Stability Estimates in the Inverse Transmission Scattering Problem

We consider the inverse transmission scattering problem with piecewise constant refractive index. Under mild a priori assumptions on the obstacle we establish logarithmic stability estimates.


Introduction
In this paper we consider the scattering of acoustic time-harmonic waves in an inhomogeneous medium. More precisely we shall consider a penetrable obstacle D and we want to recover information on its location from a knowledge of Cauchy data on the boundary of a region Ω containing the obstacle D.
Given a spherical incident wave u i (·, x 0 ) = Φ(·, x 0 ), where the point source x 0 is located on the boundary of a ball B of radius R, B such that Ω ⊂ B, and Φ denotes the fundamental solution to the Helmholtz equation we denote by G(x, x 0 ) = u i (x, x 0 ) + u s (x, x 0 ) the Green's function of the equation (1.1a) div (γ(x)∇G(x, x 0 )) + k 2 n(x)G(x, where the scattered field u s satisfies the Sommerfeld radiation condition Here k > 0 is the wave number and r = |x|. We shall study equation (1.1a) with piecewise constant coefficients, in particular we shall consider γ and n to be of the following form where λ and δ are given constants. We refer to Is06] for basic information on scattering problem of this type. The unique determination of D from a knowledge of the far field data has been established by Isakov [Is90]. The purpose of the present paper is to establish a stability result. Under reasonable mild assumptions on the regularity of ∂D we show that there is a continuous dependance of D on the Cauchy data on ∂Ω with a modulus of continuity of logarithmic type. This rate of continuity appears optimal in view of the recent paper [DC-Ro] indicating the strong ill-posedness of the inverse problem.
The main ideas employed to obtain stability rely on the study of the behavior of G(x, x 0 ) when x and x 0 get close and the use of unique continuation. These ideas go bach to [Is88] where a uniqueness result for the inverse inclusion problem is proved and it has also been used in inverse scattering theory in [Is90]. In order to apply these ideas to stability some further properties on singular solutions and quantitative estimates of unique continuation are needed. We refer to [Al-DC] where similar ideas are developed for studying the stability of the inverse inclusion problem.
The stability issue in inverse scattering theory has been considered by Isakov [Is92,Is93] for the determination of a sound-soft obstacle. Hähner and Hohage [Ha-Ho] considered equation (1.1a) with a = 1 and n(x) smooth. They showed that n depends on G(x, x 0 ), x, x 0 ∈ ∂B, with a logarithmic rate of continuity. They considered both far field data and near field data. They improve and simplify a previous result of Stefanov [St]. We finally mention a result obtained by Potthast [Po] for impenetrable obstacles which is also based on the use of singular solutions.
The plan of the paper is the following. In Section 2 we give the a priori assumptions we need and we state the stability theorem. In Section 3 the proof of the stability theorem is given based on some auxiliary results whose proofs are collected in Section 4 and Section 5. In particular, in Section 4 we establish some results on singular solutions of equation (1.1a) and in Section 5 we study quantitative estimates of unique continuation.

The Main Result
In this section we state the stability theorem. Before doing this we shall give some definitions we need and introduce the a priori assumptions on the regularity of the obstacle. For any x = (x 1 , x 2 , x 3 ) ∈ R 3 and any r > 0 we denote by B r (x) the open ball in R 3 of radius r centered in the point x, B r (0) = B r and for x = (x 1 , x 2 ) ∈ R 2 we denote by B r (x ) the open ball in R 2 of radius r centered in the point x . In places, we shall denote a point Definition 2.1. Let Ω be a bounded domain in R 3 . Given α, 0 < α ≤ 1, we shall say that a portion S of ∂Ω is of class C 1,α with constants r 0 , L > 0 if for any P ∈ S, there exists a rigid transformation of coordinates under which we have P = 0 and where ϕ is a C 1,α function on B r0 ⊂ R 2 satisfying ϕ(0) = |∇ϕ(0)| = 0 and ϕ C 1,α (B r 0 ) ≤ Lr 0 .
Definition 2.2. We shall say that a portion S of ∂Ω is of Lipschitz class with constants r 0 , L > 0 if for any P ∈ S, there exists a rigid transformation of coordinates under which we have P = 0 and where ϕ is a Lipschitz continuous function on B r0 ⊂ R 2 satisfying ϕ(0) = 0 and ϕ C 0,1 (B r 0 ) ≤ Lr 0 .
Remark 2.1. We use the convention to scale all norms in such a way that they are dimensionally equivalent to their argument. For instance, for any ψ ∈ C 1,α (B r0 ) we set Assumptions on the obstacle D For given numbers r 0 , L > 0, 0 < α < 1, we shall assume there exists a bounded domain Ω such that the obstacle D satisfies the following conditions: In the sequel we shall refer to numbers r 0 , L, α, R, a, b and k as the a priori data.
The inverse problem we are concerned with is the determination of the obstacle D from the knowledge of the Cauchy data of the singular solutions G(·, x 0 ) on ∂Ω for all points source x 0 located on ∂B.
For two possible obstacles D 1 , D 2 satisfying (2.2) we shall denote by G i , i = 1, 2, the corresponding solutions to (1.1a) satisfying the Sommerfeld radiation condition (1.1b).
Remark 2.3. We stress the fact that we don't need any assumption on k.

Proof of the Stability Theorem
We denote by G the connected component of Ω \ (D 1 ∪ D 2 ) such that ∂Ω ⊂ G and Ω D = Ω \ G. Theorem 2.2 evaluates how close the two inclusions are in term of the Hausdorff distance d H . We recall a definition of this metric.
In order to deal with the Hausdorff distance we introduce a simplified variation of it which we call modified distance.
Definition 3.1. We shall call modified distance between D 1 and D 2 the number We wish to remark here that such modified distance does not satisfy the axioms of a metric and in general does not dominate the Hausdorff distance (see [Al-Be-Ro-Ve, §3] for related arguments).
Proposition 3.1. Let D 1 , D 2 be two obstacles satisfying (2.2). Then where c depends only on the a priori assumptions.

Proof. See [Al-DC, Proposition 3.1]
With no loss of generality, we can assume that there exists a point O of ∂D 1 ∩ ∂Ω D , where the maximum in the Definition 3.1 is attained, that is We remark that G is solution to div (γ(x)∇G(x, y)) + k 2 n(x)G(x, y) = −δ(x, y).
We shall denote by G 1 and G 2 Green's functions when D = D 1 and D 2 respectively and γ i , n i , i = 1, 2, the corresponding coefficients.
Integrating by parts we have Let us define for y, w ∈ CB Thus (3.7) can be rewritten as Let us fix P ∈ ∂D. We can assume P ≡ 0. We denote by ν(P ) the outer unit normal vector to Ω D in P and we rotate the coordinate system in such a way that ν(P ) = (0, 0, −1).
Let us denote by χ + (x) the characteristic function of the half-space and by G + the Green's function of div Proposition 3.2. Let D ⊂ Ω be a bounded open set whose boundary is of class C 1,α with constants r 0 , L. Then there exist constants c 1 , c 2 depending on a, α, k and L such that for every x, y ∈ R 3 . Proof. (3.9) and (3.10) can be obtained following Proposition 3.1]. In [Al-DC] the key point is the piecewise regularity of the transmission problem. For a proof of that we refer to [DB-El-Fr] and [Li-Vo].
We shall state now two propositions that describe the behavior of f (y) and S 1 (y) when we move the singularity y toward the boundary of the inclusion. We postpone their proofs in the last Section 5. Proposition 3.3. Let D 1 , D 2 two obstacles verifying (2.2) and let y = hν(O), with O defined in (3.6). If, given ε > 0 we have where 0 < A < 1 and c, B, F > 0 are constants that depend only on the a priori data.
Proposition 3.4. Let D 1 , D 2 two obstacles verifying (2.2) and let y = hν(O), with O defined in (3.6). Then for every h, 0 < h < min{r 2 , d µ } where c 1 , c 2 , c 3 and r 2 are positive constants only depending on the a priori data.
Proof of Theorem 2.2. Let O ∈ ∂D 1 as defined (3.6), that is Then, for y = hν(O), with 0 < h < h 1 , where h 1 = min{d µ , cr 0 , r 2 /2}, using (3.9), we have Using Proposition 3.3, we have On the other hand, by Proposition 3.4 and (3.12), there exists h 0 > 0, only depending on the a priori data, such that for h, Thus we have 2F the theorem follows using Proposition 3.1. In the other case we have that is, solving for d µ , and recalling that, in this case, and, in particular when ε 1 ≤ ε < 1 Finally, using Proposition 3.1, the theorem follows.

Remarks on Singular Solutions
Proposition 4.1. Let D ⊂ R 3 be an open set with C 1,α boundary with constants r 0 , L, let P be a point in ∂D and let us denote with ν(P ) the outer normal vector to D in P . There exist positive constants c 3 , c 4 depending on a, k, α and L such that for every x ∈ D ∩B r (P ) and y = hν(P ), with 0 < r < (min{ 1 2 (8L) −1/α , 1 2 })r 0 = r 0 , 0 < h < (min{ 1 2 (8L) −1/α , 1 2 }) r0 2 . Proof. Let us fix r 1 = min{ 1 2 (8L) −1/α r 0 , r0 2 }. In the ball B r0 (P ) the boundary of D can be represented as the graph of a C 1,α function ϕ. Let us introduce now the following change of variable that transform in B r0 (P ) ∂D in the x -axis. For every r > 0, let Q r (P ) be the cube centered at P , with sides of length 2r and parallel to the coordinates axes. We have that the ball B r (P ) is inscribed into Q r (P ). We define where θ ∈ C ∞ (R) be such that 0 ≤ θ ≤ 1, θ(t) = 1, for |t| < 1, θ(t) = 0, for |t| > 2 and | dθ dt | ≤ 2. Since the C 1,α regularity of ϕ, it is possible to verify that the following inequalities hold: where c ≥ 1 depends on L and α only. Ψ is a C 1,α diffeomorphism from R 3 into itself. Let us define the cylinder C r1 as C r1 = {x ∈ R 3 : |x | < r 1 , |x n | < r 1 }. For x, y ∈ C r1 , we shall denote (4.16) G(x, y) = G(Ψ −1 (x), Ψ −1 (y)).
We estimate now the first derivative of R. To estimate the first derivative of R let us consider a cube Q ⊂ B + r1/4 (x) of side cr 1 /4, with 0 < c < 1, such that x ∈ ∂Q. The following interpolation inequality holds: where δ = 1 1+α , c depends on L only and Since, from the piecewise Hölder continuity of ∇G and of ∇G + , we have that where c depends on L only, thus we conclude where η = α 2 1+α and c depends on L only. Concerning G + we have where c depends on k, α and L only.
Proposition 4.2. Let G + and G 0 + as above, then there exist positive constants c 5 , c 6 depending on the a priori data such that for every x, y ∈ R 3 we have Hence for [Li-St-We] we have Similarly it can be evaluated the integral over B |x−y| 3 (y).
Let us consider now the integral over G. For z ∈ G we have that |z − y| ≥ |x−z| 3 , then we obtain Let us prove now (4.25). We use the interpolation inequality As in Proposition 4.1, since we obtain |∇R(x, y)| ≤ ch −2+η ≤ ch −1 .

Proof of Proposition 3.3 and 3.4
Proof of Proposition 3.3. Let us consider f (y, w), where w is a fixed point in CB. Since f , as a function of y, is a radiating solution of then by [Co-Kr, Theorem 2.14], for y ∈ CB we have whereŷ = y/|y|, Y m n is a spherical harmonic of order n and h (1) n is a spherical Hankel function of the first kind of order n. Let us consider y such that R < R 1 < |y| < R 2 . For an integer N , using Schwarz inequality and the asymptotic behavior of Hankel function (see (2.38) for some constant c depending on R, R 1 and R 2 . Thus, taking the limit as N → +∞, we can conclude that where c depends on R, R 1 and R 2 . Analogous considerations can be carried on fixing y and varying w. Thus, we can conclude that for all (y, w) ∈ B R2 B R1 2 |f (y, w)| ≤ |f |∂B×∂B | ≤ cε.
where c = c(L, R). Similarly |S 2 (y, w)| ≤ ch −2 . Then we conclude that At this stage we shall make use iteratively of the three spheres inequality (see [La, Ku]). Let u be a solution of Lu = 0 in G, let x ∈ G. There exist r 1 , r, r 2 , 0 < r 1 < r < r 2 < R and τ ∈ (0, 1) such that where c and τ depend on R, r/r 2 , r 1 /r 2 and L. Applying (5.28) to u(·) = f (·, w), with x = x ∈ B 4R B 3R , r 1 = r 0 /2, r = 3r 0 /2 and r 2 = 2r 0 we obtain For every y ∈ G h , we denote by γ a simple arc in G joining x to y. Let us define {x i }, i = 1, . . . , s as follows , otherwise let i = s and stop the process. By construction, the balls B r0/2 (x i ) are pairwise disjoint, |x i+1 − x i | = r 0 for i = 1, . . . , s − 1, |x s − y| ≤ r 0 . There exists β such that s ≤ β. An iterated application of the three spheres inequality (5.28) for f (see for instance pg. 780], [Al-DB, Appendix E]) gives that for any r, 0 < r < r 0 We can estimate the right hand side of (5.29) by (5.27) and obtain for any r, 0 < r < r 0 There exists a C 1,α neighborhood U of O in ∂Ω D with constants r 0 and L. Thus there exists a non-tangential vector field ν, defined on U such that the truncated cone where θ = arctan(1/L). Let us define λ 1 = min r 0 1 + sin θ , r 0 3 sin θ , θ 1 = arcsin sin θ 4 , θ, r 0 ). Let G = G 1 , since ρ 1 ≤ r 0 /2, we can use (5.30) in the ball B ρ1 (G) and we can approach O ∈ ∂D 1 by constructing a sequence of balls contained in the cone C(O, ν(O), θ 1 , r 0 ). We define, for k ≥ 2 Hence ρ k = χ k−1 ρ 1 , λ k = χ k−1 λ 1 and . For any r, 0 < r ≤ d(1), let k(r) be the smallest integer such that d(k) ≤ r, that is By an iterated application of the three spheres inequality over the chain of balls B ρ1 (G 1 ), . . . , B ρ k(r) (G k(r) ), we have for 0 < r < cr 0 , where c, 0 < c < 1, depends on L. Let us consider now f (y, w) as a function of w. First we observe that For y, w ∈ G h , y = w, using (3.9) Similarly for S 2 . Therefore For w ∈ B 4R B 3R and y ∈ G h , using (5.32), we have |f (y, w)| ≤ ch −A εβ τ k(r)−1 .
Proceeding as before, let us fix y ∈ G such that dist(y, Ω D ) = h andw ∈ B 4R B 3R such that dist(w, ∂B R ) = R/2. Taking r = R/2, r 1 = 3r, r 2 = 4r, w 1 = O + λ 1 ν and using iteratively the three spheres inequality, we have where τ and s are as above. Therefore where γ = τ β , with β as above, so 0 < γ < 1 and A = Aτ s − 4 + γ. Once again, let us apply the three spheres inequality over a chain of balls contained in a cone with vertex in O, choosing y = w = hν(O) we obtain We observe that, for 0 < h < cr 0 , where 0 < c < 1 depends on L, k(h) ≤ c| log h| = −c log h, so we can write Then in (5.33) we obtain Proof of Proposition 3.4. Let us define r 2 = min{r 0 , r 2 }, where r 0 is the one of Proposition 4.1 and r 2 will be fixed later. For every x, y such that |x − y| < r, with 0 < r < r 2 , the following asymptotic formula holds (cf. Proposition 4.1) We now distinguish two situations: 1) x ∈ B r ∩ (D 1 ∩ D 2 ); 2) x ∈ B r ∩ (D 1 D 2 ).
If case 1) occurs then the asymptotic formula (4.14) holds also for G 2 since the hypothesis of Proposition 4.1 are met. From [Al,Lemma 3.1] there exists r 2 , depending on the a priori data, such that (5.34) ∇G 1 (x, y) · ∇G 2 (x, y) ≥ c|x − y| −2 .