An Expansion Theorem for Two-dimensional Elastic Waves and its Application

Published online in Wiley InterScience

(www.interscience.wiley.com) DOI: 10.1002/mma.753 MOS subject classification: 35 C 10

An expansion theorem for two-dimensional elastic

waves and its application

Kun-Chu Chen

and Ching-Lung Lin

1_{Department of Mathematics, National Cheng-Kung University, Tainan 701, Taiwan} 2_{Department of Mathematics, National Chung Cheng University, Chia-Yi 62117, Taiwan}

Communicated by P. Hagedorn

SUMMARY

We prove an Atkinson–Wilcox-type expansion for two-dimensional elastic waves in this paper. The approach developed on the two-dimensional Helmholtz equation will be applied in the proof. When the elastic fields are involved, the situation becomes much harder due to two wave solutions propagating at different phase velocities. In the last section, we give an application about the reconstruction of an obstacle from the scattering amplitude. Copyright_{q 2006 John Wiley & Sons, Ltd.}

KEY WORDS: Atkinson–Wilcox; Helmholtz equation

1. INTRODUCTION

Let u(x) ∈ C2be a solution of the scalar Helmholtz equation
u + k2u= 0 in the exterior of the

ball with radius a>0 and satisfy Sommerfeld’s radiation condition. It is a well-known property [1, 2] that u, in the spherical coordinates (r,
, ), can be expressed as

u(r,
, ) = r−1eikr
∞

n=0

fn(
, )r−n (1)

where the series converges for r>a and converges absolutely and uniformly with respect to r,
,  in the domain r>a + ε>a. The series may be differentiated term-by-term in all variables. Moreover, the coefficients fn for n>0, can be constructed recursively from the far-field pattern f0(
, ).

Similar results for Maxwell’s equations and elastic equations in three dimensions were proved by

∗_{Correspondence to: Ching-Lung Lin, Department of Mathematics, National Chung Cheng University, Chia-Yi} 62117, Taiwan.

†_{E-mail: cllin@math.ccu.edu.tw}

Contract/grant sponsor: National Science Council and National Center for Theoretical Science

Wilcox[3] and by Dassios [4], respectively. In two dimensions, a convergent expansion theorem for the scalar radiation solution was established by Karp[5]. However, a similar expansion theorem for two-dimensional elastic waves is still missing. The present paper is an attempt to fill this gap. In three dimensions, the way of driving expansion theorems for radiation solutions to the Helmholtz, Maxwell’s and elastic equations relies on integral representations of radiation solutions and the fundamental solution eikr/r to the scalar Helmholtz equation. One of the key points is that d(r−1exp(ikr))/dr = r−1exp(ikr)(ik − r−1) (2) As for two dimensions, the fundamental solution for scalar Helmholtz equation is H₀(1)(kr), where H₀(1)(z) is the Hankel function of the first kind, of order zero. From Reference [6, p. 74], we found that

d(H₀(1)(kr))/dr = −k H₁(1)(kr)

where H₁(1)(z) is the Hankel function of the first kind, of order 1. Unfortunately, H₁(1)(z) cannot be expressed by H₀(1)(z) as we had for eikr/r in (2). So in two dimensions, if an expansion theorem does hold, it will not be as neat as (1). In fact, Karp[5] showed that a radiation solution

u= u(r,
) ∈ C2to the scalar Helmholtz equation in the region r>a admits the following expansion:

u= H₀(1)(kr)
∞ n=0r −n_F n(
) + H₁(1)(kr) ∞
n=0r −n_G n(
) (3)

where the series converges absolutely and uniformly in r
a + ε>a and can be differentiated term by term with respect to r and
. The coefficients F0 and G0 are determined from the formulas

F0(
) = [ f0(
) + f0(
+ )]/2

−iG0(
) = [ f0(
) − f0(
− )]/2

(4) where f0(
) is the so-called far-field pattern or amplitude. Furthermore, Fn and Gn for n>0 are

constructed, respectively, from F0 and G0.

Unlike the approach for the three-dimensional case, Karp took a different route to obtain the expansion formula (3) by expressing u in the form

u=
∞

n=0H (1)

n (kr)(ancos n
+ bnsin n
)

and writing H_n(1)(kr) as a linear combination of H₀(1)(kr) and H₁(1)(kr) with coefficients which are polynomials in 1/r. These polynomials are Lommel’s polynomials. We shall follow Karp’s approach for two-dimensional elastic waves. The starting point is to decompose the elastic wave into longitudinal part upand the transverse part us. We then apply Karp’s results to upand us, separately. Most of efforts are devoted to driving formulas of determining coefficients in the expansion.

2. EXPANSION THEOREM FOR 2D ELASTIC WAVES

In this paper, we consider the time-harmonic elastic wave equation in two dimensions

where and  are Lam´e constants satisfying >0, +>0 and >0 is the density. It is well-known that u of (5) can be decomposed into u= up+ us and up, us satisfy

up_{+ k}2

pup= 0,
us+ ks2us= 0 (6)

and

∇⊥· up_{= 0, ∇ · u}s_{= 0 (7)}

where k2_p= 2/( + 2), k2_s= 2/ and ∇⊥= (−*x2, *x1). Moreover, a solution to (5) is called

radiation if it satisfies the Kupradze radiation conditions lim

r→∞

√

r(*up/*r − ikpup) = 0 and _r→∞lim

√

r(*us/*r − iksus) = 0

where up is the longitudinal field and us is the transverse field. In other words, u is a radiation solution of the elastic equation (5) if and only if each component of up and us is a radiation solution of the scalar Helmholtz equation and condition (7) holds.

Now, we directly apply Karp’s results to each component of up and us to yield

Theorem 1

Let u= u(r,
) ∈ C2be a radiation of (5) in r>a>0. Then u admits the following convergent series expansion: u= up+ us = H₀(1)(kpr) ∞
n=0r −n_Fp n(
) + H₁(1)(kpr) ∞
n=0r −n_Gp n(
) + H₀(1)(ksr) ∞
n=0r −n_Fs n(
) + H1(1)(ksr) ∞
n=0r −n_Gs n(
) (8)

for r>a and the series converges absolutely and uniformly in r
a + ε>a. It also can be differen-tiated term-by-term with respect to r ,
any number of times and the resulting series all converge absolutely and uniformly.

From the Betti integral representation formula and the asymptotic behavior of H₀(1)(z), we can see that any radiation solution u of (5) has the asymptotic form

u=√2 exp(−i/4)(kpr)−1/2exp(ikpr)u∞p(
) ˆr

+√2 exp(−i/4)(ksr)−1/2exp(iksr)us_∞(
)ˆ
+ O(r−3/2) (9)

as r→ ∞ uniformly in all directions
. Here the pair (u_∞p , us_∞) is called the far-field pattern of the radiation solution u. We observe that the far-field pattern coming from the longitudinal part

up is normal to the unit circle, while the far-field associated with the transverse part us is tangent to the unit circle.

Our next task is to determine the coefficients in expansion (8) from the far-field pattern

(up

∞, us∞) of the radiation solution u. Detailed computations will be carried out in the following

section.

3. DETERMINATION OF COEFFICIENTS

To begin, we would like to put related differential operators in the polar coordinates. Let F(r,
) =

fr(r,
)ˆr + f
(r,
)ˆ
be a vector field and f (r,
) be a scalar function where ˆr = (cos
, sin
)tand

ˆ
= (− sin
, cos
)t_{. In the polar coordinates, we know that}

∇ · F = * fr/*r + r−1fr+ r−1* f
/*
(10)

and

∇ f = (* f/*r)ˆr + r−1(* f/*
)ˆ
(11) On the other hand, we can derive that

f = *2

f/*r2+ r−1* f/*r + r−2*2f/*
2 (12) To determine the coefficients in (8), we replace u in (5) by (8). Before the computations, it is useful to note that

H₀(1)(z)= −H₁(1)(z) and H₁(1)(z)= H₀(1)(z) − z−1H₁(1)(z) (13) Furthermore, we use the following notations:

Fnp(
) = fn,rp (
)ˆr + f_n,
p (
)ˆ
= fn,rp ˆr + f_n,
p ˆ

Gnp(
) = gn,rp (
)ˆr + g_n,
p (
)ˆ
= gn,rp ˆr + g_n,
p ˆ

F_ns(
) = f_n,rs (
)ˆr + f_n,
s (
)ˆ
= f_n,rs ˆr + f_n,
s ˆ
Gs_n(
) = g_n,rs (
)ˆr + g_n,
s (
) ˆ
= g_n,rs ˆr + g_n,
s ˆ

Making use of (13) and (8), we get from (5) that

S_0,rp H₀(1)(kpr)ˆr + S_0,p
H₀(1)(kpr)ˆ
+ S_1,rp H₁(1)(kpr)ˆr + S_1,
p H₁(1)(kpr)ˆ
+ Ss 0,rH0(1)(ksr)ˆr + S s 0,
H0(1)(ksr)ˆ
+ S s 1,rH1(1)(ksr)ˆr + S s 1,
H1(1)(ksr)ˆ
= 0 (14)

where S_0,rp = 
∞ n=0{n 2_r−n−2_fp n,r− 2kpnr−n−1gnp,r − k2pr−nf p n,r+ r−n−2( fnp,r)} + ( + )
∞ n=0{(n 2_{− 1)r}−n−2_fp n,r− 2kpnr−n−1gn,rp } + ( + )
∞ n=0{−k 2 pr−nf p n,r− (n + 1)r−n−2( f_n,p
)+ kpr−n−1(g_n,
p )} + 2
∞ n=0r −n_fp n,r (15) S_0,
p = 
∞ n=0{n 2_r−n−2_fp n,
− 2kpnr−n−1g_n,
p − k2pr−nf p n,
+ r−n−2( fn,
p )} + ( + )
∞ n=0{(1 − n)r −n−2_{( f}p n,r)+ kpr−n−1(gn,rp )+ r−n−2( f_n,p
)} + 2
∞ n=0r −n_fp n,
(16) S_1,rp = 
∞ n=0{2kpnr −n−1_fp n,r+ (n + 1)2r−n−2g_n,rp } + 
∞ n=0{r −n−2_(gp n,r)− k2pr−ng p n,r} + 2 ∞
n=0 r−ng_n,rp + ( + )
∞ n=0{2nkpr −n−1_fp n,r− k2pr−ng p n,r− kpr−n−1( f_n,
p )} + ( + )
∞ n=0{n(n + 2)r −n−2_gp n,r− (n + 2)r−n−2(g_n,
p )} (17) S_1,
p = 
∞ n=0{2kp nr−n−1f_n,p
+ (n + 1)2r−n−2g_n,
p − k2_pr−ng_n,p
} + 
∞ n=0{r −n−2_(gp n,
)} + 2 ∞
n=0r −n_gp n,
+ ( + )
∞ n=0{−kpr −n−1_{( f}p n,r)− nr−n−2(gn,rp )+ r−n−2(gnp,
)} (18) S_0,rs =  ∞
n=0{n 2_r−n−2_fs n,r− 2ksnr−n−1gs_n,r− k2sr−nfn,rs + r−n−2( fn,rs )} + ( + )
∞ n=0{(n 2_{− 1)r}−n−2_fs n,r− 2ksnr−n−1g_n,rs }

+ ( + )
∞ n=0{−k 2 sr−nfn,rs − (n + 1)r−n−2( f_n,
s )+ ksr−n−1(g_n,s
)} + 2
∞ n=0r −n_fs n,r (19) S_0,
s = 
∞ n=0{n 2_r−n−2_fs n,
− 2ksnr−n−1g_n,
s − k2sr−nf_n,
s + r−n−2( fn,s
)} + ( + )
∞ n=0{(1 − n)r −n−2_{( f}s n,r)+ ksr−n−1(gns,r)+ r−n−2( fn,
s )} + 2
∞ n=0r −n_fs n,
(20) S_1,rs = 
∞ n=0{2ks nr−n−1f_n,rs + (n + 1)2r−n−2gs_n,r} + 
∞ n=0{r −n−2_(gs n,r)− ks2r−ngn,rs } + 2 ∞
n=0 r−ng_n,rs + ( + )
∞ n=0{2nksr −n−1_fs n,r− ks2r−ngsn,r− ksr−n−1( f_n,s
)} + ( + )
∞ n=0{n(n + 2)r −n−2_gs n,r− (n + 2)r−n−2(gn,
s )} (21) and S_1,
s = 
∞ n=0{2ks nr−n−1f_n,
s + (n + 1)2r−n−2g_n,
s − k_s2r−ngs_n,
} + 
∞ n=0{r −n−2_(gs n,
)} + 2 ∞
n=0 r−ng_n,
s + ( + )
∞ n=0{−ksr −n−1_{( f}s n,r)− nr−n−2(gn,rs )+ r−n−2(gsn,
)} (22)

It should be noted that (14) implies

S_0,rp = S_0,
p = S_1,rp = S_1,p
= S_0,rs = S_0,s
= S_1,rs = S_1,s
= 0

Now, collect the terms of r0in (16), it admits that 0= − k2_pf_0,p
(
) + 2f_0,p
(
)

= − 2_{( + 2)}−1_fp

0,
(
) + 2f0,
p (
)

= 2_{( + )( + 2)}−1_fp 0,
(
)

which implies f_0,
p (
) = 0. Similarly, evaluate the coefficients of r0 in (18), (19 and (21), we have

f_0,
p (
) = g_0,p
(
) = f_0,rs (
) = g_0,rs (
) = 0 (23) From (23) and (8), the components up and us verify the following expansions

up= H₀(1)(kpr) ∞
n=1r −n_Fp n(
) + H1(1)(kpr) ∞
n=1r −n_Gp n(
) + H₀(1)(kpr) f_0,rp(
)ˆr + H₁(1)(kpr)g_0,rp (
)ˆr (24) and us= H₀(1)(ksr) ∞
n=1r −n_Fs n(
) + H1(1)(ksr) ∞
n=1r −n_Gs n(
) + H₀(1)(kpr) f_0,
s (
)ˆ
+ H₁(1)(kpr)gs_0,
(
)ˆ
(25)

where up and us also satisfy the Helmholtz equation (6). Following the same arguments in Reference [5] to up and us, respectively, these terms f_0,rp(
), g₀p_,r(
), f_0,
s (
) and g_0,s
(
) are related to the radiation pattern by the formulas

f_0,rp(
) = [u_∞p (
) + u_∞p (
+ )]/2 g_0,rp (
) = i[u_∞p(
) − u_∞p(
+ )]/2

f_0,
s (
) = [us_∞(
) + us_∞(
+ )]/2 g_0,
s (
) = i[us_∞(
) − us_∞(
+ )]/2

(26)

Collecting the terms of r−1 in (16), (18), (19) and (21), it implies that

( + )kp(g_0,rp )= (k2p− 2) f p 1,
= −( + 2)−1( + )2f1,
p ( + )kp( f_0,rp)= (2− k2p)g p 1,
= ( + 2)−1( + )2g1,
p ( + )ks(g_0,s
)= [( + 2)ks2− 2] f1,rs = −1( + )2f1,rs ( + )ks( f_0,s
)= [2− ( + 2)ks2]gs1,r= −−1( + )2g1,rs (27)

From (23), (26) and (27), we obtain that f_1,
p (
), g_1,
p (
), f_1,rs (
) and g_1,rs (
) can be expressed in terms of the radiation pattern (u_∞p, us_∞). Now we are devoted to giving expressions for all coefficients in terms of F_0,
p (
), G_0,
p (
), F_0,s
(
), Gs_0,
(
), f_1,
p , g_1,
p , f_1,rs and g_1,rs . Therefore, all coefficients can be represented by means of the radiation pattern (u_∞p, us_∞). To begin with,

take (15), (17), (20) and (22) into consideration, it implies the following equalities: ( + ){(n2_{− 1) f}p n,r− 2kp(n + 1)g_n+1,rp − (n + 1)( f_n,
p )+ kp(g_n+1,p
)} + {n2_fp n,r− 2kp(n + 1)g_n+1,rp + ( fn,rp )} = 0 (28) ( + ){2(n + 1)kpf_n+1,rp − kp( f_n+1,p
)+ n(n + 2)gn,rp − (n + 2)(g_n,
p )} + {2kp(n + 1) f_n+1,rp + (n + 1)2gn,rp + (gn,rp )} = 0 (29) {n2_fs n,
− 2ks(n + 1)gs_n+1,
+ ( f_n,s
)} + ( + ){(1 − n)( f_n,rs )} + ( + ){( fs n,
)+ ks(g_n+1,rs )} = 0 (30) {2ks(n + 1) f_n+1,
s + (n + 1)2gs_n,
+ (gs_n,
)} − ( + ){ks( f_n+1,rs )} + ( + ){(gs n,
)− n(gn,rs )} = 0 (31) for n
0.

In view of (16), (18), (19) and (21), we have the identities in the following:

{n2_fp n,
− 2kp(n + 1)g_n+1,
p − k2pf p n+2,
+ ( fn,
p )} + 2fn+2,
p + ( + ){(1 − n)( fp n,r)+ kp(g_n+1,rp )+ ( f_n,p
)} = 0 (32) {2kp(n + 1) f_np_+1,
+ (n + 1)2g_np_,
− k2pg p n+2,
+ (g p n,
)} +  2_gp n+2,
+ ( + ){−kp( f_n+1,rp )− n(gn,rp )+ (g_n,
p )} = 0 (33) ( + ){(n2_{− 1) f}s n,r− 2ks(n + 1)gs_n+1,r− k2sfn+2,rs − (n + 1)( fn,s
)} + ( + )ks(g_n+1,s
)+ {n2f_n,rs − 2ks(n + 1)g_n+1,rs } + {( fs n,r)− ks2fn+2,rs } + 2fn+2,rs = 0 (34) {2ks(n + 1) f_n+1,rs + (n + 1)2gs_n,r+ (g_n,rs )− ks2gn+2,rs } + 2gn+2,rs + ( + ){2(n + 1)ksf_n+1,rs − ks2gn+2,rs − ks( f_n+1,s
)} + ( + ){n(n + 2)gs n,r− (n + 2)(gsn,
)} = 0 (35) for n
0.

More precisely, Equations (28)–(35) imply that g_n+1,rp = ( + )[2kp( + 2)(n + 1)]−1{(n2− 1) fn,rp − (n + 1)( f_n,p
)} + [2kp( + 2)(n + 1)]−1{n2fn,rp + ( fn,rp )} + kp( + )(g_n+1,
p )[2kp( + 2)(n + 1)]−1 (36) f_n+1,rp = −( + )[2kp( + 2)(n + 1)]−1{n(n + 2)gnp,r− kp( f_n+1,
p )} − [2kp( + 2)(n + 1)]−1{(n + 1)2gn,rp + (gn,rp )} − (n + 2)( + )(gp n,
)[2kp( + 2)(n + 1)]−1 (37) gs_n+1,
= ( + )[2ks(n + 1)]−1{(1 − n)( f_n,rs )+ ( f_n,
s )+ ks(g_n+1,rs )} + [2ks(n + 1)]−1{n2f_n,
s + ( f_n,
s )} (38) f_n+1,
s = ( + )[2ks(n + 1)]−1{ks( f_n+1,rs )− (g_n,s
)+ n(g_n,rs )} − [2ks(n + 1)]−1{(n + 1)2g_n,
s + (gs_n,
)} (39) f_n+2,
p = −( + 2)[( + )2]−1{n2f_n,p
− 2kp(n + 1)g_n+1,p
+ ( f_n,p
)} − ( + 2)(2₎−1_{{(1 − n)( f}p n,r)+ kp(g_n+1,rp )+ ( f_n,p
)} (40) g_n+2,
p = − ( + 2)[( + )2]−1{2kp(n + 1) f_n+1,
p + (n + 1)2g_n,
p } − ( + 2)(2₎−1_{(gp n,
)− kp( f p n+1,r)− n(gn,rp )} − ( + 2)(gp n,
)[( + )2]−1 (41) f_n+2,rs = ( + )[( + )2]−1{(n2− 1) f_n,rs − 2ks(n + 1)g_n+1,rs } + 2_{[( + )}2_]−1_{n2_fs n,r− 2ks(n + 1)g_n+1,rs + ( f_n,rs )} + ( + )[( + )2_]−1_{k s(gs_n+1,
)− (n + 1)( f_n,s
)} (42) g_ns_+2,r= 2[( + )2]−1{2ks(n + 1) fns+1,r+ (n + 1)2gn,rs + (gn,rs )} + ( + )[( + )2_]−1_{{2(n + 1)k} sf_n+1,rs + n(n + 2)gs_n,r} − ( + )[( + )2_]−1_{k s( f_n+1,s
)+ (n + 2)(g_n,s
)} (43) where n
0.

Through the recursion formulas (36)–(43), all coefficients of u in (8) can be expressed by

F_0,
p (
), G_0,p
(
), F_0,
s (
), Gs_0,
(
), f_1,
p , g_1,
p , f_1,rs and g_1,rs inductively. Therefore, we can use the radiation pattern(u_∞p, us_∞) to determine all coefficients of u in (8).

4. APPLICATION TO INVERSE PROBLEMS

Let a sound-hard obstacle ∈ R2be an open subset with C2boundary. We assume that R2\¯ is connected. Furthermore, we suppose that is contained in the open ball BR= B(0, R). It should

be noted that may consist of finitely many bounded domains. We are cared about the scattering problem for the inhomogeneous isotropic elasticity system. Define the elastic tensor C= (Ci j kl) by

Ci j kl= (x)i jkl+ (x)jli k+ (x)ilj k (44)

where(x)>0 and (x) + (x)>0. Moreover, C(x) = C for |x|>R, where C is a homogeneous isotropic elastic tensor.

To apply an Atkinson–Wilcox-type expansion for two-dimensional elastic wave, we consider an inverse scattering problem in the following. Let u(x) ∈ C2satisfy

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ Lu + 2_{u= f in R}2_\¯ T(D, )u = 0 on  u(x) = up(x) + us(x) lim r→∞ √

r(*up/*r − ikpup) = 0 and _r→∞lim

√

r(*us/*r − iksus) = 0

(45)

where Lu = div(C(x)∇u), f ∈ L2_comp(R2\¯) and T (D, ) is boundary traction operator defined by

(T (D, ))i k= jlCi j kl j*l

with being the unit outer normal of . Moreover, for any matrix E = (Ekl), we define that

(C E)i j= klCi j klEkl

4.1. Inverse problem

Reconstruct  from the far-field pattern (u_∞p , us_∞) of the radiation solution u to (45) at a fixed>0.

To apply an Atkinson–Wilcox-type expansion for two-dimensional elastic wave derived above, we shall reconstruct by means of the far-field pattern. Some three-dimensional obstacle scattering problems were proved in [7, 8]. Nevertheless, the situation becomes much harder when elastic fields are involved, since we must deal with two wave solutions propagating at different phase velocities. Dassios and Rigou[9] established some basic results containing a Runge’s-type theorem in three-dimensional elastic scattering. Recently, Alves and Kress[10] applied the so-called linear sampling methods to three-dimensional elastic obstacle scattering. Using the same idea, Arens [11] established some results in the two-dimensional inverse elastic wave scattering. On the other hand, the probe method predicts whether a point descending from the boundary of the body along a given path hits the discontinuity surface and when it occurs. This method was applied to reconstruct the obstacles in References[7, 8]. However, the approaches developed in References

[7, 8] essentially rely on an Atkinson–Wilcox-type expansion for three-dimensional Helmholtz equation. Having an Atkinson–Wilcox-type expansion for two-dimensional elastic wave at hand, the same arguments in References[7, 8, 12, 13] can be applied to the two-dimensional inverse elastic wave scattering.

Algorithm of the reconstruction

Step 1: We assume that 0 is not a Dirichlet eigenvalue ofL+2in BR. Having an Atkinson–

Wilcox-type expansion for two-dimensional elastic wave derived in Section 3, we can solve upand

us inR2\ BR from the far-field pattern (u∞p, us∞) of the radiation solution u to (45). Therefore,

the Green function G(x, y) of us on|x| = |y| = R can be determined, where G(x, y) satisfies

us(x) =



R2_\¯G(x, y) f (y) dy

Step 2: As the same arguments in Reference[13] where the inhomogeneous anisotropic elasticity

system is considered, we show that the Dirichlet-to-Neumann map on *BR can be constructed

by the measurements G(x, y) on |x| = |y| = R. For the reader’s convenience, the similar nota-tions in Reference [13] will be used. Define the Dirichlet-to-Neumann map _ : H1/2(*BR) →

H−1/2(*BR) by

(g) = T (D, x/ x )v|*BR

wherev is the solution of ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Lv + 2_{v = 0 in B} R\¯ T(D, )v = 0 on v = g ∈ H1/2_(*B R) on *BR (46)

Letve be the solution of ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Lve_{+ }2_ve_{= 0 inR}2_\B R ve_{= g ∈ H}1/2_(*B R) on *BR

ve _{satisfies the radiation conditions}

(47)

Therefore, we can define the Dirichlet-to-Neumann mape : H1/2(*BR) → H−1/2(*BR) by

e_{(g) = T (D, x/ x )v|} *BR

Now for g∈ H1/2(*BR), define Mg∈ Hcomp−1 (R2\¯) by

Mg,  = g, |_*BR ∀ ∈ Hcomp1 (R2\¯)

On the other hand, letvg be the scattering solution of (45) with the source term—Mg. Define

g(x) = vg|_*BR i.e. g(x) = −  *BR G(x, y)g(y) ds, x ∈ *BR

The following key lemma verifies that the Dirichlet-to-Neumann map_ can be constructed by

G(x, y) on |x| = |y| = R.

Lemma 2 (Nakamura et al. [13, Lemma 5.3])

− e is injective and(_− e) = I .

It should be noted that is determined by G(x, y) and e can also be constructed. With the aid of Lemma 2, we can get_ by

= e− −1

Step 3: We convert our problem to construct  by the Dirichlet-to-Neumann map _. The problem is the same as Inverse Problem 2 on p. 209 of Reference [8] and Section 5.2 on p. 608 of Reference[13]. We will not repeat the proof again and refer the readers to the above articles.

ACKNOWLEDGEMENTS

We would like to thank Professor Jenn-Nan Wang for bringing the problem to our attention and for many stimulating discussions. The authors are partially supported by the National Science Council and National Center for Theoretical Science of Taiwan.

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