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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:
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
we can get
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
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:
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
It is clear that the equivalent boundary value problem is to find a solution 
of the complex equation (1.4) with the boundary conditions
and the relation
in which 
and 
It is easy to see that
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
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:
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:
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
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
in which 
. Suppose that 
in 
. Obviously 
in 
. We consider the complex equation
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:
For simplicity we can only consider the following integral equation
later on.
Lemma 2.1.
If 
, then
Proof.
It suffices to prove that for any small positive number 
, there exists a sufficiently large positive number 
such that
In fact, noting that 
, and using the Hölder inequality, we have
where 
. Now we estimate the integral
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
where 
. Moreover noting that 
, if 
, and then
Thus we obtain
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
in which 
is a positive constant.
Proof.
First of all, we verify that any solution 
of (2.2) satisfies the boundedness estimate
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
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
Noting that 
we can derive the estimate
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
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
where 
is a unique solution of the equation
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,
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
where 
in 
.
Proof.
On the basis of the results as in [10, Section 5, Chapter III], we know that the integral equations
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
with the boundary condition
one can obtain
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
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
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
obviously 
satisfy the system of first-order complex equations
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:
where
Proof.
In the following we verify the (3.5). From (3.4), we have
In addition, from (3.5) it follows that
It is easy to see that
satisfy the system of complex equations
with the conditions 
and 
for sufficient large 
, and 
are the solutions of the complex equation
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:
where
Proof.
Now we verify that (3.12) and (3.13) are true. Denote
we see that 
satisfy the system of first-order complex equations
such that 
and 
as 
Thus we have
In addition, from (3.12) it follows that
It is obvious that
satisfy the system of complex equations
with the conditions 
and 
for sufficient large 
, and 
are the solutions of the complex equation
From (3.6) and Lemma 3.3 below, the functions 
on 
can be obtained, then
Here we use the Green formula
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
We first prove one lemma (see [7]).
Lemma 3.4.
The function 
is a solution of the integral equations
if and only if it is a solution of the integral equation
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
(see [10, Chapters I and III]), we have
where 
on 
. Moreover by using the Plemelj-Sokhotzki formula for Cauchy type integral (see [12, 13])
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
hence
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
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
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
On the basis of the result in [10], we can get
in which 
and 
are positive constants only dependent on 
and 
. Similarly, we can obtain the second estimate in (3.32).
In addition, for
we have
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
in which 
.
Proof.
From the formula (3.4), we can get
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
for the complex equation
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
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
in which 
are real functions of 
, and 
is a positive constant. Moreover define 
in 
. Denote
and we can get
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
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
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
Proof.
It is sufficient to prove the equality
in which 
and 
with the condition 
on 
. We first consider the complex form of the Green formula about 
with 
.
If 
are the above functions, by using the Green formula, we have
thus
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
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
From (4.11), we have
If we define 
then 
, and denote
one gets
Setting that 
obviously 
is a real function, and
where 
on 
is derived from Theorem 2.5.
Finally we use the Carleman estimate for 
and (4.16), and can get
Taking into account 
, and choosing
it is easy to see that 
in 
, and then
Consequently
this shows the coefficients 
of equations in (4.1) in 
.
References
Kirsch A: An Introduction to the Mathematical Theory of Inverse Problems, Applied Mathematical Sciences. Volume 120. Springer, New York, NY, USA; 1996:x+282.
Isakov V: Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences. Volume 127. 2nd edition. Springer, New York, NY, USA; 2006:xiv+344.
Sung L-Y: An inverse scattering transform for the Davey-Stewartson II equations—I. Journal of Mathematical Analysis and Applications 1994, 183(1):121–154. 10.1006/jmaa.1994.1136
Sung L-Y: An inverse scattering transform for the Davey-Stewartson II equations—II. Journal of Mathematical Analysis and Applications 1994, 183(2):289–325. 10.1006/jmaa.1994.1145
Sung L-Y: An inverse scattering transform for the Davey-Stewartson II equations—III. Journal of Mathematical Analysis and Applications 1994, 183(3):477–494. 10.1006/jmaa.1994.1155
Brown RM, Uhlmann GA: Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions. Communications in Partial Differential Equations 1997, 22(5–6):1009–1027. 10.1080/03605309708821292
Cheng J, Yamamoto M: The global uniqueness for determining two convection coefficients from Dirichlet to Neumann map in two dimensions. Inverse Problems 2000, 16(3):L25-L30. 10.1088/0266-5611/16/3/101
Tamasan A: On the scattering method for the -equation and reconstruction of convection coefficients. Inverse Problems 2004, 20(6):1807–1817. 10.1088/0266-5611/20/6/007
Tong ZL, Cheng J, Yamamoto M: A method for constructing the convection coefficients of elliptic equations from Dirichlet to Neumann map. Science in China. Series A 2004, 34: 752–766.
Vekua IN: Generalized Analytic Functions. Pergamon Press, Oxford, UK; 1962.
Wen G, Begehr HGW: Boundary Value Problems for Elliptic Equations and Systems, Pitman Monographs and Surveys in Pure and Applied Mathematics. Volume 46. Longman Scientific & Technical, Harlow, UK; 1990:xii+411.
Mushelishvili NI: Singular Integral Equation. P. Noordhoff, Groningen, The Netherlands; 1953.
Wen G: Conformal Mappings and Boundary Value Problems, Translations of Mathematical Monographs. Volume 106. American Mathematical Society, Providence, RI, USA; 1992:viii+303.
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