PROPAGATION OF POLARIZATION IN ELASTODYNAMICS WITH RESIDUAL STRESS AND TRAVEL TIMES

SÖNKE HANSEN AND GUNTHER UHLMANN

1. Introduction

Consider an elastic medium which occupies a bounded domain Ω ⊂ ℝ^{3} with smooth boundary ∂Ω and exterior normal ν. Displacement is a time-dependent vector field u(t, ·) on \overline{Ω}. Small displacements satisfy, in a source-free medium, the equations for (linearized) elastodynamics,

(1)  ρ ∂^{2}u/∂t^{2} = ∇ · S with S = R + ∇u R + CE.

Here 0 < ρ ∈ C^{∞}(\overline{Ω}) denotes the density, S is the Piola-Kirchhoff stress tensor which obeys the relation SF^{T} = F S^{T} where F = I + ∇u is the deformation gradient. Divergence and transpose are taken with respect to the Euclidean metric | · |. The elasticity tensor C maps infinitesimal strain tensors E = (∇u+∇u^{T})/2 to symmetric stress tensors CE. C represents the material properties on the elastic medium. R(x), the residual stress tensor, is a symmetric 3×3-matrix, C^{∞} on \overline{Ω}. It satisfies ∇·R = 0. See [Gur84, Sect. 23], [Hog86]. We call

(2)  P u = −ρ ∂^{2}u/∂t^{2} + ∇ · (R + ∇u R + CE)

the operator for elastodynamics. P is isotropic if the elasticity tensor is as follows,

(3)  CE = λ tr(E)I + 2μE with 0 < λ, μ ∈ C^{∞}(\overline{Ω}).

λ and μ are the Lamé parameters of the elastic medium.

The inverse problem for operators of elastodynamics is to recover as much as possible of the elasticity tensor C and of the residual stress tensor R from measurements performed at the space-time boundary ℝ × ∂Ω. See [ML87] for the beginnings of an acoustoelastic theory of residual stress determination based on wave propagation methods.

We deal with an inverse problem for subclasses L(L, ε) of operators for isotropic elastodynamics. Here L, ε > 0, and, by definition, P ∈ L(L, ε) if and only if

(4)  λ(x) + 2μ(x), 1/μ(x), 1/ρ(x) ≤ L when x ∈ \overline{Ω}

and

(5)  |R(x)| ≤ εμ(x) when x ∈ \overline{Ω}.

The first author thanks the Department of Mathematics at the University of Washington for its hospitality during his visit in fall 2000. The second author is partly supported by NSF grant DMS-0070488 and a John Simon Guggenheim fellowship. The second author also thanks MSRI for partial support and for providing a very stimulating environment during the inverse problems program in fall 2001.

If ε > 0 is sufficiently small the initial boundary value problem is well-posed and microlocal parametrices exist. Assumptions like (5) with ε small have been introduced before to ascertain well-posedness. See, e.g., [Rob97] for the static case.

The (hyperbolic) Dirichlet-to-Neumann map

(6)  Λ : u|_{ℝ×∂Ω} ↦ ν · S|_{ℝ×∂Ω}

encodes boundary measurements. Here u solves (1) with zero initial data. We say that a property of an operator for elastodynamics from a given class is determined by boundary measurements if the property is the same for any two operators in the class with identical Dirichlet-to-Neumann maps.

A useful approach to inverse problems consists in using high-frequency waves u generated by boundary data with singularities. From travel times of singularities of u recorded at ∂Ω one then aims to recover the requested properties. The latter problem is called an inverse kinematic problem. Obviously, an important step is to prove that travel times are in fact determined by boundary measurements. The main goal of this paper is provide a result of this kind which is applicable also when caustics may develop.

We study the propagation of polarization in the sense of Dencker [Den82] for the initial boundary problem of the operator for isotropic elastodynamics, P. In Proposition 4.1 we show that P is a system of real principal type if the residual stress R satisfies

(7)  μ(x)|ξ|^{2} + R(x)ξ · ξ > 0 when (x, ξ) ∈ T^{⋆}(\overline{Ω}) \ 0.

If (7) holds then

(8)  ⟨ξ, ξ⟩_{S} = (μ(x)|ξ|^{2} + R(x)ξ · ξ)/ρ(x),
(9)  ⟨ξ, ξ⟩_{P} = ((λ(x) + 2μ(x))|ξ|^{2} + R(x)ξ · ξ)/ρ(x),

(x, ξ) ∈ T^{⋆}(\overline{Ω}), are the duals g_{S/P}^{−1} of Riemannian metrics g_{S/P} on \overline{Ω}. The characteristic variety of P is the union of the subvarieties τ^{2} − ⟨ξ, ξ⟩_{S} = 0 and τ^{2} − ⟨ξ, ξ⟩_{P} = 0 which correspond to shear and compressional waves, respectively.

The lens map or scattering relation S of a metric g on \overline{Ω} is defined as follows. Consider bicharacteristic curves, γ : [a, b] → T^{⋆}(\overline{Ω} × ℝ), of the Hamilton function H(t, x, τ, ξ) = τ^{2} −g^{−1}(x, ξ) which satisfy the following: γ(]a, b[) lies over the interior, γ intersects the boundary non-tangentially at γ(a) and γ(b), and time increases along γ. By definition, S is the subset of (T^{⋆}(ℝ × ∂Ω) \ 0)^{2} obtained by projecting endpoint pairs (γ(b), γ(a)). It is well-known that S is a homogeneous canonical relation on T^{⋆}(ℝ × ∂Ω) \ 0. (See [Gui77] for the concept of a scattering relation.) S is a diffeomorphism between open subsets of T^{⋆}(ℝ × ∂Ω) \ 0. We denote by S_{S} (resp. S_{P}) the lens map of g_{S} (resp. g_{P}) and call it the shear (resp. compressional) lens map.

Our main result is the following.

Theorem 1.1. Given L > 0 there exists ε > 0 such that in the class L(L, ε) the shear and the compressional lens maps are determined by boundary measurements.

Note that travel times of shear and compressional waves are recovered separately from boundary measurements.

Let g be Riemannian metric on \overline{Ω}. Denote by D the open subset of ∂Ω × ∂Ω which consists of the pairs (x, y) of boundary points which can be joined by a geodesic which passes through the interior except for the endpoints x and y where it intersects ∂Ω transversally. By definition, the boundary distance function of (\overline{Ω}, g) is the function d : D → [0, ∞[ which assigns to (x, y) ∈ D the geodesic distance, i.e., the infimum of the lengths of such geodesics. If (\overline{Ω}, g) is strictly convex then D is the complement of the diagonal and d is smooth. Geodesics of g are projections of bicharacteristic curves of τ^{2} − g^{−1}(x, ξ) = 0. Geodesic distances equal travel times. When d is smooth it is a generating function of (a subset of) the lens maps S of g, i.e., ((t_{1}, x_{1}, τ, ξ_{1}), (t_{0}, x_{0}, τ, ξ_{0})) ∈ S if t_{1} − t_{0} = d(x_{1}, x_{0}) and ξ_{j} = −τ∂d(x_{1}, x_{0})/∂x_{j} for j = 0, 1. (See [Car35], [GS77].) Clearly, S determines d. Hence we have the following corollary of Theorem 1.1. Here we call the boundary distance functions of the metrics g_{S} and g_{P} the shear and the compressional boundary distance functions d_{S} and d_{P}, respectively.

Corollary 1.2. Given L > 0 there exists ε > 0 such that in the class L(L, ε) the shear and the compressional boundary distance functions are determined by boundary measurements.

Rachele [Rac00b] has a similar result under additional assumptions which exclude conjugate points. Note that our result allows the presence of conjugate points.

In the case R = 0 the metrics g_{S} and g_{P} are conformal to the Euclidean metric. Mukhometov [Muk82] solved the inverse kinematic problem for conformal classes of metrics under assumptions which exclude conjugate points. Corollary 1.2 and, e.g., Croke’s theorem [Cro91, Theorem C] imply the following uniqueness result of Rachele.

Corollary 1.3. [Rac00a, Theorem 1] In the class of operators of isotropic elastodynamics with vanishing residual stresses, and with (\overline{Ω}, g_{S}), (\overline{Ω}, g_{P}) strictly convex, the compressional speeds and shear speeds, c_{P} = \sqrt{(λ + 2μ)/ρ} and c_{S} = \sqrt{μ/ρ}, are determined by boundary measurements.

If residual stresses do not vanish the metrics become anisotropic. From Corollary 1.2 and a result of Stefanov-Uhlmann on the anisotropic inverse kinematic problem [SU98, Theorem 1.1] we deduce the following result.

Corollary 1.4. There is a C^{12}(\overline{Ω}) neighbourhood U of the euclidean metric such that the following holds. Let P^{(1)} and P^{(2)} be operators of isotropic elastodynamics. Assume Λ^{(1)} = Λ^{(2)}. Assume \overline{Ω} strictly convex with respect to the metrics g_{S}^{(j)} and g_{P}^{(j)}. If g_{S}^{(j)}, g_{P}^{(j)} ∈ U then g_{S}^{(1)} = Ψ_{S}^{⋆}g_{S}^{(2)}, g_{P}^{(1)} = Ψ_{P}^{⋆} g_{P}^{(2)} with diffeomorphisms Ψ_{S}, Ψ_{P} : \overline{Ω} → \overline{Ω} which leave the boundary fixed, i.e., Ψ_{S}(x) = Ψ_{P} (x) = x if x ∈ ∂Ω.

In [SU98, Theorem 1.1] an additional flatness assumption at the boundary of Ω is made. This assumption is superfluous in view of [LSU01, Theorem 2.1].

We prove Theorem 1.1 in section 6. The facts needed about propagation of singularities and polarizations in non-glancing boundary problems for systems of real principal type are proved in section 2 for first order systems. These are applied to second order systems and to elastodynamics in sections 3 and 4, respectively. In particular, section 4 contains an analysis of the Dirichlet-to-Neumann map Λ and its pseudo-differential properties.

2. Singularities of First Order Boundary Problems

We summarize some facts from the microlocal theory of boundary problems. The results are due to Dencker [Den82], Gérard [Gér85], Melrose [Mel81], and Taylor [Tay75].

Let Z an open subset of half-space \overline{ℝ_{+}} × ℝ^{n} equipped with coordinates x ≥ 0 and y = (y_{1}, …, y_{n}). ξ and η = (η_{1}, …, η_{n}) are the dual coordinates in cotangent space. Denote the boundary and the interior of Z by Y and Z^{∘}, respectively. D′(Z) denotes the space of extendible distributions on Z^{∘}. Pseudo-differential operators of order at most m on Y and on Z acting along Y are written A(y, D_{y}) ∈ Ψ^{m}(Y) and B(x, y, D_{y}) ∈ Ψ_{t}^{m}(Z), respectively. S^{m} = S^{m}(Y × ℝ^{n}) and S_{t}^{m} = S^{m}(Z × ℝ^{n}) are the corresponding symbol spaces. Elements of Ψ_{t}^{m}(Z) are called tangential pseudodifferential operators. Symbols are always assumed polyhomogeneous (classical). Pseudo-differential operators will always be chosen properly supported. We denote by r_{Y} u = u|_{Y} the restriction of u ∈ D′(Z), when defined.

We consider u ∈ D′(Z)^{K} such that P u ∈ C^{∞}(Z)^{K} where P is a K × K system of pseudo-differential operators which are differential with respect to x,

(10)  P = ∑_{j=0}^{m} P_{j}D_{x}^{j} with P_{j} = P_{j}(x, y, D_{y}) ∈ Ψ_{t}^{j}(Z), P_{0} = Id_{K} .

The boundary wavefront set WF_{b}(u), defined in [Mel81], is a closed subset of the compressed cotangent bundle T̃^{⋆}(Z). If the P_{j} are differential operators then P u ∈ C^{∞} is a non-characteristic boundary problem and hence u is normally regular in the sense of Melrose [Mel81, II.9]. Recall from [Mel81] or [Hör85, 18.3] the following properties of a normally regular distribution u. u ∈ C^{∞}([0, ε[, D′(ℝ^{n})) locally near Y. Au is normally regular if A is a tangential pseudo-differential operator. The boundary wavefront set WF_{b}(u) ⊂ T^{⋆}(Y)∪T^{⋆}(Z^{∘}) ⊂ T̃^{⋆}(Z). (y, η) ∈ T^{⋆}(Y)\WF_{b}(u) if and only if Au ∈ C^{∞}(Z) for some operator A = A(x, y, D_{y}) ∈ Ψ_{t}^{m}(Z) which is non-characteristic at (0, y, η). The polarization set WF_{pol}^{(s)}(u) is, by definition the intersection of the sets

N_{A} = {(x, ξ; w) ∈ T^{⋆}(Z^{∘}) × ℂ^{K} ; σ(A)(x, ξ)w = 0}

where A ∈ Ψ^{0} runs over all 1 × K systems such that Au ∈ H^{(s)}(Z^{∘}). See [Den82] and [Gér85] for the precise definition and for results on the propagation of polarization along Hamilton orbits.

Let P as in (10) with principal symbol p. Following Dencker [Den82, Definition 3.1] we say that P is of real principal type if, microlocally near a given point, the characteristic variety is given by q = 0 with a scalar symbol q of real principal type and if there exists a matrix-valued symbol, p̃ such that p̃p = q Id_{K}. If we assume P of real principal type then H = H_{q} is a Hamilton field of the characteristic variety V = q^{−1}(0) of P. A point (y, η) ∈ T^{⋆}Y \ 0 is called glancing for P if, with respect to the natural projection, its preimage in V ∩ T_{Y}^{⋆}Z contains a point where Hx = 0, else (y, η) is called non-glancing for P. Bicharacteristics intersect the boundary transversally at non-glancing points.

We now specialize to first order systems, m = 1. Let G = G(x, y, D_{y}) ∈ Ψ_{t}^{1}(Z) be an N × N matrix of tangential pseudo-differential operators with homogeneous principal symbol g. We assume that D_{x} Id_{N} −G is of real principal type. We are interested in the singularities of normally regular solutions of

(11)  D_{x}w − G(x, y, D_{y})w ≡ 0 mod C^{∞}(Z)^{N}.

Let (y^{(0)}, η^{(0)}) ∈ T^{⋆}(Y) \ 0 non-glancing for D_{x} Id_{N} −G.

The following decoupling lemma is due to Taylor [Tay75] in the case of simple real characteristics and to Gérard [Gér85] in the case of real principal type systems.

Lemma 2.1. In a conic neighbourhood Γ of (0, y^{(0)}, η^{(0)}), the algebraic and geometric multiplicities of the real eigenvalues of g(x, y, η) are equal and constant. There are homogeneous real-valued μ_{1}, …, μ_{J} ∈ S_{t}^{1} which enumerate, in Γ, the distinct real eigenvalues of g. Let N_{j} denote the multiplicity of μ_{j}. There is an elliptic N × N matrix S ∈ Ψ_{t}^{0} such that microlocally near (0, y^{(0)}, η^{(0)}),

(12)  (D_{x} Id_{N} −G)S ≡ S(D_{x} Id_{N} −H) mod Ψ_{t}^{−∞}.

H ∈ Ψ_{t}^{1} is a block matrix with non-zero entries only on the diagonal,

(13)  H = ( \begin{matrix}μ_{1}(x, y, D_{y}) Id_{N_{1}} \\ & . & . & . \\ & & & & μ_{J}(x, y, D_{y}) Id_{N_{J}} \\ & & & & & E_{+} \\ & & & & & & E_{−} \end{matrix} ) .

The imaginary parts of the eigenvalues of the principal symbols of E_{+}, E_{−} ∈ Ψ_{t}^{1} are positive and negative, respectively.

Proof.The following constructions hold in some conic neighbourhood Γ of (0, y^{(0)}, η^{(0)}). Γ may become smaller as the proof proceeds.

Since A = D_{x} Id_{N} −G is of real principal type its characterictic variety is V = q^{−1}(0) with a scalar real principal type symbol q. The non-glancing assumption implies ∂q/∂ξ ≠ 0 at points (0, y^{(0)}, ξ, η^{(0)}) ∈ V. By the implicit function theorem, the real eigenvalues of g(x, y, η) are smooth homogeneous functions μ_{1}(x, y, η) < … < μ_{J}(x, y, η) in Γ. We extend them as homogeneous real valued symbols μ_{1}, …, μ_{J} ∈ S_{t}^{1}(Z × ℝ^{n}).

Let μ be a real eigenvalue of g^{(0)} = g(0, y^{(0)}, η^{(0)}). We show that the geometric multiplicity of μ equals its algebraic multiplicity,

(14)  ker ((μ − g^{(0)})^{r}) = ker (μ − g^{(0)}), ∀r ∈ ℕ.

Let a = ξ − g denote the principal symbol A. By the non-glancing hypothesis ∂/∂ξ is transversal to the characteristic variety det a = 0 at (y^{(0)}, η^{(0)}). The intrinsic characterisation of real principal type [Den82, Prop. 3.2] shows that ∂a/∂ξ = Id maps the kernel of a isomorphically onto the cokernel of a at ξ = μ. Hence

(15)  ker(μ − g^{(0)}) ∩ im(μ − g^{(0)}) = 0.

Equation (14) easily follows from (15).

Let γ_{1}, …, γ_{J} be non-intersecting closed positively oriented Jordan curves in the complex plane such that γ_{j} encloses μ_{j}(0, y^{(0)}, η^{(0)}) but no other eigenvalue of g^{(0)}. (To enclose means that the winding number is non-zero.)

(16)  π_{j}(x, y, η) = ∫_{γ_{j}} (λ − g(x, y, η)/|η|)^{−1} \frac{dλ}{2πi}

is the spectral projector onto the sum of generalized eigenspaces associated with the eigenvalues enclosed by γ_{j} of g(x, y, η)/|η|. Clearly, ker (μ_{j} − g^{(0)}) ⊂ im π_{j}. By (14) equality holds at (0, y^{(0)}, η^{(0)}). By [Den82, Prop. 3.2] the dimension of ker (μ_{j} − g^{(0)}) is constant. Also the rank of π_{j} is constant in Γ. Hence

(17)  ker(μ_{j} − g) = im π_{j} in Γ.

It follows from (17) that the geometric and the algebraic multiplicities of the real eigenvalues of g coincide everywhere in Γ. Therefore we can find an elliptic N × N matrix s(x, y, η) ∈ S_{t}^{0} such that s^{−1}gs ∈ S_{t}^{1} has, in Γ, the block structure of the principal symbol of the operator H claimed in (13).

Choose S with principal symbol equal to s. We obtain (12) with the error class Ψ_{t}^{−∞} replaced by Ψ_{t}^{0}, however. We use the uncoupling technique of [Tay75] to obtain K ∈ Ψ_{t}^{−1} such that the error is Ψ_{t}^{−∞} if we replace S by S(Id +K). After doing this, however, H will only satisfy a weaker form than (13) with μ_{j}(x, y, D_{y}) Id_{N_{j}} is replaced by μ_{j}(x, y, D_{y}) Id_{N_{j}} +M_{j} with some M_{j} ∈ Ψ_{t}^{0}. By [Gér85, Lemme 2.1.] there exist elliptic N_{j} × N_{j} matrices E_{j} ∈ Ψ_{t}^{0} such that

((D_{x} − μ_{j}(x, y, D_{y})) Id_{N_{j}} −M_{j})E_{j} ≡ E_{j}((D_{x} − μ_{j}(x, y, D_{y})) Id_{N_{j}} )

holds modulo operators in Ψ^{−∞}. Let E ∈ Ψ_{t}^{0}, N × N, denote the diagonal block matrix with blocks E_{1}, …, E_{J} and, in the lower right corner, Id. Finally, to remove the M_{j}’s, we replace S by SE. □

Let B be a K × N matrix in Ψ_{t}^{0} with homogeneous principal symbol b. Given h ∈ D′(Y)^{K} we wish to solve equation (11) under the boundary condition specified by B and h,

(18)  \begin{aligned}D_{x}w − Gw & ≡ 0 mod C^{∞}(Z)^{N}, \\ Bw|_{Y} & ≡ h mod C^{∞}(Y)^{K}. \end{aligned}

Let M_{+}∪M_{−} = {1, …, J} be a disjoint union decomposing the set of real eigenvalues of g(0, y^{(0)}, η^{(0)}) into two parts. We call the eigenvalue μ_{j} forward (resp. backward) if j ∈ M_{+} (resp. j ∈ M_{−}). Correspondingly, we call characteristics and bicharacteristic curves forward or backward. In case D_{x} Id_{N} −G is hyperbolic with respect to a time variable t(x, y) such a decomposition arises as follows. A bicharacteristic γ issuing from the boundary into the interior is forward (resp. backward) if t increases (resp. decreases) along γ

We shall find a microlocal parametrix of the boundary problem (18) if a condition of Lopatinski type holds. Define, for (y, η) sufficiently close to (y^{(0)}, η^{(0)}), the forward Lopatinski space as the following linear subspace of ℂ^{N},

(19)  L_{g}^{+}(y, η) = im ∫_{γ^{+}} (λ − g(0, y, η))^{−1} dλ.

γ^{+} is a closed positively oriented Jordan curve in the complex plane which encloses the eigenvalues of g^{(0)} = g(0, y^{(0)}, η^{(0)}) which are real and forward or which have positive imaginary part. γ^{+} encloses no other eigenvalues of g^{(0)}.

Proposition 2.2. Assume that b(0, y^{(0)}, η^{(0)}) maps L_{g}^{+}(y^{(0)}, η^{(0)}) onto ℂ^{K}. Then there exists a conic neighbourhood Γ ⊂ T^{⋆}(Y) of (y^{(0)}, η^{(0)}) and an operator W : D′(Y)^{K} → D′(Z)^{N} such that the following holds. For every h ∈ D′(Y)^{K} with WF(h) ⊂ Γ the distribution w = W h ∈ D′(Z)^{N} is normally regular and solves (18). WF(w|_{Z^{∘}}) is contained in the union of the forward bicharacteristics which issue from WF(h). W_{0} := r_{Y} ∘ W ∈ Ψ^{0}(Y) is a N × K pseudo-differential operator. The principal symbol w_{0}(y, η) of W_{0} maps ℂ^{K} into L_{g}^{+}(y, η) and satisfies bw_{0} = Id in Γ.

Proof.Let S and H as in Lemma 2.1, and denote their principal symbols by s and h, respectively. Clearly, gs = sh. Hence L_{g}^{+} = sL_{h}^{+}. The block structure of H and the partitioning into forward and backward eigenvalues defines a projector Π on ℂ^{N}. Π projects onto the subspace corresponding to the blocks μ_{j}(x, y, D_{y}) Id_{N_{j}}, j ∈ M_{+}, and E_{+} of H along the subspace corresponding to the blocks μ_{j}(x, y, D_{y}) Id_{N_{j}}, j ∈ M_{−}, and E_{−} of H. Notice that L_{h}^{+} = Πℂ^{N}. By assumption

(20)  ℂ^{K} = bL_{g}^{+} = bsL_{h}^{+} = bsΠℂ^{N}.

The Cauchy problems

(D_{x} − μ_{j}(x, y, D_{y}))v ∈ C^{∞}(Z), v|_{Y} ∈ C^{∞}(Y),

are solved using scalar Fourier integral operators V_{j}, [Dui73]. The wavefront set of the solution v = V_{j}f is contained in the image of the bicharacteristics associated with ξ − μ_{j}(x, y, η) = 0 which issue from WF(f). The parabolic system

D_{x}v − E_{+}v ∈ C^{∞}(Z), v|_{Y} ∈ C^{∞}(Y),

is solved using a Poisson operator V_{+}, [Tay75]. The solution v = V_{+}f has no singularities in Z^{∘}. Therefore we may construct an operator V : D′(Y)^{N} → D′(Z)^{N} such that the following holds for any f ∈ D′(Y)^{N}. v = V f is normally regular, D_{x}v − Hv ∈ C^{∞}(Z)^{N}, and WF(v|_{Z^{∘}}) is contained in the union of the forward bicharacteristics which issue from WF(f). Furthermore, modulo C^{∞}(Y)^{N}, v|_{Y} ≡ Πf.

r_{Y} BSV is a K × N system of pseudo-differential operators on Y. Its principal symbol bsΠ is, close to (y^{(0)}, η^{(0)}), surjective by (20). Choose a N × K operator C ∈ Ψ^{0}(Y) which is a right inverse, r_{Y} BSV C ≡ Br_{Y} SV C ≡ Id. W = SV C satisfies the claims. □

Remark 1. If the boundary data f is a Lagrangian distribution then the solution w = W f is Lagrangian with respect to the forward characteristics. Röhrig [Röh] derives the transport equations for the principal symbol of w along the bicharacteristics.

To prepare waves with specified polarization we need the following result about propagation of polarization at the boundary. Essentially this is a corollary of [Gér85, Théorème 6.1].

Proposition 2.3. Let w ∈ D′(Z)^{N} normally regular such that (y^{(0)}, η^{(0)}) ∉ WF_{b}(D_{x}w− Gw). Assume w|_{Y} ∈ H^{(s−1)}(Y), s > 1. Let μ ∈ S^{1} be a real eigenvalue of g(0, ·) in a conic neighbourhood of (y^{(0)}, η^{(0)}). Let Q ∈ Ψ^{0}(Y) with principal symbol equal to, in a neighbourhood of (y^{(0)}, η^{(0)}), the spectral projector on the eigenspace of the eigenvalue μ. Then (y^{(0)}, η^{(0)}) ∈ WF^{(s)}(Qw|_{Y}) if and only if WF_{pol}^{(s)}(w) contains a Hamilton orbit above the μ-bicharacteristic which issues from (y^{(0)}, η^{(0)}).

Proof.Choose a parametrix S^{−1} of S in Lemma 2.1 and put w′ = S^{−1}w. The hypotheses of the Proposition still hold with w replaced by w′ and with G replaced by H of (13). Let Q_{μ} denote the projection to the components of the block which corresponds to μ in the block decomposition (13). Then Q − Q_{μ} ∈ Ψ^{−1}(Y) and, using the assumption on w|_{Y}, WF^{(s)}(Q_{μ}w|_{Y}) = WF^{(s)}(Qw|_{Y}). v = Q_{μ}w solves the diagonal system (y^{(0)}, η^{(0)}) ∉ WF_{b}(D_{x}v − μ(x, y, D_{y})v). The assertion follows from well-known results on propagation of singularities in the Cauchy problem for scalar strictly hyperbolic equations and from [Den82, Theorem 4.2]. □

3. Second Order Boundary Problems

Here we reduce the Dirichlet problem for second order real principal systems to a boundary problem for a first order real principal type system.

Let P = D_{x}^{2} Id_{K} +P_{1}(x, y, D_{y})D_{x} + P_{2}(x, y, D_{y}) be a K × K matrix of differential operators of second order. We are interested in the Dirichlet problem

(21)  \begin{aligned}P u & ≡ 0 mod C^{∞}(Z)^{K}, \\ u|_{Y} & ≡ f mod C^{∞}(Y)^{K}. \end{aligned}

Any solution u is normally regular.

We associate with (21) an equivalent first order boundary problem (18) as follows. Set N = 2K, G ∈ Ψ_{t}^{1} the N × N matrix

(22)  G = ( \begin{matrix}0 & ⟨D_{y}⟩ Id_{K} \\ −P_{2} ⟨D_{y}⟩^{−1} & −P_{1} \end{matrix} ) ,

and B ∈ Ψ_{t}^{0} the K ×N matrix with Bw = w_{1}, w = (w_{1}, w_{2}). Here ⟨D_{y}⟩ ∈ Ψ_{t}^{1} denotes the operator with full symbol ⟨η⟩ = (1 + |η|^{2})^{1/2} ∈ S_{t}^{1}.

Lemma 3.1. Let f ∈ D′(Y)^{K} and h = ⟨D_{y}⟩f. Solutions u of (21) and w = (w_{1}, w_{2}) of (18) are related as follows. If u solves (21) then w = (w_{1}, w_{2}) = (⟨D_{y}⟩u, D_{x}u) solves (18). Conversely, if w solves (18) then u = ⟨D_{y}⟩^{−1}w_{1} solves (21).

Proof.The first statement follows immediately from the definition of G and B. For the proof of the converse statement let w be a solution of (18). The first row of D_{x}w ≡ Gw and the ellipticity of ⟨D_{y}⟩ imply D_{x}u ≡ w_{2}. Hence the second row implies P u ≡ 0. By our choice of B and h the boundary conditions are equivalent: ⟨D_{y}⟩u = Bw ≡ h = ⟨D_{y}⟩f. □

Let (x, y, ξ, η) ∈ T^{⋆}(Z), η ≠ 0. Let p = ξ^{2} + p_{1}ξ + p_{2} denote the principal symbol of P. Then the principal symbol of G is

g = ( \begin{matrix}0 & |η| \\ −p_{2}/|η| & −p_{1} \end{matrix} ) .

Lemma 3.2. Let η ≠ 0. Then

(23)  (ξ − g′)(ξ − g) = ( \begin{matrix}p \\ & p \end{matrix} ) where g′ = ( \begin{matrix}−p_{1} & −|η| \\ p_{2}/|η| & 0 \end{matrix} )

and

(24)  ker(ξ − g) = ( \begin{matrix}|η| Id_{K} \\ ξ Id_{K} \end{matrix} ) ker(p).

The characteristic varieties of P and D_{x} Id_{N} −G are equal. If P is of real principal type then so is D_{x} Id_{N} −G.

Proof.Equation (23) is verified by direct computation. Clearly, (ξ − g)w = 0 with w = (w_{1}, w_{2}), holds if and only if ξw_{1} = |η|w_{2} and pw_{2} = 0. To prove the last assertion assume there is a K × K matrix of symbols, p̃, such that p̃p = q Id_{K} holds with a scalar real principal type smbol q. Then, using (23), we obtain a N × N matrix of symbols, ã, such that ã(ξ − g) = q Id_{N}. □

Remark 2. Assume P of real principal type. Let C_{0}u = (⟨D_{y}⟩u|_{Y} , D_{x}u|_{Y}) denote the Cauchy data of a solution of P u ≡ 0. It follows from Proposition 2.3 and Lemma 3.2 that WF_{pol}^{(s+1)}(u) contains a Hamilton orbit above a given bicharacteristic issuing from γ = (y, η) ∈ T^{⋆}(Y)\0 if and only if γ ∈ WF^{(s)}(QC_{0}u) where Q ∈ Ψ^{0}(Y) with principal symbol equal to the spectral projector onto the eigenspace {(⟨η⟩a, ξa); p(0, y, ξ, η)a = 0} which corresponds to the given characteristic.

We give sufficient conditions for the existence of a microlocal parametrix for the boundary problem (21).

Proposition 3.3. Assume P of real principal type. Let (y^{(0)}, η^{(0)}) ∈ T^{⋆}Y \ 0 be nonglancing for P. Let γ^{+} be a closed positively oriented Jordan curve which does not meet the poles of λ ↦ p(0, y^{(0)}, λ, η^{(0)})^{−1} and which has winding number 1 (resp. 0) with respect to the poles with positive (resp. negative) imaginary part. Assume that

(25)  K ≥ rank ∫_{γ^{+}} (λ − g(0, y^{(0)}, η^{(0)}))^{−1} dλ,
(26)  K ≤ rank ∫_{γ^{+}} p(0, y^{(0)}, λ, η^{(0)})^{−1} dλ.

Then there exists a conic neighbourhood Γ ⊂ T^{⋆}(Y) of (y^{(0)}, η^{(0)}) and an operator U : D′(Y)^{K} → D′(Z)^{K} such that for any f ∈ D′(Y)^{K} with WF(f) ⊂ Γ the distribution u = Uf ∈ D′(Z)^{K} is normally regular and solves (21). WF(u|_{Z^{∘}}) is contained in the union of the forward bicharacteristics which issue from WF(f). U′ := r_{Y} D_{x}U ∈ Ψ^{1}(Y) is a K ×K pseudo-differential operator with principal symbol u′ which satisfies

(27)  u′(y^{(0)}, η^{(0)}) ∫_{γ^{+}} p(0, y^{(0)}, λ, η^{(0)})^{−1} dλ = ∫_{γ^{+}} λ p(0, y^{(0)}, λ, η^{(0)})^{−1} dλ.

Proof.We use the equivalence, stated in Lemma 3.1, of (21) with the first order boundary problem (18).

A real eigenvalue of g(0, y^{(0)}, η^{(0)}) is, by definition, forward if it is enclosed by γ^{+}. First we show that our assumptions imply the following formula for the Lopatinski space,

(28)  L_{g}^{+}(y^{(0)}, η^{(0)}) = im ( \begin{matrix}∫_{γ^{+}} |η^{(0)}| p(0, y^{(0)}, λ, η^{(0)})^{−1} dλ \\ ∫_{γ^{+}} λ p(0, y^{(0)}, λ, η^{(0)})^{−1} dλ \end{matrix} ) .

From (23) we infer that the resolvent of g is

(29) (λ − g)^{−1} = ( \begin{matrix}∗ & |η|p(λ)^{−1} \\ ∗ & λp(λ)^{−1} \end{matrix} ) where p(λ) = λ^{2} + p_{1}λ + p_{2}, λ ∈ ℂ.

Hence the right hand side in (28) is contained in the left hand side. Equality follows from the dimension assumptions (25) and (26).

The principal symbol of B is b = (Id_{K}, 0). Therefore (28), (25), and (26) imply bL_{g}^{+} = ℂ^{K} at (0, y^{(0)}, η^{(0)}). Proposition 2.2 applies to give a solution operator of (18), W = (W_{1}, W_{2}) with W_{1} = BW. Define U = ⟨D_{y}⟩^{−1}W_{1}⟨D_{y}⟩. It follows from Lemma 3.1 that P U ≡ 04s, and r_{Y} U ≡ Id, and W_{2}⟨D_{y}⟩ ≡ D_{x}U. Hence (⟨D_{y}⟩, U′) ≡ r_{Y} W ⟨D_{y}⟩ ∈ Ψ^{1}(Y). The principal symbol of r_{Y} W, (Id_{K}, |η|^{−1}u′(y, η)), maps ℂ^{K} into the Lopatinski space L_{g}^{+}(y, η). Now we can read the formula (27) off the equation (28).

The bound on WF(u|_{Z^{∘}}) follows from the bound on WF(w|_{Z^{∘}}) in Proposition 2.2. □

4. Isotropic Elastodynamic Equations

In following P denotes an operator for isotropic elastodynamics introduced in (2) and (3) such that (7) holds.

The boundary problem P u = g in ℝ × Ω, and u = 0 on ℝ × ∂Ω, has the variational

formulation: (ρü, v) + a(u, v) + (g, v) = 0, ∀v ∈ (H^{(1)}^{∘}(Ω))^{3}. Here a = a_{0} + a_{R}, a_{0}(u, v) = ∫_{Ω} tr (CE(u) E(v)^{T} ) dx, and a_{R}(u, v) = ∫_{Ω} tr ((∇u) R (∇v)^{T} ) dx. It follows from Korn’s inequality that a_{0} satisfies a coerciveness estimate |a_{0}(u, u)| ≥ c∥u∥_{1}^{2} with a positive constant depending only on Ω and on a lower bound on the Lamé coefficient μ, [DL76]. Given L > 0 there exists ε > 0 such that a is coercive if P ∈ L(L, ε). In fact, a_{R} is absorbed into the coerciveness estimate if (5) is assumed with 0 < ε = ε(L) sufficiently small. We use [DL76, Thm. III.4.1.] to conclude that the initial boundary value problem for P with Dirichlet boundary conditions in H_{c}^{(s)}(ℝ × ∂Ω)^{3}, s ≥ 3, is well-posed. In particular, the Dirichlet-to-Neumann (DN) map (6) is defined,

(30)  Λ : H_{c}^{(s+1)}(ℝ × ∂Ω)^{3} → H^{(s)}(ℝ × ∂Ω)^{3} if s ≥ 2.

Let (t, x, τ, ξ) denote a generic point in T^{⋆}(ℝ × \overline{Ω}) \ 0. The Euclidean metric ξ^{2} = ξ · ξ is used to identify tangent and cotangent vectors of \overline{Ω}. For η, ζ ∈ ℂ^{3} the dot product is the analytic (non-Hermitian) extension, η · ζ = η_{1}ζ_{1} + η_{2}ζ_{2} + η_{3}ζ_{3}. π = π(ξ) = (ξ ⊗ ξ)/(ξ · ξ) denotes the orthogonal projection onto a nonzero direction if ξ ∈ ℝ^{3}.

As a consequence of (7) the metrics defined in (8) and (9) satisfy

(31)  0 < ⟨ξ, ξ⟩_{S} < ⟨ξ, ξ⟩_{P} if ξ ≠ 0.

The norms associated with these metrics are denoted |ξ|_{S/P} = \sqrt{⟨ξ, ξ⟩_{S/P}}.

Proposition 4.1. The scalar symbols q_{S/P} (t, x, τ, ξ) = ρ(x)(τ^{2} − ⟨ξ, ξ⟩_{S/P} ) and their product q_{S}q_{P} are of real principal type. P is a system of real principal type with principal symbol

(32)  p = q_{S}(Id_{3} −π) + q_{P} π.

Proof.A straightforward computation gives the principal symbol p of P at (t, x, τ, ξ) ∈ T^{⋆}(ℝ × \overline{Ω}) as follows:

p = ρτ^{2} Id_{3} −(λ + μ)(ξ ⊗ ξ) − μξ^{2} Id_{3} −(ξ · Rξ) Id_{3} .

Hence p has the asserted form. It follows from (31) that q_{S} and q_{P} are of real principal type. Furthermore

(33)  q_{P} (t, x, τ, ξ) < q_{S}(t, x, τ, ξ) if ξ ≠ 0.

Hence also q = q_{S}q_{P} is a scalar symbol of real principal type. Now q^{−1}(0) = (det p)^{−1}(0) and p̃p = q Id for p̃ = q_{P} (Id_{3} −π) + q_{S}π. According to [Den82, Definition 3.1] P is a system of real principal type with characteristic variety q = 0. □

Remark 3. Man [Man98] proposes for elastodynamics with residual stress R a more general constituitive law S = R + ∇u · R + CE where the elasticity tensor C also depends linearly on R. In the isotropic case CE consists of the right-hand side in (3) plus the R dependent terms

(34) β_{1} tr(E) tr(R)I + β_{2} tr(R)E + β_{3}( tr(E)R + tr(ER)I) + β_{4}(ER + RE).

In the inverse problem for real media the additional terms should not be neglected since typically R is much larger than the stress S. A straightforward calculation shows that the elastodynamic operator P with this isotropic stress-strain relation is still of real principal type in case β_{3} = β_{4} = 0, λ + 2μ + β_{1} tr(R) > μ + β_{2} tr(R)/2, and (μ(x) + β_{2}(x) tr(R(x))/2)|ξ|^{2} + R(x, ξ) > 0 when (x, ξ) ∈ T^{⋆}(\overline{Ω}) \ 0.

We recall some notions of the microlocal theory of boundary problems and apply them to the system of elastodynamics. Let γ = (t, x, τ, ξ_{|}) ∈ T^{⋆}(ℝ × ∂Ω) \ 0. This means that there is given (t, x, τ, ξ) ∈ T^{⋆}(ℝ × \overline{Ω}) with x ∈ ∂Ω and ξ_{|} = ξ|_{T_{x}(∂Ω)}. γ is called an elliptic, a hyperbolic, or a glancing point of S/P mode if the following quadratic equation in z,

q_{S/P} (t, x, τ, ξ − zν(x)) = 0,

has no real roots, two distinct real roots, or a double real root, respectively. T^{⋆}(ℝ × ∂Ω) \ 0 decomposes into the disjoint union of the elliptic region E_{S/P}, the hyperbolic region H_{S/P}, and the glancing hypersurface G_{S/P} of the S/P mode. Because of (33) we have E_{S} ⊂ E_{P} and H_{P} ⊂ H_{S}. T^{⋆}(ℝ×∂Ω)\0 is the disjoint union of the hyperbolic region H_{P}, the mixed region E_{P} ∩ H_{S}, the elliptic region E_{S}, and the glancing set G = G_{S} ∪ G_{P}. The lens maps satisfy S_{S/P} ⊂ H_{S/P} × H_{S/P}.

A simple real root z is called forward (resp. backward) if the bicharacteristic curve starting in ξ − zν enters ℝ × Ω when time increases (resp. decreases). Characteristics and bicharacteristics are called forward or backward correspondingly. Observe from Hamiltons equations that a characteristic ξ − zν, z real, of q_{S/P} is forward (resp. backward) if τ⟨ξ − zν, ν⟩_{S/P} is positive (resp. negative). We denote by z_{S/P} = z_{S/P} (t, x, τ, ξ, ν) the forward real root z or the complex root z with positive imaginary part of q_{S/P} (t, x, τ, ξ − zν) = 0. We shall use the abbreviation ξ_{S/P} = ξ − z_{S/P} ν(x).

Given δ > 0 we define

(35)  Γ_{δ} = {(t, x, τ, ξ_{|}) ∈ T^{⋆}(ℝ × ∂Ω) \ 0 ; |τ| ≥ δ|ξ_{|}|}.

Here |ξ_{|}| = |ξ| if ξ · ν = 0. Λ is pseudo-differential microlocally at nonglancing points in Γ_{δ}.

Proposition 4.2. Let L, δ > 0. Assume (4). There exists 0 < ε = ε(L, δ) such that under the assumption (5) the following holds. Given γ ∈ Γ_{δ} \G the DN map Λ equals, in a microlocal neighbourhood of (γ, γ), a first order pseudo-differential operator with principal symbol given as follows.

(36)  σ(Λ) : a ↦ λ(a · ξ′)ν + μ(a · ν)ξ′ + μ(ξ′ · ν)a + (Rξ′ · ν)a

when a ∈ ℂξ′, resp. a · ξ′ = 0, with ξ′ = ξ_{P}, resp. ξ′ = ξ_{S}.

We the need the following fact about the characteristics of P.

Lemma 4.3. Let L, δ > 0. Assume (4). There exists 0 < ε = ε(L, δ) such that the following is true if (5) holds. For γ = (t, x, τ, ξ_{|}) ∈ Γ_{δ} \ G we have z_{S} ≠ z_{P}, ξ_{S}^{2} ≠ 0, ξ_{P}^{2} ≠ 0, and

(37)  ξ_{S} · ξ_{P} ≠ 0.

We prove Lemma 4.3 in section 7.

Proof of Proposition 4.2. Choose ε as in Lemma 4.3. Let γ = (t, x, τ, ξ_{|}) ∈ Γ_{δ} \ G. Flatten the boundary ∂Ω near x with a change of coordinates such that the differential at x is orthogonal.

We consider the symbol p = p(t, x, ·). Its inverse is p^{−1} = q_{S}^{−1}(Id_{3} −π) + q_{P}^{−1}π. Let γ^{+} be a closed Jordan curve enclosing z_{S} and z_{P} but no other roots of q(τ, ξ−zν) = 0. Observe c_{S/P} := (d/dz)q_{S/P} (τ, ξ − zν)|_{z=z_{S/P}} ≠ 0. By the residue theorem

\begin{aligned}A_{j} : & = ∫_{γ^{+}} z^{j} p(τ, ξ − zν)^{−1} \frac{dz}{2πi} \\ & = (z_{S}^{j}/c_{S}) (Id_{3} −π(ξ_{S})) + (z_{P}^{j} /c_{P}) π(ξ_{P}) \end{aligned} ; (38)

for every non-negative integer j. We show that A_{0} is non-singular. Assume A_{0}w = 0. Then 0 = ξ_{S} ·A_{0}w = (ξ_{S} ·ξ_{p})(ξ_{P} ·w)/(c_{P} ξ_{P}^{2} ). Applying Lemma 4.3 we infer ξ_{P} ·w = 0. Therefore w = π(ξ_{S})w and 0 = (ξ_{P} · ξ_{S})(ξ_{S} · w). Applying Lemma 4.3 again we get ξ_{S} · w = 0. Hence w = π(ξ_{S})w = 0.

The invertibility of A_{0} implies that (26) holds with K = 3. Inequality (25) holds because the rank is bounded by N = 6 minus the dimension of the eigenspaces corresponding to eigenvalues not enclosed by γ^{+} which is 3.

We can now apply Proposition 3.3. We find, microlocally near γ, a parametrix U for the initial boundary problem. Hence γ ∉ WF (Λf − (ν · S(Uf))|_{ℝ×∂Ω}) if WF f is contained in a small conic neighbourhood of γ. Here S(u) denotes the stress tensor which corresponds to the displacement u. The displacement-to-traction map u ↦ ν · S(u) is a first order differential operator with principal symbol

(39)  s(x, ξ) := λ(ν ⊗ ξ) + μ(ξ ⊗ ν) + μ(ξ · ν) Id_{3} +(Rξ · ν) Id_{3} .

It then follows from Proposition 3.3 that Λ ∈ Ψ^{1} in a conic neigbourhood of γ.

It remains to prove the formula (36) for the principal symbol of Λ. From (38) we obtain

(40)  A_{1}v = \begin{cases}z_{S}A_{0}v ∈ ξ_{S}^{⊥} & if v ∈ ξ_{P}^{⊥}, \\ z_{P} A_{0}v ∈ ℂξ_{P} & if v ∈ ℂξ_{S}. \end{cases}

From (27) we get a formula for the principal symbol of the normal derivative −D_{ν}U followed by restriction to ℝ × ∂Ω. Using this, (39), and (40) we deduce (36). □

Remark 4. Given L > 0 we choose δ > 0 such that T^{⋆}(ℝ × ∂Ω) \ Γ_{δ} ⊂ E_{S} holds for every operator in L(L, 1/2). We then choose 0 < ε ≤ 1/2 such that for every P ∈ L(L, ε) the initial boundary problem with Dirichlet boundary conditions is well-posed and the assertions in Proposition 4.2 hold. It follows from the proof of Proposition 4.2 that microlocal forward and backward parametrices exist in the hyperbolic, in the mixed, and in part of the elliptic region.

5. Propagation of Polarization

Let L > 0. Choose 0 < δ, ε as in Remark 4 at the end of section 4. In the following we assume P ∈ L(L, ε).

We analyze the polarization of solutions of P u = 0 by applying polarization filters, i.e., certain approximate projection operators, to the Cauchy data of u. Fix E ∈ Ψ^{1}(ℝ × ∂Ω) scalar elliptic with principal symbol e. The Cauchy data of a solution of P u ∈ C^{∞}(ℝ × \overline{Ω}) are, by definition, Cu = (Eu|_{ℝ×∂Ω}, ν · S|_{ℝ×∂Ω}). Notice that the Cauchy data of a solution u of the initial boundary value problem P u = 0, u = f at ℝ × ∂Ω, and u = 0 initially, are represented using the DN map as follows: Cf = (Ef, Λf) = Cu. Here, abusing notation, we also defined Cf.

We now describe, on the principal symbol level, the spaces onto which polarization filters project. Let γ = (t, x; τ, ξ_{|}) ∈ T^{⋆}(ℝ × ∂Ω) \ 0. Set

B_{S/P}^{±} (γ) = ( \begin{matrix}e(γ) Id_{3} \\ s(x, ξ − z_{S/P}^{±} ν) \end{matrix} ) ker p(t, x, τ, ξ − z_{S/P}^{±} ν) ⊂ ℂ^{6}

if γ ∈ H_{S/P}. Here z_{S/P}^{+} and z_{S/P}^{−} are the forward and backward roots of q_{S/P} (t, x, τ, ξ− zν) = 0, respectively. Also define the linear subspaces

B_{S/P} (γ) = ∑ ( \begin{matrix}e(γ) Id_{3} \\ s(x, ξ − zν) \end{matrix} ) ker p(t, x, τ, ξ − zν)

where the sum ranges over the roots of q_{S/P} (t, x, τ, ξ − zν) = 0. Clearly, B_{S/P} (γ) = B_{S/P}^{+} (γ) + B_{S/P}^{−} (γ) if γ ∈ H_{S/P}. The disjoint unions

B_{S/P}^{±} = ˙∪_{γ∈H_{S/P}} B_{S/P}^{±} (γ) and B_{S/P} = ˙∪_{γ∉G_{S/P}} B_{S/P} (γ)

are subsets of the trivial bundles, ℂ^{6}.

Lemma 5.1. B_{S}^{±}, resp. B_{P}^{±}, are vector subbundles of ℂ^{6} over H_{S}, resp. H_{P}, of ranks 2, resp. 1. B_{P} is a vector subbundle of ℂ^{6} over H_{P} ∪ E_{P} of rank 2. Furthermore

(41) ℂ^{6} = B_{S}^{+} ⊕ B_{S}^{−} ⊕ B_{P}^{+} ⊕ B_{P}^{−} over H_{P}, ℂ^{6} = B_{S}^{+} ⊕ B_{S}^{−} ⊕ B_{P} over H_{S} ∩ E_{P}.

Proof.Given a point in ∂Ω we introduce coordinates x = (x_{1}, …, x_{n}) = (x′, x_{n}) such that Ω and ∂Ω correspond to x_{n} > 0 and x_{n} = 0, respectively. Let ξ = (ξ′, ξ_{n}) denote the dual variables. We also arrange that, at the given point in the coordinates, formula (39) still holds. At x_{n} = 0,

( \begin{matrix}e Id_{3} \\ s \end{matrix} ) = ( \begin{matrix}h_{1} Id_{3} & 0 \\ ∗ & ∂s/∂ξ_{n} \end{matrix} ) ( \begin{matrix}(τ^{2} + ξ′^{2})^{1/2} Id_{3} \\ ξ_{n} Id_{3} \end{matrix} ) .

λ, μ ≥ 0 and (7) imply the ellipticity of ∂s/∂ξ_{n}. Hence

(42)  h(t, x′, τ, ξ′) = ( \begin{matrix}h_{1} Id_{3} & 0 \\ ∗ & ∂s/∂ξ_{n} \end{matrix} )

defines an elliptic 6 × 6 symbol of order 0. Therefore, it suffices to prove the Lemma when, in the definitions of the vector spaces B_{S/P}^{∗} (γ), the symbols e and s(x, ξ) are replaced by (τ^{2}+ξ′^{2})^{1/2} Id_{3} and ξ_{n} Id_{3}, respectively. Having made this replacement the assertions follow from spectral decomposition of the first order symbol g associated with p in Lemma 3.2 and from (24) together with the known dimensions of ker p. □

Let π_{S/P}^{±} and π_{S/P} denote the projectors associated with the decompositions (41). In the following Π_{S/P}^{±} and Π_{S/P} denote 6 × 6 systems of pseudo-differential operators of order 0 having principal symbols π_{S/P}^{±} and π_{S/P}. The operators are defined microlocally where the symbols are.

We now state how polarization in solutions can be tested on the Cauchy data.

Proposition 5.2. Let u ∈ D′(ℝ × \overline{Ω})^{3} such that P u ∈ C^{∞}(ℝ × \overline{Ω})^{3}, Cu ∈ H^{(s−1)}(ℝ × ∂Ω)^{6}. Let γ ∈ T^{⋆}(ℝ × ∂Ω) \ 0. If γ ∈ H_{P} then γ ∈ WF^{(s)}(Π_{P}^{±}Cu) if and only if WF_{pol}^{(s+1)}(u) contains a Hamilton orbit above the forward/backward compressional wave bicharacteristic which issues from γ into the interior. If γ ∈ H_{S} \ G_{P} then γ ∈ WF^{(s)}(Π_{S}^{±}Cu) if and only if WF_{pol}^{(s+1)}(u) contains a Hamilton orbit above the forward/backward shear wave bicharacteristic which issues from γ into the interior.

Proof.Introduce coordinates x = (x_{1}, …, x_{n}) as in the proof of Lemma 5.1. Abbreviate D_{x} = (D′, D_{n}). The simplified Cauchy data

C_{0}u = (⟨D_{t}, D′⟩u|_{x_{n}=0}, D_{n}u|_{x_{n}=0})

are related to the Cauchy data Cu as follows: Cu ≡ HC_{0}u where H ∈ Ψ^{0}(ℝ×∂Ω)^{6×6} with principal symbol equal to h of equation (42). It now suffices to prove the Proposition with Cu replaced by C_{0}u and the operators Π_{S/P}^{±} replaced by H^{−1}Π_{S/P}^{±} H. The assertions now follow from Proposition 2.3 if we recall the argument in Remark 2 of section 3. □

Curves which are bicharacteristics over the interior and reflected at non-glancing boundary points, with or without conversion between shear and compressional mode, are called broken bicharateristics. The propagation of singularities in the Cauchy data is stated recursively as follows.

Proposition 5.3. Let f ∈ H_{c}^{(s)}(ℝ × ∂Ω), s ≥ 3, with WF^{(s+1)}(f) ⊂ Γ_{δ}. Let T ∈ ℝ such that no forward broken bicharacteristic which issues from

WF^{(s+1)}(f) ∩ ( WF^{(s)}(Π_{S}^{+}Cf) ∪ WF^{(s)}(Π_{P}^{+}Cf))

intersects G ∩ {t ≤ T }. Then, after intersection with {t ≤ T },

WF^{(s)}(Cf) = WF^{(s+1)}(f) ∪ S_{S}( WF^{(s)}(Π_{S}^{+}Cf))
∪ S_{P} ( WF^{(s)}(Π_{P}^{+}Cf)).

(43)

Proof.Let u denote the solution of P u = 0 with Dirichlet boundary value f and zero initial data. By [Den82, Theorem 4.2] and [Gér85] the polarization set WF_{pol}^{(s+1)}(u) ∩ {t ≤ T } is contained in the union of Hamilton orbits which lie above the broken bicharacteristics which issue from WF^{(s+1)}(f). We apply Proposition 5.2 at both endpoints of bicharacteristics which connect boundary points. We obtain S_{S/P} ( WF^{(s)}(Π_{S/P}^{+} Cf)) ⊂ WF^{(s)}(Π_{S/P}^{−} Cf). Therefore the right hand side of (43) is contained in the left hand side. To prove the opposite inclusion let γ ∈ WF^{(s)}(Cf) \ WF^{(s+1)}(f). Then γ is the endpoint of a forward bicharacteristic contained in WF^{(s+1)}(u) and, by Proposition 5.2, issued from WF^{(s)}(Π_{S}^{+}Cf) or WF^{(s)}(Π_{P}^{+}Cf). □

Proposition 5.2 combined with the following result permits us to specify, without having to know the coefficients of P, sources for which compressional singularities are muted.

Proposition 5.4. Choose M ∈ Ψ^{0}(ℝ×∂Ω)^{3×3} such that its principal symbol m equals at every (t, x, τ, ξ_{|}) ∈ T^{⋆}(ℝ × ∂Ω) the orthogonal projector onto the one-dimensional subspace of ℝ^{3} which is orthogonal to ξ and ν(x). Let γ ∈ H_{P}. There is a conic neighbourhood Γ ⊂ H_{P} of γ such that the following inclusion holds for every f ∈ H_{c}^{(s)}(ℝ × ∂Ω)^{3}, s ≥ 3, with WF(f) ⊂ Γ:

(44)  WF^{(s)} (Π_{P}^{+}Cf) ⊂ WF^{(s+1)}(f − Mf).

Proof.First we show

(45)  ( \begin{matrix}m & 0 \\ 0 & m \end{matrix} ) ℂ^{6} ⊂ B_{S}^{+} + B_{S}^{−}.

To see this let γ = (t, x, τ, ξ_{|}) and a ∈ ℂ^{3} with ξ · a = ν · a = 0, ν = ν(x). In view of (32) a belongs to the kernel of p(t, x, τ, ξ_{S}^{±}). In view of the definition (39) we have s(x, ξ_{S}^{±})a = ⟨ξ_{S}^{±}, ν⟩_{S}a. Hence (e(γ)a, ⟨ξ_{S}^{±}, ν⟩_{S}a) ∈ B_{S}^{±}(γ). ⟨ξ_{S}^{+} − ξ_{S}^{−}, ν⟩_{S} ≠ 0 because z_{S}^{+} ≠ z_{S}^{−}. Therefore we obtain (0, a), (a, 0) ∈ B_{S}^{+}(γ) + B_{S}^{−}(γ) proving (45).

π_{P} = π_{P}^{+} +π_{P}^{−} vanishes on B_{S}^{+} +B_{S}^{−}. Therefore (45) and the symbol calculus imply

(46)  Π_{P} ( \begin{matrix}M & 0 \\ 0 & M \end{matrix} ) ∈ Ψ^{−1}.

Choose the conic neighbourhood Γ of γ in such a way that the DN map Λ is a pseudo-differential operator in Γ × Γ. Shrinking Γ if necessary we may assume that every solution of P u = 0 which has zero initial data and Dirichlet data f with WF(f) ⊂ Γ does not contain backward bicharacteristics issuing from Γ in its wavefront set.

Observe from formula (36) that the principal symbol of Λ maps the space onto which m projects into itself. Hence ΛM − MΛM ∈ Ψ^{0} and therefore

(47)  ( \begin{matrix}EM \\ ΛM \end{matrix} ) ≡ ( \begin{matrix}M & 0 \\ 0 & M \end{matrix} ) ( \begin{matrix}E Id_{3} \\ ΛM \end{matrix} )  mod Ψ^{0}.

Let f ∈ H_{c}^{(s)}(ℝ × ∂Ω)^{3}, s ≥ 3, with WF(f) ⊂ Γ. Equations (46) and (47) imply

Π_{P} ( \begin{matrix}EMf \\ ΛMf \end{matrix} ) ∈ H^{(s)}(ℝ × ∂Ω).

Assume γ ∉ WF^{(s+1)}(f − Mf). Then, recalling Cf = (Ef, Λf), we obtain

(48)  γ ∉ WF^{(s)}(Π_{P} Cf).

Proposition 5.2 and our choice of Γ imply that (48) holds with Π_{P} replaced by Π_{P}^{−}. Hence (48) also holds with Π_{P} replaced by Π_{P}^{+}. □

6. Proof of Theorem 1.1

The idea is to recover the lens maps from the elements of WF(Λf) \ WF(f) with least time where f ranges over point sources.

Let L > 0. Choose 0 < δ, ε as in Remark 4 at the end of section 4. Let P^{(1)}, P^{(2)} ∈ L(L, ε). Assume Λ^{(1)} = Λ^{(2)}. We show that the shear and compressional lens maps are equal: S_{S}^{(1)} = S_{S}^{(2)} and S_{P}^{(1)} = S_{P}^{(2)}. Let S̃ denote the union of the sets S_{S/P}^{(j)} ∩ ((G^{(k)} × Γ_{δ}) ∪ (Γ_{δ} × G^{(k)})). Observe that S_{S/P}^{(j)} \ S̃ is open and dense in S_{S/P}^{(j)} . We first prove that the lens maps agree outside S̃.

Fix s ≥ 3. Below we choose, given γ^{in} ∈ Γ_{δ} nonglancing, sources f with the properties

(49)  f ∈ H_{c}^{(s)}(ℝ × ∂Ω)^{3}, WF^{(s+1)}(f) = ℝ_{+}γ^{in}.

We study the singularities of the Cauchy data Cf = (Ef, Λ^{(1)}f) = (Ef, Λ^{(2)}f). Because of the zero initial condition the backward bicharacteristics issuing from γ^{in} are disjoint from WF^{(s+1)}(u). By Proposition 5.2 γ^{in} ∉ WF^{(s)}(Π_{S/P}^{−(j)}Cf). γ^{in} ∈ WF^{(s)}(Cf) by assumption on f since E is elliptic. Therefore γ^{in} ∈ WF^{(s)}(Π_{S}^{+(j)}Cf)∪ WF^{(s)}(Π_{P}^{+(j)}Cf) ∪ E_{P}^{(j)}.

Let (γ^{out}, γ^{in}) ∈ S_{S}^{(1)} \ S̃. Choose f with (49) and γ^{in} ∉ WF^{(s+1)}(f − Mf). Proposition 5.4 implies γ^{in} ∉ WF^{(s)}(Π_{P}^{+(j)}Cf) for j = 1 and j = 2. Consequently, γ^{in} ∈ WF^{(s)}(Π_{S}^{+(j)}Cf). Proposition 5.3 with P^{(1)} implies

(50)  WF^{(s)}(Cf) ∩ {t ≤ t(γ^{out})} = ℝ_{+}γ^{in} ∪ ℝ_{+}γ^{out}.

(50) and Proposition 5.3 with P^{(2)} imply (aγ^{out}, γ^{in}) ∈ S_{S}^{(2)} for some a > 0. The covariable τ is constant along bicharacteristics. Therefore a = 1. Thus we have shown S_{S}^{(1)} \ S̃ ⊂ S_{S}^{(2)}. Interchanging P^{(1)} with P^{(2)} we obtain

(51)  S_{S}^{(1)} \ S̃ = S_{S}^{(2)} \ S̃.

Let (γ^{out}, γ^{in}) ∈ S_{P}^{(1)} \ S̃. Choose f with (49) and

(52)  γ^{in} ∈ WF^{(s)}(Π_{P}^{+(1)}Cf) \ WF^{(s)}(Π_{S}^{+(1)}Cf).

Proposition 5.3 with P^{(1)} implies (50). (50) and Proposition 5.3 with P^{(2)} imply (aγ^{out}, γ^{in}) ∈ S_{S}^{(2)} ∪ S_{P}^{(2)} with a > 0. Again a = 1 follows. If (γ^{out}, γ^{in}) ∈ S_{S}^{(2)} then also (γ^{out}, γ^{in}) ∈ S_{S}^{(1)} by (51). Hence (γ^{out}, γ^{in}) ∈ S_{P}^{(2)} if (γ^{out}, γ^{in}) ∉ S_{S}^{(1)}. Suppose (γ^{out}, γ^{in}) ∈ S_{S}^{(1)}. Choose f as above but now with projectors Π_{S/P}^{+(1)} in (52) replaced by Π_{S/P}^{+(2)}. Equation (50) still follows from Proposition 5.3 with P^{(1)} because (γ^{out}, γ^{in}) ∈ S_{S}^{(1)} ∩ S_{P}^{(1)}. γ^{in} ∉ WF^{(s)}(Π_{S}^{+(2)}Cf), (50), and Proposition 5.3 with P^{(2)} imply (γ^{out}, γ^{in}) ∈ S_{P}^{(2)}. Thus we have shown S_{P}^{(1)} \ S̃ ⊂ S_{P}^{(2)}. Interchanging P^{(1)} with P^{(2)} we obtain

(53)  S_{P}^{(1)} \ S̃ = S_{P}^{(2)} \ S̃.

It remains to show that (51) and (53) hold with S̃ replaced by the empty set. Assume (γ^{out}, γ^{in}) ∈ S_{S}^{(1)}. We use a limit argument to prove (γ^{out}, γ^{in}) ∈ S_{S}^{(2)}. First we observe that γ^{in}, γ^{out} ∉ G_{S}^{(2)}. Suppose not. Then a neighbourhood of (γ^{out}, γ^{in}) in S_{S}^{(1)} contains a point which is in (Γ_{δ} ×E_{S}^{(2)})\S̃ or in (E_{S}^{(2)} ×Γ_{δ})\S̃. This point cannot be in S_{S}^{(2)}. This contradicts (51). Choose a sequence (γ_{k}^{out}, γ_{k}^{in}) ∈ S_{S}^{(1)} ∩ S_{S}^{(2)}, k ∈ ℕ, which converges to (γ^{out}, γ^{in}). Let γ_{k} denote the shear wave bicharacteristic for P^{(2)} with γ_{k}(0) = γ_{k}^{in} and γ_{k}(t_{k}) = γ_{k}^{out}, t_{k} > 0. The length sequence of the corresponding sequence of geodesics is bounded. By compactness there is a limit geodesic and thus a bicharacteristic γ : [0, T] → T^{⋆}(ℝ × \overline{Ω}) of P^{(2)} with γ(0) = γ^{in} and γ(T) = γ^{out}. It suffices to show γ(t) lies over the interior when 0 < t < T. Suppose we had γ(t^{∗}) above the boundary for some 0 < t^{∗} < T. Then every neighbourhood of (γ(t^{∗}), γ^{in}) has non-empty intersection with S_{S}^{(2)} hence, by (51), also with S_{S}^{(1)}. This contradicts the continuity of the map S_{S}^{(1)} at γ^{in}. Hence we have shown S_{S}^{(1)} ⊂ S_{S}^{(2)}. The other inclusions are proved in the same way.

7. Proof of Lemma 4.3

Let γ = (t, x, τ, ξ_{|}) ∈ Γ_{δ} \ G. To ease notation we drop the coordinates (t, x).

Recall the definitions of the symbols q_{S/P} (τ, ξ)/ρ = τ^{2} −|ξ|_{S/P}^{2} , the metrics ρ|ξ|_{P}^{2} = ρ|ξ|_{S}^{2} + (λ + μ)|ξ|^{2}, ρ|ξ|_{S}^{2} = μ|ξ|^{2} + Rξ · ξ, and the characteristics ξ_{S/P} = ξ − z_{S/P} ν, q_{S/P} (τ, ξ_{S/P}) = 0. We have |ν| = 1.

The equation q_{S}(τ, ξ − zν) − q_{P} (τ, ξ − zν) = (λ + μ)(ξ − zν)^{2} holds for z ∈ ℂ. It implies ξ_{S} · ξ_{P} = ξ_{S/P}^{2} = 0 if z_{S} = z_{P}. Also it implies q_{P} (τ, ξ_{S}) = 0 if ξ_{S}^{2} = 0. So if we had ξ_{S}^{2} = 0 then z_{S} ∉ ℝ because of the real principal type property of q_{S}q_{P}. Since Im z_{S/P} ≥ 0 this can only happen if z_{P} = z_{S}. Therefore ξ_{S}^{2} = 0 implies ξ_{S} · ξ_{P} = 0. In the same way ξ_{P}^{2} = 0 implies ξ_{S} · ξ_{P} = 0. Therefore, it suffices to prove the inequality (37).

Choose 0 < ε ≤ 1/2. ε > 0 will be decreased further depending on L and δ only. The smallness assumption (5) on the residual stress tensor implies

(54)  1 − ε ≤ \frac{ρ|η|_{S}^{2}}{μ|η|^{2}} , \frac{ρ|η|_{P}^{2}}{(λ + 2μ)|η|^{2}} ≤ 1 + ε if η ≠ 0.

Consider the elliptic case, γ ∈ E_{S} ⊂ E_{P}. Without loss of generality we assume ξ · ν = 0. Then ξ_{S} · ξ_{P} = |ξ|^{2} + z_{S}z_{P} and |ξ| = |ξ_{|}| ≤ |τ|/δ. (λ + 2μ)/ρ ≤ L^{2} by (4). Therefore, we assume

(55)  ρτ^{2} > 2ε(λ + 2μ)|ξ|^{2} ≥ 2εμ|ξ|^{2}.

z = z_{S/P} is the solution with positive imaginary part of the quadratic equation

|ν|_{S/P}^{2} z^{2} − 2bz + (|ξ|_{S/P}^{2} − τ^{2}) = 0 where b = Rξ · ν.

Notice that the signs of the real parts of z_{S} and z_{P} are equal to the sign of b. Hence z_{S}z_{P} ∉ ℝ and thus ξ_{S} · ξ_{P} ≠ 0 if b ≠ 0. Assume b = 0. We solve the quadratic equations and then estimate using (54) and (55):

|z_{S}z_{P} |^{2} = \frac{ρ(|ξ|_{S}^{2} − τ^{2})}{ρ|ν|_{S}^{2}} · \frac{ρ(|ξ|_{P}^{2} − τ^{2})}{ρ|ν|_{P}^{2}}

< \frac{(μ(1 + ε) − 2εμ)|ξ|^{2}}{(1 − ε)μ|ν|^{2}} · \frac{((λ + 2μ)(1 + ε) − 2ε(λ + 2μ))|ξ|^{2}}{(1 − ε)(λ + 2μ)|ν|^{2}}

≤ |ξ|^{4}.

Hence |ξ|^{2} + z_{S}z_{P} ≠ 0, i.e., (37) holds.

In the mixed case, γ ∈ E_{P} ∩ H_{S}, we have z_{S} ∈ ℝ and Im z_{P} > 0. This implies (37).

Consider the hyperbolic case, γ ∈ H_{P} ⊂ H_{S}. Without loss of generality we assume

(56)  ⟨ξ, ν⟩_{P} = 0.

Then the roots of 0 = τ^{2} − |ξ − zν|_{P}^{2} = −|ν|_{P}^{2} z^{2} − |ξ|_{P}^{2} + τ^{2} have opposite signs. Since q_{P} < q_{S} this is also true for the roots of 0 = τ^{2} − |ξ − zν|_{S}^{2}. Furthermore 0 < |z_{P} | < |z_{S}|. Since z_{S} and z_{P} are both forward they have the same sign. For simplicity we assume 0 < z_{P} < z_{S}. We shall use the estimate

(57)  ξ_{S} · ξ_{P} ≥ |ξ|^{2} − εz_{S}|ξ| + z_{S}z_{P} .

(λ + 2μ)ξ · ν + Rξ · ν = 0 is equation (56) restated. From this and (5) we deduce 2|ξ · ν| ≤ ε|ξ|. Hence |(z_{S} + z_{P})(ξ · ν)| ≤ 2z_{S}|ξ · ν| ≤ εz_{S}|ξ|. (57) follows.

The equation q_{S}(τ, ξ − z_{S}ν) − q_{P} (τ, ξ − z_{P} ν) = 0 is equivalent to

(58)  z_{S}^{2}(|ν|_{S}^{2} − t^{2}|ν|_{P}^{2} ) − 2z_{S}⟨ξ, ν⟩_{S} − ((λ + μ)/ρ)|ξ|^{2} = 0 where t = z_{P} /z_{S}.

We estimate the root z_{S} of this quadratic equation. Using the Cauchy-Schwarz inequality and (54) to estimate |ν|_{S} from below we deduce from (58)

|z_{S}|/4 ≤ |ξ|_{S} + |ξ|\sqrt{(λ + μ)/μ} if 2t|ν|_{P} ≤ |ν|_{S}.

Decreasing ε > 0 if necessary, we assume εz_{S} ≤ |ξ| if 2t|ν|_{P} ≤ |ν|_{S}. Inserting this estimate into (57) we get ξ_{S} · ξ_{P} ≥ z_{S}z_{P} > 0. It remains to prove (37) when 2t|ν|_{P} > |ν|_{S}. From (54) and (4) we get |ν|_{P} /|ν|_{S} ≤ 3(λ + 2μ)/μ ≤ 3L^{2}. Decreasing ε > 0 if necessary, we assume ε^{2}|ν|_{P} ≤ 2|ν|_{S}. Hence ε^{2}z_{S}^{2}/4 < z_{S}z_{P}. We estimate the right hand side of (57) from below and get ξ_{S} · ξ_{P} > (|ξ| − εz_{S}/2)^{2} ≥ 0.

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Sönke Hansen, Fachbereich Mathematik-Informatik, Universität Paderborn, 33095 Paderborn, Germany

Gunther Uhlmann, Department of Mathematics, University of Washington, Seattle, WA 98195-4350, USA