Revisiting the probe and enclosure methods

This paper is concerned with reconstruction issue of inverse obstacle problems governed by partial differential equations and consists of two parts. (i) The first part considers the foundation of the probe and enclosure methods for an impenetrable obstacle embedded in a medium governed by the stationary Schr\"odinger equation. Under a general framework, some natural estimates for a quantity computed from a pair of the Dirichlet and Neumann data on the outer surface of the body occupied by the medium are given. The estimates enables us to derive almost immediately the necessary asymptotic behaviour of indicator functions for both methods. (ii) The second one considers the realization of the enclosure method for a penetrable obstacle embedded in an absorbing medium governed by the stationary Schr\"odinger equation. The unknown obstacle considered here is modeled by a perturbation term added to the background complex-valued $L^{\infty}$-potential. Under a jump condition on the term across the boundary of the obstacle and some kind of regularity for the obstacle surface including Lipschitz one as a special case, the enclosure method using infinitely many pairs of the Dirichlet and Neumann data is established.


Introduction
The probe and enclosure methods initiated by the author in [9], [8], [11], [10], [13], [12] and [14] are methodologies in reconstruction issue for inverse obstacle problems governed by partial differential equations. Nowadays we have various applications, see the recent survey paper [19] for the results and references. However, looking back to some of those applications, one has not yet fully developed its possibility.
In this paper, we consider the following two topics on inverse obstacle problems governed by the stationary Schrödinger equation: (i) the foundation of the probe and enclosure methods for an impenetrable obstacle; (ii) the enclosure method for a penetrable obstacle embedded in an absorbing medium.
To explain the content of the first part, consider an inverse obstacle problem governed by the Laplace equation in a bounded domain Ω of R 3 with smooth boundary. The problem is motivated by a possibility of application to nondestructive testing and formulated as follows.
Given an arbitrary solution v = v(x) of the Laplace equation in the whole domain Ω, let u = u(x) be a solution of where D is an open subset of R 3 with, say, smooth boundary and satisfies that D ⊂ Ω and Ω \ D is connected; ν denotes the unit outward normal vector to ∂D. The D is a mathematical model of a defect occurred in a body Ω. Consider the pair of the Dirichlet and Neumann data u| ∂Ω = v| ∂Ω and ∂u ∂ν | ∂Ω , where ν denotes also the unit outward normal vector to ∂Ω. The problem is to extract information about the geometry of D from a set of the pairs.
The probe and enclosure methods, say, see section 2 in [18] and section 5 in [12], enables us to extract D itself and the convex hull of D, respectively from the indicator functions calculated from specially chosen infinitely many pairs of the Dirichlet and Neumann data.
The one of common key points for both methods is the fundamental estimates for the pair: where , denotes the dual pairing between H − 1 2 (∂Ω) and H 1 2 (∂Ω) and C a positive constant independent of v.
However, from the beginning of those methods, it was not known whether (1.2) type estimates is valid for the Helmholtz equation with a fixed wave number or not. So to realize the both methods in [11] and [12] for inverse obstacle problems governed by the Helmholtz equation the author developed some technical argument. The argument worked well for the probe method. However, for the enclosure method [12] we needed the argument combined with some additional restriction on the curvature of the surface of the obstacle. Later, in [26], [27] Sini-Yoshida developed an argument to remove such restriction and obtained the estimates: where u is a solution of and v is an arbitrary solution of equation ∆v + k 2 v = 0 in Ω. It is assumed that k 2 is not a Dirichlet eigenvalue of −∆ in D nor eigenvalue of −∆ in Ω \ D with the Dirichlet and Neumann boundary conditions on ∂Ω and ∂D, respectively. Note that the validity of the upper estimate part on (1.3) is easily deduced and already known at the beginning of the probe and enclosure methods. Thus, the real problem was to establish the lower estimate part. The essence of their proof on the lower estimate is as follows. First we have the integral expression of the middle term on (1.3) from [12]: where w = u − v. By dropping the third term we have This is the common part as the previous approach in [12] and thus the problem is to give a good estimate for w L 2 (Ω\D) which is weaker than ∇v L 2 (D) . For this the author in [12] made use of a character of specially chosen input v under some restriction, see Lemma 4.2 in [12] 1 . In contrast to this, they deduced that, for a fixed s ∈ ] 3 2 , 1[ independent of general input v w L 2 (Ω\D) ≤ C v H s (D) . (1.5) They combined this with the additional estimate (cf. Theorem 1.4.3.3 on page 26 in [5]) where C ǫ is a positive number depending on an arbitrary small positive number ǫ and independent of v. From (1.4), (1.5) and (1.6) for a sufficiently small ǫ, one gets the lower estimate on (1.3). The proof of (1.5) which is one of the main parts of their paper employs the potential theoretic approach on fractional order Sobolev spaces and quite involved. Especially its central part, Lemma 4.3 in [26] states the invertibility of an operator involving the adjoint of the double layer potential on a negative fractional order Sobolev space on ∂D.
In the first part of this paper we give an alternative and elementary approach, however, which yields the sharper estimate than (1.5), that is, From this we immediately obtain the lower estimate on (1.3)(without using (1.6)). It is a special case of the result under a general framework. The major advantage of our approach is: we do not need any detailed knowledge of the governing equation for the background medium in contrast to their approach in which the fundamental solution of the Helmholtz equation played the central role. Thus it covers inverse problems for impenetrable obstacles governed by the stationary Schrödinger equation almost immediately. As a direct consequence we obtain an extension and improvement of the author's previous application of the enclosure method to an obstacle with impedance boundary condition in [17].
The description above is the comparison of the methods for the proof of the lower estimate on (1.3). From a technical point of view, it should be pointed out that, the proof in [26,27] mentioned above which is based on the potential theory covers the case when ∂D is Lipschitz. Our proof even in this simple situation needs a higher regularity ∂D ∈ C 1,1 . This is because of the trace theorem of H 2 -functions. See Lemma 2.2.
The content of the second part is as follows. The unknown obstacle considered here is a penetrable one and modeled by a perturbation term added to the background complex-valued L ∞ -potential of the stationary Schrödinger equation. Under a condition on the jump of the real or imaginary part of the term across the boundary of the obstacle and some kind of regularity for the obstacle surface including Lipschitz one as a special case, the enclosure method using infinitely many pairs of the Dirichlet and Neumann data is established. This is a full extension of a result in the unpublished manuscript [15] (see also [19]) in which an inverse obstacle problem for a non-absorbing medium governed by the stationary Schrödiner equation was considered. Now let us formulate the two problems mentioned above more precisely and describe statements of the results.

Impenetrable obstacle
Let Ω be a bounded domain of R 3 with smooth boundary. Let v ∈ H 1 (Ω) be a weak solution of where V 0 ∈ L ∞ (Ω). This means that, for all ψ ∈ H 1 (Ω) with ψ = 0 on ∂Ω in the sense of the trace we have − Then the bounded linear functional ∂v ∂ν | ∂Ω ∈ H − 1 2 (∂Ω) is well defined by the formula where η is an arbitrary element in H 1 (Ω) such that η = f ∈ H 1 2 (∂Ω). Note that the welldefinedness means that the value (1.9) does not change when η is replaced with any other one having the same trace on ∂Ω. This is because of (1.8). The boundedness is a consequence a special choice of η such that η H 1 (Ω) ≤ C f H 1/2 (∂Ω) with a positive constant C independent of f , whose existence is ensured by the trace theorem about the lifting (cf., Theorem 1.5.1.3 in [5]). The functional ∂v ∂ν | ∂Ω is boundary local in the sense that it can be determined by the value of v in U ∩ Ω, where U is an arbitrary small neighbourhood of ∂Ω. This is also a consequence of (1.8).
Let D be a nonempty open set of R 3 such that D ⊂ Ω, Ω \ D is connected and ∂D is Lipschitz. Let λ ∈ L ∞ (∂D). The λ can be a complex valued function.
Given a weak solution v of (1.7), let u ∈ H 1 (Ω \ D) be a weak solution of (1. 10) This means that, for all ϕ ∈ H 1 (Ω \ D) with ϕ = 0 on ∂Ω in the sense of the trace we have and u = v on ∂Ω in the sense of trace. Note that u depends on v on ∂Ω.
Then the bounded linear functional ∂u ∂ν | ∂Ω ∈ H − 1 2 (∂Ω) is well defined by the formula . Needless to say, for the well-definedness and boundedness, we have made use of (1.12) and the trace theorem for the lifting [5], respectively.
Our starting point is the following representation formula.
Proof. The w satisfies w = 0 on ∂Ω in the sense of the trace. From (1.9) for η = v and (1.12) for φ = v, respectively we have Thus one gets Here using (1.11) for ϕ = w we have Therefore we have obtained the desired formula. ✷ Note that Proposition 1.1 is valid under the existence of u. We never make use of any estimate on u.
To go further hereafter we impose an assumption concerning with the uniqueness and existence of u together with some estimates on the reflected solution w = u − v. (1.14) and that the unique solution satisfies where C is a positive constant independent of F . This means that p satisfies p = 0 on ∂Ω in the sense of the trace and, for all φ ∈ H 1 (Ω \ D) with φ = 0 on ∂Ω in the sense of the trace, we have (1.16) From this with φ = p we have Here using Theorem 1.5.1.10 in [5], we have, for all ǫ > 0 Thus, choosing a small ǫ and using (1.16), we conclude that and (1.15) yields p H 1 (Ω\D) ≤ C F L 2 (Ω\D) . (1.17) By Assumption 1 together with a standard lifting argument, we know that, given v which is an arbitrary weak solution of (1.7) there exists a unique weak solution u ∈ H 1 (Ω \ D) of (1.10) in the sense mentioned above. Here we impose the following assumption 2 .
Thus taking the real part of the both sides on (1.18), we obtain Now we state one of the main results of this paper.
Theorem 1.1. Assume that ∂D is C 1,1 and λ ∈ C 0,1 (∂D). Under Assumptions 1 and 2 we have where C 1 , C 2 and C 3 are positive numbers independent of v.
Some remarks are in order. (i) The validity of the system of inequalities on (1.20), especially the lower estimate is independent of the signature of the real or imaginary part of λ.
(ii) Taking the imaginary part of the both sides on (1.18), we obtain It would be not suitable for us to extract the information about the location of D itself from this equation since in Theorem 1.1, there is no restriction on the signature of Im λ.

Penetrable obstacle in an absorbing medium
Let Ω be bounded domain of R 3 with C 1,1 -boundary. In this subsection we consider an inverse obstacle problem governed by the stationary Schrödinger equation where both V 0 and V are complex valued L ∞ (Ω) functions. Let u ∈ H 1 (Ω) be a weak solution of (1.21). As usual, define the linear functional ∂u ∂ν | ∂Ω on H 1 2 (∂Ω) by the formula where η is an arbitrary element in H 1 (Ω) such that η = f ∈ H 1 2 (∂Ω) and ν denotes the unit outward normal vector field to ∂Ω. Note that this right-hand side is invariant to any η as long as the condition η = f on ∂Ω is satisfied. Thus choosing a special lifting for f , we see that this functional is bounded, that is, belongs to H − 1 2 (∂Ω). We consider the following problem.
Problem 2. Assume that V 0 is known. Extract information about the discontinuity, roughly speaking the place where V (x) = 0 from the Neumann data ∂u ∂ν | ∂Ω for the weak solution of (1.21) with some known Dirichlet data u| ∂Ω .
Our aim is not to reconstruct the full knowledge of V or V 0 + V itself unlike [25] and [23].
Here we employ the enclosure method introduced in [13] 3 .
The following assumption for V 0 + V corresponds to Assumption 1.

Assumption 3.
Given an arbitrary complex valued function F ∈ L 2 (Ω) there exists a unique weak solution p ∈ H 1 (Ω) of and that the unique solution satisfies where C is a positive constant independent of F .
As a simple consequence of the weak formulation of (1.22), the estimate (1.23) is equivalent to the following estimate: where the positive constant C ′ is independent of F . Thus Assumption 3 means that the homogeneous Dirichlet problem (1.22) is well-posed. Then, by using a lifting argument and (1.24), we see that, given an arbitrary complex valued function g ∈ H 1/2 (∂Ω) there exists a unique weak solution u ∈ H 1 (Ω) of equation (1.21) with u = g on ∂Ω in the trace sense. Succeeding to Assumption 3, to clarify the meaning of discontinuity in Problem 2 we introduce the following assumption. In contrast to impenetrable case we do not impose Assumption 2. The set D is a candidate of the discontinuity to be detected. Note that the condition D ⊂ Ω contains the case ∂D ∩ ∂Ω = ∅. So it is different from the usual condition D ⊂ Ω.
First we show that under Assumptions 3-4 and Re V has a kind of jump condition across ∂D specified later, the information about the convex hull of D can be obtained by knowing ∂u ∂ν | ∂Ω for g = v| ∂Ω , where v is an explicit solution of equation (1.7) which is just the so called complex geometrical optics solution constructed in [25] to give an answer to the uniqueness issue of the Calderón problem [2] given by the following steps.
1. Given real unit vector ω choose ϑ an arbitrary real unit vector perpendicular to ω.
andṼ 0 denotes the zero extension of V 0 outside Ω. The distribution e x·z G z (x) on R 3 is called the Faddeev Green's function at z · z = 0.
This pointwise-estimate is the key for our argument since no regularity condition about V 0 and V more than their essential boundedness is imposed. Now having v = v(x, z) given by (1.26), for all τ >> 1 we introduce the indicator function in the enclosure method Note that, in general v * = v and still v * satisfies (1.7). This is the key point to treat complexvalued potential V 0 .
Definition 1.1. We say that the a real-valued function K(x), x ∈ Ω has a positive/negative jump on ∂D from the direction ω if there exist C = C(ω) > 0 and δ = δ(ω) > 0 such that Let ω be a unit vector. Then, for almost all s ∈ R, the set is Lebesgue measurable with respect to the two dimensional Lebesgue measure µ 2 on the plane We say that D is p-regular with respect to ω if there exist positive numbers C and δ such that the set S ω (s) satisfies This type of notion has been used in [13] for the original version of the enclosure method for the equation ∇ · γ∇u = 0. 5 More precisely he proved the estimate The decaying order is not important for our purpose.
Our first result of this subsection under Assumptions 3-4 is the following.
Theorem 1.2. Let 1 ≤ p < ∞ and ω be a unit vector. Assume that D is p-regular with respect to ω. If Re V has a positive or negative jump on ∂D from direction ω, then we have, for all and Re V has a negative jump Besides, we have the one-line formula Note that, as a direct consequence we have the classical characterization In [13] it is pointed out that if D is a nonempty open set of R 3 such that ∂D is Lipschitz or satisfies the interior cone condition, then D is 3-regular with respect to an arbitrary unit vector ω. Thus one gets the following corollary. Concerning with the extraction of discontinuity of Im V which is the main subject of this paper, under Assumptions 3 and 4 we obtain the following result. Theorem 1.3. Let 1 ≤ p < ∞ and ω be a unit vector. Assume that D is p-regular with respect to ω. If Im V has a positive or negative jump on ∂D from direction ω, then we have, for all and Im V has a negative jump Besides, we have the one-line formula This paper is organized as follows. In Section 2 the proof of Theorem 1.1 is given. It is based on (1.19) and Lemmas 2.1-2.2 in which the estimetes of H 1 (Ω \ D) and L 2 (Ω \ D)-norms of the reflected solution w = u − v are given. The most emphasized one is Lemma 2.2, which ensures w L 2 (Ω\D) ≤ C v L 2 (D) for a positive constant independent of v. This estimate has not been known in the previous studies in the probe and enclosure methods. Section 3 is devoted to the proof of Theorems 1.2-1.3. It is based on the Alessandrini identity and Lemmas 3.1-3.2 in which an upper bound of the L 2 (Ω)-norm of the reflected solution w = u − v and the comparison of e τ x·ω L 1 (D) relative to e τ x·ω L 2 (D) as τ → ∞, respectively are given. In particular, the proof of Lemma 3.2 is simple, however, covers a general obstacle compared with the previous unpublished work [15] and no restriction on the direction ω. Concerning with the proof of Lemma 3.2, in Section 4 an approach in [26] for Lipschitz obstacle case is presented. It is pointed out that their approach does not work in the Lipschitz obstacle case and needs higher regularity. Section 5 is devoted to applications of Theorem 1.1 to the probe and enclosure methods and an important example covered by Theorems 1.2 and 1.3. In Section 6, concerning with Assumption 2 and Sini-Yoshida's another result in [26], we describe two open problems on the enclosure method.
2 Proof of Theorem 1.1

Two lemmas
In this subsections we give two estimates for w = u − v which are crucial for establishing the fundamental inequalities.
Proof. The w satisfies w = 0 on ∂Ω in the sense of the trace. Let F = w and p be the weak solution in Assumption 2. Since u = w + v satisfies (1.11) for all ϕ ∈ H 1 (Ω \ D) with ϕ = 0 on ∂D, one can substitute ϕ = p into it, we obtain On the other hand p satisfies (1. 16 Thus we have the expression Here we claim that .

(2.2)
This is proved as follows. By the lifting in the trace theorem (Theorem 1.5.1.3 in [5]), one can find ap ∈ H 1 (D) such that p =p on ∂D in the sense of the trace and By the trace theorem, we have Z ∈ H 1 (Ω) and Z = 0 on ∂Ω in the sense of the trace. Substituting ψ = Z into (1.8), we obtain Then (1.17) with F = w yields the rough estimate (2.6) Using similar argument for the derivation of (2.2), we have .
Thus p ′ coincides with p ′′ almost everywhere in Ω \ D and satisfies Then (1.15) yields ηp H 2 (Ω\D) ≤ C F L 2 (Ω\D) . (2.10) Now we are ready to prove a more accurate estimate than that of Lemma 2.1.
Here, by the interior elliptic regularity of v in Ω we have v| D ∈ H 2 (D). Then integration by parts [5] yields Thus (2.1) becomes Therefore (2.13) becomes where ǫ 1 and ǫ 2 are arbitrary positive numbers with 0 < ǫ j < 1, j = 1, 2. And similarly Thus from (1.19) one gets Choosing ǫ 1 and ǫ 2 so small, one obtain Now applying Lemma 2.2 to this right-hand side, we obtain the lower bound on (1.20). ✷ 3 Proof of Theorems 1.2 and 1.3

Two lemmas
First we give an L 2 -estimate of the so-called reflected solution in terms of the L 1 -norm of v which is an arbitrary solution of equation (1.7) 6 . The proof is essentially the same as that of Lemma 3.1 in [19] (taken from unpublished manuscript [15]) in which the case V 0 (x) ≡ k 2 is treated. Here for reader's convenience we present its proof.
Proof. Set w = u − v. By Assumption 3, one can find the unique weak solution p ∈ H 1 (Ω) of (1.22) with F = w. By (1.23) and V 0 + V 1 ∈ L ∞ (Ω), we have ∆p ∈ L 2 (Ω) and ∆p L 2 (Ω) ≤ C 0 w L 2 (Ω) . The elliptic regularity up to boundary, for the Laplace operator with Dirichlet boundary condition yields p ∈ H 2 (Ω) and p H 2 (Ω) ≤ C 1 w L 2 (Ω) 7 . By the Sobolev imbedding theorem, we have p L ∞ (Ω) ≤ C 2 p H 2 (Ω) . Thus one gets It follows from (1.21) that the function w is the weak solution of Using this and the weak form of (1.22) with F = w, we have the expression This yields w 2 L 2 (Ω) ≤ p L ∞ (Ω) V v L 1 (Ω) and thus from (3.2) we obtain the desired estimate. ✷ Remark 3.1. By applying the same argument for the derivation of (3.2) to (3.3), we obtain Next we describe another lemma which is originally stated in the unpublished manuscript [15] and Lemma 3.2 in [19] provided D is an open set of R 3 and ∂D ∈ C 2 and ω satisfies: The proof presented here is completely different from the original one and removes such assumption.
Lemma 3.2. Assume that D is p-regular with respect to ω. We have where v 0 (x) = e τ x·ω .
Proof. It suffices to prove the formula Let ǫ be an arbitrary positive number. We have Thus (3.6) yields By the p-regularity of D with respect to ω and Fubini's theorem, for all τ >> 1 we have Thus one gets This together with (3.7) yields (3.9) Now set ǫ = τ −γ with an arbitrary fixed number γ ∈ ]0, 1[. Then as τ → ∞ we have A combination of this and (3.9) yields (3.5).

✷
If D is open and ∂D is Lipschitz, then D is 3-regular with respect to all unit vectors ω. Thus Lemma 3.2 is valid also for such D without any restriction on ω. The advantage of the proof above is: we do not make use of any local coordinate system of D in a neighbourhood of ∂D.

Finishing the proof
Let us continue to prove Theorem 1.2. We give a proof only the case when Re V has a positive jump since another case can be treated similarly.
By the Alessandrini identity and Assumption 3, from (1.32) we have
The proof of Theorem 1.3 can be done in the same way, so its description is omitted.
Remark 3.2. The point of the derivation of (3.11) is the estimate of the following type where Q ∈ L ∞ (Ω) and satisfies Q(x) = 0 a. e. x ∈ Ω \ D. If D ⊂ Ω instead of D ⊂ Ω, one can use the local interior estimate (3.4). In fact, we have Note that the upper estimate on (1.27) is also used.

Proof of Lemma 3.2 via Sini-Yoshida's approach
First we present a formal application of their approach in the proof of a lemma in [26] to the proof of Lemma 3.2 in the case when ∂D is Lipschitz. Second the author would like to point a problem out whether their idea really can cover the case when ∂D is Lipschitz.
Given a sufficiently small positive number δ they cover the compact set {x ∈ ∂D | x · ω = h D (ω)} by a finite set {C j } j=1,···,N of the rotation of around point x j ∈ ∂D, j = 1, · · · , N with x j · ω = h D (ω) and translation to x j of a cubic domain centered at the origin (0, 0, 0) which corresponds to point x j 8 in R 3 in such a way that • each C j ∩ D has the expression where y ′ = (y 1 , y 2 ), l j (y ′ ) is a Lipschitz continuous function on R 2 and satisfies 0 ≤ l j (y ′ ) ≤ C|y ′ | with a positive constant C and A j is an orthogonal matrix. If necessary, choosing a smaller δ one may assume that where π ω denotes the plane x · ω = h D (ω). The point is the choice of A j in each C j ∩ D. For x = x j + A j y, we have From their computation in [26], it is clear that they choose each A j in such a way that Thus we have Since C j ∩ D ⊂ D, j = 1, · · · , N one has the lower and upper bound: where The point is, in both the lower and upper bound the same estimator M (τ ) satisfying M (2τ ) ≤ CM (τ ) appears. Besides by using the local expression of C j ∩ D and (4.3) the each term in M (τ ) one has where l j (y ′ ) is a Lipschitz continuous function and satisfying 0 ≤ 0 ≤ l j (y ′ ) ≤ C|y ′ | with a positive constant C. Note that this yields a rough lower estimate By virtue of the Schwarz inequality the M (τ ) has the property: A combination of this, the right-hand side on (4.4) and (4.5) gives Thus one gets (3.5) with convergence rate is O(τ − 1 2 ). The author thinks that their idea of proof has an advantage since it never makes use of any concrete upper bound of M (τ ) as τ → ∞ which needs some additional condition about the behaviour of l j (y ′ ) as y ′ → (0, 0).

What is a problem on their approach?
The problem is the choice of A j in such a way that A j satisfies (4.2) and C j ∩D has the expression (4.1) at the same time.

This yields the expression
(4.6) Thus this expression becomes (4.3) type, that is e −τ (h D (ω)−x·ω) = e −τ y 2 if and only if θ = 0. Then we see that C(0) ∩ D = D never coincides with a domain having the form of the two dimensional version of (4.1). Of course, by choosing another θ one can make C(θ) ∩ D such as the domain of (4.1) type, however, in that case one can not obtain the expression e −τ (h D (ω)−x·ω) = e −τ y 2 for x = A(θ) ∈ C(θ) ∩ D.
In three-dimensional case, consider a domain having a cone or convex polyhedron as a part, then one encounters a more complicated situation. Therefore in the Lipschitz case, their approach does not work as they wished. In this sense our approach is better to cover a broad class of boundary. Note that the comment also works for the articles [20] and [21] following their paper.

Revisiting the enclosure method
First we consider the simplest case. Let k be a fixed positive number. We identify the set of all real unit vectors with the unit sphere S 2 . Given an arbitrary ω ∈ S 2 and ϑ ∈ S 2 with ω · ϑ = 0 define v(x, z) = e x·z , x ∈ R 3 where z = τ ω + i √ τ 2 + k 2 ϑ and τ > 0. Since the complex vector z satisfies the equation z · z = −k 2 , the v = v( · ; z) is an entire solution of the Helmholtz equation ∆v + k 2 v = 0.
Define the indicator function of the enclosure method by the formula: where v = v( · ; z) and u is the weak solution of (1.10) with V 0 (x) ≡ k 2 . The following result is an extension of the author's previous result Theorem E in [17].
Besides, we have the one-line formula Proof. Theorem 1.1 gives Since ∂D is C 1,1 and thus Lipschitz. Then ∂D is 3-regular with respect to all ω ∈ S 2 and this yields 9 e −2τ h D (ω) v 2 L 2 (D) ≥ C 6 τ −3 . Noting the trivial estimate e −2τ h D (ω) v L 2 (D) ≤ C 7 , we finally obtain Now from this all the assertions of Theorem 5.1 are valid. ✷ In [26] they proved a corresponding result to Theorem 5.1 for the case when λ = 0 and ∂D is Lipschitz. Needless to say it is almost a direct consequence of (1.3) as we have seen above 10 .
Next consider the general case. Let V 0 ∈ C 0,1 (Ω). A combination of Theorem 6.2.4 on p.277 in [5] and a cut-off function, one has aṼ 0 ∈ C 0,1 (R 3 ) with compact support such that V 0 (x) =Ṽ 0 (x) for all x ∈ Ω. ReplaceṼ 0 in equation (1.25) by this one and solve it. Using the solution, we define v = v( · , z) with τ >> 1 be the same form as (1.26). Since the newṼ 0 belongs to H 1 (R 3 ) with compact support, from the Ramm estimate we have This implies that not only estimate (1.27) for v but also for ∇v for all τ >> 1: Let u be the weak solution of (1.10) with v = v( · , z) and define the indicator function I ω,ϑ (τ ) as the exactly same form as above. Then as a direct corollary of Theorem 1.1 combined with (1.27) and (5.2) we immediately obtain the following result.

A simple proof of the side A of the probe method
In this subsection it is assumed that V 0 (x) ≡ k 2 with a fixed positive number k satisfies Assumption 1, ∂D is C 1,1 and λ ∈ C 0,1 (∂D). By a standard lifting argument, given y ∈ Ω \ D there exists the unique weak solution w y ∈ H 1 (Ω \ D) of Recall the indicator function in the side A of the probe method, Definition 2.3 in [17]: Given an arbitrary point y ∈ Ω a needle σ with a tip at y is a non self-intersecting piecewise linear curve with a parameter t ∈ [0, 1] such that σ(0) ∈ ∂Ω, σ(1) = y, and σ(t) ∈ Ω for all t ∈ ]0, 1[. Let ξ = {v n } be an arbitrary needle sequence for (y, σ), that is, each v n ∈ H 1 (Ω) is a weak solution of the Helmholtz equation in Ω and, for all compact set K of R 3 satisfying K ⊂ Ω \ σ it holds that It is known at the early stage of the probe method [11](see also [16]) that under the additional assumption that k 2 is not a Dirichlet eigenvalue of −∆ in Ω, there exists a needle sequence for an arbitrary (y, σ) by the Runge approximation property of the Helmholtz equation. Besides, if σ is given by a part of a line which is called a straight needle, then one can give an explicit needle sequence without any restriction on k 2 , see [18].
Recall also the indicator sequence, Definition 2.2 in [17]: where ξ = {v n } is a needle sequence for (y, σ) and u n is the weak solution of The following result gives us (i) a computation procedure of the indicator function from the indicator sequence; (ii) qualitative behaviour of the computed indicator function at the places away from and near obstacle. It is an extension of the result in [11] when ∂D is C 2 and λ(x) ≡ 0 and the result in [17] when ∂D ∈ C 2 and λ ∈ C 1 (∂D) with min x∈∂D Im λ(x) > 0 11 to the case ∂D ∈ C 1,1 and λ ∈ C 0,1 (∂D).
Theorem 5.3. Let k be an arbitrary fixed positive number satisfying Assumption 1 with V 0 (x) ≡ k 2 . Assume that ∂D is C 1,1 and λ ∈ C 0,1 (∂D). ✷ It should be emphasized that, by virtue of Theorem 1.1 the proof given here is so simple compared with the previous one for impenetrable obstacle. For more information about the previous studies see [17], Subsections 3.2-3.3 and, in particular, Section 2 in which a remark on a proof in [3] under the previous formulation of the probe method [9] and [11], is given. And it would be possible to extend Theorem 5.1 to general real-valued V 0 by replacing G y with a singular solution for the Schrödinger equation (1.7).

Cases covered by Theorems 1.2 and 1.3
In this subsection, we consider an inverse obstacle problem governed by the equation This is the case when V 0 and V in (1.21) are given by where a, a 0 , b and b 0 are real-valued and belong to L ∞ (Ω) (no further regularity) and k > 0. The equation (5.3) is coming from the time harmonic solution of the wave equation See also [4] for more information about the background. We see that Assumptions 3 and 4 for this example are equivalent to the following ones. Assumption 3'. Given F ∈ L 2 (Ω) there exists a unique weak solution p ∈ H 1 (Ω) of and that the unique solution satisfies where C is a positive constant independent of F . (ii) b(x) − b 0 (x) has a positive or negative jump across ∂D from ω.
Note that the place where b(x) − b 0 (x) = 0 corresponds to an unknown absorbing region added to the background absorbing medium. Theorem 1.3 gives us a method of estimating such a place from above.

Problems remaining open
This section describes some open issues related to enclosure method only.

Impenetrable obstacle embedded in an absorbing medium
Develop the enclosure method for an impenetrable obstacle embedded in an absorbing medium, that is, the governing equation is given by the stationary Schrödinger equation with a fully complex-valued background potential.
More precisely, let v and v * be given by (1.26) and (1.29), respectively.
Open Problem 1. Remove Assumption 2 and use the same form (1.28) as the indicator function for impenetrable obstacle. Clarify, as τ → ∞ the asymptotic behaviour of this indicator function.
Similarly to (1.13) we have 1) where u * is the weak solution of (1.10) for v replaced with v * , w = u − v and w * = u * − v * . Further, using the weak formulation of the governing equation of w, we have Thus (6.1) has another expression: Thus the main task is to study on the asymptotic behaviour of ∇w · ∇w * over Ω \ D as τ → ∞ or/and the first term of the right-hand side on (6.2).

Extending Sini-Yoshida's another result on a penetrable obstacle
First one has to mention another result for a penetrable obstacle in [26]. The governing equation is given by ∇ · γ(x)∇u + k 2 n(x)u = 0, x ∈ Ω, 1, x ∈ Ω \ D, So this is the case when V 0 (x) ≡ k 2 . It is assumed that both γ D and n D are real-valued, γ D ∈ L ∞ (D), n D ∈ L ∞ (D) and ess.inf x∈D γ D (x) > 0; D is an open subset of R 3 such that D ⊂ Ω and ∂D is Lipschitz; 0 is not a Dirichlet eigenvalue for equation (6.3).
Note that γ has a jump across ∂D, however, any jump condition for n across ∂D is not imposed. The D is a mathematical model of a penetrable obstacle embedded in a known homogeneous background medium and appears as the perturbation term added to the leading coefficient of the governing equation.
Given a solution v of the Helmholtz equation ∆v +k 2 v = 0 in Ω, they developed an argument to derive the following lower estimate for a fixed p < 2: where u is the solution of (6.3) with u = v on ∂Ω, w = u − v and γ ∂u ∂ν | ∂Ω is defined by an analogous way as ∂v ∂ν | ∂Ω . It is a consequence of the two estimates below. (i) The well known lower estimate which goes back to [7] − ∂v ∂ν | ∂Ω − γ ∂u ∂ν | ∂Ω , v ≥ (ii) For some p < 2 w L 2 (Ω) ≤ C v W 1,p (D) , (6.5) where C is a positive constant independent of v. So their contribution is an argument for the proof of (6.5). It is divided into two parts. Step 1. Prepare two facts.
Thus the problem becomes to give a good estimate of ∇w L p (Ω) from above in the sense that the possible upper bound is weaker than ∇v L 2 (D) .
Step 2. For the purpose they employ Theorem 1 on page 198 in [22] for the L p -norm of the gradient of the solution of the elliptic problem      ∇ · γ∇r = ∇ · G + H, x ∈ Ω, where, p is in a neighbourhood of p = 2, both G and H belong to L p (Ω). Note that the term k 2 n(x)r is dropped, however, for our purpose it is no problem. Applying this to a decomposition of w, one gets ∇w L p (Ω) ≤ C v W 1,p (D) with a p < 2 satisfying 6 5 ≤ p. This is a brief story of the idea for the proof of (6.5).
Once we have (6.4) (and the trivial upper bound of the left-hand side on (6.4) by C v 2 H 1 (D) ), letting v = e x·z the same as that of Subsection 5.1, showing lim τ →∞ ∇v L p (D) ∇v L 2 (D) = 0 (6.6) and using ∇v 2 L 2 (D) ∼ τ 2 v 2 L 2 (D) , they established lower and upper bounds for the indicator function defined by in terms of ∇v 2 L 2 (D) . Then some lower and upper bounds of norm ∇v 2 L 2 (D) enabled them to extract the value h D (ω) from the asymptotic behaviour of the indicator function. This gives an extension of Theorem 1.1 in [13] which is the case k = 0. However, it should be noted that, as pointed out in Section 4, their proof of (6.6) given in Lemma 3.7 in [26] employs the same approach explained in Section 4 and does not work for the Lipschitz obstacle case 12 .
Open Problem 2. It would be interested to consider the case when n(x) is given by x ∈ Ω \ D, where both n 0 and n D are essentially bounded in Ω and complex-valued.
More precisely, in this case same as Theorems 1.2 and 1.3 one has to replace the indicator function (6.7) with the new one defined by where v * = v(x, z) and v = v(x, z) is given by (1.26) with V 0 (x) ≡ n 0 (x). Thus the problem is to clarify the aymptotic behaviour of this new indicator function. Clearly, we will encounter the same problem as impenetrable obstacle case.
Those two open problems are in a new situation not covered by previous studies. Such study together with the idea in Theorems 1.2 and 1.3 shall open a possibility of the application of the enclosure method to the stationary Maxwell system for an obstacle embedded in an absorbing medium. Note that, for a non-absorbing medium, some results realizing the enclosure method exist, see [21] and [20], and [28] which are based on the author's previous argument in [12].