Open Access

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq1_HTML.gif and a bounded domain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq2_HTML.gif in https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq5_HTML.gif . Different boundary conditions are given on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq6_HTML.gif and https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq10_HTML.gif and a bounded domain https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq12_HTML.gif and a bounded domain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq13_HTML.gif in https://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 https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq19_HTML.gif and a bounded domain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq20_HTML.gif in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq22_HTML.gif has smooth boundary https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq23_HTML.gif (e.g., https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq24_HTML.gif ), and the crack https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq26_HTML.gif enclosing a bounded domain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq27_HTML.gif such that the normal vector https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq28_HTML.gif on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq30_HTML.gif which we again denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq31_HTML.gif . The bounded domain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq32_HTML.gif is located inside the domain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq33_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq35_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq36_HTML.gif .

Suppose that

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq37_HTML.gif is an injective piecewise https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq38_HTML.gif function. We denote the outside of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq40_HTML.gif and the inside by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq41_HTML.gif . Here we suppose that the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq42_HTML.gif is divided into two parts https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq43_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq45_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq46_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq47_HTML.gif lead to the following problem:

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq48_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq49_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq50_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq51_HTML.gif . The total field https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq53_HTML.gif , and the unknown scattered field https://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

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

uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq55_HTML.gif with https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq57_HTML.gif in the following.

Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq59_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq60_HTML.gif to denote the usual Sobolev spaces, https://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

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

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

and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq69_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq72_HTML.gif , we want to show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq73_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq74_HTML.gif .

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq75_HTML.gif (with boundary https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq79_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq80_HTML.gif :

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

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

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

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq89_HTML.gif , https://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

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

Hence

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

So, from [13, Theorem https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq92_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq93_HTML.gif .

We use https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq94_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq95_HTML.gif to denote the jump of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq96_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq97_HTML.gif across the crack https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq100_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq102_HTML.gif , by Green representation formula

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

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

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

where

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq104_HTML.gif , and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq107_HTML.gif from inside https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq108_HTML.gif , we obtain (for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq109_HTML.gif )

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

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

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

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

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

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

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

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

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

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

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

Define

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

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

https://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

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

where

https://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

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

where the operator https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq127_HTML.gif and evaluated on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq128_HTML.gif , with analogous definition for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq129_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq130_HTML.gif , and https://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])

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq132_HTML.gif to boundary https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq133_HTML.gif we have

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

or

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

Like previous, define

https://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

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

where

https://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

https://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:

https://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),

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

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

Then using (2.36),

https://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

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

Restricting (2.38) to https://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

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

where

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

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

Define

https://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

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

or

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

where the operators https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq136_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq137_HTML.gif , and https://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:

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq139_HTML.gif from inside https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq140_HTML.gif we obtain (for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq141_HTML.gif )

https://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

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

Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq142_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ48_HTML.gif
(2.48)

Proof.

Denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq144_HTML.gif a sufficiently large ball with radius https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq145_HTML.gif containing https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq146_HTML.gif and use Green formula inside https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq147_HTML.gif . Furthermore noticing https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq148_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq150_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq151_HTML.gif we have

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

where

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

Define

https://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

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

Similarly, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq152_HTML.gif and https://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:

https://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

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

If we define

https://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:

https://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

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

and its dual space

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq154_HTML.gif maps https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq155_HTML.gif continuously into https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ59_HTML.gif
(3.3)
such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ60_HTML.gif
(3.4)

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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ61_HTML.gif
(3.5)
Similarly, the operators https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq161_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ62_HTML.gif
(3.6)
such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ63_HTML.gif
(3.7)

Define https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq163_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq164_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq165_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq167_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq168_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq169_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq170_HTML.gif be the extension by zero to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq171_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq172_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq173_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq174_HTML.gif respectively.

Denote https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq176_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq177_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq178_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq179_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq180_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq182_HTML.gif as the following:

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

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

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

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

whence the operator https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq198_HTML.gif has a trivial kernel if https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq201_HTML.gif . To this end let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq203_HTML.gif , and we want to prove that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq204_HTML.gif .

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

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

This potential https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq209_HTML.gif satisfies Helmholtz equation in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq211_HTML.gif inside https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq212_HTML.gif and approaching the boundary https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq213_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq214_HTML.gif ), we have

https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ74_HTML.gif
(3.18)
Similarly, considering the potential https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq215_HTML.gif inside https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq216_HTML.gif and approaching the boundary https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq217_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq218_HTML.gif ), then restricting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq219_HTML.gif to the partial boundary https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq220_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ75_HTML.gif
(3.19)
and restricting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq221_HTML.gif to the partial boundary https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq222_HTML.gif , we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq223_HTML.gif in the region https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq224_HTML.gif and approach the boundary https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq225_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq226_HTML.gif ), and then restricting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq227_HTML.gif to the partial boundary https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq228_HTML.gif , similar to (3.19), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ77_HTML.gif
(3.21)
Refering to (3.20),
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq229_HTML.gif satisfies the following boundary value problem:
https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ81_HTML.gif
(3.25)

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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_Equ82_HTML.gif
(3.26)
Notice that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F524846/MediaObjects/13661_2009_Article_853_IEq233_HTML.gif , and so
https://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
https://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 " https://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 https://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  https://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.