The Probe Method and Its Applications II Masaru IKEHATA Department of Mathematics, Faculty of Engineering Gunma University, Kiryu 376-8515, JAPAN 29 March 2005 Abstract The probe method gives a general idea to extract information about unknown objects embedded in a known background medium from the Dirichlet-to-Neumann map on the boundary of the medium. Recently the author gave a new formulation of the probe method, raised new questions about the probe method itself and gave answers to some of them in concrete inverse boundary value problems. In this paper an explanation of the idea in a typical inverse boundary value problem for the Laplace equation is given. Some unsolved problems also are mentioned. AMS: 35R30 KEY WORDS: Dirichlet-to-Neumann map, probe method, cavity, crack, obstacle, inverse boundary value problem, Laplace equation, elasticity, indicator sequence, needle, impedance boundary condition
1
Introduction
In 1997 April when the author was getting on the train for Asakusa from Kiryu, he got a general idea to extract information about unknown objects embedded in a known background medium from the Dirichlet-to-Neumann map on the boundary of the medium. The author named the idea the probe method. In the same year Eighth international colloquium on differential equations was held at Plovdiv, Bulgaria, August 18-23. In the colloquium the author asked the organizer to change the contents of the talk and title and presented the probe method. This means that the year when the probe method was firstly presented in front of the third parsons abroad is 1997. The method has been applied to several inverse boundary value problems and scattering problems ([3], [5], [10, 11, 12, 14], [18], [19]). In [7, 4] numerical testings of the probe method have been done. The appearance of these numerical works is what the author dreamed of for many years. The dream came true! In order to give an explanation for an observation obtained in [7], quite recently the author gave a new formulation of the probe method, raised new questions about the probe method itself and gave answers to some of them in concrete inverse boundary value problems [15]. The aim of this expository paper is to: reconsider further the probe method by considering a typical inverse boundary value problem for the Laplace equation; present some of results in the reconsidering project of the previous applications of the original probe 1
method to an inverse obstacle scattering problem with impedance boundary condition ([16]) and the so-called inverse crack problem ([17]). We consider this paper the second version of the expository paper [13] and thus added the number II in the title. In the last section we mention also some unsolved problems. 1.1. A prototype for inverse boundary value problems Let Ω be a bounded domain in Rm (m = 2, 3) with Lipschitz boundary. Let D be an open subset of Ω and satisfy that D ⊂ Ω; Ω \ D is connected. We always assume that D is given by a union of finitely many bounded Lipschitz domains D1 , · · · , DN such that Dj ∩ Dl = ∅ if j 6= l. We denote by ν the unit outward normal relative to Ω \ D. Given f ∈ H 1/2 (∂Ω) let u ∈ H 1 (Ω \ D) denote the weak solution of the elliptic problem 4u = 0 in Ω \ D, ∂u = 0 on ∂D, ∂ν
(1.1)
u = f on ∂Ω. Define
∂u |∂Ω . ∂ν We set ΛD = Λ0 in the case when D = ∅. ΛD is called the Dirichlet-to-Neumann map. Here we consider the problem of extracting information about the shape and location of D from ΛD or its partial knowledge. 1.2. A new formulation of the probe method The probe method gives us a reconstruction formula of ∂D by using (Λ0 − ΛD )f for infinitely many f . Given a point x ∈ Ω let Nx denote the set of all piecewise linear curves σ : [0, 1] 7−→ Ω such that : • σ(0) ∈ ∂Ω, σ(1) = x and σ(t) ∈ Ω for all t ∈]0, 1[; • σ is injective. We call σ ∈ Nx a needle with tip at x. Choose an arbitrary fundamental solution G of the Laplace equation in Rm and fix the solution. For the new formulation of the probe method we need the following. Definition 1.1. Let σ ∈ Nx . We call the sequence ξ = {vn } of H 1 (Ω) solutions of the Laplace equation a needle sequence for (x, σ) if it satisfies, for each fixed compact set K of Rm with K ⊂ Ω \ σ([0, 1]) ΛD f =
lim (kvn ( · ) − G( · − x)kL2 (K) + k∇{vn ( · ) − G( · − x)}kL2 (K) ) = 0.
n−→∞
Needless to say, the existence of the needle sequence is a consequence of the Runge approximation property of the Laplace equation. In [15] we clarified the behaviour of the needle sequence on the needle as n −→ ∞(however, see also Section 3). For the description we make a definition. Let b be a nonzero vector in Rm . Given x ∈ Rm , ρ > 0 and θ ∈]0, π[ the set V = {y ∈ Rm | |y − x| < ρ and (y − x) · b > |y − x||b| cos(θ/2)} is called a finite cone of height ρ, axis direction b and aperture angle θ with vertex at x. 2
The two lemmas given below are the core of the new formulation of the probe method (see [15] for the proof). Lemma 1.1. Let x ∈ Ω be an arbitrary point and σ be a needle with tip at x. Let ξ = {vn } be an arbitrary needle sequence for (x, σ). Then, for any finite cone V with vertex at x we have Z lim |∇vn (y)|2 dy = ∞. n−→∞ V ∩Ω
Lemma 1.2. Let x ∈ Ω be an arbitrary point and σ be a needle with tip at x. Let ξ = {vn } be an arbitrary needle sequence for (x, σ). Then for any point z ∈ σ(]0, 1[) and open ball B centered at z we have Z
lim
n−→∞ B∩Ω
|∇vn (y)|2 dy = ∞.
Definition 1.2. Given x ∈ Ω, needle σ with tip x and needle sequence ξ = {vn } for (x, σ) define Z {(Λ0 − ΛD )f n }fn dS, n = 1, 2, · · · I(x, σ, ξ)n = ∂Ω
where fn (y) = vn (y), y ∈ ∂Ω. {I(x, σ, ξ)n }n=1,2,··· is a sequence depending on ξ and σ ∈ Nx . We call the sequence the indicator sequence. The following theorem is a special version of Theorem 2.1 in [15] which should be the prototype of the statement using the new formulation of the probe method. Theorem 1.3. Given x ∈ Ω and needle σ with tip at x we have: (A) if x ∈ Ω \ D and σ(]0, 1]) ∩ D = ∅, then for any needle sequence ξ = {vn } for (x, σ) the sequence {I(x, σ, ξ)n } is convergent; (B) if x ∈ Ω \ D and σ(]0, 1]) ∩ D 6= ∅, then for any needle sequence ξ = {vn } for (x, σ) we have limn−→∞ I(x, σ, ξ)n = ∞; (C) if x ∈ D, then for any needle sequence ξ = {vn } for (x, σ) we have limn−→∞ I(x, σ, ξ)n = ∞. Moreover, we have: a point x ∈ Ω belongs to Ω \ D if and only if the exists a needle σ with tip at x and a needle sequence ξ for (x, σ) such that the sequence {I(x, σ, ξ)n } is bounded from above. Proof. Integration by parts yields Z ∂Ω
Z
Z
{(Λ0 − ΛD )f }f dS =
Ω\D
|∇(u − v)|2 dy +
D
|∇v|2 dy
(1.2)
where u solves (1.1) with f = v|∂Ω and v ∈ H 1 (Ω) is an arbitrary solution of the Laplace equation in Ω. This identity is a special version of the corresponding one established in [12]. Thus we have Z I(x, σ, ξ)n ≥ |∇vn |2 dy. D
Then from Lemmas 1.1 to 1.2 one obtains the desired conclusions in two cases (B) and (C). The case (A) is trivial and we omit the proof of the remaining part of the statement(Corollary 2.1 of [15]). 3
2 Note that the limit of the indicator sequence in the case (A) does not depend on the choice of σ whenever σ(]0, 1]) ∩ D = ∅ and coincides with the indicator function I defined by the formula Z
I(x) =
Z
|∇wx |2 dy +
Ω\D
D
|∇G(y − x)|2 dy, x ∈ Ω \ D
where wx ∈ H 1 (Ω \ D) is the unique weak solution of the problem: 4w = 0 in Ω \ D, ∂w ∂ = − (G( · − x)) on ∂D, ∂ν ∂ν w = 0 on ∂Ω. We called wx the reflected solution by D. I(x)(≥ 0) has two important properties: it is bounded for x ∈ Ω \ D with dis (x , ∂D) > ² for each fixed ² > 0; for all a ∈ ∂D limx−→a I(x) = ∞. This is the essence of the previous formulation of the probe method. When one considerers a similar problem for the crack, the volume of D becomes zero. Thus one can easily guess that one can not make use of the second term in (1.2) for showing the blowup of the indicator sequence. For the purpose we should find an idea to make use of the first term of (1.2). Quite recently in [17] the author found the idea in the problem related to the crack and established a corresponding result to Theorem 1.3. In Section 2 we describe the idea in the present case.
2
Blowup of the reflected solution
Lemma 2.1. Let v ∈ H 1 (Ω) be a solution of the equation 4v = 0 in Ω. Let u ∈ H 1 (Ω\D) be the weak solution of the problem (1.1) for f = v|∂Ω . Set w = u − v. Then w satisfies that the trace of w onto ∂Ω vanishes and, for all ϕ ∈ H 1 (Ω \ D) with ϕ = 0 on ∂Ω Z
Z
Ω\D
∇w · ∇ϕdy =
∂D
∂v ϕdS. ∂ν
(2.1)
We say that w in this lemma is the reflected solution of v by D. One can easily check (2.1) and so, we omit the proof. In this section we study of the blowup property of the sequence of the reflected solutions w1 , w2 · · · of v1 , v2 , · · · by D where {vn } is a given needle sequence for (x, σ) satisfying σ(]0, 1]) ∩ D 6= ∅. We focus on the blowup of the sequence of the energy Z Ω\D
|∇wn |2 dy.
Lemma 2.2. Let v ∈ H 1 (Ω) be a solution of the equation 4v = 0 in Ω. Let w ∈ H 1 (Ω\D) be the reflected solution of v by D. Then we have Z
D
|∇v|2 dy
kvkH 1 (D)
≤ C1 C2 k∇wkL2 (Ω\D) .
4
(2.2)
Proof. From the trace theorem we know that there exists p ∈ H 1 (Ω \ D) such that p = v on ∂D, p = 0 on ∂Ω. This p satisfies kpkH 1 (Ω\D) ≤ C1 kv|∂D kH 1/2 (∂D)
(2.3)
where C1 = C1 (Ω \ D) > 0 and independent of v. Integration by parts gives Z
Z 2
D
|∇v| dy =
∂D
∂v vdS. ∂ν
(2.4)
Then a combination of (2.1) for ϕ = p and (2.4) yields Z
Z 2
∇w · ∇pdy.
(2.5)
kΨ|∂D kH 1/2 (∂D) ≤ C2 kΨkH 1 (D) .
(2.6)
D
|∇v| dy =
Ω\D
Let C2 = C2 (D) > 0 satisfy, for all Ψ ∈ H 1 (D)
Taking the real part of (2.5), we have Z D
|∇v|2 dy ≤ k∇wkL2 (Ω\D) k∇pkL2 (Ω\D) ≤ C1 k∇wkL2 (Ω\D) kv|∂D kH 1/2 (∂D) ≤ C1 C2 k∇wkL2 (Ω\D) kvkH 1 (D)
This completes the proof of (2.2). 2 Using a variant of the Poincar´e inequality one can prove Lemma 2.3([17]). Let x ∈ Ω and σ be a needle with tip at x. Let ξ = {vn } be a needle sequence for (x, σ). If Z lim |∇vn |2 dy = ∞, n−→∞ D
then there exists a natural number n0 such that the sequence Z
{Z
D
D
|vn |2 dy
|∇vn |2 dy
}n≥n0 ,
is bounded. The following theorem is a direct consequence of Lemmas 2.2 and 2.3 and gives us a characterization of Ω \ D using the sequence {k∇wn k2L2 (Ω\D) }. Theorem 2.4. Given x ∈ Ω and needle σ with tip at x we have: 5
• if x ∈ Ω \ D and σ(]0, 1]) ∩ D = ∅, then for any needle sequence ξ = {vn } for (x, σ) the sequence {k∇wn k2L2 (Ω\D) } is convergent; • if x ∈ Ω \ D and σ(]0, 1]) ∩ D 6= ∅, then for any needle sequence ξ = {vn } for (x, σ) we have limn−→∞ k∇wn k2L2 (Ω\D) = ∞; • if x ∈ D, then for any needle sequence ξ = {vn } for (x, σ) we have limn−→∞ k∇wn k2L2 (Ω\D) = ∞. Moreover, we have: a point x ∈ Ω belongs to Ω \ D if and only if the exists a needle σ with tip at x and a needle sequence ξ for (x, σ)such that the sequence {k∇wn k2L2 (Ω\D) } is bounded from above. Clearly Theorem 2.4 gives an alternative proof of Theorem 1.3. And also limn−→∞ wn = wx in H 1 (Ω \ D) in the case (A) where wx is the reflected solution by D(see Section 1). From (2.2) we have Z D
|∇G(y − x)|2 dy
kG( · − x)kH 1 (D) and thus conclude that
≤ C1 C2 k∇wx kL2 (Ω\D) .
Z
lim
x−→a Ω\D
|∇wx (y)|2 dy = ∞
for each a ∈ ∂D. Thus we could cover also the previous version of the probe method by using only the first term of (1.2).
3
Solved and unsolved problems
3.1. The complete description of the blowup set In [15] we introduced the notion of the blowup set of the sequence of H 1 (Ω \ D) functions. Definition 3.1. We say that the sequence {gn } of H 1 (Ω \ D) functions blows up at the point z ∈ Ω \ D if for any open ball B centered at z it holds that Z
lim
n−→∞ B∩(Ω\D)
|∇gn (y)|2 dy = ∞.
We call the set of all points z ∈ Ω \ D such that {gn } blows up at z the blowup set of {gn }. Given x ∈ Ω and σ ∈ Nx let ξ = {vn } be an arbitrary needle sequence for (x, σ). Let wn be the reflected solution of vn by D. It follows from Theorem 2.4 that if σ(]0, 1]) ∩ D = ∅, then the blowup set of the sequence {wn } is empty. In [15] we raised the question: determine the blowup set of {wn }. Therein we considered special D, Ω and σ with σ(]0, 1]) ∩ D 6= ∅ and showed that the blowup set of {wn } for an arbitrary needle sequence ξ for (x, σ) is given by transforming σ(]0, 1]) ∩ D. However, the complete description of the blowup set in general case remains open. 3.2. Crack Let Σ be a (m − 1)-dimensional closed submanifold of Rm of class C 0 with boundary. Σ is divided into two parts: the interior and the boundary denoted by Int Σ and ∂Σ, respectively.
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We assume that that there exists an open subset D with Lipschitz boundary of Ω, having finitely many connected components and satisfying the following: D ⊂ Ω;
(?) Ω \ D is connected; Σ ⊂ ∂D.
We denote by ν the unit outward normal relative to D unless otherwise specified. Then given f ∈ H 1/2 (∂Ω) one can find a unique solution u in a suitable function space, of an weak formulation of the elliptic problem 4u = 0 in Ω \ Σ, ∂u = 0 on Σ, ∂ν
(3.1)
u = f on ∂Ω. The solution does not depend on the choice of D satisfying (?). And one can define the Dirichlet-to-Neumann map ΛΣ which is also invariant for the choice of D satisfying (?). In [17] we considered the problem: extract information about the shape and location of Σ from ΛΣ . We applied the probe method to this problem. The key points are: • the representation formula of the Dirichlet-to-Neumann map Z ∂Ω
Z
{(Λ0 − ΛΣ )f }f dS =
Ω\Σ
|∇(u − v)|2 dy
(3.2)
where u solves (3.1) with f = v|∂Ω and v ∈ H 1 (Ω) is an arbitrary solution of the Laplace equation in Ω; • invariance of u − v with respect to the choice of D satisfying (?); • an explicit lower bound of k∇(u − v)k2L2 (Ω\Σ) involving a suitable modification D0 of the original D and v on D0 which corresponds to (2.2). (3.2) corresponds to (1.2) and is well known. Using those, we studied the behaviour of the corresponding indicator sequence and gave also an alternative simple proof of the application of the previous version of the probe method to the problem in [18]. It maybe possible to apply the method presented in this paper to also the corresponding problem for the system of equations in the theory of linear elasticity. This will be done in a forthcoming paper. In [19] the application of the previous version of the probe method to this problem is given. For the study of the uniqueness issue of the inverse problems for crack see [1, 6, 8] and references therein. We also point out the paper [2] which is an application of Kirsch’s factorization method to an inverse problem for crack. 3.3. Obstacle with impedance boundary condition Let k ≥ 0. Let D be an open subset with Lipschitz boundary of Ω and satisfy that D ⊂ Ω; Ω \ D is connected. We always assume that 0 is not a Dirichlet eigenvalue of 4 + k 2 in Ω. 7
Let λ ∈ L∞ (∂D) and satisfy the condition: there exists a positive constant C > 0 such that Im λ(x) ≥ C for almost all x ∈ ∂D. This assumption is motivated by a possibility of application to inverse scattering problems of electromagnetic/acoustic wave. Given f ∈ H 1/2 (∂Ω) one can find a unique solution u ∈ H 1 (Ω \ D), of an weak formulation of the elliptic problem 4u + k 2 u = 0 in Ω \ D, ∂u + λu = 0 on ∂D, ∂ν u = f on ∂Ω. Using this solution, one can define the Dirichlet-to-Neumann map Λ(D, λ) . The problem considered in [16] is : extract information about the shape and location of D from Λ(D, λ) . Note that: in [3] the previous version of the probe method has been applied to the problem and a reconstruction formula of ∂D is given. In their paper they state that the reconstruction formula of ∂D by the probe method is valid under the assumption:λ is just essentially bounded. However, reviewing their proof, one knows that their argument does not cover this case and works in the case when, say, λ ∈ C 1 (∂D). The reader can see [16] for the detail. Note that therein two alternative simple proofs of the result in [3] are also given. Thus one may propose: give the reconstruction formula by the previous version of the probe method under the regularity condition λ ∈ L∞ (∂D). It should be pointed out that their result is an extension of the corresponding results in [12], in which we considered the case when λ −→ 0(sound-hard)/λ −→ ∞(sound-soft). Also note that, in [11] we considered inverse obstacle scattering problems with a fixed frequency and established a way of reconstructing sound-hard/sound-soft obstacle from the scattering data. The main result of [16] is: if both kλkL∞ (∂D) and k 2 are so small, then one has a theorem similar to Theorem 1.3. This is an extension of a result in [15]. For the proof of the main result we employ two types of the Poincar´e inequalities and the following lemma which is a special case of Theorem 1.5.1.10 in page 41 of [9]. Lemma 3.1. Let W be a bounded open subset of Rm with a Lipschiz boundary Γ. Then there exists a positive constant K(W ) such that Z
Z 2
Γ
|u| dS ≤ K(W )(²
Z 2
W
−1
|∇u| dy + ²
W
|u|2 dy)
for all u ∈ H 1 (W ) and ² ∈]0, 1[. The smallness conditions on k and λ involve K(W ) for W = D, Ω \ D and the constants in the Poincar´e inequalities. Finally we propose: drop the smallness conditions on k and λ. This remains open. 3.4. Co-response spectrum and Co-response set
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One can extend the wrapping cloth of the probe method to cover more general case. Let Λ : H 1/2 (∂Ω) −→ H −1/2 (∂Ω) be an arbitrary bounded linear operator. Definition 3.2. Given open subset U of Ω define N (U ) = ∪x∈U {x} × Nx . We say that a point (x, σ) ∈ N (Ω) does not belong to the set CRS(Λ; Λ0 ) if, for any needle sequence ξ = {vn } for (x, σ) the indicator sequence {I(x, σ, ξ)n } defined by replacing ΛD in Definition 1.2 with Λ is not bounded. Thus CRS(Λ; Λ0 ) is the set of all points (x, σ) ∈ N (Ω) such that, for a suitable needle sequence ξ for (x, σ) the indicator sequence is bounded. We name this set the co-response spectrum of Λ relative to Λ0 . The co-response spectrum is the all what we can extract from Λ − Λ0 by using the probe method. Definition 3.3. Define CR(Λ; Λ0 ) = π(CRS(Λ; Λ0 )) where π denotes the projection N (Ω) 3 (x, σ) 7→ x ∈ Ω. We name this set the co-response set of Λ relative to Λ0 . The reader should know that the relationship between the co-response spectrum and the co-response set is analogous to the one between the wave front set and the singular support of distributions. The following is a direct translation of Theorem 1.3 and describes a relationship between the co-response spectrum/set of ΛD relative to Λ0 and D. Corollary 3.1. The set CRS(ΛD ; Λ0 ) is contained in N (Ω \ D). Moreover, the formula CR(ΛD ; Λ0 ) = Ω \ D, is valid.
Acknowledgement This research was partially supported by Grant-in-Aid for Scientific Research (C)(2) (No. 15540154) of Japan Society for the Promotion of Science.
References [1] Alessandrini, G. and DiBenedetto, E., Determining 2-Dimensional cracks in 3dimensional bodies: uniqueness and stability, Indiana Univ. Math. J., 46(1997), 1-82. [2] Br¨ uhl, M., Hanke, M. and Pidcock, M., Crack detection using electrostatic measurements, Mathematical Modelling and Numerical Analysis, 35(2001), 595-605.
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[3] Cheng, J., Liu, J. J. and Nakamura, G., Recovery of the shape of an obstacle and the boundary impedance from the far-field pattern, J. Math. Kyoto Univ., 43(2003), no. 1, 165-186. [4] Cheng, J., Liu, J. J. and Nakamura, G., The numerical realization of the probe method for the inverse scattering problems from the near field data, Inverse Problems, 21(2005), 839-855. [5] Daido, Y., Ikehata, M. and Nakamura, G., Reconstruction of inclusions for the inverse boundary value problem with mixed type boundary condition, Appl. Anal., 83(2004), 109-124. [6] Eller, M., Identification of cracks in three-dimensional bodies by boundary measurements, Inverse Problems, 12(1996), 395-408. [7] Erhard, K. and Potthast, R., A numerical study of the probe method, submitted. [8] Friedman, A. and Vogelius, M., Determining cracks by boundary measurements, Indiana Univ. Math. J., 38(1989), 527-556. [9] Grisvard, P., Elliptic problems in nonsmooth domains, Pitman, Boston, 1985. [10] Ikehata, M., Reconstruction of the shape of the inclusion by boundary measurements, Comm. PDE., 23(1998), 1459-1474. [11] Ikehata, M., Reconstruction of an obstacle from the scattering amplitude at a fixed frequency, Inverse Problems, 14(1998), 949-954. [12] Ikehata, M., Reconstruction of obstacle from boundary measurements, Wave Motion, 30(1999), 205-223. [13] Ikehata, M., The probe method and its applications, Inverse problems and related topics, Nakamura, G., Saitoh, S., Seo, J. K. and Yamamoto, M. editors, CRC Press UK, 2000, 57-68. [14] Ikehata, M., Reconstruction of inclusion from boundary measurements, J. Inv. IllPosed Problems, 10(2002), 37-65. [15] Ikehata, M., A new formulation of the probe method and related problems, Inverse Problems, 21(2005), 413-426. [16] Ikehata, M., Characterization of the shape of an obstacle with impedance boundary condition using the Dirichlet-to-Neumann map, preprint, December, 2004. [17] Ikehata, M., Inverse crack problem and probe method, submitted, 2005. [18] Ikehata, M. and Nakamura, G., Reconstruction formula for identifying cracks, J. Elasticity, 70(2003), 59-72. [19] Nakamura, G., Uhlmann, G. and Wang, J. N., Reconstruction of cracks in an inhomogeneous anisotropic elastic medium, J. Math. Pures Appl., 82(2003), 1251-1276.
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e-mail address
[email protected]
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