The Problem of Scattering by a Mixture of Cracks and Obstacles
© Guozheng Yan. 2009
Received: 8 September 2009
Accepted: 2 November 2009
Published: 30 November 2009
Consider the scattering of an electromagnetic time-harmonic plane wave by an infinite cylinder having an open crack and a bounded domain in as cross section. We assume that the crack is divided into two parts, and one of the two parts is (possibly) coated on one side by a material with surface impedance . Different boundary conditions are given on and . Applying potential theory, the problem can be reformulated as a boundary integral system. We obtain the existence and uniqueness of a solution to the system by using Fredholm theory.
Crack detection is a problem in nondestructive testing of materials which has been often addressed in literature and more recently in the context of inverse problems. Early works on the direct and inverse scattering problem for cracks date back to 1995 in  by Kress. In that paper, Kress considered the direct and inverse scattering problem for a perfectly conducting crack and used Newton's method to reconstruct the shape of the crack from a knowledge of the far-field pattern. In 1997, M nch considered the same scattering problem for sound-hard crack , and in the same year, Alves and Ha Duong discussed the scattering problem but for flat cracks in . Later in 2000, Kress's work was continued by Kirsch and Ritter in  who used the factorization method to reconstruct the shape of the crack from the knowledge of the far-field pattern. In 2003, Cakoni and Colton in  considered the direct and inverse scattering problem for cracks which (possibly) coated on one side by a material with surface impedance . Later in 2008, Lee considered an inverse scattering problem from an impedance crack and tried to recover impedance function from the far field pattern in . However, studying an inverse problem always requires a solid knowledge of the corresponding direct problem. Therefore, in the following we just consider the direct scattering problem for a mixture of a crack and a bounded domain , and the corresponding inverse scattering problem can be considered by similar methods in [1, 2, 4–12] and the reference therein.
Briefly speaking, in this paper we consider the scattering of an electromagnetic time-harmonic plane wave by an infinite cylinder having an open crack and a bounded domain in as cross section. We assume that the cylinder is (possibly) partially coated on one side by a material with surface impedance . This corresponds to the situation when the boundary or more generally a portion of the boundary is coated with an unknown material in order to avoid detection. Assuming that the electric field is polarized in the mode, this leads to a mixed boundary value problem for the Helmholtz equation defined in the exterior of a mixture in .
Our aim is to establish the existence and uniqueness of a solution to this direct scattering problem. As is known, the method of boundary integral equations has widely applications to various direct and inverse scattering problems (see [13–17] and the reference therein). A few authors have applied such method to study the scattering problem with mixture of cracks and obstacles. In the following, we will use the method of boundary integral equations and Fredholm theory to obtain the existence and uniqueness of a solution. The difficult thing is to prove the corresponding boundary integral operator which is a Fredholm operator with index zero since the boundary is a mixture and we have complicated boundary conditions.
The outline of the paper is as follows. In Section 2, the direct scattering problem is considered, and we will establish uniqueness to the problem and reformulate the problem as a boundary integral system by using single- and double-layer potentials. The existence and uniqueness of a solution to the corresponding boundary integral system will be given in Section 3. The potential theory and Fredholm theory will be used to prove our main results.
2. Boundary Integral Equations of the Direct Scattering Problem
Consider the scattering of time-harmonic electromagnetic plane waves from an infinite cylinder with a mixture of an open crack and a bounded domain in as cross section. For further considerations, we suppose that has smooth boundary (e.g., ), and the crack (smooth) can be extended to an arbitrary smooth, simply connected, closed curve enclosing a bounded domain such that the normal vector on coincides with the outward normal vector on which we again denote by . The bounded domain is located inside the domain , and .
where is an injective piecewise function. We denote the outside of with respect to the chosen orientation by and the inside by . Here we suppose that the is divided into two parts and and consider the electromagnetic field E-polarized. Different boundary conditions on , and lead to the following problem:
The problems (2.5) and (2.3) have at most one solution.
Suppose that (with boundary ) is a sufficiently large ball which contains the domain . Obviously, to the Helmholtz equation in (2.5), the solution satisfies the following transmission boundary conditions on the complementary part of :
So, from [13, Theorem ] and a unique continuation argument we obtain that in .
The proof of this lemma can be found in .
From (2.13)–(2.18), we have
By using the boundary conditions in (2.5), we rewrite (2.21) as
Furthermore, we modify (2.24) as
Like previous, define
Then we can rewrite (2.29) as
Similar to (2.26), we modify (2.31) as
and we have mapping properties:
Combining (2.13) and (2.14),
Using (2.17) and (2.18),
Then using (2.36),
From (2.29), we have
and using the notation in previous, we can rewrite (2.40) as
The last term in (2.46) can be reformulated as
Since and in (2.47), we have the following result (see ).
and then we can rewrite (2.49) as
Combining (2.52), (2.26), (2.33), and (2.44), we have
If we define
then (2.54) can be rewritten as a boundary integral system:
3. Existence and Uniqueness
Based on the Fredholm theory, we show the existence and uniqueness of a solution to the integral system (2.56).
and its dual space
For convenience, in the following discussion we define
The uniqueness result Theorem 2.1 in Section 2 implies that
So we complete the proof of the theorem.
Combining Theorems 3.1 and 3.2, we have the following
The boundary integral system (2.56) has a unique solution.
If we remove the condition that " is not Dirichlet eigenvalue of the Laplace operator in ," instead of it by the assumption that Im , then Theorem 2.1 in Section 2 and Theorem 3.3 in Section 3 are also true.
This research is supported by NSFC Grant no. 10871080, Laboratory of Nonlinear Analysis of CCNU, COCDM of CCNU.
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