotropy. It turns out we are able to establish the UCP for (1.4) provided the residual stress is sufficiently small. Our main idea comes from Weck’s recent article [21] where he proved the UCP for the isotropic elasticity system with zeroth or first order perturbations which contains the results previous obtained by [1], [5]. Weck actually proved something more, namely, he established the UCP for a rather general system of second order differential inequalities with the Laplacian principal part. Like many literature on the UCP, the key step in [21] is to prove appropriate Carleman estimates. Here we will adopt Weck’s approach to (1.4) with small residual stress, but we have to work a little harder to derive the desired Carleman estimates because we need to deal with variable coefficients second order principal parts due to the presence of residual stress. As indicated previously, having established the UCP, we can prove the Runge approximation property for (1.4) with constraints on Dirichlet data. With this tool at hand, we can solve the inverse problem of identifying inclusions or cavities inside an elastic body with small residual stress by the localized Dirichlet-to-Neumann map using the methods in [9] and [10].

This paper is organized as follows. In Section 2, we state and prove the UCP for (1.4) with small residual stress based on suitable Carleman estimates. The derivation of these Carleman estimates is given in Section 3. In Section 4, we will discuss the application of UCP for (1.4) to the aforementioned inverse problem. In the paper, C stands for a generic constant and its value may vary from line to line.

2. Unique continuation

To begin, let us denote v_{i} = u_{i} for 1 ≤ i ≤ n and v_{n+1} = ∂_{i}u_{i}. Then, it follows from (1.4) that

\begin{aligned}0 & = (Lu)_{i} \\ & = (μ̃Δ + t_{kj}∂_{j}∂_{k})v_{i} + (λ̃ + μ̃)∂_{i}v_{n+1} + (∂_{j}t_{kj})∂_{k}v_{i} + (∂_{i}λ̃)v_{n+1} \\ & +(∂_{j}μ̃)(∂_{i}v_{j} + ∂_{j}v_{i}) + ω^{2}ρv_{i} \\ & = (μ̃Δ + t_{kj}∂_{j}∂_{k})v_{i} + R_{i}^{(1)}(v_{1}, ⋯, v_{n}, v_{n+1}) in Ω, 1 ≤ i ≤ n, \end{aligned} (2.1)

where R_{i}^{(1)}’s are some first order differential operators. Next, by taking the divergence of (1.4), we obtain that

\begin{aligned}0 & = ∂_{i}(Lu)_{i} \\ & = ((λ̃ + 2μ̃)Δ + t_{kj}∂_{j}∂_{k})v_{n+1} + 2(∂_{i}μ̃)Δv_{i} + (∂_{i}t_{kj})∂_{j}∂_{k}v_{i} + 2∂_{i}(λ̃ + μ̃)∂_{i}v_{n+1} \\ & +(∂_{j}t_{kj})∂_{k}v_{n+1} + (∂_{i}∂_{j}t_{kj})∂_{k}v_{i} + (Δλ̃)v_{n+1} + (∂_{i}∂_{j}μ̃)(∂_{i}v_{j} + ∂_{j}v_{i}) + ω^{2}(∂_{i}ρ)v_{i} + ω^{2}ρv_{n+1} \\ & = ((λ̃ + 2μ̃)Δ + t_{kj}∂_{j}∂_{k})v_{n+1} + R^{(2)}(v_{1}, ⋯, v_{n}) + R_{n+1}^{(1)}(v_{1}, ⋯, v_{n+1}), \end{aligned} (2.2)

where R^{(2)} is a pure second order (containing only second derivatives) and R_{n+1}^{(1)} is a first order differential operators, respectively. It should be mentioned that R^{(2)} acts only on v_{1}, ⋯, v_{n}. In view of (1.3), we can see that if

max_{kj}∥t_{kj}∥_{L^{∞}(Ω)} < ε (2.3)

with ε ≪ 1, then

μ̃ > δ′ > 0 and λ̃ + 2μ̃ > δ′ > 0 ∀x ∈ Ω.

With the equations (2.1) and (2.2) in mind, motivated by Weck’s paper [21], we will prove the UCP for the following system of differential inequalities

\begin{matrix}|A_{1}(x, ∂)u^{1}| ≤ CQ(u^{1}, u^{2})^{1/2}, \\ |A_{2}(x, ∂)u^{2}| ≤ C\{∑_{ijk}|∂_{i}∂_{j}u_{k}^{1}| + Q(u^{1}, u^{2})^{1/2}\}, \end{matrix} (2.4)

where u^{l} : Ω → ℝ^{m_{l}}, m_{l} ∈ ℤ_{+}(psitive integers) and A_{l}(x, ∂) = a_{ij}^{l}∂_{i}∂_{j} with real symmetric matrix (a_{ij}^{l}), l = 1, 2 and Q(u^{1}, u^{2}) = ∑_{ikl}(|∂_{i}u_{k}^{l}|^{2} + |u_{k}^{l}|^{2}).

Theorem 2.1. Let a_{ij}^{l} ∈ W^{1,∞}(Ω) and (u^{1}, u^{2}) ∈ H_{loc}^{2}(Ω) × H_{loc}^{2}(Ω) satisfy (2.4). Then there exists an ε > 0 such that if

max_{ij}∥a_{ij}^{l}(x) − δ_{ij}∥_{L^{∞}(Ω)} < ε, (2.5)

then (u^{1}, u^{2}) vanishes identically in Ω if it vanishes in a non-empty open subset of Ω.

Theorem 2.1 immediately implies the UCP for (1.4) with small residual stress.

Corollary 2.1. Let coefficients λ, μ, β_{1}, β_{2}, t_{kj} belong to W^{2,∞}(Ω) and ρ be in W^{1,∞}(Ω). Then there exists an ε > 0 such that if (2.3) is satisfied with this ε, then the system (1.4) possesses the UCP.

The proof of Theorem 2.1 relies on the following Carleman estimates.

Proposition 2.1. Assume that the differential operators A_{1} and A_{2} satisfy the assumptions in Theorem 2.1. Let r_{0} < 1 and U_{r_{0}} = {u ∈ C_{0}^{∞}(ℝ^{n} \ {0}) : supp(u) ⊂ B_{r_{0}}}, where B_{r_{0}} is the ball centered at the origin with radius r_{0}. Then there exist positive constants β_{0} and ε_{0} such that if (2.5) is satisfied with ε ≤ ε_{0}, then for all β ≥ β_{0} and u ∈ U_{r_{0}} we have that

∫ r^{−σ}ψ^{2}∑_{ij}|∂_{i}∂_{j}u|^{2}dx ≤ C ∫ r^{−σ}ψ^{2}(β^{2}r^{−2β−2}|∇u|^{2} + |A_{l}u|^{2})dx (2.6)

and

β^{2} ∫ r^{−σ−β−1}ψ^{2}(|∇u|^{2} + |u|^{2})dx ≤ C ∫ r^{−σ}ψ^{2}|A_{l}u|^{2}dx (2.7)

for l = 1, 2, where r = |x|, ψ = exp(r^{−β}) and σ = σ_{0} + cβ with σ_{0}, c ∈ ℝ.

The proof of Proposition 2.1 is postponed until the next section. Here we want to prove Theorem 2.1 based on this proposition.

Proof of Theorem 2.1: It suffices to prove the theorem for m_{1} = m_{2} = 1 case. Let (u^{1}, u^{2}) vanish in a neighborhood of x_{0} ∈ Ω. Without loss of generality, we assume x_{0} = 0. We set r̃ = min{1/2, dist(0, ∂Ω)}. Now let χ ∈ C_{0}^{∞}(ℝ^{n}) be a cut-off function satisfying 0 ≤ χ ≤ 1, χ|_{B_{r̃/2}} = 1 and supp (χ) ⊂ Br̃. Denote v_{l} = χu^{l}, l = 1, 2. From (2.4) we have that

\begin{matrix}|A_{1}v_{1}| ≤ C(e(v_{1}) + e(v_{2}))^{1/2} + f_{1}, \\ |A_{2}v_{2}| ≤ C[∑_{ij}|∂_{i}∂_{j}v_{1}| + (e(v_{1}) + e(v_{2}))^{1/2}] + f_{2}, \end{matrix} (2.8)

where e(v) = |∇v|^{2} + |v|^{2} and f_{l} is supported in Br̃ \ B_{r̃/2} for l = 1, 2. It follows from (2.8) that

I := γ ∫ r^{−β}ψ^{2}|A_{1}v_{1}|^{2}dx + ∫ rψ^{2}|A_{2}v_{2}|^{2}dx ≤ C(F + G + ∫ rψ^{2}∑_{ij}|∂_{i}∂_{j}v_{1}|^{2}dx), (2.9)

where

\begin{matrix}F = γ ∫ r^{−β}ψ^{2}f_{1}^{2}dx + ∫ rψ^{2}f_{2}^{2}dx, \\ G = ∫ (r + γr^{−β})ψ^{2}(e(v_{1}) + e(v_{2}))dx. \end{matrix}

Here γ is a large positive parameter which will be chosen later on. By the standard approximation argument, we can see that v_{1} and v_{2} satisfy estimates (2.6) and (2.7). Taking σ = −1 in the estimate (2.6) for l = 1 and substituting it into (2.9) yield

I ≤ C(F + G + ∫ rψ^{2}|A_{1}v_{1}|^{2}dx + β^{2} ∫ r^{−2β−1}ψ^{2}|∇v_{1}|^{2}dx). (2.10)

Replacing the last term of (2.10) with the help of (2.7) for σ = β and l = 1, we obtain that

I ≤ C(F + G + ∫ r^{−β}ψ^{2}|A_{1}v_{1}|^{2}dx). (2.11)

Now taking γ sufficiently large, we can absorb the last term of (2.11) and get

I ≤ C(F + G). (2.12)

From now on we fix the parameter γ.

Next using σ = β in (2.7) for l = 1 and σ = −1 in (2.7) for l = 2, we find that

\begin{aligned}H : & = β^{2} ∫ r^{−2β−1}ψ^{2}e(v_{1})dx + β^{2} ∫ r^{−β}ψ^{2}e(v_{2})dx \\ & ≤ C(∫ r^{−β}ψ^{2}|A_{1}v_{1}|^{2}dx + ∫ rψ^{2}|A_{2}v_{2}|^{2}dx). \end{aligned} (2.13)

Combining (2.12) and (2.13) gives

H ≤ C(F + G) ≤ C(F + ∫ (r + γr^{−β})ψ^{2}(e(v_{1}) + e(v_{2}))dx). (2.14)

Now observing that r < r^{−β} < βr^{−β} < βr^{−2β−1} when r ≤ r̃ and β > 1, we obtain from (2.14) that

H ≤ C(F + β ∫ r^{−2β−1}ψ^{2}e(v_{1})dx + β ∫ r^{−β}ψ^{2}e(v_{2})dx). (2.15)

Taking β sufficiently large in (2.15), we get that

H ≤ CF,

i.e.

β^{2} ∫ r^{−2β−1}ψ^{2}e(v_{1})dx + β^{2} ∫ r^{−β}ψ^{2}e(v_{2})dx ≤ C(∫ r^{−β}ψ^{2}f_{1}^{2}dx + ∫ rψ^{2}f_{2}^{2}dx)

from which we immediately have

β^{2} ∫_{B_{r̃/2}} r^{−β}ψ^{2}(v_{1}^{2} + v_{2}^{2})dx ≤ C ∫_{Br̃\B_{r̃/2}} r^{−β}ψ^{2}(f_{1}^{2} + f_{2}^{2})dx. (2.16)

Since r^{−β}ψ^{2} is a strictly decreasing function, (2.16) implies that

β^{2} ∫_{B_{r̃/2}}(v_{1}^{2} + v_{2}^{2})dx ≤ C ∫_{Br̃\B_{r̃/2}}(f_{1}^{2} + f_{2}^{2})dx

and therefore (v_{1}, v_{2}) = 0 on B_{r̃/2} if we choose β sufficiently large. Clearly, (u^{1}, u^{2}) must be zero throughout Ω. □

3. Proof of Carleman estimates

This section is devoted to the proof of Proposition 2.1. It suffices to prove (2.6) and (2.7) for A_{1}. Therefore, we denote a_{ij}^{1} = a_{ij} and A_{1} = A. To prove (2.6), we first recall the following estimate in [21]

∫ r^{−σ}ψ^{2}∑_{ij}|∂_{i}∂_{j}u|^{2}dx ≤ C ∫ r^{−σ}ψ^{2}(β^{2}r^{−2β−2}|∇u|^{2} + |Δu|^{2})dx

(see [21, Lemma 2]) from which we have that

\begin{aligned}∫ r^{−σ}ψ^{2}∑_{ij}|∂_{i}∂_{j}u|^{2}dx & ≤ C ∫ r^{−σ}ψ^{2}(β^{2}r^{−2β−2}|∇u|^{2} + |Au|^{2} + |Δu − Au|^{2})dx \\ & ≤ C ∫ r^{−σ}ψ^{2}(β^{2}r^{−2β−2}|∇u|^{2} + |Au|^{2} + ε^{2}∑_{ij}|∂_{i}∂_{j}u|^{2})dx. \end{aligned}

Thus, choosing ε small enough immediately implies the estimate (2.6).

The proof of (2.7) is lengthy. Here we will adopt some techniques from [18], [21] and [22]. Let ϕ = ψ^{−1} and u = r^{τ/2}ϕz, then

\begin{aligned}r^{−σ/2}ψAu & = r^{−σ/2}ψA(r^{τ/2}ϕz) \\ & = r^{−σ/2}ψ[r^{τ/2}ϕAz + 2a_{ij}∂_{i}z∂_{j}(r^{τ/2}ϕ) + zA(r^{τ/2}ϕ)]. \end{aligned}

By virtue of the inequality (a + b + c)^{2} ≥ 2ab + 2bc, we have that

\begin{aligned}∫ r^{−σ}ψ^{2}|Au|^{2}dx & ≥ 4 ∫ r^{−σ}ψ^{2}a_{ij}∂_{i}z∂_{j}(r^{τ/2}ϕ)r^{τ/2}ϕAzdx \\ & +4 ∫ r^{−σ}ψ^{2}a_{ij}∂_{i}z∂_{j}(r^{τ/2}ϕ)zA(r^{τ/2}ϕ)dx. \end{aligned} (3.1)

With the choice of τ = σ + β + 2, we can compute

\begin{matrix}I := ∫ r^{−σ}ψ^{2}a_{ij}∂_{i}z∂_{j}(r^{τ/2}ϕ)r^{τ/2}ϕAzdx \\ = β ∫ a_{ij}∂_{i}zx_{j}Azdx + τ/2 ∫ r^{β}a_{ij}∂_{i}zx_{j}Azdx. \end{matrix}

It is readily seen that the leading term (for large β) of I is β ∫ a_{ij}∂_{i}zx_{j}Azdx. Repeated integration by parts shows that

\begin{aligned}2 ∫ a_{ij}∂_{i}zx_{j}Azdx & = 2 ∫ a_{ij}∂_{i}zx_{j}a_{kl}∂_{k}∂_{l}zdx \\ & = − ∫ ∂_{i}z∂_{l}(a_{kl}a_{ij}x_{j})∂_{k}zdx + ∫ ∂_{k}z∂_{i}(a_{kl}a_{ij}x_{j})∂_{l}zdx \\ & − ∫ ∂_{l}z∂_{k}(a_{kl}a_{ij}x_{j})∂_{l}zdx. \end{aligned} (3.2)

Using (3.2) we obtain that

\begin{matrix}|I| ≤ Cβ| ∫ ∂_{i}z∂_{l}(a_{kl}a_{ij}x_{j})∂_{k}zdx| \\ ≤ Cβ∥∇z∥^{2} \\ ≤ Cβ(∥∇(r^{−τ/2}ψ)u∥^{2} + ∥r^{−τ/2}ψ∇u∥^{2}) \\ ≤ C(β^{3} ∫ r^{−σ−3β−4}ψ^{2}|u|^{2}dx + β ∫ r^{−σ−β−2}ψ^{2}|∇u|^{2}dx). \end{matrix} (3.3)

Next we observe that

J := ∫ r^{−σ}ψ^{2}a_{ij}∂_{i}z∂_{j}(r^{τ/2}ϕ)zA(r^{τ/2}ϕ)dx
= β ∫ r^{−σ+τ/2−β−2}ψa_{ij}∂_{i}zx_{j}zA(r^{τ/2}ϕ)dx
+τ/2 ∫ r^{−σ+τ/2−2}ψa_{ij}∂_{i}zx_{j}zA(r^{τ/2}ϕ)dx.

Straightforward calculations show that

∂_{i}∂_{j}ϕ = (β^{2}x_{i}x_{j}r^{−2β−4} + βδ_{ij}r^{−β−2} − β(β + 2)x_{i}x_{j}r^{−β−4})ϕ

and

∂_{i}∂_{j}r^{τ/2} = (τ/2)(τ/2 − 2)r^{τ/2−4}x_{i}x_{j} + (τ/2)r^{τ/2−2}δ_{ij}.

So the leading term of J is

β^{3} ∫ r^{−2β−4}a_{ij}∂_{i}zx_{j}a_{kl}x_{k}x_{l}zdx.

Note that we have chosen τ = σ + β + 2. Performing the integration by parts, we can see that

β^{3} ∫ r^{−2β−4}a_{ij}∂_{i}zx_{j}a_{kl}x_{k}x_{l}zdx
= −2β^{3} ∫ z∂_{i}(r^{−2β−4}a_{ij}x_{j}a_{kl}x_{k}x_{l})zdx
≥ (4 − o(β))β^{4} ∫ r^{−2β−6}a_{ij}x_{i}x_{j}a_{kl}x_{k}x_{l}|z|^{2}dx
≥ (4 − o(β))β^{4}(1 − O(ε)) ∫ r^{−2β−2}|z|^{2}dx
≥ (4 − o(β))β^{4}(1 − O(ε)) ∫ r^{−σ−3β−4}ψ^{2}|u|^{2}dx,

where o(β) → 0 as β → ∞ and O(ε) is a constant bounded by Cε. In other words, we have that

J ≥ (4 − o(β))β^{4}(1 − O(ε)) ∫ r^{−σ−3β−4}ψ^{2}|u|^{2}dx. (3.4)

Notice that we need to keep track the leading constant here in order to obtain the desired estimate. Combining (3.1), (3.3) and (3.4) gives

\begin{matrix}∫ r^{−σ}ψ^{2}|Au|^{2}dx + C(β^{3} ∫ r^{−σ−3β−4}ψ^{2}|u|^{2}dx + β ∫ r^{−σ−β−2}ψ^{2}|∇u|^{2}dx) \\ ≥ 4(4 − o(β))β^{4}(1 − O(ε)) ∫ r^{−σ−3β−4}ψ^{2}|u|^{2}dx \end{matrix}

from which we can derive that

∫ r^{−σ}ψ^{2}|Au|^{2}dx + Cβ ∫ r^{−σ−β−2}ψ^{2}|∇u|^{2}dx ≥ 4(4 − o(β))β^{4}(1 − O(ε)) ∫ r^{−σ−3β−4}ψ^{2}|u|^{2}dx. (3.5)

By the ellipticity condition and performing the integration by parts, we can get that

\begin{matrix}(1 − O(ε)) ∫ r^{−σ−β−2}ψ^{2}|∇u|^{2}dx \\ ≤ ∫ r^{−σ−β−2}ψ^{2}a_{ij}∂_{i}u∂_{j}udx \\ ≤ | ∫ u∂_{i}(r^{−σ−β−2}ψ^{2})a_{ij}∂_{j}udx| + | ∫ r^{−σ−β−2}u∂_{i}(a_{ij})∂_{j}udx| \\ +| ∫ r^{−σ−β−2}ua_{ij}∂_{i}∂_{j}udx| \\ := K_{1} + K_{2} + K_{3}. \end{matrix} (3.6)

Using the relation |ab| ≤ (a^{2} + b^{2})/2, we can estimate

\begin{aligned}K_{1} & = | ∫ u∂_{i}(r^{−σ−β−2}ψ^{2})a_{ij}∂_{j}udx| \\ & ≤ ∫ 2βr^{−σ−2β−4}ψ^{2}|ua_{ij}x_{i}∂_{j}u|dx \\ & ≤ (2 + o(β))β^{2}(1 + O(ε)) ∫ r^{−σ−3β−4}ψ^{2}|u|^{2}dx + (1 + O(ε))/2 ∫ r^{−σ−β−2}ψ^{2}|∇u|^{2}dx. \end{aligned} (3.7)

Likewise, for K_{1} and K_{2}, we have that

K_{2} ≤ C(r_{0}^{β} ∫ r^{−σ−3β−4}ψ^{2}|u|^{2}dx + r_{0}^{β+2} ∫ r^{−σ−β−2}ψ^{2}|∇u|^{2}dx) (3.8)

and

K_{3} ≤ C(r_{0}^{β}β^{2} ∫ r^{−σ−3β−4}ψ^{2}|u|^{2}dx + β^{−2} ∫ r^{−σ}ψ^{2}|Au|^{2}dx). (3.9)

Plugging (3.7), (3.8) and (3.9) into (3.6) and multiplying the new inequality by β^{2}, we obtain that

\begin{matrix}β^{2}(1 − O(ε)) ∫ r^{−σ−β−2}ψ^{2}|∇u|^{2}dx \\ ≤ (2 + o(β))β^{4}(1 + O(ε)) ∫ r^{−σ−3β−4}ψ^{2}|u|^{2}dx + β^{2}(1 + O(ε))/2 ∫ r^{−σ−β−2}ψ^{2}|∇u|^{2}dx \\ +C(r_{0}^{β}β^{2} ∫ r^{−σ−3β−4}ψ^{2}|u|^{2}dx + r_{0}^{β+2}β^{2} ∫ r^{−σ−β−2}ψ^{2}|∇u|^{2}dx) \\ +C(r_{0}^{β}β^{4} ∫ r^{−σ−3β−4}ψ^{2}|u|^{2}dx + ∫ r^{−σ}ψ^{2}|Au|^{2}dx). \end{matrix} (3.10)

Adding (3.10) to (3.5) and taking β sufficiently large and ε small enough, we conclude that

β^{4} ∫ r^{−σ−3β−4}ψ^{2}|u|^{2}dx + β^{2} ∫ r^{−σ−β−2}ψ^{2}|∇u|^{2}dx ≤ C ∫ r^{−σ}ψ^{2}|Au|^{2}dx

which immediately implies (2.7).

4. Applications to inverse problems

In this section we will discuss the application of the UCP for (1.4) to the inverse problem of identifying inclusions or cavities by boundary measurements. To begin, assume that D is an open subset of Ω with Lipschitz boundary such that Ω \ D̄ is connected. The domain D stands for region of the inclusion or cavity embedded in Ω. Let the reference elasticity tensor ℂ(x) with components C_{ijkl}(x) be defined by (1.5), i.e.

C_{ijkl} = λ̃δ_{ij}δ_{kl} + (μ̃δ_{jl} + t_{jl})δ_{ik} + μ̃δ_{il}δ_{jk},

where λ̃ = λ + β_{1}(trT) and μ̃ = μ + (1/2)β_{2}(trT). Here we require that the Lamé moduli satisfy the strong convexity condition

μ(x) > δ > 0 and nλ(x) + 2μ(x) > δ > 0 ∀ x ∈ Ω (4.1)

which is equivalent to

ℂ(x)E · E ≥ κE_{ij}E_{ij} = κ|E|^{2}, κ > 0, ∀ x ∈ Ω (4.2)

for all matrix E provided that ε in (2.3) is sufficiently small. It is obvious that (4.1) implies (1.3). Next we assume that ℂ̃ is some fourth-rank tensor such that ℂ + χ_{D}ℂ̃ satisfies the strong convexity condition (4.2), where χ_{D} denotes the characteristic function of D. Moreover, suppose that ℂ̃ satisfies the following jump condition

∀ x ∈ ∂D, ∃ C_{x} > 0, ∃ δ_{x} > 0 such that ℂ̃(y)E · E ≥ C_{x}|E|^{2} or ℂ̃(y)E · E ≤ −C_{x}|E|^{2} (4.3)

for almost all y ∈ B_{δ_{x}}(x) ∩ D and all real matrices E. Let all components of ℂ(x) and ℂ̃(x) be in L^{∞}(Ω), then it is easy to show that there exists a unique solution u ∈ H^{1}(Ω) to

\begin{cases}∇ · ((ℂ + χ_{D}ℂ̃)∇u) = 0 in Ω, \\ u = f on ∂Ω \end{cases}

for any f ∈ H^{1/2}(∂Ω). In this case, the domain D is an inclusion. So we can define the Dirichlet-to-Neumann (displacement-to-traction) map Λ_{I} : H^{1/2}(∂Ω) → H^{−1/2}(∂Ω) by

Λ_{I}(f) = (ℂ∇u)ν|_{∂Ω}.

We are interested in the following inverse problem

IP.A Reconstruct the inclusion D from the knowledge of Λ_{I}(f)_{Γ_{0}} for infinitely many f ∈ H^{1/2}(∂Ω) with supp (f) ⊂ Γ_{0}, where Γ_{0} is a non-empty subset of ∂Ω.

Likewise, in the extreme case, if the tensor ℂ̃ becomes −ℂ, then the domain D corresponds to a cavity. In the same way, we can prove that there exists a unique solution u ∈ H^{1}(Ω \ D̄) to the following boundary value problem

\begin{cases}∇ · (ℂ∇u) = 0 in Ω \ D̄, \\ (ℂ∇u)ν = 0 on ∂D, (ℂ∇u)ν = g on ∂Ω \end{cases}

for any g ∈ H^{1/2}(∂Ω). Therefore, we can define the Dirichlet-to-Neumann map Λ_{C} : H^{1/2}(∂Ω) → H^{−1/2}(∂Ω) by

Λ_{C}(g) = (ℂ∇u)ν|_{∂Ω}.

Similarly, we will consider the inverse problem

IP.B Reconstruct the cavity D from the knowledge of Λ_{C}(g)|_{Γ_{0}} for infinitely many g with supp (g) ⊂ Γ_{0}.

Note that explicit reconstruction algorithms for recovering the inclusion or cavity embedded in an elastic body have been developed in [9] and [10] where the reference medium is assumed to be either inhomogeneous isotropic or anisotropic with homogeneous or analytic elasticity tensors. Here we want to extend their results to the elasticity system with residual stress (1.4). To this end, we will need the Runge approximation property with constraint for (1.4) which is a consequence of the UCP (see Corollary 2.1). Its proof can be found in [10].

Proposition 4.1. Assume that all coefficients of ℂ are in W^{2,∞}(Ω) and the residual stress satisfies (2.3) with ε given in Corollary 2.1. Let U and Ω be two open bounded domains with Lipschitz and C^{2} boundaries, respectively, such that Ū ⊂ Ω. Denote Γ_{0} a subset of the boundary ∂Ω. Let u ∈ H^{1}(U) satisfy

∇ · (ℂ∇u) = 0 in U.

Then for any compact subset K ⊂ U such that Ω \ K is connected and any ε̃ > 0 there exists w ∈ H^{1}(Ω) satisfying

∇ · (ℂ∇w) = 0 in Ω

with supp (w|_{∂Ω}) ⊂ Γ_{0} such that

∥w − u∥_{H^{1}(K)} < ε̃.

Remark: The reason for using C^{2} boundary on Ω is that we want to extend all coefficients of ℂ into a larger domain Ω̃ and the newly extended coefficients have the same regularity W^{2,∞} in Ω̃.

Having the Runge approximation property Proposition 4.1 at hand, we now can apply the methods in [9] and [10] to solve IP.A and IP.B. It should be pointed out that the reference elasticity tensor in [9] and [10] satisfies the full symmetry properties, i.e.

C_{ijkl} = C_{klij} = C_{jikl}.

Nevertheless, it is not hard to check that the proofs in [9] and [10] are still valid if we only assume C_{ijkl} = C_{klij} which is the case for the elasticity system with residual stress (1.4). Precisely, we can prove that

Theorem 4.1 (Identification of inclusion). Let the domain Ω have C^{2} boundary. Assume that the elasticity tensor ℂ given by (1.5) possesses W^{2,∞}(Ω) coefficients satisfying (4.1). Furthermore, the residual stress tensor T in ℂ satisfies the smallness condition described in Corollary 2.1. Let (D_{1}, ℂ̃_{1}) and (D_{2}, ℂ̃_{2}) be two inclusions such that ℂ + χ_{D}ℂ̃_{i} and ℂ̃_{i} satisfy (4.2) and (4.3), respectively, and Ω \ D̄_{i} is connected, i = 1, 2. If

Λ_{I_{1}}(f) = Λ_{I_{2}}(f) on Γ_{0}

for all f ∈ H^{1/2}(∂Ω) with supp (f) ⊂ Γ_{0}, then

D_{1} = D_{2}.

Theorem 4.2 (Identification of cavity). Let the assumptions in Theorem 4.1 on Ω and ℂ hold. Assume that D_{1} and D_{2} are two cavities and Ω \ D̄_{1} and Ω \ D̄_{2} are connected. Let

Λ_{C_{1}}(f) = Λ_{C_{2}}(f) on Γ_{0}

for all f ∈ H^{1/2}(∂Ω) with supp (f) ⊂ Γ_{0}, then D_{1} = D_{2}.

Remarks:

1. We do not assume ℂ̃_{1} = ℂ̃_{2} in Theorem 4.1. Also, the regularity of the medium inside of the inclusions is only assumed to be essentially bounded.

2. In [10], a reconstruction algorithm for recovering the cavity D is given. The same algorithm can be applied to Theorem 4.2 as well.

References

[1] D.D. Ang, M. Ikehata, D.D. Trong and M. Yamamoto, Unique continuation for a stationary isotropic Lamé system with varaiable coefficients, Comm. in PDE, 23(1998), 371-385.

[2] G. Alessandrini and A. Morassi, Strong unique continuation for the Lamé system of elasticity, preprint.

[3] L. Bers, F. John, and M. Schechter, ”Partial Differential Equations”, John Wiley, New York 1964.

[4] L. De Carli and T. Ōkaji, Strong unique continuation property for the Dirac equation, Publ. RIMS, Kyoto Univ., 35(1999), 825-846.

[5] B. Dehman and L. Rabbiano, La propriété du prolongement unique pour un système elliptique: le système de Lamé, J. Math. Pures Appl., 72(1993), 475-492.

[6] M. Eller, V. Isakov, G. Nakamura and D. tataru, Uniqueness and stability in the Cauchy problem for Maxwell’ and elasticity systems, preprint.

[7] M. Ikehata, Identification of the shape of the inclusion having essentially bounded conductivity, J. Inv. Ill-Posed Prob., 7(1999), 533-540.

[8] M. Ikehata, Reconstruction of the shape of the inclusion by boundary measurements, Comm. in PDE, 23(1998), 1459-1474.

[9] 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.

[10] M. Ikehata and G. Nakamura, Reconstruction of cavity from boundary measurements, preprint.

[11] V. Isakov, On uniqueness of recovery of a discontinuous conductivity coefficients, Comm. Pure Appl. Math., 41(1988), 865-877.

[12] V. Isakov, A non-hyperbolic Cauchy problem for □_{b}□_{c} and its applications to elasticity theory, Comm. Pure Appl. Math., 39(1986), 747-767.

[13] D. Jerison, Carleman inequalities for the Dirac and Laplace operator and unique continuation, Adv. Math., 63(1986), 118-134.

[14] R. Kohn and M. Vogelius, Determining conductivity by boundary measurements II, interior results, Comm. Pure Appl. Math., 38(1985), 643-667.

[15] P. Lax, A stability theorem for solutions of abstract differential equations and its application to the study of local behavior of solutions of elliptic equations, Comm. Pure Appl. Math., 9(1956), 747-766.

[16] C.-S. Man, Hartig’s law and linear elasticity with initial stress, Inverse Prob., 14(1998), 313-319.

[17] T. Ōkaji, Strong unique continuation property ffor time harmonic Maxwell equations, Preprint.

[18] M. Protter, Unique continuation for elliptic equations, Trans. AMS, 95(1960), 81-91.

[19] V. Vogelsang, Absence of embedded eigenvalues of the Dirac equation for long range potentials, Analysis, 7(1987), 259-274.

[20] V. Vogelsang, On the strong continuation principle for inequalities of Maxwell type, Math. Ann., 289(1991), 285-295.

[21] N. Weck, Unique continuation for systems with Lamé principal part, Math. Methods Appl. Sci., 24(2001), 595–605.

[22] N. Weck, Unique continuation for some systems of partial differential equations, Appl. Anal., 13(1982), 53-63.

Gen Nakamura, Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan

  E-mail address: gnaka@math.hokudai.ac.jp

Jenn-Nan Wang, Department of Mathematics, National Cheng-Kung University, Tainan 701, Taiwan

  E-mail address: jnwang@msri.org