Integrating the probe and singular sources methods: III. Mixed obstacle case
-
Masaru Ikehata
Abstract
The main purpose of this paper is to develop further the integrated theory of the probe and singular sources methods (IPS) which may work for a group of inverse obstacle problems. Here as a representative and typical member of the group, an inverse obstacle problem governed by the Helmholtz equation with a fixed wave number in a bounded domain is considered. It is assumed that the solutions of the Helmholtz equation outside the set of unknown obstacles satisfy the homogeneous Dirichlet or Neumann boundary conditions on each surface of obstacles. This is the case when two extreme types of obstacles are embedded in a medium. By considering this case, not only a concise technique for IPS is introduced but also a general correspondence principle from IPS to the probe method is suggested. Besides, as a corollary it is shown that the probe method together with the singular sources method reformulated in terms of the probe method has the Side B under a smallness conditions on the wave number k, which is the blowing up property of a sequence computed from the associated Dirichlet-to-Neumann map.
1 Introduction
Both the probe method of Ikehata [8, 7] (later reformulated in [11]) and singular sources method of Potthast [20, 19] now become well-known classical analytical methods for reconstruction issue of inverse obstacle problems governed by partial differential equations. This paper is concerned with the integrated theory of the probe and singular sources methods (IPS), which is initiated by the author himself in [16, 17]. In particular, we focus on the role of IPS in deriving the probe and singular sources methods together with introducing a technique to treat some kind of inverse obstacle problems governed by partial differential equations. For the purpose we consider a prototype inverse obstacle problem governed by the Helmholtz equation with a fixed wave number.
Now let us formulate the prototype problem.
Let Ω be a bounded domain of
Let
This means that,
Then the bounded linear functional
where
In this paper, by considering the prototype inverse obstacle problem mentioned below, we further develop a technique to the integrated theory of the probe and singular sources methods.
Problem.
Extract information about the geometry of
For the problem to have meaning we impose a restriction on k:
Assumption 1.
The boundary value problem (1.1) with
Under Assumption 1 it is well known that the weak solution u of (1.1) exists and unique. Then the map
is well defined by
This is called the Dirichlet-to-Neumann map.
So Problem becomes the extraction problem of information about the geometry
of
It follows from the definition we have the symmetry:
for all
Besides, if
In this paper, we always consider k such that Assumption 1 is satisfied and, unless otherwise stated the
In [16] by considering the case that
It should be pointed out that, using the original probe method [9], this type of problem itself has been considered by Cheng, Liu, Nakamura and Wang [2]. However, they do not have the view point of the IPS developed in this paper. See also Remark 4 for more detailed comparison. Note that, for mixed obstacles placed in the whole space there are some applications of the factorization method [5, 18, 3], monotonicity method [1] and both [4].
1.1 The IPS function
The IPS in this paper starts with introducing a family of singular solutions for the back ground medium.
Let
where
and
Note that
Since the imaginary part of
as the distribution of
Definition 1.
Given
We call the function
Hereafter we simply write
Since the system (1.4) is linear, we have the natural and trivial decomposition of the solution as
where
and
The unique solvability of the boundary value problems (1.6) and (1.7) are also a consequence of Assumption 1.
Needless to say, from (1.5) we have
We call this the outer decomposition or natural decomposition of IPS function.
The IPS for Problem is based on the discovery of the following two representation formulae.
Theorem 1.
Let
We have the expression focused on the Neumann obstacle
(1.9)
where
(1.10)We have the expression focused on the Dirichlet obstacle
(1.11)
where
(1.12)
Introducing the functions
It shoulder be pointed out that the two expressions (1.9) and (1.11) contain the common terms of integrals
Except for those, the expression of the right-hand side on (1.11) coincides with the one on (1.9) multiplied by
The following corollary is a direct consequence of the facts listed below:
the property that for any finite cone V with vertex at the origin
Corollary 1.
The IPS function
For each
Proof.
In what follows we denote by
Here, by Lemma 2 in Appendix we have, as
Then (c) together with Fatou’s lemma yields (i).
Next consider the case when
Again, Lemma 2 yields, as
This yields the validity of (ii). The validity of statement (iii) is almost clear. ∎
Therefore, using the asymptotic behavior of IPS function
Remark 1.
In this paper, about the choice of the family
Under the assumption that
Theorem 1 yields the following corollary.
Corollary 2.
Choose
We have the expression focused on the Neumann obstacle
where
We have the expression focused on the Dirichlet obstacle
where
Remark 2.
In particular, if
Note that the integrand of this right-hand side is absolutely integrable. Thus, the formulae in Corollary 2 become
and
And also we have:
Corollary 3.
Choose
We have the expression focused on the Neumann obstacle
where
We have the expression focused on the Dirichlet obstacle
where
1.2 IPS function to DN map
In this section, we assume that Assumption 1 for the case
plays the central role, where v is an arbitrary solution of the Helmholtz equation
The idea of the method of complementing function mentioned above suggests us the decomposition formulae for the form (1.15) stated below.
Theorem 2.
Let
(1.16)
and
(1.17)
where w,
The formulae (1.16) and (1.17) are new and useful for establishing the probe method for Problem.
Those should be considered as the generalization of the well known decomposition formula in the case when
It should be emphasized that the proof of Theorem 2 given in Section 3.2 is independent of IPS. Besides, the decomposition formulae (1.16) and (1.17) themselves would be valid also in the context of the weak solution. However, formulae (1.16) and (1.17) should be considered as a byproduct of introducing the IPS function at first. Without seeking the energy decomposition of IPS as done in [16] one could never find the idea of the method of complementing functions to form (1.15).
Organization of the paper.
In Section 2 the proof of Theorem 1 is given.
The proof is based on integration by parts and clarifies the meaning of introducing the complementary
functions
In Appendix we describe two lemmas which yield the upper estimate of the
2 Proof of Theorem 1
First we show that the
Lemma 1.
It holds that
(2.1)
and
(2.2)
Proof.
We start with the standard expression
(2.3)
Applying the boundary conditions on
(2.4)
Since x and y outside both
Thus (2.4) becomes
(2.5)
Besides, we have
(2.6)
and
From (2.6) one has
From (2.7) one has
Thus one obtains the two representation of a single integral as follows.
From (2.8) we have
(2.10)
From (2.9) we have
(2.11)
Substituting (2.10) and (2.11) into (2.5), we obtain (2.1) and (2.2). ∎
Next we show that the
Lemma 2.
We have
(2.12)
Proof.
To explain the reason for the introduction of the function
First same as (2.3) we start with the standard expression
Here we impose the boundary conditions
Then (2.13) becomes
(2.15)
Here we have
Thus (2.15) becomes
(2.16)
Here using the boundary condition of
Besides we have
That is,
Note that we made use of the first boundary condition on
Therefore (2.16) becomes
Here we impose the boundary condition of
Note that the set of boundary conditions (2.14) and (2.19) coincides with that of (1.7).
From (2.1), (2.2) and (2.12) we immediately obtain the following two expressions for
Proposition 1.
It holds that
(2.20)
and
(2.21)
Now let us explain the role of introducing the solutions of (1.10) and (1.12). It is concerned with the last terms on (2.20) and (2.21). We call this technique the method of complementing function.
Lemma 3.
We have
(2.22)
and
(2.23)
Proof.
First we rewrite the integral
Using the equation (1.10), we have
Interchanging x and y, we obtain the expression (2.22).
Second we rewrite the integral
Using the equation (1.12), we have
Interchanging x and y, we obtain the expression (2.23). ∎
Thus (2.20) together with (2.22) yields
(2.24)
And (2.21) together with (2.23) yields
(2.25)
Letting
(2.26)
and
(2.27)
Here rewrite
and
Rewrite also as
and
Then from (2.26), (2.27), (2.28), (2.29), (2.30) and (2.31) we obtain (1.9) and (1.11) of Theorem 1.
Remark 1.
Assume that
It is easy to see that equation (2.32) combined with (i) of Corollary 2 makes the representation of IPS function so simple:
This is an extension of the expression of IPS function given in [16, Remark 1.7] to the case
3 Integrated theory
3.1 IPS to Side A of probe method
In this subsection we derive the probe method via the integrated theory of the probe and singular sources methods.
We fix an arbitrary
First we recall the notion of a needle. Given
Definition 1.
Given
is called the indicator sequence for the probe method.
Note that, hereafter, unless otherwise specified we assume that Assumption 1 for the case
The Side A of the probe method starts with the convergence property of the indicator sequence as described below.
Theorem 1.
Let
where
Proof.
Fix
Define
The form of
That is,
(3.5)
By Definition 1 and the assumption
Since
Therefore the second term of the right-hand side on (3.6) is also convergent and its limit satisfies
Using this and the trivial expression
we obtain
This is nothing but (3.3).
Next we show that the right-hand side of the formula (3.3) coincides with that of formula (3.2).
Recalling (2.12) of Lemma 2 with
Besides from equation (1.7) we have
(3.8)
From (3.7) and (3.8) together with (1.8) we see that
(3.9)
A combination of (3.3) and (3.9) yields the desired conclusion. ∎
Definition 2.
The function
The formula (3.1) should be understood as a computation formula of the indicator function by using
The second and third terms of the right-hand side on (3.2)
are bounded when x is away from
Then it follows from Theorem 1, (3.2) and (3.8) that the
(3.10)
and
(3.11)
Then we can easily check that, for each
So the convergence of the indicator sequence (3.1), blowing up property of the
The point that should be emphasized
is: from IPS we obtained that
Remark 1.
If
Definition 3.
For an arbitrary point
In principle, it is possible to calculate
In particular, if
Finally, by (3.2) and (3.13) we have the inner decomposition of IPS function:
The equations (1.8) and (3.14) give us two ways of decomposition of IPS function.
3.2 Proof of Theorem 2 and a corollary
Proof of Theorem 2.
First we prove the validity of (1.16). We have
(3.15)
Besides we have
(3.16)
Here we have
Now from (3.15), (3.16) and (3.17) we obtain (1.16).
Next we have
(3.18)
Note that once we have found the equation to prove, the proof is just a calculation.
The point of Theorem 1 is the introduction of complementing functions
has two energy integral expressions given by (3.15) and (3.18).
As a direct corollary, we obtain:
Corollary 1.
Let
(3.19)
and
(3.20)
where
It should be emphasized that the point is the idea or the principle of the derivation of the things to be proved, like (1.16) and (1.17) or, (3.19) and (3.20). It is not a trivial fact as we have already seen. It is based on the correspondence principle mentioned below.
Principle.
Replace the singular solution
Note that conversely (3.19) and (3.20) yield immediately (3.10) and (3.11) by taking the limit and the formula (3.1), respectively.
3.3 Side B of probe method
The Side B of the probe method is concerned with the blowing up property of the indicator sequence. It is based on Theorem 2 or Corollary 1 and the blowing up property of the needle sequence stated below.
Proposition 1.
Given an arbitrary point
Let V be an arbitrary finite cone with vertex at x . Then we have
Let
This fact has been already established in [11].
Once we have (3.19), (3.20) and Proposition 1, one gets the following theorem which states the Side B of the probe method.
Theorem 2.
Let
Then for any needle sequence
Proof.
We describe only the case when
where C is a positive constant. Here, by Proposition 1 under (a) and (b) above we have
Remark 3.
Note that Theorem 2 does not cover all the possible cases for
The problem is the case when
Here, applying the idea described therein to formulae (3.19) and (3.20), we show a result.
We assume that
where
The assumption on k is as follows: k satisfies all the inequalities listed below:
Here the constants
Constant
Constant
Constant
Theorem 3.
All the statements of Theorem 2 for
Proof.
Consider the case σ satisfy
Thus (3.19) yields
where
By the convergence property of
here
The treatment of the case when
Remark 4.
It should be noted that in [2] they considered the probe method [9, 7] for the Helmholtz equation
3.4 Singular sources method included in IPS
The singular sources method consists of three parts listed below.
Given
(3.25)It holds that:
For each
(3.26)
The statements (b) and (c) are the direct consequence of Corollary 1, outer decomposition (1.8), (1.3) and (1.7).
Note that Remark 1 works also for (3.26) in the case when
Remark 5.
If
provided
So this is the completely integrated version of the probe and singular sources methods.
By Theorem 2 this version also has the Side B.
To distinguish from other cases we denote
3.5 Side B of singular sources method
Given
in
It is clear that the
satisfies the Helmholtz equation in Ω
and that the sequence
in
Here note that we have
Therefore we obtain
Besides, as a corollary of Theorem 2 we obtain
Corollary 2.
Let
And also as a corollary of Theorem 3 we have:
Corollary 3.
Let
Let
The function
Now let
where
Here we have the trivial decomposition
This together with the symmetry of Dirichlet-to-Neumann maps
(3.27)
Therefore, using Theorem 2 for the choice
Corollary 4.
Let
And also from Theorem 3 and Corollary 3 we have:
Corollary 5.
Let
Corollaries 4 and 5 could never be obtained using a single methodology, and show us the greatest advantage of the integrated theory.
Remark 6.
It follows from (3.27) and the existence of the needle sequence which is a consequence of the Runge approximation property for the Helmholtz equation in Ω we have the expression
where the
3.6 Additional remarks
3.6.1 Lifting
First for general
So define
Then (3.28) yields the symmetry
Besides, by (3.1) and (3.3), we have another expression for the indicator function:
This together with (3.29) yields
The inner decomposition (3.14) itself has the lifted version.
For general
where
A similar computation to (3.8) yields
Thus we have
By (3.13), this yields
Next rewrite (3.29) as
This together with (3.30) yields
This is the lifted version of inner decomposition (3.14). Note also that we have twisted decomposition
3.6.2 Uniqueness
We consider only two cases:
Note that
The expression (3.34) together with symmetry of
for each fixed
for each fixed
Here the symbols
By the unique continuation property of the Helmholtz equation, one concludes:
indicator function
where
Summing up, we have obtained the following uniqueness theorem by using needles localized, say in a small neighborhood of
Proposition 2.
Let
The key of the proof is to put the calculation process of the lifting
3.6.3 Symmetry of
I
1
(
x
,
y
)
and implications
For general
Note that for general
For general
Note that all the integrands are absolutely integrable since
Recalling the expression (1.2), we see that
Therefore from (3.35) we obtain, for
And this together with (3.33) yields twisted symmetry:
Besides, the expression (3.31) together with symmetry yields:
for each fixed
for each fixed
Besides, using (3.33) we conclude that, for each fixed
and for each fixed
Finally, from (3.36) we obtain, for each fixed
As a conclusion, we have:
Proposition 3.
Let
4 Conclusion and remarks
It became clear that the IPS function plays the central role
in deriving the probe and singular sources methods. Besides, the method of complementing function, which is introduced in the proof of Theorem 1,
makes everything so clear.
Everything about the both methods
can be derived from the knowledge of the IPS function
In this paper, we have considered only the case when the governing equation is given by the Helmholtz equation. However, the method developed here can be applied also to the same type of inverse obstacle problems governed by various partial differential equations, for example, the Navier equation, the Stokes system, the biharmonic equation, and so on. And also it would be interested to consider their time domain versions by the spirit of IPS. Those belong to our next project.
Our theory yields also an alternative simple proof of a result on the probe method described in [2], which is
nothing but the Side A called in this paper.
However, it should be pointed out that
the Side B without smallness of k is still open at the present time even for the case
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: 24K06812
Funding statement: The author was supported by Grant-in-Aid for Scientific Research (C) (No. 24K06812) of Japan Society for the Promotion of Science.
A Appendix
Note that in this appendix it is assumed that
Lemma 1.
Let
where C is positive constant independent of v.
Proof.
Set
Decompose w as
and thus
Considering
So the problem is to show that
Here we employ a slightly modified argument for the proof of [10, (4.12) in Lemma 4.1]. Solve
Then we have
Thus one gets
By elliptic regularity up to boundary, we have
and the trace theorem yields
Combining these, we obtain (A.1). ∎
Lemma 2.
The solution
Proof.
There are two ways to validate the statement.
The first one is a combination of Lemma 1 and a limiting argument based on the Runge approximation property for the Helmholtz equation
in the whole domain Ω provided
and thus
Considering
where C is a positive constant independent of
Note that we are considering general
Then assumption (1.3) and
we obtain
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- Blow-up of solutions to some classes of ill-posed operator equations and a growth quantification result
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Articles in the same Issue
- Frontmatter
- An inverse problem for non-selfadjoint Dirac operator with transmission conditions at finite interior points
- Inverse Sturm–Liouville problem with polynomials in the boundary condition and multiple eigenvalues
- Blow-up of solutions to some classes of ill-posed operator equations and a growth quantification result
- On an inverse problem with high-order overdetermination conditions
- Two regularization methods for identifying the source term of Caputo–Hadamard type time fractional diffusion-wave equation
- Integrating the probe and singular sources methods: III. Mixed obstacle case
- Stability estimate and the Tikhonov regularization method for the Kuramoto–Sivashinsky equation backward in time
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