- Research Article
- Open access
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The Inverse Problem for Elliptic Equations from Dirichlet to Neumann Map in Multiply Connected Domains
Boundary Value Problems volume 2009, Article number: 305291 (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 [1–9], 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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ1_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ2_HTML.gif)
we can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ4_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ6_HTML.gif)
It is clear that the equivalent boundary value problem is to find a solution of the complex equation (1.4) with the boundary conditions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ7_HTML.gif)
and the relation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ8_HTML.gif)
in which and
It is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ9_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ10_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ11_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ12_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ13_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ14_HTML.gif)
in which . Suppose that
in
. Obviously
in
. We consider the complex equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ15_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ16_HTML.gif)
For simplicity we can only consider the following integral equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ17_HTML.gif)
later on.
Lemma 2.1.
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ18_HTML.gif)
Proof.
It suffices to prove that for any small positive number , there exists a sufficiently large positive number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ19_HTML.gif)
In fact, noting that , and using the Hölder inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ20_HTML.gif)
where . Now we estimate the integral
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ21_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ22_HTML.gif)
where . Moreover noting that
, if
, and then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ23_HTML.gif)
Thus we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ24_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ25_HTML.gif)
in which is a positive constant.
Proof.
First of all, we verify that any solution of (2.2) satisfies the boundedness estimate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ26_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ27_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ28_HTML.gif)
Noting that we can derive the estimate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ29_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ30_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ31_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ32_HTML.gif)
where is a unique solution of the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ33_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ34_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ35_HTML.gif)
where in
.
Proof.
On the basis of the results as in [10, Section 5, Chapter III], we know that the integral equations
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ36_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ37_HTML.gif)
with the boundary condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ38_HTML.gif)
one can obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ39_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ40_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ41_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ42_HTML.gif)
obviously satisfy the system of first-order complex equations
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ43_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ44_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ45_HTML.gif)
Proof.
In the following we verify the (3.5). From (3.4), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ46_HTML.gif)
In addition, from (3.5) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ47_HTML.gif)
It is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ48_HTML.gif)
satisfy the system of complex equations
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ49_HTML.gif)
with the conditions and
for sufficient large
, and
are the solutions of the complex equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ50_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ51_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ52_HTML.gif)
Proof.
Now we verify that (3.12) and (3.13) are true. Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ53_HTML.gif)
we see that satisfy the system of first-order complex equations
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ54_HTML.gif)
such that and
as
Thus we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ55_HTML.gif)
In addition, from (3.12) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ56_HTML.gif)
It is obvious that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ57_HTML.gif)
satisfy the system of complex equations
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ58_HTML.gif)
with the conditions and
for sufficient large
, and
are the solutions of the complex equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ59_HTML.gif)
From (3.6) and Lemma 3.3 below, the functions on
can be obtained, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ60_HTML.gif)
Here we use the Green formula
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ61_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ62_HTML.gif)
We first prove one lemma (see [7]).
Lemma 3.4.
The function is a solution of the integral equations
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ63_HTML.gif)
if and only if it is a solution of the integral equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ64_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ65_HTML.gif)
(see [10, Chapters I and III]), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ66_HTML.gif)
where on
. Moreover by using the Plemelj-Sokhotzki formula for Cauchy type integral (see [12, 13])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ67_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ68_HTML.gif)
hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ69_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ70_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ71_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ72_HTML.gif)
On the basis of the result in [10], we can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ73_HTML.gif)
in which and
are positive constants only dependent on
and
. Similarly, we can obtain the second estimate in (3.32).
In addition, for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ74_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ75_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ76_HTML.gif)
in which .
Proof.
From the formula (3.4), we can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ77_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ78_HTML.gif)
for the complex equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ79_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ80_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ81_HTML.gif)
in which are real functions of
, and
is a positive constant. Moreover define
in
. Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ82_HTML.gif)
and we can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ83_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ84_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ85_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ86_HTML.gif)
Proof.
It is sufficient to prove the equality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ87_HTML.gif)
in which and
with the condition
on
. We first consider the complex form of the Green formula about
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ88_HTML.gif)
with .
If are the above functions, by using the Green formula, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ89_HTML.gif)
thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ90_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ91_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ92_HTML.gif)
From (4.11), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ93_HTML.gif)
If we define then
, and denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ94_HTML.gif)
one gets
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ95_HTML.gif)
Setting that obviously
is a real function, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ96_HTML.gif)
where on
is derived from Theorem 2.5.
Finally we use the Carleman estimate for and (4.16), and can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ97_HTML.gif)
Taking into account , and choosing
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ98_HTML.gif)
it is easy to see that in
, and then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ99_HTML.gif)
Consequently
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F305291/MediaObjects/13661_2008_Article_838_Equ100_HTML.gif)
this shows the coefficients of equations in (4.1) in
.
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Acknowledgment
The research was supported by the National Natural Science Foundation of China (10671207).
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Wen, G., Xu, Z. & Yang, F. The Inverse Problem for Elliptic Equations from Dirichlet to Neumann Map in Multiply Connected Domains. Bound Value Probl 2009, 305291 (2009). https://doi.org/10.1155/2009/305291
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DOI: https://doi.org/10.1155/2009/305291