The Problem of Scattering by a Mixture of Cracks and Obstacles

Boundary Value Problems20092009:524846

DOI: 10.1155/2009/524846

Received: 8 September 2009

Accepted: 2 November 2009

Published: 30 November 2009

Abstract

Consider the scattering of an electromagnetic time-harmonic plane wave by an infinite cylinder having an open crack http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq1_HTML.gif and a bounded domain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq2_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq3_HTML.gif as cross section. We assume that the crack http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq4_HTML.gif is divided into two parts, and one of the two parts is (possibly) coated on one side by a material with surface impedance http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq5_HTML.gif . Different boundary conditions are given on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq6_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq7_HTML.gif . 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.

1. Introduction

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 [1] 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq8_HTML.gif nch considered the same scattering problem for sound-hard crack [2], and in the same year, Alves and Ha Duong discussed the scattering problem but for flat cracks in [3]. Later in 2000, Kress's work was continued by Kirsch and Ritter in [4] 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 [5] considered the direct and inverse scattering problem for cracks which (possibly) coated on one side by a material with surface impedance http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq9_HTML.gif . 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 [6]. 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq10_HTML.gif and a bounded domain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq11_HTML.gif , and the corresponding inverse scattering problem can be considered by similar methods in [1, 2, 412] 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq12_HTML.gif and a bounded domain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq13_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq14_HTML.gif as cross section. We assume that the cylinder is (possibly) partially coated on one side by a material with surface impedance http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq15_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq16_HTML.gif mode, this leads to a mixed boundary value problem for the Helmholtz equation defined in the exterior of a mixture in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq17_HTML.gif .

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 [1317] 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq18_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq19_HTML.gif and a bounded domain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq20_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq21_HTML.gif as cross section. For further considerations, we suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq22_HTML.gif has smooth boundary http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq23_HTML.gif (e.g., http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq24_HTML.gif ), and the crack http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq25_HTML.gif (smooth) can be extended to an arbitrary smooth, simply connected, closed curve http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq26_HTML.gif enclosing a bounded domain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq27_HTML.gif such that the normal vector http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq28_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq29_HTML.gif coincides with the outward normal vector on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq30_HTML.gif which we again denote by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq31_HTML.gif . The bounded domain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq32_HTML.gif is located inside the domain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq33_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq34_HTML.gif .

In the whole paper, we assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq35_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq36_HTML.gif .

Suppose that

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ1_HTML.gif
(2.1)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq37_HTML.gif is an injective piecewise http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq38_HTML.gif function. We denote the outside of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq39_HTML.gif with respect to the chosen orientation by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq40_HTML.gif and the inside by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq41_HTML.gif . Here we suppose that the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq42_HTML.gif is divided into two parts http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq43_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq44_HTML.gif and consider the electromagnetic field E-polarized. Different boundary conditions on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq45_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq46_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq47_HTML.gif lead to the following problem:

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ2_HTML.gif
(2.2)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq48_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq49_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq50_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq51_HTML.gif . The total field http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq52_HTML.gif is decomposed into the given incident field http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq53_HTML.gif , and the unknown scattered field http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq54_HTML.gif which is required to satisfy the Sommerfeld radiation condition

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ3_HTML.gif
(2.3)

uniformly in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq55_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq56_HTML.gif .

We recall some usual Sobolev spaces and some trace spaces on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq57_HTML.gif in the following.

Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq58_HTML.gif be a piece of the boundary. Use http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq59_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq60_HTML.gif to denote the usual Sobolev spaces, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq61_HTML.gif is the trace space, and we define

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ4_HTML.gif
(2.4)

Just consider the scattered field http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq62_HTML.gif , then (2.2) and (2.3) are a special case of the following problem.

Given http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq63_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq64_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq65_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq66_HTML.gif find http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq67_HTML.gif such that

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ5_HTML.gif
(2.5)

and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq68_HTML.gif is required to satisfy the Sommerfeld radiation condition (2.3). For simplicity, we assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq69_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq70_HTML.gif .

Theorem 2.1.

The problems (2.5) and (2.3) have at most one solution.

Proof.

Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq71_HTML.gif be a solution to the problem (2.5) with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq72_HTML.gif , we want to show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq73_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq74_HTML.gif .

Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq75_HTML.gif (with boundary http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq76_HTML.gif ) is a sufficiently large ball which contains the domain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq77_HTML.gif . Obviously, to the Helmholtz equation in (2.5), the solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq78_HTML.gif satisfies the following transmission boundary conditions on the complementary part http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq79_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq80_HTML.gif :

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ6_HTML.gif
(2.6)

where " http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq81_HTML.gif " denote the limit approaching http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq82_HTML.gif from outside and inside http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq83_HTML.gif , respectively. Applying Green's formula for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq84_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq85_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq86_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq87_HTML.gif , we have

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ7_HTML.gif
(2.7)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq88_HTML.gif is directed into the exterior of the corresponding domain.

Using boundary conditions on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq89_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq90_HTML.gif and the above transmission boundary condition (2.6), we have

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ8_HTML.gif
(2.8)

Hence

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ9_HTML.gif
(2.9)

So, from [13, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq91_HTML.gif ] and a unique continuation argument we obtain that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq92_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq93_HTML.gif .

We use http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq94_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq95_HTML.gif to denote the jump of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq96_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq97_HTML.gif across the crack http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq98_HTML.gif , respectively. Then we have the following.

Lemma 2.2.

If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq99_HTML.gif is a solution of (2.5) and (2.3), then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq100_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq101_HTML.gif .

The proof of this lemma can be found in [11].

We are now ready to prove the existence of a solution to the above scattering problem by using an integral equation approaching. For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq102_HTML.gif , by Green representation formula

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ10_HTML.gif
(2.10)

and for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq103_HTML.gif

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ11_HTML.gif
(2.11)

where

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ12_HTML.gif
(2.12)

is the fundamental solution to the Helmholtz equation in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq104_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq105_HTML.gif is a Hankel function of the first kind of order zero.

By making use of the known jump relationships of the single- and double-layer potentials across the boundary http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq106_HTML.gif (see [5, 11]) and approaching the boundary http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq107_HTML.gif from inside http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq108_HTML.gif , we obtain (for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq109_HTML.gif )

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ13_HTML.gif
(2.13)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ14_HTML.gif
(2.14)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq110_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq111_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq112_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq113_HTML.gif are boundary integral operators:

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ15_HTML.gif
(2.15)

defined by (for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq114_HTML.gif )

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ16_HTML.gif
(2.16)

Similarly, approaching the boundary http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq115_HTML.gif from inside http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq116_HTML.gif we obtain (for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq117_HTML.gif )

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ17_HTML.gif
(2.17)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ18_HTML.gif
(2.18)

From (2.13)–(2.18), we have

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ19_HTML.gif
(2.19)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ20_HTML.gif
(2.20)

Restrict http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq118_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq119_HTML.gif , from (2.19) we have

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ21_HTML.gif
(2.21)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq120_HTML.gif means a restriction to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq121_HTML.gif .

Define

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ22_HTML.gif
(2.22)

Then zero extend http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq122_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq123_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq124_HTML.gif to the whole http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq125_HTML.gif in the following:

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ23_HTML.gif
(2.23)

By using the boundary conditions in (2.5), we rewrite (2.21) as

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ24_HTML.gif
(2.24)

where

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ25_HTML.gif
(2.25)

Furthermore, we modify (2.24) as

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ26_HTML.gif
(2.26)

where the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq126_HTML.gif is the operator applied to a function with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq127_HTML.gif and evaluated on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq128_HTML.gif , with analogous definition for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq129_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq130_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq131_HTML.gif . We have mapping properties (see [5, 11])

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ27_HTML.gif
(2.27)

Again from (2.13)–(2.18), restricting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq132_HTML.gif to boundary http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq133_HTML.gif we have

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ28_HTML.gif
(2.28)

or

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ29_HTML.gif
(2.29)

Like previous, define

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ30_HTML.gif
(2.30)

Then we can rewrite (2.29) as

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ31_HTML.gif
(2.31)

where

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ32_HTML.gif
(2.32)

Similar to (2.26), we modify (2.31) as

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ33_HTML.gif
(2.33)

and we have mapping properties:

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ34_HTML.gif
(2.34)

Combining (2.13) and (2.14),

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ35_HTML.gif
(2.35)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ36_HTML.gif
(2.36)

Using (2.17) and (2.18),

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ37_HTML.gif
(2.37)

Then using (2.36),

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ38_HTML.gif
(2.38)

From (2.29), we have

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ39_HTML.gif
(2.39)

Restricting (2.38) to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq134_HTML.gif and using (2.39), we modify (2.38) as

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ40_HTML.gif
(2.40)

where

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ41_HTML.gif
(2.41)

for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq135_HTML.gif .

Define

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ42_HTML.gif
(2.42)

and using the notation in previous, we can rewrite (2.40) as

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ43_HTML.gif
(2.43)

or

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ44_HTML.gif
(2.44)

where the operators http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq136_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq137_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq138_HTML.gif are restriction operators (see (2.29)). As before, we have mapping properties:

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ45_HTML.gif
(2.45)

By using Green formula and approaching the boundary http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq139_HTML.gif from inside http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq140_HTML.gif we obtain (for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq141_HTML.gif )

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ46_HTML.gif
(2.46)

The last term in (2.46) can be reformulated as

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ47_HTML.gif
(2.47)

Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq142_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq143_HTML.gif in (2.47), we have the following result (see [13]).

Lemma 2.3.

By using Green formula and the Sommerfeld radiation condition (2.3), one obtains
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ48_HTML.gif
(2.48)

Proof.

Denote by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq144_HTML.gif a sufficiently large ball with radius http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq145_HTML.gif containing http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq146_HTML.gif and use Green formula inside http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq147_HTML.gif . Furthermore noticing http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq148_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq149_HTML.gif , and the Sommerfeld radiation condition (2.3), we can prove this lemma.

Combining (2.46), (2.47), and Lemma 2.3 and restricting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq150_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq151_HTML.gif we have

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ49_HTML.gif
(2.49)

where

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ50_HTML.gif
(2.50)

Define

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ51_HTML.gif
(2.51)

and then we can rewrite (2.49) as

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ52_HTML.gif
(2.52)

Similarly, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq152_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq153_HTML.gif are restriction operators as before, and we have mapping properties:

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ53_HTML.gif
(2.53)

Combining (2.52), (2.26), (2.33), and (2.44), we have

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ54_HTML.gif
(2.54)

If we define

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ55_HTML.gif
(2.55)

then (2.54) can be rewritten as a boundary integral system:

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ56_HTML.gif
(2.56)

Remark 2.4.

If the above system (2.56) has a unique solution, our problem (2.5) with (2.3) will have a unique solution (see [13, 14]).

3. Existence and Uniqueness

Based on the Fredholm theory, we show the existence and uniqueness of a solution to the integral system (2.56).

Define

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ57_HTML.gif
(3.1)

and its dual space

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ58_HTML.gif
(3.2)

Theorem 3.1.

The operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq154_HTML.gif maps http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq155_HTML.gif continuously into http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq156_HTML.gif and is Fredholm with index zero.

Proof.

As is known, the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq157_HTML.gif is positive and bounded below up to a compact perturbation (see [18]); that is, there exists a compact operator
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ59_HTML.gif
(3.3)
such that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ60_HTML.gif
(3.4)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq158_HTML.gif denote the duality between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq159_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq160_HTML.gif .

For convenience, in the following discussion we define

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ61_HTML.gif
(3.5)
Similarly, the operators http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq161_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq162_HTML.gif are positive and bounded below up to compact perturbations (see [18]), that is, there exist compact operators
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ62_HTML.gif
(3.6)
such that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ63_HTML.gif
(3.7)

Define http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq163_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq164_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq165_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq166_HTML.gif are bounded below up and positive.

Take http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq167_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq168_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq169_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq170_HTML.gif be the extension by zero to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq171_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq172_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq173_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq174_HTML.gif respectively.

Denote http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq175_HTML.gif .

It is easy to check that the operators http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq176_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq177_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq178_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq179_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq180_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq181_HTML.gif are compact operators, and then we can rewrite http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq182_HTML.gif as the following:

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ64_HTML.gif
(3.8)
with
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ65_HTML.gif
(3.9)
where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq183_HTML.gif is compact and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq184_HTML.gif defines a sesquilinear form, that is,
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ66_HTML.gif
(3.10)

Here http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq185_HTML.gif denotes the scalar product on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq186_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq187_HTML.gif defined by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq188_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq189_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq190_HTML.gif is the scalar product on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq191_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq192_HTML.gif ).

By properties of the operators http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq193_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq194_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq195_HTML.gif , we have

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ67_HTML.gif
(3.11)
Similarly,
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ68_HTML.gif
(3.12)
So the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq196_HTML.gif is coercive, that is,
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ69_HTML.gif
(3.13)

whence the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq197_HTML.gif is Fredholm with index zero.

Theorem 3.2.

The operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq198_HTML.gif has a trivial kernel if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq199_HTML.gif is not Dirichlet eigenvalue of the Laplace operator in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq200_HTML.gif .

Proof.

In this part, we show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq201_HTML.gif . To this end let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq202_HTML.gif be a solution of the homogeneous system http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq203_HTML.gif , and we want to prove that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq204_HTML.gif .

However, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq205_HTML.gif means that

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ70_HTML.gif
(3.14)
Define a potential
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ71_HTML.gif
(3.15)
where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq206_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq207_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq208_HTML.gif have the same meaning as before and
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ72_HTML.gif
(3.16)

This potential http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq209_HTML.gif satisfies Helmholtz equation in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq210_HTML.gif and the Sommerfeld radiation condition (see [13, 14]).

Considering the potential http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq211_HTML.gif inside http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq212_HTML.gif and approaching the boundary http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq213_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq214_HTML.gif ), we have

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ73_HTML.gif
(3.17)
and (3.14) implies that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ74_HTML.gif
(3.18)
Similarly, considering the potential http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq215_HTML.gif inside http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq216_HTML.gif and approaching the boundary http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq217_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq218_HTML.gif ), then restricting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq219_HTML.gif to the partial boundary http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq220_HTML.gif :
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ75_HTML.gif
(3.19)
and restricting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq221_HTML.gif to the partial boundary http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq222_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ76_HTML.gif
(3.20)
Now, we consider the potential http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq223_HTML.gif in the region http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq224_HTML.gif and approach the boundary http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq225_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq226_HTML.gif ), and then restricting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq227_HTML.gif to the partial boundary http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq228_HTML.gif , similar to (3.19), we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ77_HTML.gif
(3.21)
Refering to (3.20),
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ78_HTML.gif
(3.22)
Combining (3.22), from (3.14) we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ79_HTML.gif
(3.23)
From (3.18)–(3.23), the potential http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq229_HTML.gif satisfies the following boundary value problem:
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ80_HTML.gif
(3.24)
and the Sommerfeld radiation condition:
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ81_HTML.gif
(3.25)

uniformly in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq230_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq231_HTML.gif .

The uniqueness result Theorem 2.1 in Section 2 implies that

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ82_HTML.gif
(3.26)
Notice that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq232_HTML.gif is not Dirichlet eigenvalue of the Laplace operator in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq233_HTML.gif , and so
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ83_HTML.gif
(3.27)
Therefore, the well-known jump relationships (see [13, 14]) imply that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ84_HTML.gif
(3.28)

So we complete the proof of the theorem.

Combining Theorems 3.1 and 3.2, we have the following

Theorem 3.3.

The boundary integral system (2.56) has a unique solution.

Remark 3.4.

If we remove the condition that " http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq234_HTML.gif is not Dirichlet eigenvalue of the Laplace operator in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq235_HTML.gif ," instead of it by the assumption that Im  http://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq236_HTML.gif , then Theorem 2.1 in Section 2 and Theorem 3.3 in Section 3 are also true.

Declarations

Acknowledgment

This research is supported by NSFC Grant no. 10871080, Laboratory of Nonlinear Analysis of CCNU, COCDM of CCNU.

Authors’ Affiliations

(1)
Department of Mathematics, Central China Normal University

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Copyright

© Guozheng Yan. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.