ON RECOVERY OF AN INHOMOGENEOUS CAVITY IN INVERSE ACOUSTIC SCATTERING

. Consider the time-harmonic acoustic scattering of an incident point source inside an inhomogeneous cavity. By constructing an equivalent integral equation, the well-posedness of the direct problem is proved in L p with using the classical Fredholm theory. Motivated by the previous work [10], a novel uniqueness result is then established for the inverse problem of recovering the refractive index of piecewise constant function from the wave ﬁelds measured on a closed surface inside the cavity.

1. Introduction. In this paper, we study an inverse scattering problem of determining an inhomogeneous cavity from many measurements inside the cavity. Precisely, let D denote the inhomogeneous cavity, which is described by a bounded connected domain in R 3 with the refractive index n(x) ∈ L ∞ (D). Let D 0 ⊂ D be a bounded connected part of the inhomogeneous cavity D with the refractive index n(x) equaling 1 in D 0 . This shows that the medium is inhomogeneous in the subdomain D 1 := D \ D 0 . Furthermore, we assume that both ∂D 0 and ∂D are all of C 2 -class.
Consider an incident field u i which is induced by a point source located at y ∈ D 0 , i.e., u i (x) := Φ(x, y) = 1 4π Then the scattering by the inhomogeneous cavity with the point source u i (x) = Φ(x, y) is modelled by the Helmholtz equation: where u s y (·) is the scattered field generated by Φ(·, y), solving equation with a Dirichlet boundary condition u s y (x) = −u i (x) on ∂D from Problem (2). If 0 is assumed not to be a Dirichlet eigenvalue for the operator + k 2 n(x) in D, we can easily show that equation (3) admits a unique solution u s y (·) ∈ H 1 (D) which depends continuously on the data Φ(·, y), using the variational approach.
In this paper we will consider the inverse cavity problem (ICP) of recovering the refractive index n(x) described by a piecewise constant function, and the shape of the interior part D 0 of the inhomogeneous cavity D from the knowledge of u s (x, y) for all x, y ∈ ∂C, corresponding to many point sources Φ(x, y). Here, C denotes an open subregion in D 0 with a C 2 -smooth boundary ∂C, i.e. C ⊂ D 0 , such that k > 0 is not a Dirichlet eigenvalue for the operator − in C. And the boundary ∂D of the inhomogeneous D is assumed to be a priori known. Recently, this class of inverse problems have received a considerable interest and we here refer to [3,5,6,7,8,9] for a detailed investigation, in which the central point related to the ICP is mainly focused on the study of qualitatively numerical methods of reconstructing the cavity in the homogeneous background medium, that is, n(x) ≡ 1 in D. For example, a factorization method was proposed in [6] with a Dirichlet or impedance condition on the boundary of the cavity and a sampling type method was proposed in [8] under the Dirichlet condition, which was later extended into the case of the Maxwell equation in [11]. However, to the best of the author's knowledge, there is much little progress for the case when D is an unknown inhomogeneous cavity in the literature. It is noticed that a related scattering problem was recently considered in [7] for a homogeneous cavity surrounded by a penetrable inhomogeneous media, where the uniqueness result was provided only for the shape of the cavity by means of the measurements inside.
Different from the previous works on the ICP, we will focus in the current paper on the uniqueness issue on the refractive index n(x) and the cavity D in Problem (2) by interior measurements generated by incident point sources (1). To this end, we first transfer Problem (2) into an equivalent integral equation by introducing a Dirichlet-Green function. With the help of this technique, we can prove that Problem (2) is uniquely solvable in the L p -sense with 1 < p ≤ 2, using the classical Fredholm theory. This may be independently interesting in its own right direction. Meanwhile, we can also provide a generalized symmetry property of the solution to Problem (2) with incident fields located at two different points. Based on these analysis, we can propose a novel technique to deal with the ICP, especially for the unique recovery of the index of refraction in the inverse cavity scattering problem.
The remainder of this paper is planed as follows. In section 2, we study the cavity scattering problem with the incident field induced by a general point source wave. The uniqueness and existence of the solution to Problem (2) is obtained in the L psense by employing an integral equation method. Section 3 is devoted to the inverse cavity scattering problem with the interior wave field measurements, corresponding to the incident point source wave (1). The uniqueness result is established for the refractive index and its support from many measurements inside the cavity.
2. Well-posedness in L p for 1 < p ≤ 2. In this section, we are devoted to study the solvability of Problem (2) with the data induced by a general point source wave Conversely, assume that u z ∈ L p (D 1 ) is a solution of (8). We conclude from [4] that w z := u z − P i z ∈ W 2,p (D 1 ) solves the equation which further means u z + k 2 n(x)u z = 0 in D 1 . Notice that w z can be extended into the inside of the domain D 0 by equation (8), as a new function in D, such that w z ∈ W 2,p (D) and w z + k 2 w z = 0 in D 0 . The function w z + P s z is thus the solution to Problem (3). The proof is now completed.
Define the integral operator T : with the density φ ∈ L p (D 1 ). For 1 < p ≤ 2, it follows from [4, Theorem 9.9] that T is bounded from L p (D) into W 2,p (D) and then the compact embedding of W 2,p (D 1 ) into L p (D 1 ) implies that T is compact on L p (D 1 ). Equation (8) is now reduced to the second kind of operator equation Theorem 2.2. For 1 < p ≤ 2, there exists a unique solution to (10) such that Proof. To prove (11), it is sufficient to show that (I − T) is injective from the Fredholm theory. Provided (I − T)ψ = 0 for some ψ ∈ L p (D 1 ), we then have ψ + k 2 n(x)ψ = 0 in D. Since 0 is not a Dirichlet eigenvalue for + k 2 n(x) in D, we conclude that ψ = 0 in D. Hence, the proof is completed. Remark 1. By the well-posedness of the problem (6), it is clearly seen that P i z has the same singularity as the incident field u i z (·) at x = z. Hence, we can obtain the estimate (11) for two different cases on the index p > 1: with l = 1, 2, 3, then it holds that 1 < p < 3 2 . Let z 1 ∈ R 3 and z 2 ∈ R 3 be in the interior of the domain D 0 with the assumption z 1 = z 2 . Based on the above analysis, we now define two functions A ε and B ε in D 0 : by the solutions u z l to Problem (2) with the incident field u i = u i z l for l = 1, 2. Here, O ε (z 1 ) and O ε (z 2 ) are two small balls centered at z 1 and z 2 , respectively, with the radii ε such that O ε (z 1 ) ⊂ D 0 and O ε (z 2 ) ⊂ D 0 . Theorem 2.3. Let A ε and B ε be defined by (12) and (13), respectively, for z 1 = z 2 . Then if either one of the limits in (14) exists.
Proof. The proof follows from a straightforward application of the second Green's theorem. To facilitate the readers to understand the theorem, we here present a detailed proof. Choosing The integral on ∂D 0 vanishes by applying the second Green's theorem in D 1 with the homogeneous Dirichlet boundary conditions for both u z1 and u z2 on ∂D.
Due to n(x) = 1 in D 0 , it follows from equation (3) that the scattered solution since u z1 and u z2 are regular, respectively, in O ε (z 2 ) and O ε (z 1 ). Combining (16) with (17), equality (15) now becomes for all sufficiently small ε > 0 from the definitions about A ε and B ε . Finally, the required equality (14) is obtained by taking the limit from the left and right terms of (18). The proof of the theorem is thus accomplished.
, the incident fields are induced by the fundamental solution to the Helmholz equation. In this case, we have p = 2 in Theorem 2.2. Moreover, we can also conclude from the singularity of the fundamental solution that the limits exist for both A ε and B ε and . Therefore, Theorem 2.3 remains valid to obtain that the limits exist for both A ε and B ε and By Theorem 2.3 and Remark 2, it is known that the solution to Problem (2) has a generalized symmetry property with two different incident sources in a general case for the index p > 1.
3. The inverse problem. In this section we aim to obtain the uniqueness on the refractive index n(x) and the support D 0 of the inhomogeneous cavity D under the assumption that the boundary ∂D is a priori known. In the inverse problem, we will assume that the refractive index n(x) is described as a piecewise constant function Here, Y = {1, 2, 3, · · · , N } is an index set with the positive fixed integer N < +∞, c j is a unknown constant for each j ∈ Y, and χ Ωj (·) denotes the characteristic function of the unknown connected subdomains Ω j , defined by 1 in Ω j and 0 otherwise. We also suppose that Ω j1 ∩ Ω j2 = ∅ for j 1 = j 2 . Furthermore, for i = j if ∂Ω i ∩ ∂Ω j has a non-empty open subset in the two-dimensional manifold of R 3 , we will naturally assume that c i = c j . The mutual boundaries between the connected subdomains Ω j , j = 1, 2, · · · , N in Y are assumed to be of C 2 -class.
Due to the high nonlinearity and severly ill-posedness of the inverse problem, the ICP is very challenging in the general case. To the best of the authors' knowledge, there are no results available in the literature, if D 0 , D 1 and n(x) are all unknown.
For convenience, let the scattered solution denoted by u s j (·, y), j = 1, 2, to Problem (2) be induced by an incident point source Φ(·, y) located at y ∈ ∂C, corresponding to the domains D (j) , D (j) 0 and the refractive index n j (·). Here, the domain C is assumed to be such that C ⊂ (D  Proof.
Step 1. We shall first prove that D 0 . Define the sequnce where ν(z * ) is the unit exterior normal at z * and δ > 0 is chosen such that z j ∈ (D 0 ) for all j ∈ N. We now consider Problem (2) with the incident waves u i of the form and let u (1) j (x) and u (2) j (x) be the solutions to Problem (2), generated by u i j (x). Using equality (20) in Remark 2 yields For y ∈ ∂C, since u s 1 (·, y)| ∂C = u s 2 (·, y)| ∂C , it follows u s 1 (·, y) = u s 2 (·, y) in D 0 ∩ D (2) 0 from uniqueness of the Dirichlet problem in combination with the unique continuation principle. Therefore, we have u 1 (·, y) = u 2 (·, y) on ∂O ε (z j ) for any sufficient small ε > 0, which further gives Inverse Problems and Imaging Volume 12, No. 2 (2018), 281-291 from (23) for l = 1, 2, 3, and j ∈ N, using a similar discussion. Due to z * ∈ ∂D in the case when f 1 = 0 and f 2 = 0. For the ITP, it was shown in [1] that the smallest real eigenvalue λ low (D z * ) trends to +∞ as the parameter ε 0 → 0. This means that for any fixed wavenumber k > 0 there exists ε 0 > 0 such that k > 0 is not an eigenvalue of the homogeneous ITP in D z * for any ε 0 ≤ ε 0 . Furthermore, we also conclude from [1] that the ITP is well posed with the L 2 -estimate assuming f p , p = 1, 2, satisfy the condition: We now define w 1,j := u (1) j and w 2,j := u (2) j in D z * . It is easily found that (w 1,j , w 2,j ) is the unique solution of the ITP with the boundary data Next we will show that the functions (f 1,j , f 2,j ) are uniformly bounded for all j ∈ N with the norm defined in (27). It is first observed from (24) that Then we can look for the following function where χ(x) ∈ C 2 (R 3 ) is a cut-off function supported in O ε1 (z * ) with ε 1 < ε 0 and χ(x) = 1 in O ε2 (z * ) with ε 2 < ε 1 . Clearly, it holds by (30) that h j = f 1,j and ∂ ν h j = f 2,j on ∂D z * . Moreover, for any fixed j ∈ N, we have h j ∈ H 1 (D z * ) since 0 . Therefore, (f 1,j , f 2,j ) satisfies the above condition between (26) and (27). Using the definition of (f 2,j , f 1,j ) yields that where we have used the fact that χ = 1 in O ε2 (z * ) and u (p) j solves the Helmholtz equation in D z * \ O ε2 (z * ) in the weak sense. Noticing that it then follows from (22) that u i j L q (D (p) ) ≤ C are uniformly bounded for all j ∈ N and p = 1, 2, with one fixed 1 < q < 3 2 . This, together with Theorem 2.2, shows that u (p) j L q (D (p) ) ≤ C is uniformly bounded for all j ∈ N and p = 1, 2, whence follows from the continuous embedding of W 2,q into H 1 , equation (10) and the boundedness of the operator T from L q into W 2,q for 6 5 < q < 3 2 . Combining (26), (32) and (33), we now arrive at where C > 0 is independent of j ∈ N.
Further, it can be easily checked that as j → ∞. Therefore, we have proved D 0 . We next show that n 1 (x) = n 2 (x) when the refractive index n j (x), j = 1, 2 satisfies the condition (21). Before going further, we first introduce some useful notations. For j = 1, 2, let X j := {Ω j,m : m = 1, 2, · · · , N j } denote the sets consisting of a finite number of subdomains of D 1 with respect to the refractive index n j . For convenience, we may suppose that ∂Ω 1,1 ∩ ∂D 0 = ∅, which means that ∂Ω 1,1 and ∂D 0 share a non-empty open subset in the two-dimensional manifold of R 3 . Thus, we can choose some point z * ∈ ∂Ω 1,1 ∩ ∂D 0 and define D δ0 := B δ0 (z * ) ∩ Ω 1,1 with sufficiently small δ 0 > 0, where B δ0 (z * ) is a ball centered at z * with the radius δ 0 > 0. Here, we remark that z * and δ 0 can be chosen such that n 1 and n 2 are all fixed constants in D δ0 , due to the fact that both n 1 and n 2 are two piecewise constant functions. In what follows, we will first in this step prove n 1 = n 2 in D δ0 . To this end, we define z j := z * − (ε/j)ν(z * ), j = 1, 2, 3, · · · with ε > 0 small enough such that z j ∈ D 0 . Now let u j ) solves the following interior transmission problem where f 1 := f 1,j and f 2 := f 2,j are defined as in (28)-(29). Arguments similar to those between (25) and (35) lead to a contradiction. Then we obtain n 1 = n 2 in D δ0 . We continue to show that n 1 = n 2 in Ω 1,1 . If Ω 1,1 ⊆ Ω 2,m0 for some m 0 ≤ N 2 , we obviously have n 1 = n 2 in Ω 1,1 . Otherwise, we will prove that that is, Ω 1,1 is just composed of a finite number of elements in X 2 . We now prove (37) by contradiction. Assuming that (37) does not hold true, then there exists some subdomain Ω 2,mj 0 in X 2 satisfying that Ω 1,1 ∩ Ω 2,mj 0 = ∅, Ω 1,1 ⊆ Ω 2,mj 0 and Ω 2,mj 0 ⊆ Ω 1,1 , see Figure 1. So we can choose a connected part A 1 ⊂ (Ω 1,1 ∩ Ω 2,mj 0 ) and another connected part A 2 ⊂ (Ω 2,mj 0 \ Ω 1,1 ) such that ∂A 2 ∩ (∂Ω 1,1 ∩ ∂Ω 1,2 ) = ∅. This means that ∂A 2 , ∂Ω 1,1 and ∂Ω 1,2 share a non-empty open subset in the two-dimensional manifold of R 3 , denoted by Σ. For convenience, we can also assume that A 2 ⊂ Ω 1,2 in X 1 due to the fact that A 2 ⊂ Ω 1,1 . Noticing that A 1 ⊂ Ω 1,1 , it then follows from the arguments in the proof of the result that n 1 = n 2 in D δ0 (see (36)) in combination with the unique continuation principle that n 1 = n 2 in A 1 . Further, from the assumptions on n j , j = 1, 2 (see (21)), we may suppose that n 1 = c 1,1 in A 1 with some constant c 1,1 , and consequently one has n 2 = n 1 = c 1,1 in A 1 . This further means n 2 = c 1,1 in A 2 since n 2 is a fixed constant in Ω 2,mj 0 . On the other hand, recalling that A 2 ⊂ Ω 1,2 we can also have n 1 = c 1,2 in A 2 with some constant c 1,2 . Next we will prove n 1 = n 2 in A 2 .
Step 4. Finally, the assertion n 2 = n 1 in Ω 1,j , 2 < j < N 1 can be derived by using the similar arguments as Step 3. This completes the proof of the theorem.