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The Inverse Problem for Elliptic Equations from Dirichlet to Neumann Map in Multiply Connected Domains

Boundary Value Problems20092009:305291

DOI: 10.1155/2009/305291

Accepted: 5 January 2009

Published: 15 January 2009

Abstract

The present paper deals with the inverse problem for linear elliptic equations of second order from Dirichlet to Neumann map in multiply connected domains. Firstly the formulation and the complex form of the problem for the equations are given, and then the existence and global uniqueness of solutions for the above problem are proved by the complex analytic method, where we absorb the advantage of the methods in previous works and give some improvement and development.

1. Formulation of the Inverse Problem for Second-Order Elliptic Equations from Dirichlet to Neumann Map

In [19], the authors posed and discussed the inverse problem of second-order elliptic equations. In this paper, by using the complex analytic method, the corresponding problem for linear elliptic complex equations of first-order in multiply connected domains is firstly discussed, afterwards the existence and global uniqueness of solutions of the inverse problem for the elliptic equations of second-order are obtained.

Let be an -connected domain bounded domain in the complex plane with the boundary , where are inside of . Consider the linear elliptic equation of second-order:
(11)

in which are real functions of and is a positive constant. Moreover let in . The above condition is called Condition . In this paper the notations are the same as those in [10] or [11].

Denote
(12)
we can get
(13)
where . We choose a conformal mapping from the above general domain onto the circular domain with the boundary , , and . In this case, the complex equation (1.3) is reduced to the complex equation
(14)

where is the inverse function of , and in is a known Hölder continuously differentiable function (see [10, Section 2, Chapter I]), hence the above requirement can be realized.

Introduce the Dirichlet boundary condition for (1.1) as follows:
(15)

where is a positive constant, which is called Problem for (1.1) or (1.4). By [10, 11], Problem has a unique solution (or ) satisfying (1.1) (or (1.4)) and the Dirichlet boundary condition (1.5). From this solution, we can define the Dirichlet to Neumann map or by .

Our inverse problem is to determine the coefficient and of (1.1) (or in (1.3)) from the map . In the following, we will transform the Dirichlet to Neumann map into a equivalent boundary condition. In fact, if we find the derivative of positive tangent direction with respect to the unit arc length parameter and of the boundary with , then
(16)
It is clear that the equivalent boundary value problem is to find a solution of the complex equation (1.4) with the boundary conditions
(17)
and the relation
(18)
in which and It is easy to see that
(19)

where is the arc length of and applying the Green formula, we can see that the function determined by the integral in (1.8) in is single-valued.

Under the above condition, the corresponding Neumann boundary condition is
(110)
where is the unit outwards normal vector of . The boundary value problem (1.1) (or (1.4)), (1.10) will be called Problem . Taking into account the partial indexes of and are equal to and and are equal to , thus the index of the above boundary value problem is . In general the above Problem is not solvable, we need to give the modified boundary conditions as follows:
(111)
where and and on on is an undetermined real constant (see [11, Chapter VI]). Hence, the Dirichlet to Neumann map can be transformed into the boundary conditions as follows:
(112)
which will be called Problem for the complex equation (1.4) (or (1.1)) with the relation (1.8), where is a complex function satisfying the condition
(113)

For any function in the Dirichlet boundary condition (1.5), there is a set of the functions of Neumann boundary condition (1.10), where is corresponding to the complex equation (1.4) one by one, namely if we know the boundary value and one complex equation in (1.4), then the boundary value can be determined. Inversely if the in (1.10) is given, then one complex equation in (1.4) can be determined, which will be verified later on. We denote the set of functions by , where is a complex number and is as stated in (1.12).

2. Some Relations of Inverse Problem for Second-Order Elliptic Equations from Dirichlet to Neumann Map

According to [10], introduce the notations
(21)
in which . Suppose that in . Obviously in . We consider the complex equation
(22)
where and is a complex number. On the basis of the Pompeiu formula (see [10, Chapters I and III]), the corresponding integral equation of the complex equation (2.2) is as follows:
(23)
For simplicity we can only consider the following integral equation
(24)

later on.

Lemma 2.1.

If , then
(25)

Proof.

It suffices to prove that for any small positive number , there exists a sufficiently large positive number such that
(26)
In fact, noting that , and using the Hölder inequality, we have
(27)
where . Now we estimate the integral
(28)
We choose two sufficiently small positive constants and , and divide the domain into three parts: , and , such that for the above positive number , we can get
(29)
where . Moreover noting that , if , and then
(210)
Thus we obtain
(211)

This shows that the formula (2.6) is true.

Lemma 2.2.

If , where is a positive constant, then the solution of (2.2) satisfies the estimate
(212)

in which is a positive constant.

Proof.

First of all, we verify that any solution of (2.2) satisfies the boundedness estimate
(213)
where is a positive constant. Suppose that (2.13) is not true, then there exists a sequence of coefficients , which satisfy the same condition of coefficient and weakly converges to , and the corresponding integral equations
(214)
possess the solutions , but are unbounded. Hence we can choose a subsequence of denoted by again, such that as , and can assume . Obviously are solutions of the integral equations
(215)
Noting that we can derive the estimate
(216)
(see [10, 11]), thus
(217)
Hence from , we can choose a subsequence denoted by again, which uniformly converges to in , it is clear that is a solution of the equation
(218)

On the basis of the result in [10, Section 5, Chapter III], the solution in , however, from , there exists a point , such that , which is impossible. This shows that (2.13) and then the estimate (2.12) are true.

Lemma 2.3.

Under the above conditions, one has
(219)
where is a unique solution of the equation
(220)

Proof.

Denote by the solution of (2.2) in . From Lemma 2.2, we know that the solution satisfies the estimate (2.12). Moreover by using (2.5), that is,
(221)

we can choose subsequences and , where as , such that in uniformly converges to as , which is a solution of (2.20) in (see [11]). The uniqueness of solutions of (2.20) can be seen from the proof of Lemma 2.4 below.

Lemma 2.4.

The solution of (2.20) can be expressed as
(222)

where in .

Proof.

On the basis of the results as in [10, Section 5, Chapter III], we know that the integral equations
(223)

have the unique solutions in and respectively, this shows that the function in can be extended in . Moreover by the result in [10, 11], the solution can be expressed as in . Note that as , and the entire function in satisfies the condition as , hence in , and then in .

Theorem 2.5.

For the inverse problem of the equation
(224)
with the boundary condition
(225)
one can obtain
(226)

which is a known function.

Proof.

From the expression (2.22) of the solution in and in , it follows that (2.26) is true.

3. The Inverse Scattering Method for Second-Order Elliptic Equations from Dirichlet to Neumann Map

For the complex equation (1.4), through the transformation , we can obtain that the function satisfies the complex equation
(31)
where and in , in this case every function in is reduced to , hence later on it suffices to discuss the complex equation (3.1) and system of complex equations
(32)

where . In the following we will find two solutions and of complex equation with the conditions and as .

Now we find two solutions and in of (3.1) with the conditions and for sufficiently large . In other words, there exist two solutions and in of (3.2) with the conditions and as . Denote
(33)
obviously satisfy the system of first-order complex equations
(34)

such that and as According to the way in [8], we can obtain the following two lemmas.

Lemma 3.1.

Under the above conditions, there exist two functions satisfying the system of complex equations:
(35)
where
(36)

Proof.

In the following we verify the (3.5). From (3.4), we have
(37)
In addition, from (3.5) it follows that
(38)
It is easy to see that
(39)
satisfy the system of complex equations
(310)
with the conditions and for sufficient large , and are the solutions of the complex equation
(311)

Later on we will verify .

Similarly to the way from (3.2) to (3.6), we can obtain the following result.

Lemma 3.2.

Under the above conditions, there exist two functions satisfying the system of complex equations:
(312)
where
(313)

Proof.

Now we verify that (3.12) and (3.13) are true. Denote
(314)
we see that satisfy the system of first-order complex equations
(315)
such that and as Thus we have
(316)
In addition, from (3.12) it follows that
(317)
It is obvious that
(318)
satisfy the system of complex equations
(319)
with the conditions and for sufficient large , and are the solutions of the complex equation
(320)
From (3.6) and Lemma 3.3 below, the functions on can be obtained, then
(321)
Here we use the Green formula
(322)

and for , and . This shows that the function for is known, and then we can solve the solutions of equations in (3.5). On the basis of Lemma 3.2, we can obtain the system of complex equations in (3.12) and the coefficient of (3.1). This is just the so-called inverse scattering method. We mention that sometimes are written as .

Lemma 3.3.

Under the above conditions, the functions as stated in (1.12) are the solutions of the system of integral equations
(323)

We first prove one lemma (see [7]).

Lemma 3.4.

The function is a solution of the integral equations
(324)
if and only if it is a solution of the integral equation
(325)

Proof.

It is clear that we can only discuss the case of . If is a solution of (3.24), then . On the basis of the Pompeiu formula
(326)
(see [10, Chapters I and III]), we have
(327)
where on . Moreover by using the Plemelj-Sokhotzki formula for Cauchy type integral (see [12, 13])
(328)

which is the formula (3.25).

On the contrary if (3.25) is true, then there exists a solution of equation in with the boundary values , thus we have (3.26), where the integral in is analytic, whose boundary value on is
(329)
hence
(330)

and the formula (3.24) is true.

Proof of Lemma 3.3.

On the basis of the theory of integral equations (see [12, 13]), we can obtain the solutions and of (3.23). From Lemma 3.4, we define the functions
(331)

which are analytic in with the boundary values on respectively, and satisfy the complex equation (3.1).

Moreover according to [6, 7], we can obtain the following two lemmas.

Lemma 3.5.

Under the above conditions, one has
(332)

where , the positive constant is only dependent on and , and is a sufficiently large positive number. Moreover the function in (3.6) satisfies . In particular, where is a non-negative number.

Proof.

From Lemma 3.1, noting that , , we have
(333)
On the basis of the result in [10], we can get
(334)

in which and are positive constants only dependent on and . Similarly, we can obtain the second estimate in (3.32).

(335)
we have
(336)

in which . It is not difficult to see that , where is a non-negative constant.

Lemma 3.6.

Under the above conditions, one can find the coefficients of the complex system of first-order equations in as follows
(337)

in which .

Proof.

From the formula (3.4), we can get
(338)

where as , hence the the formula (3.37) is true.

Theorem 3.7.

For the inverse problem of Problem for (1.3) with Condition , one can reconstruct the coefficients and .

Proof.

Similarly to [9], we will use the generalized Cauchy formula
(339)
for the complex equation
(340)
to find the function in , in which are the standard kernels of equation (3.40) (see [10, Chapter III]). In fact, denote in , and on is known from Theorem 2.5, then according to (3.39), we can find the function in . Moreover from
(341)

thus the coefficient in is obtained.

4. The Global Uniqueness Result for Inverse Problem of First-Order Elliptic Complex Equations from Dirichlet to Neumann Map

For the elliptic equation of second-order
(41)
in which are real functions of , and is a positive constant. Moreover define in . Denote
(42)
and we can get
(43)
where . As stated in Section 1, suppose that the above equations satisfy Condition , and through a conformal mapping , the complex equations in (4.3) can be reduced to the following form
(44)

where is a circular domain, and .

If are the corresponding solutions of (4.4) from the Dirichlet to Neumann maps , and , then the boundary conditions of the inverse boundary value problem for second-order elliptic equations in (4.1) from Dirichlet to Neumann map can be reduced to
(45)

where is a known complex function. In the following we will prove the uniqueness theorem as follows.

Theorem 4.1.

For the inverse problem of Problem for (1.1) (or (1.3)) with Condition , one can uniquely determine the coefficients . In other words, if for (4.1), then .

We first prove the Carleman estimate (see [7]).

Lemma 4.2.

If the complex function with the condition on , and the real function then one has the Carleman estimate
(46)

Proof.

It is sufficient to prove the equality
(47)
in which and with the condition on . We first consider the complex form of the Green formula about
(48)

with .

If are the above functions, by using the Green formula, we have
(49)
thus
(410)

This is just the formula (4.7) for . Due to the density of in , it is known that (4.7) is also true for with the condition on .

Lemma 4.3.

Under the above conditions, one can derive
(411)

Proof.

On the basis of on , and the results of Lemmas 3.1 and 3.2, it follows that the corresponding coefficients , and then in . This shows that the formula (4.11) is true.

Proof of Theorem 4.1.

Similarly to [7], we can prove
(412)
From (4.11), we have
(413)
If we define then , and denote
(414)
one gets
(415)
Setting that obviously is a real function, and
(416)

where on is derived from Theorem 2.5.

Finally we use the Carleman estimate for and (4.16), and can get
(417)
Taking into account , and choosing
(418)
it is easy to see that in , and then
(419)
Consequently
(420)

this shows the coefficients of equations in (4.1) in .

Declarations

Acknowledgment

The research was supported by the National Natural Science Foundation of China (10671207).

Authors’ Affiliations

(1)
School of Mathematical Sciences, Peking University
(2)
School of Information, Renmin University of China

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