The Inverse Problem for Elliptic Equations from Dirichlet to Neumann Map in Multiply Connected Domains
© The Author(s) 2009
Received: 10 July 2008
Accepted: 5 January 2009
Published: 15 January 2009
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.
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.
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 .
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.
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
This shows that the formula (2.6) is true.
in which is a positive constant.
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.
we can choose subsequences and , where as , such that in uniformly converges to as , which is a solution of (2.20) in (see ). The uniqueness of solutions of (2.20) can be seen from the proof of Lemma 2.4 below.
where in .
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 .
which is a known function.
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
where . In the following we will find two solutions and of complex equation with the conditions and as .
such that and as According to the way in , we can obtain the following two lemmas.
Later on we will verify .
Similarly to the way from (3.2) to (3.6), we can obtain the following result.
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 .
We first prove one lemma (see ).
which is the formula (3.25).
and the formula (3.24) is true.
Proof of Lemma 3.3.
which are analytic in with the boundary values on respectively, and satisfy the complex equation (3.1).
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.
in which and are positive constants only dependent on and . Similarly, we can obtain the second estimate in (3.32).
in which . It is not difficult to see that , where is a non-negative constant.
in which .
where as , hence the the formula (3.37) is true.
For the inverse problem of Problem for (1.3) with Condition , one can reconstruct the coefficients and .
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
where is a circular domain, and .
where is a known complex function. In the following we will prove the uniqueness theorem as follows.
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 ).
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 .
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.
where on is derived from Theorem 2.5.
this shows the coefficients of equations in (4.1) in .
The research was supported by the National Natural Science Foundation of China (10671207).
- Kirsch A: An Introduction to the Mathematical Theory of Inverse Problems, Applied Mathematical Sciences. Volume 120. Springer, New York, NY, USA; 1996:x+282.View ArticleGoogle Scholar
- Isakov V: Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences. Volume 127. 2nd edition. Springer, New York, NY, USA; 2006:xiv+344.Google Scholar
- 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.1994.1136MATHMathSciNetView ArticleGoogle Scholar
- 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.1994.1145MATHMathSciNetView ArticleGoogle Scholar
- 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.1994.1155MATHMathSciNetView ArticleGoogle Scholar
- 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. 10.1080/03605309708821292MATHMathSciNetView ArticleGoogle Scholar
- 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. 10.1088/0266-5611/16/3/101MATHMathSciNetView ArticleGoogle Scholar
- Tamasan A: On the scattering method for the -equation and reconstruction of convection coefficients. Inverse Problems 2004, 20(6):1807–1817. 10.1088/0266-5611/20/6/007MATHMathSciNetView ArticleGoogle Scholar
- 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.Google Scholar
- Vekua IN: Generalized Analytic Functions. Pergamon Press, Oxford, UK; 1962.MATHGoogle Scholar
- 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.Google Scholar
- Mushelishvili NI: Singular Integral Equation. P. Noordhoff, Groningen, The Netherlands; 1953.Google Scholar
- Wen G: Conformal Mappings and Boundary Value Problems, Translations of Mathematical Monographs. Volume 106. American Mathematical Society, Providence, RI, USA; 1992:viii+303.Google Scholar
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