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On an inverse boundary value problem for a nonlinear elastic wave equation

We survey the inverse boundary value problem for a nonlinear elastic wave equation which was considered in [de Hoop, Uhlmann, Wang. Math. Ann. (2019) doi:10.1007/s00208-018-01796-y]. We show that all the parameters appearing in the equation can be uniquely determined from boundary measurements under certain geometric assumptions. The approach is based on second order linearization and Gaussian beams.


Introduction
Consider the initial boundary value problem for the nonlinear elastic wave equation (1) We note that the above equation is quasilinear.
Here Ω ⊂ R 3 is a bounded domain with smooth boundary ∂Ω. Here we denote x = (x 1 , x 2 , x 3 ) to be the Cartesian coordinates. The stress tensor S has the form S ij =λε mm δ ij + λ ε mm ∂u i ∂x j + 2µ ε ij + ε jn ∂u i ∂x n + A ε in ε jn + B(2 ε nn ε ij + ε mn ε mn δ ij ) + C ε mm ε nn δ ij + O(u 3 ), where ε is the strain tensor defined as ε ij (u) = 1 2 and ε is the linearized strain tensor ε ij (u) = 1 2 ∂u i ∂x j + ∂u j ∂x i . are all smooth on Ω. The parameters λ and µ are called Lamé moduli and ρ is the density. This model is widely used and can be found in [22,9,8].
In this article, we study the inverse problem of recovering the elastic parameters λ, µ, ρ, A , B, C from displacement-to-traction map where ν is the exterior normal unit vector to ∂Ω.
The S -and P -wavespeeds are related with the Lamé moduli λ, µ and the density ρ in the following way c S = µ ρ , c P = λ + 2µ ρ .
Then c P > c S in Ω. Denote the Riemannian metrics associated with P/S-wave speeds to be g P/S = c −2 P/S ds 2 . Then P -and S -waves travel along geodesics in the Riemannian manifolds (Ω, g P ) and (Ω, g S ) respectively. Let diam P/S (Ω) be the diameter of Ω with respect to g P/S . For the inverse problem, one can first recover the Dirchlet-to-Neumann map Λ lin for the linear elastic wave equation (3) by first order linearization of Λ (cf. [8]) It was shown in [14] that from Λ lin one can recover the scattering relation associated to the wave speeds c P and c S . Using the result of [31] one can determine c S and c P if the foliation condition is satisfied for both metrics g P/S and T is larger that all the lengths of the geodesics of g P/S in Ω. For a more precise statement see [29,Theorem 1.4]. Recall that a Riemannian manifold (M, g) satisfies the foliation condition if it can be foliated by strictly convex hypersurfaces [33]. The foliation condition is satisfied for (Ω, g P/S ), for instance, if ∂Ω is strictly convex (with respect to g P/S ) and the wave speeds c P/S increase with depth. They are also satisfied under some additional conditions on the curvature (see [26,30]). As pointed out in [29] the foliation condition is a natural generalization of the Herglotz [15] and the Wieckert-Zoeppritz [36] conditions. The foliation condition allows for conjugate points. If the boundary is strictly convex for g P/S and there are no conjugate points for g P/S , the uniqueness of c P and c S was shown by Rachele in [27]. One can in fact determine the three parameters λ, µ, ρ, if c P = 2c S except at isolated points in Ω, under the foliation condition [4] and the no conjugate points condition [28]. We summarize the results for the linear elastic wave equation (3) in the following.
Proposition 2. Assume T > diam S (Ω), ∂Ω is strictly convex with respect to g P/S , and either of the following condition holds (1) (Ω, g P/S ) has no conjugate points; (2) (Ω, g P/S ) satisfies the foliation condition.
Then Λ lin uniquely determines µ ρ and λ ρ in Ω . Assume further λ = 2µ only at isolated points in Ω, then ρ is also uniquely determined.
In this paper we mainly focus on the determination of the nonlinear elastic parameters A , B, C . In [8], the authors proved the uniqueness of A and B, by analyzing the nonlinear interactions of distorted plane waves. The approach originated from [21], and has been successfully used to study inverse problems for nonlinear hyperbolic equations [6,20,25,34,35]. We will present an alternative approach to the proof of the uniqueness of A and B, and further extend to the uniqueness of C . Our work is still based on the higher order linearization utilized in aforementioned work, but instead of distorted plane waves we will use Gaussian beams. We note here that Gaussian beams have been used to study various inverse problems [2,3,10,11,12,13,18]. We emphasize here that Gaussian beams can be constructed allowing conjugate points.
We summarize the main theorem of this article here: , ∂Ω is strictly convex with respect to g P/S , and either of the following condition holds (1) (Ω, g P ) has no conjugate points; (2) (Ω, g P ) satisfies the foliation condition.
The rest of this paper is organized as follows. In Section 2, we carry out the second order linearization of the displacement-to-traction map and derive an integral identity, from which we can recover the parameters of interest. In Section 3, we construct Gaussian beam solutions to linear elastic wave equation, for both P -and S -waves. Finally, Section 4 is devoted to the proof of the main theorem.

Second-order linearization of displacement-to-traction map
We will apply the higher order linearization technique introduced in [21] to the displacementto-traction map Λ, and arrive at an integral identity which could be used for the recovery of the parameters. The linearization of Λ itself has already been used in [8]. Higher order linearization of Dirichlet-to-Neumann map and the resulted integral identities for semilinear and quasilinear elliptic equations are used [32,17,1,5,23,24,12,19]. Assume u solves (1) with Dirichlet boundary value Denote u (j) , j = 1, 2 to be the solution to the linearized elastic wave equation with boundary value f j , i.e., Applying ∂ 2 ∂ǫ 1 ∂ǫ 2 to (1), we obtain the equation for U (12) Here We note that (12) ) + ν · G(u (1) , u (2) ).
Assume v solves the initial boundary value problem for the backward elastic wave equation By integration by parts, we get 0 Ω ∇ · S L (U (12) )v + C∇U (12) : ∇v dxdt (1) , u (2) ) v + C∇U (12) : ∇v dxdt (1) , u (2) )v + C∇U (12) : ∇v dxdt Here we use the notation A : B = 3 i,j=1 A ij B ij for matrices A and B. Therefore, the displacement-to-traction map determines 0 Ω G(∇u (1) , ∇u (2) , ∇v) dxdt, where G(∇u (1) , ∇u (2) , ∇v) Here we use the notation ∇ T u = (∇u) T . Then we need to construct special solutions for the linear elastic wave equation and recover the parameters A , B, C from the integral (8). We emphasize here that the solutions are constructed with known coefficients λ, µ, ρ in the linearized equation.

Gaussian beam solutions
Denote M = [0, T ] × Ω. We note that M can be viewed as a Lorentzian manifold with metric −dt 2 + g P or −dt 2 + g S .
We will construct Gaussian beam solutions u to the linear elastic wave equation of the form with a large parameter ̺. The phase function ϕ is complex-valued. The principal term e i̺ϕ(t,x) a(t, x) is concentrated near a null geodesic ϑ in (M, −dt 2 + g P/S ). The remainder term R ̺ will vanish as ̺ → +∞.
Construction of Gaussian beams. We will construct approximate Gaussian beam of order N of the form in a neighborhood of ϑ, Here δ > 0 is a small parameter. The smooth function χ : R → [0, +∞) satisfies χ(t) = 1 for |t| ≤ 1 4 and χ(t) = 0 for |t| ≥ 1 2 . We refer to [13] for more details. We denote a 0 = a = (a 1 , a 2 , a 3 ) and The parameter δ is small such that a| t=0 = a| t=T = 0.
Then we construct the remainder term R ̺ . We let R ̺ be the solution to the following initial boundary value problem Here , in a neighborhood of ϑ. By (13) and [13, Lemma 2], we have By L 2 estimates for second order hyperbolic equation, we have We can take N large enough and use Sobolev imbedding to obtain We remark here that u = u ̺ + R ̺ solves the equation (3) with

3.1.
Construction of the phase. We will construct phase function ϕ = ϕ P/S such that vanishes on ϑ up to third order. In terms of Fermi coordinates z = (z 0 = τ, z 1 , z 2 , z 3 ) for g P/S = −dt 2 + g P/S we need Notice that (16) is equivalent to Thus the phase function ϕ can be constructed as in [13] in the form Here for each k, ϕ k is a complex valued homogeneous polynomial of degree k with respect to the variables z i , i = 1, 2, 3. In this paper, we will use the explicit forms of ϕ 0 , ϕ 1 , ϕ 2 , which will be constructed below. Following the lines in [11], one can take and Here H is a symmetric matrix with ℑH(τ ) > 0.

3.2.
Construction of the amplitude for P -waves. We consider the Lorentzian manifold (M, −dt 2 + g P ) and ϑ is a null-geodesic in it. For P -waves, the polarization vector a should be in parallel with the wave vector ∇ϕ on ϑ. Denote ϕ = ϕ P and take Component-wisely, the gradient of ϕ has the form By (19), we have and I 1 = (−ρ(∂ t ϕ) 2 + (λ + 2µ)|∇ϕ| 2 )A P ∇ϕ.
By the construction of the phase function ϕ, we have (13) satisfied.
Substituting the expression for b α into the equation (13) with k = 2 and |Θ| = 1 and i = 1, we end up with a transport equation for ∂ Θ ∂z Θ A P on ϑ, from which we can determine the value of ∂ Θ ∂z Θ A P . Then using again (22), we can determine b α . Finally, the equation (13) with k = 2 and |Θ| = 1 and α = 2, 3 gives us the value of ∂ Θ ∂z Θ b α on ϑ. Continuing with this process, we can have (13) satisfied with k = 2 and |Θ| ≤ N .
By calculation, we find that the equation I 2 = 0 for i = α ′ always holds on ϑ, with arbitrary choice of b α ′ .
Next we consider the equation (I 2 ) i = 0 for i = 1. We obtain the following equation on the null geodesic ϑ ∂A S ∂z α + F(ϕ, A S | ϑ ) = 0. Substitute the expression for b 1 into the equation (13) with k = 2, |Θ| = 1 and i = 1. We will end up with a transport equation for ∂ Θ ∂z Θ A S on ϑ, from which we can determine the value of ∂ Θ ∂z Θ A S . Then using again (26), we can determine b 1 . Finally, the equation (13) with k = 2 and |Θ| = 1 gives us the value of ∂ Θ ∂z Θ b 1 on ϑ. As for P -waves, we can construct lower order terms in the amplitude a such that (13) is satisfied.

Proof of the main theorem
We will prove the uniqueness of A , B, C from the displacement-to-traction Λ in this section. For the determination of A , B, we will have a pointwise recovery in the interior Ω. With A and B determined, certain type of weighted ray transform of C (along any geodesic in (Ω, g P )) can be obtained from Λ. Under some geometric conditions, the weighted ray transform is invertible.
For readers' convenience, we still include the proof here.
The above equation (with b as the unknown) always has a nonzero solution. This finishes the proof of the lemma.

Remark 2.
We only used the nonlinear interaction of P and SV waves to determine A and B. It has already been observed to be possible in [8], wherein nonlinear interactions of other types are analyzed as well. Now assume (Ω, g P ) satisfies the foliation condition. As in [33], for any point q ∈ ∂Ω, there exists a wedge-shaped neighborhood O q ⊂ Ω of q such that any geodesic in (O q , g P ) has no conjugate points. Then the three null-geodesics ϑ (1) , ϑ (2) , can intersect only at p (since c P > c S ). We can now recover A and B in O q . Then the foliation condition allows a layer stripping scheme to recover the two parameters in the whole domain Ω. For more details, we refer to [33]. 4.2. Determination of C . Finally, we recover the parameter C .
Here K = K i j ∂ ∂y i ⊗ dy j , K i j = g ik R kj , R is the Ricci tensor. As in [11], we now have the Jacobi weighted ray transform of the first kind (cf. [12]) of C c −9/2 P ρ −3/2 =: f along the geodesic γ in (Ω, g P ) passing through x 0 = γ(t 0 ), By [12,Proposition 3], J Y f uniquely determines f (x 0 ) if (Ω, g P ) has no conjugate points. Therefore, we can recover C (x 0 ) since c P and ρ are already known. Here we use only the ray transform along this single geodesic. It is possible because we have the integral of f along the geodesic with a class of weights.
If (Ω, g P ) satisfies the foliation condition. We can adopt a layer stripping method as previously used. We can also directly use the the invertibility of weighted geodesic ray transform with a single weight established in [26].