Reconstruction of penetrable obstacles in the anisotropic acoustic scattering

We develop reconstruction schemes to determine penetrable obstacles in a region of \mathbb{R}^{2} or \mathbb{R}^{3} and we consider anisotropic elliptic equations. This algorithm uses oscillating-decaying solutions to the equation. We apply the oscillating-decaying solutions and the Runge approximation property to the inverse problem of identifying an inclusion in an anisotropic elliptic differential equation.


1.
Introduction. In the study of inverse problems, we are interested in the special type of solutions for elliptic equations or systems which play an essential role since the pioneer work of Caldéron. Sylvester and Uhlmann [13] introduced complex geometrical optics (CGO) solutions to solve the inverse boundary value problems of the conductivity equation. Based on CGO solutions, Ikehata proposed the so called enclosure method to reconstruct the impenetrable obstacle, for more details, see [2,3,4]. There are many results concerning this reconstruction algorithm, such as [9,15]. The researchers constructed CGO-solutions with polynomial-type phase function of the Helmholtz equation ∆u + k 2 u = 0 or the elliptic system with the Laplacian as the principal part.
When the medium is anisotropic, we need to consider more general elliptic equations, such as anisotropic scalar elliptic equation in a bounded domain Ω ⊂ R 3 , where A 0 (x) = (a 0 ij (x)), a 0 ij (x) = a 0 ji (x), and we assume the uniform ellipticity condition, that is, for all ξ = (ξ 1 , ξ 2 , · · · ξ n ) ∈ R n , λ 0 |ξ| 2 ≤ i,j a 0 ij (x)ξ i ξ j ≤ Λ 0 |ξ| 2 and x ∈ Ω. In two dimensional case, we can transform (1.1) to an isotropic equation by using isothermal coordinates, then we can apply the CGO-solutions for this case, which can be found in [14]. When Ω ⊂ R 3 , we cannot directly transform (1.1) to an isotropic equation as we do in R 2 , thus we need to use the oscillating-decaying solutions in our reconstruction algorithm. In [10], the author introduced oscillatingdecaying solutions for the conductivity equation ∇·(γ(x)∇u) = 0 with the isotropic conductivity.
We make the following assumptions. 1. Let Ω ⊂ R 3 be a bounded C ∞ -smooth domain and assume that D is an unknown obstacle with Lipschitz boundary such that D Ω ⊂ R 3 with an inhomogeneous index of refraction subset of a larger domain Ω. 2. Let A(x) = (a ij (x)) and A 0 (x) = (a 0 ij (x)) be symmetric matrices with a ij (x) = a 0 ij (x)+ a ij (x)χ D , where each a 0 ij (x) is bounded C ∞ -smooth, A(x) = ( a ij (x)) ∈ L ∞ (D) is regarded as a perturbation in the unknown obstacle D and A(x)ξ · ξ ≥ λ|ξ| 2 for any ξ ∈ R 3 and x ∈ D with some λ > 0. Further A(x) satisfies λ|ξ| 2 ≤ A(x)ξ · ξ ≤ Λ|ξ| 2 for some constants 0 < λ ≤ Λ. Now, let k > 0 and consider the steady state anisotropic acoustic wave equation with Dirichlet boundary condition For the unperturbed case, we have In this paper, we assume that k 2 is not a Dirichlet eigenvalue of the operator −∇ · (A∇•) and −∇ · (A 0 ∇•) in Ω. It is known that for any f ∈ H 1/2 (∂Ω), there exists a unique solution u to (1.2). We define the Dirichlet-to-Neumann map Λ D : H 1/2 (∂Ω) → H −1/2 (∂Ω) in the anisotropic case as the following.
is a unit outer normal on ∂Ω.
Inverse problem. Identify the location and the convex hull of D from the DN-map Λ D . The domain D can also be considered as an inclusion embedded in Ω. The aim of this work is to give a reconstruction algorithm for this problem. Note that the information on the medium parameter A(x) = ( a ij (x)) inside D is not known a priori.
The main tool in our reconstruction method is the oscillating-decaying solutions for the second order anisotropic elliptic differential equations. We use the results from the paper [11] to construct the oscillating-decaying solution. In section 2, we will construct the oscillating-decaying solutions for anisotropic elliptic equations. Note that even if k = 0, which means the equation is ∇ · (A(x)∇u) = 0, we do not know of any CGO-type solutions. Roughly speaking, given a hyperplane, an oscillating-decaying solution is oscillating very rapidly along this plane and decaying exponentially in the direction transverse to the same plane. Oscillating-decaying solutions are special solutions with the imaginary part of the phase function nonnegative. Note that the domain of the oscillating-decaying solutions is not over the whole Ω, so we need to extend such solutions to the whole domain. Fortunately, the Runge approximation property provides us a good approach to extend this special solution in Section 3.
In Ikehata's work, the CGO-solutions are used to define the indicator function (see [4] for the definition). In order to use the oscillating-decaying solutions to the inverse problem of identifying an inclusion, we employ the Runge approximation property to redefine the indicator function. It was Lax [5] that first recognized the Runge approximation property is a consequence of the weak unique continuation property. In our case, it is clear that the anisotropic elliptic equation has the weak unique continuation property if the leading part is Lipschitz continuous. Finally, the main theorem and reconstruction algorithm will be presented in Section 4. We remark that the reconstruction algorithm in this paper is weaker than the standard enclosure method for instance, in the sense that our method does not explain what happens to the indicator function after the probing hyperplane has met the obstacle. The results in Section 4 only imply that the indicator function is zero when the hyperplane has not touched the obstacle, and becomes nonzero at the touching point.

2.
Construction of oscillating-decaying solutions. In this section, we follow the paper [11] to construct the oscillating-decaying solution in the anisotropic elliptic equations. In our case, since we only consider a scalar elliptic equation, its construction is simpler than that in [11]. Consider the anisotropic Helmholtz type equation Note that the oscillating-decaying solutions of ∇ · (A(x)∇u) = 0 in Ω will have the same form as the equation (2.1), which means the lower order term k 2 u will not affect the representation of the oscillating-decaying solutions, the following are the construction details. Now, we assume that the domain Ω is an open, bounded smooth domain in R 3 and the coefficients A(x) = (a ij (x)) is a symmetric 3×3 matrix satisfying uniformly elliptic condition, which means Assume that , ∀α ∈ Z 3 + } is the anisotropic coefficients. Note that A(x) ∈ B ∞ already implies that A is Lipschitz continuous and the Lipschitz continuity property of A(x) will apply the weak unique continuation property of (2.1) (see [1] for example).
We give several notations as follows. Assume that Ω ⊂ R 3 is an open set with smooth boundary and ω ∈ S 2 is given. Let η ∈ S 2 and ζ ∈ S 2 be chosen so that {η, ζ, ω} forms an orthonormal system of R 3 . We then denote x = (x · η, x · ζ). Let t ∈ R, Ω t (ω) = Ω ∩ {x · ω > t} and Σ t (ω) = Ω ∩ {x · ω = t} be a non-empty open set. We consider a scalar function u χt,b,t,N,ω (x, τ ) := u(x, τ ) ∈ C ∞ (Ω t (ω)\Σ t (ω)) ∩ C 0 (Ω t (ω)) with τ 1 satisfying: where ξ ∈ S 2 laying in the span of η and ζ and fixed is a nonzero smooth function and 0 = b ∈ C 3 . Moreover, β χt,b,t,N,ω (x , τ ) is a smooth function supported in supp(χ t ) satisfying: for some constant c > 0. From now on, we use c, c and their capitals to denote general positive constants whose values may vary from line to line. As in the paper [11], u χt,b,t,N,ω can be written as and r χtb,t,N,ω satisfying is a complex function with its real part ReA t (x ) > 0, and γ χt,b,t,N,ω is a smooth function supported in supp(χ t ) satisfying where a > 0 is some constant depending on A t (x ). Without loss of generality, we consider the special case where t = 0, ω = e 3 = (0, 0, 1) and choose η = (1, 0, 0), ζ = (0, 1, 0). The general case can be obtained from this special case by change of coordinates. Define L = L A and M · = e −iτ x ·ξ L(e iτ x ·ξ ·), where x = (x 1 , x 2 ) and ξ = (ξ 1 , ξ 2 ) with |ξ | = 1, then M is a differential operator. To be precise, by using a jl = a lj , we calculate M to be given by 33 M . Now, we use the same idea in [11], define e, f = ij a ij e i f j , where e = (e 1 , e 2 , e 3 ), f = (f 1 , f 2 , f 3 ) and denote e, f 0 = e, f | x3=0 . Let P be a differential operator, and we define the order of P , denoted by ord(P ), in the following sense: is a smooth complex function with its real part greater than 0 and ϕ(x ) ∈ C ∞ 0 (R 2 ). In this sense, similar to [11], we can see that τ , ∂ 3 are of order 1, ∂ 1 , ∂ 2 are of order 0 and x 3 is of order -1. Now according to this order, the principal part M 2 (order 2) of M is: . Note that the principal part M 2 does not involve the lower order term k 2 ·, so we can follow all the constructions in the same procedures as in [11] and we omit details.
3. Tools and estimates. In this section, we introduce the Runge approximation property and a very useful elliptic estimate: Meyers L p -estimates.
[5] Let L be a second order elliptic operator, solutions of an equation Lu = 0 are said to have the Runge approximation property if, whenever K and Ω are two simply connected domains with K ⊂ Ω, any solution in K can be approximated uniformly in compact subsets of K by a sequence of solutions in Ω.
There are many applications for Runge approximation property in inverse problems. Similar results for some elliptic operators can be found in [5], [6]. The following theorem is a classical result for Runge approximation property for second order elliptic equations.
Then for any compact subset K ⊂ O and any > 0, there exists U ∈ H 1 (Ω) Proof. The proof is standard and it is based on the weak unique continuation property for the anisotropic second order elliptic operator L 0 and the Hahn-Banach theorem. For more details, how to derive the Runge approximation property from the weak unique continuation, we refer readers to [5] 3.2. Elliptic estimates and some identities. We need some estimates for solutions to some Dirichlet problems which will be used in next section. Recall that, for f ∈ H 1/2 (∂Ω), let u and u 0 be solutions to the Dirichlet problems (1.2) and (1.3), respectively. Note that a ij (x) = a 0 ij (x) + a ij (x)χ D and we set w = u − u 0 , then w satisfies the Dirichlet problem where A(x) = (a ij (x)), A 0 (x) = (a 0 ij (x)) and A(x) = ( a ij (x)). Then in the following lemmas, we give some estimates for w. Lemma 3.3. There exists a positive constant C independent of w such that we have Proof. The proof follows from [12] by Freidrich's inequality, see [7] p.258 and use a standard elliptic regularity.
We next prove some useful identities.
for any ϕ ∈ H 1 (Ω). Since u = u 0 = f on ∂Ω, the left hand side of the identity has the same value whether we take ϕ = u or ϕ = u 0 , and it is equal to The right hand side of the above identity is real. Hence, by taking the real part, we have The estimates in the following lemma play an important role in our reconstruction algorithm.
Lemma 3.6. We have the following identities: In particular, we have where C > 0 is a constant depending on A(x) and c is a constant depending on A, A 0 and A.
Proof. Multiplying the identity ∇ · (A(x)∇w) + k 2 w + ∇ · ( Aχ D ∇u 0 ) = 0 byw and integrating over Ω, we get A∇u 0 · ∇u 0 dx, and use (3.9) we can obtain Similarly, multiplying the identity ∇ · ( Aχ D ∇u) + ∇ · (A 0 ∇w) + k 2 w = 0 byw and integrating over Ω, we get and use (3.9) again, we can obtain For the remaining part, (3.12) is an easy consequence of (3.10) Finally, for the lower bound, we use Aχ D is a positive definite matrix by our previous assumptions in section 1.

4.
Detecting the convex hull of the unknown obstacle.

Main theorem.
Recall that we have constructed the oscillating-decaying solutions in section 2, and note that this solution can not be defined on the whole domain, that is, the oscillating-decaying solutions u χt,b,t,N,ω (x, τ ) only defined on Ω t (ω) Ω. Nevertheless, with the help of the Runge approximation property, we can only determine the convex hull of the unknown obstacle D byΛ D f for infinitely many f . We define B to be an open ball in R 3 such that Ω ⊂ B. Assume that Ω ⊂ R 3 is an open smooth domain with B ⊂ Ω. As in the section 2, set ω ∈ S 2 and {η, ζ, ω} forms an orthonormal basis of R 3 . Suppose t 0 = inf x∈D x · ω = x 0 · ω, where x 0 = x 0 (ω) ∈ ∂D. For any t ≤ t 0 and > 0 small enough, we can construct to be the oscillating-decaying solution for ∇·(A 0 (x)∇·)+k 2 · in B t− (ω) = B∩{x·ω > t− }, where χ t− (x ) ∈ C ∞ 0 (R 2 ) and b ∈ C. Note that in section 2, we have assumed the leading coefficient A 0 (x) ∈ B ∞ (R 3 ). Similarly, we have the oscillating-decaying solution u χt,b,t,N,ω (x, τ ) = χ t (x )Q t e iτ x·ξ e −τ (x·ω−t)At(x ) b + γ χt,b,t,N,ω (x, τ ) + r χt,b,t,N,ω for L A 0 in B t (ω). In fact, for any τ , u χt− ,b,t− ,N,ω (x, τ ) → u χt,b,t,N,ω (x, τ ) in an appropriate sense as → 0. For details, we refer readers to consult all the details and results in [11], and we list consequences in the following.
Proof. We follow the argument in [12]. We only prove (4.2) and (4.3) and the proof of (4.4) and (4.5) are similar arguments.