Shape identification in inverse medium scattering problems with a single far-field pattern

Consider time-harmonic acoustic scattering from a bounded penetrable obstacle $D\subset \mathbb R^N$ embedded in a homogeneous background medium. The index of refraction characterizing the material inside $D$ is supposed to be H\"older continuous near the corners. If $D\subset \mathbb R^2$ is a convex polygon, we prove that its shape and location can be uniquely determined by the far-field pattern incited by a single incident wave at a fixed frequency. In dimensions $N \geq 3$, the uniqueness applies to penetrable scatterers of rectangular type with additional assumptions on the smoothness of the contrast. Our arguments are motivated by recent studies on the absence of non-scattering wavenumbers in domains with corners. As a byproduct, we show that the smoothness conditions in previous corner scattering results are only required near the corners.


Introduction and main results
Assume a time-harmonic incident wave is incident onto a bounded penetrable obstacle D ⊂ R N (N ≥ 2) embedded in a homogeneous medium.The incident field u in may be any non-trivial solution in L 2 loc (R N ) of the Helmholtz equation where k > 0 is the wavenumber.For instance, the incident wave is allowed to be a plane wave exp(ikx • d) with incident direction d ∈ S N −1 := {x ∈ R N : |x| = 1}, or a Herglotz wave of the form In this paper we suppose the scatterer D to be a convex polygon in R 2 or a convex polyhedron in R N .The physical properties of the inhomogeneous medium D can be characterized by the refractive index function (or potential) q(x).Without loss of generality we suppose q(x) = 1 for x ∈ D e = R N \D due to the homogeneity of the background medium.
Denote by u = u in + u sc the total field generated by u in , where u sc is the outgoing scattered field which satisfies the Helmholtz equation (∆ + k 2 )u sc = 0 in D e and the Sommerfeld radiation condition uniformly in all directions.The propagation of the total wave is governed by the Helmholtz equation ∆u(x) + k 2 q(x)u(x) = 0 in R N . (1.2) Across the interface ∂D, we assume the continuity of the total field and its normal derivative (already implicitly contained in the formulation (1.2)), i.e., Here the superscripts (•) ± stand for the limits taken from outside and inside, respectively, and ν ∈ S N −1 is the unit normal on ∂D pointing into D e .The unique solvability of the scattering problem (1.1)-(1.3) in H 2 loc (R N ) is well-known if q ∈ L ∞ (R N ) (see e.g.[CK98,Chapter 8] or [Ki11,Chapter 6]).In particular, the Sommerfeld radiation condition (1.1) leads to the asymptotic expansion uniformly in all directions x := x/|x|, x ∈ R N .The function u ∞ (x) is a real-analytic function defined on S N −1 and is referred to as the far-field pattern or the scattering amplitude for u in .The vector x ∈ S N −1 is the observation direction of the far field.This paper concerns the uniqueness in recovering the boundary ∂D (or equivalently, the convex hull of the support of the contrast q − 1) from the far-field pattern generated by one incident wave at a fixed frequency.The study on global uniqueness with a single incident plane or point source wave is usually difficult and challenging.For sound-soft or sound-hard obstacles, such uniqueness results have been obtained within the class of polyhedral or polygonal scatterers; see e.g., [AR05,CY03,EY06,EY08,HL14,LZ06].The proofs rely heavily on the reflection principles for the Helmholtz equation with respect to a Dirichlet or Neumann hyperplane and on properties of the incident wave (for instance, plane waves do not decay at infinity and point source waves are singular).However, the approach of using reflection principles does not apply to penetrable scatterers due to the lack of "reflectible" (Dirichlet or Neumann) boundary conditions for the Helmholtz equation.To the best of our knowledge, uniqueness with one incident wave is still unknown within the class of non-convex polyhedral obstacles of impedance type.
Earlier uniqueness results on shape identification in inverse medium scattering were derived by sending plane waves with infinitely many directions at a fixed frequency (see e.g., [EH11,Is08,Is90,KG08,Ki93]), which results in overdetermined inverse problems.Uniqueness with a single far-field pattern has been verified in two cases: D is a ball (not necessarily centered at the origin) and q ≡ q 0 = 1 is a constant in D [HLL15], or D is a convex polygon or polyhedron and q is real-analytic on D satisfying |q − 1| > 0 on ∂D [EH15].The unique determination of a variable index of refraction q in R N from knowledge of the far-field patterns of all incident plane waves at fixed frequency, or by measuring the Dirichlet-to-Neumann map of the Helmholtz equation, has also been intensively studied.We refer to [SU87,HN87,Na88] and the survey [Uh14] for results for N ≥ 3 and to recent results [Bu08,BIY15] for N = 2.
The purpose of this article is to remove the real-analyticity assumption made in [EH15] on the refractive index.To do this, we employ a different method that is motivated by the recent studies [BPS14,PSV14] on the absence of non-scattering wavenumbers in corner domains.This method relies on the construction of suitable complex geometrical optics (CGO) solutions to the Helmholtz equation.Recall that k is called a non-scattering wavenumber if there is a nontrivial incident wave whose far-field pattern vanishes identically.If k is a non-scattering wavenumber, the functions w = u in | D and u| D solve the interior transmission eigenvalue problem Thus each non-scattering wavenumber is an interior transmission eigenvalue (see the survey [CH13]).On the other hand, if k 2 is an interior transmission eigenvalue and if the non-trivial solution w of (1.5) has a real-analytic extension from D to R N , then k 2 is also a non-scattering wavenumber.This implies that, when k 2 is a non-scattering wavenumber, the Cauchy data of the total field u on ∂D coincide with the Cauchy data of a real-analytic function which satisfies the Helmholtz equation in a neighborhood of D. A similar phenomenon can be observed around a corner point, if two distinct convex polygons or polyhedra generate the same far-field pattern.Therefore, the argument for proving the absence of non-scattering wavenumber can be used for justifying uniqueness in determining the shape of a penetrable scatterer.Let D j for j = 1, 2 be two penetrable scatterers with contrasts q j .Denote by u ∞ j the far-field pattern of the scattered field caused by a fixed incoming wave u in incident onto D j with fixed wavenumber k > 0. The first uniqueness result is in two dimensions, and applies to convex polygons.
Theorem 1.1.Let D j ⊂ R 2 for j = 1, 2 be bounded convex polygons.Assume that q j ∈ L ∞ (R 2 ) are contrasts such that q j ≡ 1 in D e j , and each vertex of D j has some neighborhood U j such that q j | D j ∩U j is C α for some α > 0. Furthermore, assume that The next result applies in dimensions N ≥ 3 but requires that the scatterers are closed rectangular boxes, i.e. sets of the form [0, a 1 ] × • • • × [0, a N ] for some a j > 0 up to rotations and translations.We write H s,p for the fractional L p Sobolev space with smoothness index s.
Theorem 1.2.Let D j ⊂ R N for j = 1, 2 be two rectangular boxes.Assume that q j ∈ L ∞ (R N ) are contrasts such that q j ≡ 1 in D e j , and each corner of D j has some neighborhood U j such that q j | D j ∩U j has regularity X as specified below.Furthermore, assume that q j (O) = 1 for each corner O of D j .Then the relation u ∞ 1 = u ∞ 2 on S N −1 implies that D 1 = D 2 , provided that one of the following assumptions holds.
(b) N ≥ 3 and X = H s,p for some s, p with 1 < p ≤ 2 and s > N/p.Theorems 1.1 and 1.2 are valid for Hölder or Sobolev potentials and avoid the realanalyticity assumption required in [EH15].The results in dimensions N ≥ 3 are confined to penetrable scatterers of rectangular type.It is still open how to prove Theorem 1.2 for general convex polyhedra with Hölder continuous contrasts.We remark that the above results remain valid for a large class of incident waves which do not vanish identically in a neighborhood of the scatterer.For instance, u in is also allowed to be a spherical point source emitted from some source position located in D e .
Our technique improves the regularity conditions of the corner scattering results of [BPS14,PSV14].Namely, regularity is only required in a small neighborhood of the corner point, and otherwise the contrasts are only required to be L ∞ .We state the results on the absence of non-scattering wavenumbers as follows.Throughout the paper we write B r := {x ∈ R N : |x| < r} for r > 0.
Theorem 1.3.Let q ∈ L ∞ (R 2 ), and let W ⊂ R 2 be a closed sector with angle < π and with vertex at O. Suppose that q ≡ 1 in W e , that q − 1 is compactly supported, and that q| W ∩Br is C α for some α > 0 and r > 0. Finally, assume that q(O) = 1.Then, with q as the contrast, for any incoming wave u in ≡ 0, the far-field pattern u ∞ can not vanish identically.
Theorem 1.4.Let q ∈ L ∞ (R N ).Suppose that q ≡ 1 in W e , that q − 1 is compactly supported, and that q| W ∩Br has regularity X for some r > 0, where one of the following conditions holds: and X = H s,p for some s, p with 1 < p ≤ 2 and s > N/p.Finally, assume that q(O) = 1.Then, with q as the contrast, for any incoming wave u in ≡ 0, the far-field pattern u ∞ can not vanish identically.
The subsequent Section 2 is devoted to the proofs of the two-dimensional results, i.e., Theorems 1.1 and 1.3.The unique determination of a rectangular box in any dimension N ≥ 3, Theorem 1.2, will be proved in Section 3. The result of Theorem 1.4 on nonscattering wavenumbers can be derived by using the same argument as in Theorem 1.3 and we omit its proof.

Proofs in two dimensions
Denote by (r, ϕ) the polar coordinates in R 2 , and by B R the disk centered at the origin O with radius R > 0. For ϕ 0 ∈ (0, π/2), define W ⊂ R 2 as the infinite sector between the half-lines Γ ± := {(r, ϕ) : ϕ = ±ϕ 0 }.The closure of W will be denoted by W , which is a closed cone in R 2 .Set (see Figure 1) Figure 1: Geometrical settings.
The following two lemmas are the essential ingredients in the proofs.The first one concerns the construction of suitable Complex Geometrical Optics (CGO) solutions to the Schrödinger equation in R 2 .For convenience we employ the common notation x := (1 + |x| 2 ) 1/2 throughout the paper.
Lemma 2.1 is the special case N = 2 of Lemma 3.1 in [PSV14], the proof of which was based on the uniform Sobolev estimates of Kenig, Ruiz and Sogge [KRS87].Relying on the construction of CGO solutions of the form (2.2), we next verify a result for the transmission problem between the Schrödinger equations with constant and piecewise Hölder continuous potentials in a finite polygonal cone.
subject to the transmission conditions (2.4) From Lemma 2.2 it follows that the Cauchy data of non-trivial solutions of the Schrödinger equations with constant and piecewise Hölder continuous potentials cannot coincide on two intersecting lines, if the potentials involved do not coincide on the intersection.The same result was verified in [EH15] but restricted to real-analytic potentials.Making use of classical corner regularity results for the Laplace equation in the plane (see e.g., [MNP00, Chapter 1.2], [Gr92, Chapter 2] or [Da88, Example 16.12]), the approach of Taylor expansion [EH15] can be generalized only to infinitely smooth potentials on S R .Hence, the above Lemma 2.2 has significantly relaxed the regularity assumption used in [EH15].Below we carry out the proof of Lemma 2.2 which is valid only when the corner of S R is convex, i.e., ϕ 0 < π/2.Proof of Lemma 2.2.We shall follow the approach from [PSV14, Section 4] but modified to be applicable to a polygonal convex cone with finite height.For clarity we divide the proof into three steps.
Step 3. End of the proof.Repeating the arguments of [PSV14, Section 5], one can deduce from (2.14), taking both signs ±, that H ≡ 0. This implies that v 1 ≡ 0 in B R .As a consequence, the Cauchy data of v 2 on Γ ± R vanish due to the transmission conditions (2.4).Finally, we get v 2 ≡ 0 by the unique continuation of solutions to the Schrödinger equation.This finishes the proof of Lemma 2.2.
We are now ready to prove Theorems 1.1 and 1.3 for general incident waves, including point source waves.
If ∂D 1 = ∂D 2 , without loss of generality we may assume there exists a corner O ∈ R 2 of ∂D 2 such that O / ∈ D 1 .We suppose further that this corner point coincides with the origin and we pick a fixed number R > 0 such that B R ⊂ D e 1 .Since D 2 is a convex polygon, rotating coordinate axes if necessary, we may assume that D 2 ∩ B R = {(r, ϕ) : |ϕ| < ϕ 0 } for some ϕ 0 ∈ (0, π/2); see Figure 3. From (2.15), it follows that where the superscripts (•) − , (•) + stand for the limits taken from D 2 and D e 2 , respectively.On the other hand, the function u 1 satisfies the Helmholtz equation with the wave number k 2 in B R , while u 2 fulfills the Schrödinger equation Since q 2 (O) = 1, applying Lemma 2.2 leads to u 1 = u 2 ≡ 0 in B R .Moreover, by unique continuation we obtain u 1 = u 2 ≡ 0 in R 2 .This implies that the scattered fields satisfy ), but since u in solves the free Helmholtz equation, unique continuation implies u in ≡ 0. This contradiction gives
Proof of Theorem 1.3.Let us consider the incident wave u in ≡ 0 with the total wave u, so that we have If for this incident wave u ∞ ≡ 0, then Rellich's lemma tells us that u ≡ u in in R 2 \ W , and Lemma 2.2 applied with v 1 = u in and v 2 = u gives u in ≡ 0 in B R .By unique continuation we get u in ≡ 0, which is a contradiction.
The proof of the above lemma is also based on the uniform Sobolev estimates of Kenig, Ruiz and Sogge [KRS87].In to avoid repeating the arguments presented in [PSV14], we shall verify Lemma 3.1 by indicating the changes necessary to the proof of [PSV14, Theorem 3.1].For this purpose we need to know into which Sobolev spaces the characteristic function of a cube belongs.Below we will write χ Q for the characteristic function of a set Q, and denote by C ∞ c (R 3 ) the space of smooth functions with compact support.
Proof.Without loss of generality, we consider Thus, we may estimate The last integral is finite when τ < 1/2.Thus, By interpolation, the mapping f → ψf maps H s,2 (R 3 ) into H s,p (R 3 ) for s ∈ ]0, 1[, and The construction of CGO solutions for a cube is proved as follows.
Proof of Lemma 3.1.We may assume that α < 1/2.The proof of the complex geometric optics construction in [PSV14] is mostly independent of the shape of W .For W = [0, ∞[ 3 we only need to check that V := χ W (1− q) has the pointwise Sobolev multiplier property of Proposition 3.4 in [PSV14], i.e., we need to check that for some constant C > 0 for arbitrarily small fixed ε > 0. The desired multiplier property in turn follows immediately, if we can show that (cf.[PSV14, Proposition 3.7]) Applying Lemma 3.2 we know that the function χ Λ belongs to H τ,p (R 3 ).This leads to the relation (3.1) by changing variables and scaling the Sobolev norm; see the proof for Proposition 3.7 in [PSV14].
To continue the proof of Theorem 1.2 under the assumption (a), we again introduce some notation.Let S R = W ∩ B R , S e R = B R \ S R , and Γ R = ∂W ∩ B R .As in two dimensions, we will employ a result of the following type.
Denote by H the lowest degree nontrivial homogeneous polynomial in the Taylor expansion of v 1 around the origin, and consider the the Laplace transform F (z) of H (see (2.10)) for z ∈ C 3 such that Re (z) • a > cos(β/2).If H has degree n, similarly as before we obtain F (ρ) = O(τ −n−3−δ ) as τ → ∞, if q(O) = 1.This estimate involves using the Hölder inequality so that the L 4 -norm of ψ appears.In the other direction, by homogeneity, we have Taking τ → ∞ gives then F (τ ω + iτ ω ⊥ ) = 0 for all τ ∈ R + and all admissible ω and ω ⊥ .From [BPS14, Theorem 2.5] we obtain the conclusion H(x) ≡ 0, which implies Proof of Theorem 1.2 under assumption (a).The result follows by the same arguments as in the proof of Theorem 1.1, except that Lemma 3.3 is used instead of Lemma 2.2.
Next we indicate how to prove Theorem 1.2 under the assumption (b).Now we let W = [0, ∞[ N where N ≥ 3. The required complex geometrical optics solutions were constructed in [BPS14] and they are given by the following result.Lemma 3.4.Let q ≡ 1 in R N \ (W ∩ B R ) for some R > 0, and let q | W ∩B R be in H s,p where 1 < p ≤ 2 and s > N/p.Let also D ⊂ R N be a bounded open set, and let 2 ≤ r < ∞.If ρ ∈ C N satisfies ρ • ρ = 0 and |Im (ρ)| is sufficiently large, then there exists a solution of the Helmholtz equation Proof.We can write q = 1 − χ K ϕ for some cube K = [0, a] N and for some ϕ ∈ H s,p c (R N ) by the conditions on q and the Sobolev extension theorem on Lipschitz domains.Writing m = χ K ϕ, the equation that we need to solve is The result would then follow from [BPS14, Theorem 2.3], except that this theorem was proved under the condition ϕ ∈ C ∞ instead of ϕ ∈ H s,p .
Inspecting the proof in [BPS14] we see that it is enough that the function where the spaces are as in [BPS14].Now we can write so that f ∈ H s,p c (R N ).We use [BPS14, Lemma 4.3 and preceding discussion] to conclude that when supp(q) ⊂ B R , one has . It follows that (3.2) will be satisfied if the function f defined above satisfies f ∈ L 1 ∩ L r .Since f ∈ L 1 c (R N ), we have f ∈ L ∞ (R N ) and it is enough to check that f ∈ L 1 .But if (ψ j (ξ)) ∞ j=0 is a Littlewood-Paley partition of unity and if 1 ≤ p ≤ 2, we obtain by the Hölder and Hausdorff-Young inequalities that where ψ j (D) is the Fourier multiplier with symbol ψ j (ξ) and the last norm is a Besov norm.Since H s,p ⊂ B s p,∞ ⊂ B N/p p,1 for s > N/p, we get f ∈ L 1 as required.This shows that (3.2) is satisfied, which concludes the proof.
Proof of Theorem 1.2 under assumption (b).It is enough to use Lemma 3.4 to prove an analogue of Lemma 3.3, and then argue as in the proof of Theorem 1.1.

Concluding remarks
In two dimensions, we have verified the uniqueness in identifying a convex penetrable scatterer of polygonal type with a single far-field pattern, provided the refractive index is discontinuous at the corner points but C α -Hölder continuous inside near the corners.
In higher dimensions, the uniqueness applies to convex polyhedra with additional assumptions on the geometrical shape (i.e., boxes) and on the smoothness of the contrast.In this study, the assumption is required only near the corner points.
Our future efforts will be devoted to the uniqueness proof in 3D for convex polyhedra with general C α -Hölder (α > 0) continuous potentials.Since the CGO solutions can be constructed with plenty of generality, the 3D proof essentially requires to evaluate the Laplace transform of a harmonic homogeneous polynomial over a general threedimensional corner domain and then to prove the vanishing of this polynomial through novel techniques.Another possible approach would be to analyze the corner and edge singularities of an elliptic equation with analytical Cauchy data in weighted Hölder spaces.Further results will be presented in a forthcoming publication.

Figure 2 :
Figure 2: Configurations of Λ R/2 and D ,R in the proof of Lemma 2.2.