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An inverse problem of the wave equation for a general doubly connected region in R2 with a finite number of piecewise smooth Robin boundary conditions
Applied Mathematics and Computation 132 (2002) 187–204 www.elsevier.com/locate/amc
An inverse problem of the wave equation for a general doubly connected region in R2 with a finite number of piecewise smooth Robin boundary conditions E.M.E. Zayed *, I.H. Abdel-Halim Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt
Abstract The spectral distribution 1 X b l ðtÞ ¼ expðitEx1=2 Þ; x¼1
where
fEx g1 x¼1 D ¼
are the eigenvalues of the negative Laplacian 2 2 X o k¼1
oxk
1 2 in the pffiffiffiffiffiffiðx ffi ; x Þ-plane, is studied for a variety of domains, where 1 < t < 1 and i ¼ 1. The dependences of b l ðtÞ on the connectivity of a bounded domain and the Robin boundary conditions are analyzed. Particular attention is given to a general annular bounded domain X in R2 with a smooth inner boundary oX1 and a smooth outer boundary oX2 where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components CJ ðJ ¼ 1; . . . ; mÞ of oX1 and on the piecewise S smooth components S CJ ðJ ¼ m þ 1; . . . ; nÞ of oX2 are considered, such that oX1 ¼ mJ¼1 CJ and oX2 ¼ nJ ¼mþ1 CJ . Some geometrical properties of X (e.g., the area of X, the total lengths of its boundary, the curvature of the boundary, the number of holes of X, etc.) are determined, from the asymptotic expansions of b l ðtÞ for jtj ! 0. Ó 2002 Elsevier Science Inc. All rights reserved.
*
Corresponding author. E-mail addresses: [email protected] (E.M.E. Zayed), [email protected] (I.H. Abdel-Halim). 0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 1 ) 0 0 1 8 6 - 2
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1. Introduction Let X be a simply connected bounded domain in R2 with a smooth boundary oX. Consider the two-dimensional Robin problem Du ¼ Eu in X; o þ c u ¼ 0 on oX; on
ð1:1Þ ð1:2Þ
where o=on denotes differentiation along the inward pointing normal to oX and c is a positive constant impedance, with u 2 C 2 ðXÞ \ CðXÞ. Denote its eigenvalues, counted according to multiplicity, by 0 < E1 6 E2 6 E3 6 6 E x 6 ! 1
as x ! 1:
ð1:3Þ
At the beginning of this century the principal problem was that of investigating the asymptotic behavior of the eigenvalues (1.3). It is well known (see [1]) that if N ðEÞ is the number of these eigenvalues less than E, then N ðEÞ
jXj E 4p
as E ! 1
ðH: Weyl; 1912Þ
and N ðEÞ ¼
jXj E þ OðE1=2 log EÞ 4p
as E ! 1
ðR: Courant; 1920Þ;
where jXj is the area of the domain X. In order to obtain further information about the geometry of X, one studies certain functions of the spectrum. The most useful to date comes from the heat equation or the wave equation. Accordingly, let etD denote the heat operator. Then we can construct the trace function HðtÞ ¼ trðetD Þ ¼
1 X
etEx ;
ð1:4Þ
x¼1
which converges for all positive t. 1=2 Let eitD be the wave operator. Then an alternative to (1.4) is to study the trace function 1=2
b l ðtÞ ¼ trðeitD Þ ¼
1 X x¼1
1=2
eitEx ;
ð1:5Þ
pffiffiffiffiffiffiffi which represents a tempered distribution for 1 < t < 1 and i ¼ 1. In the present paper, we shall concentrate our efforts on a study of the asymptotic expansion of the tempered distribution b l ðtÞ for small jtj. Zayed et al. [20] have discussed problem (1.1), (1.2) for small/large impedance c by using the wave equation approach and have determined some geo-
E.M.E. Zayed, I.H. Abdel-Halim / Appl. Math. Comput. 132 (2002) 187–204
189
metrical properties of the general bounded domain X from the asymptotic expansion of b l ðtÞ as jtj ! 0. Zayed [25,27] has discussed the Robin problem (1.1), (1.2) in the following cases: Case 1.1: c ¼ 0 (The Neumann problem) b l ðtÞ ¼
jXj joXj H ðjtjÞ þ sign t þ a0 jtj þ Oðt2 sign tÞ 2pt 8
as jtj ! 0:
ð1:6Þ
as jtj ! 0;
ð1:7Þ
Case 1.2: c ! 1 (The Dirichlet problem) b l ðtÞ ¼
jXj joXj H ðjtjÞ sign t þ a0 jtj þ Oðt2 sign tÞ 2pt 8
where H ðjtjÞ is the Heaviside unit function and 8 if t > 0; <1 sign t ¼ 0 if t ¼ 0; : 1 if t < 0: In these formule, jXj is the area of X, and joXj is the total length of oX. The constant term a0 has geometric significance, e.g., if X is smooth and convex, then a0 ¼ 16 and if X is permitted to have a finite number of smooth convex holes ‘‘h’’, then a0 ¼ 16 ð1 hÞ. Case 1.3 (The mixed problem) If L1 is the length of a component C1 of the boundary oX with the Neumann boundary conditions, and if L2 is the length of the remaining component C2 ¼ oX n C1 of oX with the Dirichlet boundary conditions, then with reference to the articles [20,25,27], we get b l ðtÞ ¼
jXj ðL1 L2 Þ H ðjtjÞ þ sign t þ a0 jtj þ Oðt2 sign tÞ 2pt 8
as jtj ! 0: ð1:8Þ
Further, the order term Oðt2 sign tÞ in (1.6)–(1.8) is yet undetermined. So, in the present paper, we discuss what geometric quantities are contained in this order term. The object of this paper is to discuss the following more general inverse problem: Let X be a general annular bounded domain in R2 consisting of a simply connected bounded inner domain X1 with a smooth boundary oX1 and a simply connected bounded outer domain X2 X1 with a smooth boundary oX2 where X1 ¼ X1 [ oX1 . Suppose that the eigenvalues (1.3) are known exactly for the Helmholtz equation
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D/ ¼ E/
ð1:9Þ
in X;
together with the piecewise smooth Robin boundary conditions o ðJ ¼ 1; . . . ; m; m þ 1; . . . ; nÞ; þ cJ / ¼ 0 on ocJ onJ
ð1:10Þ
where the inner boundary oX1 of X consists of a finite number of piecewise Sm smooth components CJ ðJ ¼ 1; . . . ; mÞ such that oX1 ¼ J ¼1 CJ , while the outer boundary oX2 of X consists of a finite number of piecewise smooth S components CJ ðJ ¼ m þ 1; . . . nÞ such that oX2 ¼ nJ ¼mþ1 CJ and cJ are positive constant impedances. The basic problem is to determine some geometrical quantities of X associated with the main problem (1.9), (1.10) from the asymptotic expansions of b l ðtÞ for small jtj, which does not seem to have been investigated elsewhere. We close this section with the remark that the special cases of the main problem (1.9), (1.10) have been discussed by many authors, see for example [16–20,23,25,27]. Thus, this problem can be considered as a more general one which has not been discussed before. 2. Statement of results Suppose that the inner boundary oX1 of the annular region X is given locally by the equations xa ¼ y a ðr1 Þ ða ¼ 1; 2Þ in which r1 is the arc-length of the counterclockwise oriented inner boundary oX1 , where y a ðr1 Þ 2 C 1 ðoX1 Þ. Suppose that the outer boundary oX2 of the annular region X is given locally by the equations xa ¼ y a ðr2 Þ ða ¼ 1; 2Þ in which r2 is the arc-length of the counterclockwise oriented outer boundary oX2 , where y a ðr2 Þ 2 C 1 ðoX2 Þ. Let LJ ðJ ¼ 1; . . . ; mÞ be the lengths of the components CJ ðJ ¼ 1; . . . ; mÞ, of the inner boundary oX1 and let K1 ðr1 Þ be the curvature of oX1 . Let LJ ðJ ¼ m þ 1; . . . ; nÞ be the lengths of the components CJ ðJ ¼ m þ 1; . . . ; nÞ, of the outer boundary oX2 and let K2 ðr2 Þ be the curvature of oX2 . Let hJ > 0 ðJ ¼ 1; . . . ; nÞ be sufficiently small numbers. Let nJ ðJ ¼ 1; . . . ; nÞ be the minimum distances from a point x ¼ ðx1 ; x2 Þ of the region X to the components CJ ðJ ¼ 1; . . . ; nÞ, respectively. Let n ðr1 Þ denote the inward drawn unit normals to the components CJ ðJ ¼ 1; . J. . ; mÞ of the inner boundary oX1 and let n ðr2 Þ denote the inward drawn unit normals to the components CJ ðJ ¼ J m þ 1; . . . ; nÞ of the outer boundary oX2 , respectively. Then we note that the coordinates in the neighborhood of the components CJ ðJ ¼ 1; . . . ; mÞ and its diagrams are in the same form as in Section 5.2 of [29] with the interchanges n1 $ nJ ; h1 $ hJ ; I1 $ IJ . DðI1 Þ $ DðIJ Þ and d $ dJ ðJ ¼ 1; . . . ; mÞ. Thus, we have the same formulae (5.2.1)–(5.2.5) of Section 5.2 in [29] with the interchanges n1 $ nJ , n ðr1 Þ $ n ðr1 Þ and tðr1 Þ $ tJ ðr1 Þ ðJ ¼ 1; . . . ; mÞ. 1
J
Similarly, the coordinates in the neighborhood of the components oXJ ðJ ¼
E.M.E. Zayed, I.H. Abdel-Halim / Appl. Math. Comput. 132 (2002) 187–204
191
m þ 1; . . . ; nÞ and its diagrams are in the same form as in Section 5.1 of [29] with the interchanges n2 $ nJ ; h2 $ hJ ; I2 $ IJ . DðI2 Þ $ DðIJ Þ and d2 $ dJ ðJ ¼ m þ 1; . . . ; nÞ. Thus, we have the same formulae (5.1.1)–(5.1.5) of Section 5.1 in [29] with the interchanges n2 $ nJ , n ðr2 Þ $ n ðr2 Þ and tðr2 Þ $ 2 J tJ ðr2 Þ ðJ ¼ m þ 1; . . . ; nÞ. Theorem 2.1. With the assumptions stated above, the asymptotic expansion of the spectral distribution b l ðtÞ for small jtj of the main problem (1.9), (1.10) can be written in the form: a1 b l ðtÞ ¼ H ðjtjÞ þ a2 sign t þ a3 jtj þ a4 t2 sign t þ Oðt3 sign tÞ t as jtj ! 0; ð2:1Þ where if 0 < cJ 1 ðJ ¼ 1; . . . ; bÞ; cJ 1 ðJ ¼ b þ 1; . . . ; mÞ; 0 < cJ 1 ðJ ¼ m þ 1; . . . ; cÞ, and cJ 1 ðJ ¼ c þ 1; . . . ; nÞ, then the coefficients at ðt ¼ 1; . . . ; 4Þ have the forms: a1 ¼ jXj=2p; (" # b m X X
1 1 a2 ¼ LJ LJ þ 2pcJ 8 J ¼1 J ¼bþ1 " #) c n X X
1 LJ LJ þ 2pcJ þ ; J ¼mþ1
a3 ¼
b X J ¼1
cJ L J
J ¼cþ1 c X
cJ L J
ð2:2Þ
ð2:3Þ
!, 2p;
ð2:4Þ
J ¼mþ1
( b Z X 1 64 pcJ 7 K12 ðr1 Þ c2J dr1 a4 ¼ 512 7 LJ CJ J ¼1 3 # Z " m X 2p 2 dr1 þ K1 ðr1 Þ c1 J L J C J J ¼bþ1 Z c X 64 pcJ 2 2 þ7 K2 ðr2 Þ cJ dr2 7 LJ J ¼mþ1 CJ " # ) 3 Z n X 2p 2 þ K2 ðr2 Þ c1 dr2 : J L J J ¼cþ1 CJ
ð2:5Þ
With reference to formulae (1.6)–(1.8) and to the articles [19,23,27], the asymptotic expansions (2.1) may be interpreted as follows: 1. X is a general annular bounded domain in R2 and we have the piecewise smooth Robin boundary conditions (1.10) with small/large impedances cJ ðJ ¼ 1; . . . ; nÞ.
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2. For the first four terms, X is a general annular bounded domain in R2 of area jXj, it has " !# c b X X 3 h¼ 1þ c LJ cJ LJ p J ¼mþ1 J J ¼1 holes, provided ‘‘h’’ is a positive Pb integer, where the components CJ ðJ ¼ 1; . . . ; bÞ of oX1 are of lengths J ¼1 LJ and of curvatures 1=2 64 pcJ 2 2 K1 ðr1 Þ cJ 7 LJ together with the Neumann boundary conditions, and P the remaining components CJ ðJ ¼ b þ 1; . . . ; mÞ of oX1 are of lengths mJ¼bþ1 ðLJ þ 2pc1 J Þ and of curvatures " 3 #1=2 2p 2 K1 ðr1 Þ c1 J LJ together with the Dirichlet boundary conditions, while the components Pc CJ ðJ ¼ m þ 1; . . . ; cÞ of oX2 are of lengths J ¼mþ1 LJ and of curvatures 1=2 64 pcJ 2 2 K2 ðr2 Þ cJ 7 LJ together with the Neumann boundary conditions, and P the remaining components CJ ðJ ¼ c þ 1; . . . ; nÞ of oX2 are of lengths nJ ¼cþ1 ðLJ þ 2pc1 J Þ and of curvatures " 3 #1=2 2p 2 K2 ðr2 Þ c1 J LJ together with the Dirichlet boundary conditions.
3. The Proof of Theorem 2.1 b ðtÞ asWith reference to the articles [16,19,23,27], it is easy to show that l sociated with the main problem (1.9), (1.10) is given by Z Z b l ðt Þ ¼ Gðx; x ; tÞ dx; ð3:1Þ X
where Gðx1 ; x2 ; tÞ is the Green function for the wave equation
E.M.E. Zayed, I.H. Abdel-Halim / Appl. Math. Comput. 132 (2002) 187–204
o2 D 2 Gðx1 ; x2 ; tÞ ¼ 0 ot
in X f 1 < t < 1g;
193
ð3:2Þ
subject to the piecewise smooth Robin boundary conditions (1.9) and the initial conditions oGðx1 ; x2 ; tÞ lim Gðx1 ; x2 ; tÞ ¼ 0;
t!0
lim
t!0
¼ dðx1 x2 Þ;
ot
ð3:3Þ
where dðx1 x2 Þ is the Dirac delta function located at the source point x2 . Let us write Gðx1 ; x2 ; tÞ ¼ G0 ðx1 ; x2 ; tÞ þ ,ðx1 ; x2 ; tÞ;
ð3:4Þ
where
G0 ðx1 ; x2 ; tÞ ¼
H ðjtj jx1 x2 jÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2p t2 jx1 x2 j
ð3:5Þ
is the ‘‘fundamental solution’’ of the wave equation (3.2) while ,ðx1 ; x2 ; tÞ is the ‘‘regular solution’’ chosen in such a way that Gðx1 ; x2 ; tÞ satisfies the piecewise smooth Robin boundary conditions (1.9). On setting x1 ¼ x2 ¼ x , we find that
jXj b H ðjtjÞ þ KðtÞ; l ðtÞ ¼ 2pt
ð3:6Þ
where KðtÞ ¼
Z Z X
ð3:7Þ
,ðx; x ; tÞ dx:
In what follows we shall use Fourier transforms with respect to 1 < t < 1 and use 1 < g < 1 as the Fourier transform parameter. Thus, we define Z Z þ1 b e2pigt Gðx1 ; x2 ; tÞ dt: ð3:8Þ G ðx1 ; x2 ; gÞ ¼
1
An application of the Fourier transform to the wave equation (3.2) shows that b ðx1 ; x2 ; gÞ satisfies the reduced wave equation G
b ðx1 ; x2 ; gÞ ¼ dðx1 x2 Þ in X; ðD þ 4p2 g2 Þ G
ð3:9Þ
together with the piecewise smooth Robin boundary conditions (1.9). The asymptotic expansion of KðtÞ, for small jtj, may then be deduced dib ðgÞ, for large jgj, where rectly from the asymptotic expansion of K
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b ðgÞ ¼ K
Z Z X
b , ðx; x ; gÞ dx:
ð3:10Þ
It is well known (see for example [19,20,23]) that Eq. (3.9) has the fundamental b 0 ðx1 ; x2 ; gÞ ¼ 1 Y0 ð2pgrx x Þ where rx x ¼ jx1 x2 j is the distance solution G 1 2 1 2 2 between the points x1 and x2 of the region X, while Y0 is the Bessel function of the second kind and of zero order. The existence of this solution enables us to b ðx1 ; x2 ; gÞ satisfying the piecewise smooth construct integral equations for G Robin boundary conditions (1.9) for small/large impedances cJ , where 0 < cJ 1 ðJ ¼ 1; . . . ; bÞ, cJ 1 ðJ ¼ b þ 1; . . . ; mÞ, 0 < cJ 1 ðJ ¼ m þ 1; . . . ; cÞ, and cJ 1 ðJ ¼ c þ 1; . . . ; nÞ. Therefore, the Green theorem gives the following integral equation: b ðx1 ; x2 ; gÞ G
2 3 b Z X 1 1 b ðx1 ; y ; gÞ4 o þ cJ 5Y0 ð2pgr y x Þ dy ¼ Y0 ð2pgrx1 x2 Þ G 2 2 2 J ¼1 CJ onJy 3 2 32 Z m 1 X o o b ðx1 ; y ; gÞ541 þ c1 5Y0 ð2pgr y x Þ dy 4 þ G 2 J 2 J ¼bþ1 CJ onJy onJy 2 3 Z c 1 X o b ðx1 ; y ; gÞ4 þ cJ 5Y0 ð2pgr y x2 Þ dy þ G 2 J ¼mþ1 CJ onJy 3 2 32 Z n 1 X o o b ðx1 ; y ; gÞ541 þ c1 5Y0 ð2pgr y x Þ dy : 4 G 2 J 2 J ¼cþ1 CJ onJy onJy
ð3:11Þ On applying the iteration methods (see, for example [18–20,23]) to the inb ðx1 ; x2 ; gÞ, which has a tegral equation (3.11), we obtain the Green function G regular part of the form !, 20 X b , ðx1 ; x2 ; gÞ ¼ Am 4; ð3:12Þ
m¼1
where A1 ¼
b Z X J ¼1
CJ
2
3 o þ cJ 5Y0 ð2pgr y x2 Þ dy ; Y0 ð2pgrx1 y Þ4 onJy
E.M.E. Zayed, I.H. Abdel-Halim / Appl. Math. Comput. 132 (2002) 187–204
A2 ¼
Z m X
A4 ¼
3 o þ cJ 5Y0 ð2pgr y x2 Þ dy ; Y0 ð2pgrx1 y Þ4 onJy
J ¼cþ1
CJ
2
Z n X CJ
2
Z c X J ¼mþ1
195
3
32
4 o Y0 ð2pgrx y Þ541 þ c1 o 5Y0 ð2pgr y x Þ dy ; 1 2 J onJy onJy
CJ
J ¼bþ1
A3 ¼
2
3
32
4 o Y0 ð2pgrx y Þ541 þ c1 o 5Y0 ð2pgr y x Þ dy ; 1 2 J onJy onJy
82 9 3 > > Z Z < b = X 6 o 7 0 þ cJ 5Y0 ð2pgr 0 Þ dy dy 0 ; Y0 ð2pgrx1 y ÞM1 ðy; y Þ 4 A5 ¼ y x2 > > CJ CJ : onJy0 ; J ¼1 3 o 4 A6 ¼ Y0 ð2pgrx1 y Þ5M2 ðy; y 0 Þ on Jy C C J J J ¼bþ1 9 82 3 > > = < 6 1 o 7 4 1 þ cJ 5Y0 ð2pgr y0 x Þ dy dy 0 ; > 2 > onJy 0 : ;
A7 ¼
2
Z m X
Z
Z
Z
c X J ¼mþ1
CJ
CJ
82 9 3 > > < = 6 o 7 0 þ cJ 5Y0 ð2pgr 0 Þ dy dy 0 ; Y0 ð2pgrx1 y ÞM3 ðy ; yÞ 4 y x2 > > : onJy0 ;
3 o 4 A8 ¼ Y0 ð2pgrx1 y Þ5M4 ðy; y 0 Þ on Jy C C J J J ¼cþ1 82 9 3 > > < = 6 1 o 7 4 1 þ cJ 5Y0 ð2pgr y0 x Þ dy dy 0 ; > 2 > onJy 0 : ; Z n X
Z
2
8 9 2 3 Z b Z m < X = X o 4 A9 ¼ Y0 ð2pgrx1 y Þ5M5 ðy; y 0 Þ dy ; onJy CJ : J ¼bþ1 CJ J ¼1 82 9 3 > > < = 6 o 7 þ cJ 5Y0 ð2pgr 0 Þ dy 0 ; 4 y x2 > > : onJy0 ;
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E.M.E. Zayed, I.H. Abdel-Halim / Appl. Math. Comput. 132 (2002) 187–204
A10 ¼
( Z b Z c X X CJ
J ¼1
) 0
Y0 ð2pgrx1 y ÞM1 ðy; y Þ dy CJ
J ¼mþ1
82 9 3 > > < = 6 o 7 þ cJ 5Y0 ð2pgr y 0 x2 Þ dy 0 ; 4 > > : onJy0 ; 8 9 2 3 Z b Z n < X = X o 4 Y0 ð2pgrx1 y Þ5M5 ðy; y 0 Þ dy A11 ¼ ; onJy CJ : J ¼cþ1 CJ J ¼1 82 9 3 > > < = 6 o 7 þ cJ 5Y0 ð2pgr 0 Þ dy 0 ; 4 y x2 > > : onJy0 ; A12 ¼
Z (X m b Z X J ¼bþ1
CJ
) 0
Y0 ð2pgrx1 y ÞM6 ðy ; yÞ dy CJ
J ¼1
82 9 3 > > < = o 7 6 0 41 þ c1 Y ð2pgr Þ dy 0 ; 5 0 y x2 J > > onJy 0 : ; A13 ¼
Z ( X Z m c X J ¼bþ1
CJ
J ¼mþ1
) 0
Y0 ð2pgrx1 y ÞM6 ðy; y Þ dy
CJ
82 9 3 > > < = o 7 6 0 41 þ c1 dy 0 ; Y ð2pgr Þ 5 0 J y x2 > > 0 on Jy : ; 8 9 2 3 Z < X Z = m n X o 4 A14 ¼ Y0 ð2pgrx1 y Þ5M2 ðy; y 0 Þ dy : J ¼cþ1 CJ onJy ; J ¼bþ1 CJ 82 9 3 > > < = 6 1 o 7 4 1 þ cJ 5Y0 ð2pgr y0 x Þ dy 0 ; > 2 > onJy 0 : ; A15 ¼
Z (X c b Z X J ¼mþ1
CJ
J ¼1
) 0
Y0 ð2pgrx1 y ÞM3 ðy; y Þ dy CJ
82 9 3 > > < = 6 o 7 þ cJ 5Y0 ð2pgr y 0 x2 Þ dy 0 ; 4 > > : onJy0 ;
E.M.E. Zayed, I.H. Abdel-Halim / Appl. Math. Comput. 132 (2002) 187–204
8 9 2 3 Z < X Z = c m X o 4 A16 ¼ Y0 ðx1 ; y ; gÞ5M7 ðy; y 0 Þ dy : J ¼bþ1 CJ onJy ; J ¼mþ1 CJ 82 9 3 > > < = 6 o 7 þ cJ 5Y0 ð2pgr 0 Þ dy 0 ; 4 y x2 > > : onJy0 ; 8 9 2 3 Z < X Z c n = X o 4 A17 ¼ Y0 ðx1 ; y ; gÞ5M7 ðy; y 0 Þ dy : J ¼cþ1 CJ onJy ; J ¼mþ1 CJ 82 9 3 > > < = 6 o 7 4 þ cJ 5Y0 ð2pgr 0 Þ dy 0 ; y x2 > > : onJy0 ; A18 ¼
Z (X n b Z X
) 0
Y0 ðx1 ; y ; gÞM8 ðy; y Þ dy 9 2 3 > > < = o 7 6 0 dy 0 ; 41 þ c1 Y ð2pgr Þ 5 0 J y x2 > > 0 on Jy : ; J ¼cþ1 8
CJ
J ¼1
CJ
8 9 2 3 Z < X Z = n m X o 4 A19 ¼ Y0 ðx1 ; y ; gÞ5M4 ðy; y 0 Þ dy : J ¼bþ1 CJ onJy ; J ¼cþ1 CJ 82 9 3 > > < = 6 1 o 7 4 1 þ cJ 5Y0 ð2pgr y0 x Þ dy 0 ; > 2 > onJy 0 : ; A20 ¼
Z ( X Z n c X
) 0
Y0 ðx1 ; y ; gÞM4 ðy; y Þ dy 9 2 3 > > < = o 7 6 0 41 þ c1 Y ð2pgr Þ dy 0 ; 5 0 y x2 J > > 0 on Jy : ; J ¼cþ1 8
where M1 ðy ; y 0 Þ ¼
M2 ðy ; y 0 Þ ¼
M3 ðy ; y 0 Þ ¼
CJ
CJ
J ¼mþ1
1 X t ðtÞ ð1Þ K1 ðy 0 ; y Þ;
t¼0 1 X
ðtÞ
K2 ðy 0 ; y Þ;
t¼0 1 X t¼0
ðtÞ
K1 ðy 0 ; y Þ;
197
198
E.M.E. Zayed, I.H. Abdel-Halim / Appl. Math. Comput. 132 (2002) 187–204
M4 ðy ; y 0 Þ ¼
1 X t ðtÞ ð1Þ K2 ðy 0 ; y Þ;
t¼0
M5 ðy ; y 0 Þ ¼
1 X t ðtÞ ð1Þ K3 ðy 0 ; y Þ;
t¼0
M6 ðy ; y 0 Þ ¼
1 X
1 X
ðtÞ
ðtÞ
K3 ðy 0 ; y Þ;
t¼0
M8 ðy ; y 0 Þ ¼
1 X t ðtÞ ð1Þ K4 ðy 0 ; y Þ;
t¼0
and
K4 ðy 0 ; y Þ;
t¼0
M7 ðy ; y 0 Þ ¼
2
3
16 o 7 ð0Þ K1 ðy 0 ; y Þ ¼ 4 þ cJ 5Y0 ð2pgr 0 Þ; yy 2 onJy 0
2
3 2
16 o o 7 ð0Þ þ c1 K2 ðy 0 ; y Þ ¼ 4 5Y0 ð2pgr y 0 y Þ; J 2 onJy onJy 0 onJy
2
3
16 o 7 ð0Þ K3 ðy 0 ; y Þ ¼ 41 þ c1 5Y0 ð2pgr y 0 y Þ; J 2 onJy 0
2
3
16 o2 o 7 ð0 K4 ðy 0 ; y Þ ¼ 4 þ c1 5Y0 ð2pgr y 0 y Þ: J 2 onJy 0 onJy onJy 0
ðtÞ
In these formulae, we note for example, that Ki ðy 0 ; y Þ ði ¼ 1; . . . ; 4Þ are the
ð0Þ
iterates of Ki ðy 0 ; y Þ ði ¼ 1; . . . ; 4Þ, respectively. On the basis of (3.12), the function b , ðx1 ; x2 ; gÞ will be estimated for large jgj together with small/large impedances cJ . The case when x1 and x2 lie in the neighborhood of the com ponents CJ ðJ ¼ 1; . . . ; mÞ of the inner boundary oX1 or in the neighborhood of the components CJ ðJ ¼ m þ 1; . . . ; nÞ of the outer boundary oX2 of the annular region X is particularly interesting. For this case, we use the local expansions of the functions Y0 ð2pgrx y Þ;
o Y0 ð2pgrx y Þ onJy
ðJ ¼ 1; . . . ; nÞ;
ð3:13Þ
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199
when the distance between x and y is small, which are very similar to that obtained in Sections 4, 5 of [19]. Consequently, the local behavior of the kernels ð0Þ
ð0Þ
ð0Þ
ð0Þ
K1 ðy 0 ; y Þ; K2 ðy 0 ; y Þ; K3 ðy 0 ; y Þ; K4 ðy 0 ; y Þ;
when the distance between y and y 0 is small, and for small/large impedances cJ follows directly from knowledge of the local expansions of (3.13). Definition 1. Let n1 and n2 be points in the upper half-plane n2 > 0. Then, we define qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q12 ¼ ðn11 n12 Þ2 þ ðn21 þ n22 Þ2 : An ek ðn1 ; n2 ; gÞ-function is defined for points n1 n2 belonging to sufficiently small domains DðIJ Þ except when n1 ¼ n2 2 IJ ðJ ¼ 1; . . . ; nÞ and k is called the degree of this function. For every positive integer K, it has the expansion: !‘ !m X o o 1 2 p1 2 p2 k e ðn1 ; n2 ; gÞ ¼ f ðn1 Þðn1 Þ ðn2 Þ Y0 ð2pgq12 Þ on11 on21 þ RK ðn1 ; n2 ; gÞ;
P
where denotes a sum of a finite number of terms in which f ðn11 Þ is an infinitely differentiable function (see [9,11,19,20]). In this expansion p1 ; p2 ; ‘; m are integers, p1 P 0; p2 P 0; ‘ P 0; k ¼ minðp1 þ p2 qÞ and q ¼ ‘ þ m. The remainder RK ðn1 ; n2 ; gÞ has continuous derivatives of order k 6 K satisfying
Dk RK ðn1 ; n2 ; gÞ ¼ OðgK eAJ gq12 Þ as jgj ! 1;
where AJ ðJ ¼ 1; . . . ; nÞ are positive constants. Thus, using methods similar to that obtained in Sections 6–10 of [19], we can show that the functions (3.13) are ek -functions with degrees k ¼ 0; 1, respectively. Consequently the kernels Ki ðy 0 ; y Þ ði ¼ 1; . . . ; 4Þ are ek -functions with degrees k ¼ 0; 0; 1; 1, respectively. Definition 2. Let x1 and x2 be points in large domains X þ CJ ðJ ¼ 1; . . . ; nÞ. Then we define r1 ¼ minðrx1 y þ rx2 y Þ y
r2 ¼ minðrx1 y þ rx2 y Þ y
if y 2 CJ ðJ ¼ 1; . . . ; bÞ;
if y 2 CJ ðJ ¼ b þ 1; . . . ; mÞ;
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r3 ¼ minðrx1 y þ rx2 y Þ y
if y 2 CJ ðJ ¼ m þ 1; . . . ; cÞ;
r4 ¼ minðrx1 y þ rx2 y Þ y
if y 2 CJ ðJ ¼ c þ 1; . . . ; nÞ:
An Ek ðx1 ; x2 ; gÞ-function is defined and infinitely differentiable with respect to x1 and x2 when these points belong to large domains X þ CJ ðJ ¼ 1; . . . ; nÞ. Thus the Ek -function has a similar local expansion of the ek -function (see [9,19]). With the help of Sections 8, 9 in [19] it is easily seen that formula (3.12) is an b ðx1 ; x2 ; gÞ can be estimated as follows: E0 ðx1 ; x2 ; gÞ-function and consequently G
b X # $ b ðx1 ; x2 ; gÞ ¼ G O ½1 þ j logð2pgr1 jeAJ gr1
þ þ þ
J ¼1 m X
$ # O ½1 þ j logð2pgr2 jeAJ gr2
J ¼bþ1 c X
$ # O ½1 þ j logð2pgr3 jeAJ gr3
J ¼mþ1 n X
$ # O ½1 þ j logð2pgr4 jeAJ gr4 ;
ð3:14Þ
J ¼cþ1
which is valid for jgj ! 1 and for small/large impedances cJ ðJ ¼ 1; . . . ; nÞ where AJ ðJ ¼ 1; . . . ; nÞ are positive constants. The estimate (3.14) shows that b ðx1 ; x2 ; gÞ is exponentially small for large jgj. This prove that G b ðx1 ; x2 ; gÞ G converges for jgj ! 1. With reference to Section 10 in [19], if the ek -expansions of the functions Ki ðy 0 ; y Þ ði ¼ 1; . . . ; 4Þ is introduced into (3.12) and if we use formulae similar to (6.4) and (6.9) of Section 6 in [19], we obtain the following local behavior of the regular part b , ðx1 ; x2 ; gÞ when r1 ; r2 ; r3 and r4 are small, which is valid for jgj ! 1 and for small/large impedances cJ ðJ ¼ 1; . . . ; nÞ: b , ðx1 ; x2 ; gÞ ¼
n X J ¼1
b , J ðx1 ; x2 ; gÞ;
ð3:15Þ
where (a) if x1 and x2 belong to sufficiently small domains DðIJ Þ ðJ ¼ 1; . . . ; bÞ, then 8 !1 9 < = 1 o b , J ðx1 ; x2 ; gÞ ¼ 1 cJ Y ð2pgq1 2Þ 2 ; 0 4: on1 þ Ofg1 expðAJ gq1 2Þg as jgj ! 1:
ð3:16Þ
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201
(b) if x1 and x2 belong to sufficiently small domains DðIJ Þ ðJ ¼ b þ 1; . . . ; mÞ, then ( !) 1 o 1 b 1 cJ , j ðx1 ; x2 ; gÞ ¼ Y0 ð2pgq1 2Þ 4 on21 þ Ofg1 expðAJ gq1 2Þg
as jgj ! 1:
ð3:17Þ
(c) if x1 and x2 belong to sufficiently small domains DðIJ Þ ðJ ¼ m þ 1; . . . ; cÞ, then 8 !1 9 < = 1 o b 1 cJ Y ð2pgq1 2Þ , J ðx1 ; x2 ; gÞ ¼ ; 0 4: on21 þ Ofg1 expðAj gq1 2Þg
as jgj ! 1:
ð3:18Þ
(d) if x1 and x2 belong to sufficiently small domains DðIJ Þ ðJ ¼ c þ 1; . . . ; nÞ, then ( !) 1 o 1 b 1 cJ , J ðx1 ; x2 ; gÞ ¼ Y0 ð2pgq1 2Þ 4 on21 þ Ofg1 expðAJ gq1 2Þg as jgj ! 1:
ð3:19Þ
When r1 P dJ > 0 ðJ ¼ 1; . . . ; bÞ, r2 P dJ > 0 ðJ ¼ b þ 1; . . . ; mÞ, r3 P dJ > 0 ðJ ¼ m þ 1; . . . ; cÞ, and r4 P dJ > 0 ðJ ¼ c þ 1; . . . ; nÞ the function b , ðx1 ; x2 ; gÞ is of order OfeBg g as jgj ! 1, B ¼ constant > 0. Thus, since lim ra =q12 ¼ 1 ða ¼ 1; . . . ; 4Þ (see [19,20]) then we have the asymptotic forra !0 mulae (3.16)–(3.19) with q12 in the small domain DðIJ Þ ðJ ¼ 1; . . . ; nÞ are replaced by ra ða ¼ 1; . . . ; 4Þ in the large domains X þ oXJ ðJ ¼ 1; . . . ; nÞ, respectively. Since for n2 P hJ > 0 ðJ ¼ 1; . . . ; nÞ the functions b , J ðx; x ; gÞ are of order Ofexpð2gAJ hJ Þg, the integral over the general annular bounded domain X of the function b , ðx; x ; gÞ can be approximated in the following way (see (3.10)): Z hJ Z LJ n X b b , J ðx; x; gÞf1 K2 ðn1 Þn2 g dn1 dn2 K ðgÞ ¼ J ¼mþ1 m X
þ
J ¼1 n X
n2 ¼0
Z
hJ n2 ¼0
n1 ¼0
Z
LJ
n1 ¼0
b , J ðx; x; gÞf1 þ K1 ðn1 Þn2 g dn1 dn2
Ofexpð2gAJ hJ Þg as jgj ! 1:
ð3:20Þ
J ¼1
b J ðx; x; gÞ ðJ ¼ 1; . . . ; nÞ (see [19,20]) are introduced into If the ek -expansion of , (3.20), and with the help of formula (11.3) of Section 11 in [19], we deduce after
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inverting Fourier transforms and using (3.6) that the result (2.1) has been constructed, and the proof of Theorem 2.1 is completed.
4. Discussions and conclusions The problem of determining some geometric quantities of the general annular bounded domain X in R2 with a finite number of piecewise smooth Robin boundary conditions (1.10), from complete knowledge of its eigenvalues has been discussed in the present paper by using the wave equation approach. It is well known that the wave equation approach has given very strong result; the definite one is that of Hormander [6] who has studied the distribution tr½expðitP Þ near to t ¼ 0 for an elliptic positive semi-definite Pseduodifferential operator P of order m in Rn . Recently, the wave equation approach in solving particular problems has been discussed by Zayed and Abdel-Halim [16–18], Zayed and Hassan [19,20], Zayed and Younis [23] and Zayed [25,27] who have studied the spectral distribution b l ðtÞ ¼
1 X
expðitEx1=2 Þ as jtj ! 0;
x¼1
for some different bounded domains in R2 with certain boundary conditions. An alternative method is the heat equation approach, which has been investigated by many authors, see for example, [2–5,7–14,21,22,24,26,28– 34]. They have studied the asymptotic expansion of the trace of the heat kernel HðtÞ ¼
1 X
expðtEx Þ
as t ! 0;
x¼1
for some bounded domains in R2 with certain boundary conditions. With reference to the above articles one can ask a question, is it possible just by listening with a perfect ear to hear the shape of the bounded domain X even if we can not see it? This question has been put rather nicely by Kac [7], who simply asked, ‘‘Can one hear the shape of a drum?’’ Recently Gordon et al. [2] have shown that the answer of Kac’s question is negative. They have shown explicitly two domains that although having different shapes, have the same eigenvalues (i.e., isospectral domains). This theoretical result was experimentally verified in [15] by employing thin microwave cavities shaped in the form of two different domains known to be isospectral. In particular, the present paper provides useful techniques to inverse problem methods via the spectral distribution of the Laplacian.
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203
References [1] R. Courant, D. Hilbert, Methods of Mathematical Physics, vol. 1, Wiley–Interscience, New York, 1953. [2] C. Gordon, D.L. Webb, S. Wolpert, One can not hear the shape of a drum, Bull. Amer. Math. Soc. 27 (1992) 134–138. [3] H.P.W. Gottlieb, Hearing the shape of an annular drum, J. Austral. Math. Soc. Ser. B 34 (1983) 435–438. [4] H.P.W. Gottlieb, Eigenvalues of the Laplacian with Neumann boundary conditions, J. Austral. Math. Soc. Ser. B 26 (1985) 293–309. [5] H.P.W. Gottlieb, Eigenvalues of the Laplacian for rectilinear regions, J. Austral. Math. Soc. Ser. B 29 (1988) 270–281. [6] L. Hormander, The spectral function of an elliptic operator, Acta Math. 121 (1968) 193– 218. [7] M. Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966) 1–23. [8] H.P. McKean, I.M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geom. 1 (1967) 43–69. [9] A. Pleijel, A study of certain Green’s functions with applications in the theory of vibrating membranes, Ark. Mat. 2 (1954) 553–569. [10] M.H. Protter, Can one hear the shape of a drum? revisited, SIAM Rev. 29 (1987) 185–197. [11] B.D. Sleeman, E.M.E. Zayed, An inverse eigenvalue problem for a general convex domain, J. Math. Anal. Appl. 94 (1983) 78–95. [12] B.D. Sleeman, E.M.E. Zayed, Trace formulae for the eigenvalues of the Laplacian, Z. Angew. Math. Phys. 35 (1984) 106–115. [13] L. Smith, The asymptotics of the heat equation for a boundary value problem, Invent. Math. 63 (1981) 467–493. [14] K. Stewartson, R.T. Waechter, On hearing the shape of a drum: further results, Proc. Camb. Phil. Soc. 69 (1971) 353–363. [15] S. Sridhar, A. Kudrolli, Experiments on Not hearing the shape of drums, Phys. Rev. Lett. 72 (1994) 2175–2178. [16] E.M.E. Zayed, I.H. Abdel-Halim, Short-time asymptotics of the trace of the wave operator for a general annular drum in R2 with Robin boundary conditions, Indian J. Pure Appl. Math. 32 (2001) 493–500. [17] E.M.E. Zayed, I.H. Abdel-Halim, The wave equation approach to the two-dimensional inverse problem for a general bounded domain with piecewise smooth mixed boundary conditions, Acta Math. Sci. 21 (2) (2001). [18] E.M.E. Zayed, I.H. Abdel-Halim, The wave equation approach to an inverse eigenvalue problem for an arbitrary multiply connected drum in R2 with Robin boundary conditions, Internat. J. Math. Math. Sci. 25 (2001) 717–726. [19] E.M.E. Zayed, A.A.M. Hassan, The wave equation approach to an inverse problem for a general convex domain, Bull. Calcutta. Math. Soc. 81 (1989) 139–156. [20] E.M.E. Zayed, A.A.M. Hassan, The wave equation approach to an inverse problem for a region in R2 with two impedance boundary conditions, J. Institute. Math. Comput. Sci. (Math. Ser.) 2 (1989) 153–167. [21] E.M.E. Zayed, A.I. Younis, An inverse problem for a general convex domain with impedance boundary conditions, Quart. Appl. Math. 48 (1990) 181–188. [22] E.M.E. Zayed, A.I. Younis, On hearing the shape of rectilinear regions, J. Math. Phys. 35 (1994) 3490–3496. [23] E.M.E. Zayed, A.I. Younis, The wave equation approach to an inverse problem for a general convex domain with impedance boundary conditions, Bull. Greek. Math. Soc. 37 (1995) 123– 130.
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[24] E.M.E. Zayed, An inverse eigenvalue problem for the Laplace operator, Lecture Notes in Mathematics, vol. 964, Springer, Berlin, 1982, pp. 718–726. [25] E.M.E. Zayed, The wave equation approach to inverse problems: an extension to higher dimensions, Bull. Calcutta Math. Soc. 78 (1986) 281–291. [26] E.M.E. Zayed, Eigenvalues of the Laplacian for the third boundary value problem, J. Austral. Math. Soc. Ser. B 29 (1987) 79–87. [27] E.M.E. Zayed, The wave equation approach to Robin inverse problem for a doubly connected region, Bull. Calcutta Math. Soc. 80 (1988) 331–342. [28] E.M.E. Zayed, Hearing the shape of a general convex domain, J. Math. Anal. Appl. 142 (1989) 170–187. [29] E.M.E. Zayed, Heat equation for an arbitrary doubly-connected region in R2 with mixed boundary conditions, J. Appl. Math Phys. (ZAMP) 40 (1989) 339–355. [30] E.M.E. Zayed, On hearing the shape of an arbitrary doubly-connected region in R2 , J. Austral. Math. Soc. Ser. B 31 (1990) 472–483. [31] E.M.E. Zayed, Heat equation for an arbitrary multiply connected region in R2 with impedance boundary conditions, IMA J. Appl. Math. 45 (1990) 233–241. [32] E.M.E. Zayed, An inverse problem for a general multiply connected bounded domain, Appl. Anal. 59 (1995) 121–145. [33] E.M.E. Zayed, Short-time asymptotics of the heat kernel of the Laplacian of a bounded domain with Robin boundary conditions, Houston J. Math. 24 (1998) 377–385. [34] E.M.E. Zayed, The asymptotics of the heat semigroup for a general bounded domain with mixed boundary conditions, Acta Math. Sinica (English Ser.) 16 (2000) 627–636.
An inverse problem of the wave equation for a general doubly connected region in R2 with a finite number of piecewise smooth Robin boundary conditions E.M.E. Zayed *, I.H. Abdel-Halim Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt
Abstract The spectral distribution 1 X b l ðtÞ ¼ expðitEx1=2 Þ; x¼1
where
fEx g1 x¼1 D ¼
are the eigenvalues of the negative Laplacian 2 2 X o k¼1
oxk
1 2 in the pffiffiffiffiffiffiðx ffi ; x Þ-plane, is studied for a variety of domains, where 1 < t < 1 and i ¼ 1. The dependences of b l ðtÞ on the connectivity of a bounded domain and the Robin boundary conditions are analyzed. Particular attention is given to a general annular bounded domain X in R2 with a smooth inner boundary oX1 and a smooth outer boundary oX2 where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components CJ ðJ ¼ 1; . . . ; mÞ of oX1 and on the piecewise S smooth components S CJ ðJ ¼ m þ 1; . . . ; nÞ of oX2 are considered, such that oX1 ¼ mJ¼1 CJ and oX2 ¼ nJ ¼mþ1 CJ . Some geometrical properties of X (e.g., the area of X, the total lengths of its boundary, the curvature of the boundary, the number of holes of X, etc.) are determined, from the asymptotic expansions of b l ðtÞ for jtj ! 0. Ó 2002 Elsevier Science Inc. All rights reserved.
*
Corresponding author. E-mail addresses: [email protected] (E.M.E. Zayed), [email protected] (I.H. Abdel-Halim). 0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 1 ) 0 0 1 8 6 - 2
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1. Introduction Let X be a simply connected bounded domain in R2 with a smooth boundary oX. Consider the two-dimensional Robin problem Du ¼ Eu in X; o þ c u ¼ 0 on oX; on
ð1:1Þ ð1:2Þ
where o=on denotes differentiation along the inward pointing normal to oX and c is a positive constant impedance, with u 2 C 2 ðXÞ \ CðXÞ. Denote its eigenvalues, counted according to multiplicity, by 0 < E1 6 E2 6 E3 6 6 E x 6 ! 1
as x ! 1:
ð1:3Þ
At the beginning of this century the principal problem was that of investigating the asymptotic behavior of the eigenvalues (1.3). It is well known (see [1]) that if N ðEÞ is the number of these eigenvalues less than E, then N ðEÞ
jXj E 4p
as E ! 1
ðH: Weyl; 1912Þ
and N ðEÞ ¼
jXj E þ OðE1=2 log EÞ 4p
as E ! 1
ðR: Courant; 1920Þ;
where jXj is the area of the domain X. In order to obtain further information about the geometry of X, one studies certain functions of the spectrum. The most useful to date comes from the heat equation or the wave equation. Accordingly, let etD denote the heat operator. Then we can construct the trace function HðtÞ ¼ trðetD Þ ¼
1 X
etEx ;
ð1:4Þ
x¼1
which converges for all positive t. 1=2 Let eitD be the wave operator. Then an alternative to (1.4) is to study the trace function 1=2
b l ðtÞ ¼ trðeitD Þ ¼
1 X x¼1
1=2
eitEx ;
ð1:5Þ
pffiffiffiffiffiffiffi which represents a tempered distribution for 1 < t < 1 and i ¼ 1. In the present paper, we shall concentrate our efforts on a study of the asymptotic expansion of the tempered distribution b l ðtÞ for small jtj. Zayed et al. [20] have discussed problem (1.1), (1.2) for small/large impedance c by using the wave equation approach and have determined some geo-
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189
metrical properties of the general bounded domain X from the asymptotic expansion of b l ðtÞ as jtj ! 0. Zayed [25,27] has discussed the Robin problem (1.1), (1.2) in the following cases: Case 1.1: c ¼ 0 (The Neumann problem) b l ðtÞ ¼
jXj joXj H ðjtjÞ þ sign t þ a0 jtj þ Oðt2 sign tÞ 2pt 8
as jtj ! 0:
ð1:6Þ
as jtj ! 0;
ð1:7Þ
Case 1.2: c ! 1 (The Dirichlet problem) b l ðtÞ ¼
jXj joXj H ðjtjÞ sign t þ a0 jtj þ Oðt2 sign tÞ 2pt 8
where H ðjtjÞ is the Heaviside unit function and 8 if t > 0; <1 sign t ¼ 0 if t ¼ 0; : 1 if t < 0: In these formule, jXj is the area of X, and joXj is the total length of oX. The constant term a0 has geometric significance, e.g., if X is smooth and convex, then a0 ¼ 16 and if X is permitted to have a finite number of smooth convex holes ‘‘h’’, then a0 ¼ 16 ð1 hÞ. Case 1.3 (The mixed problem) If L1 is the length of a component C1 of the boundary oX with the Neumann boundary conditions, and if L2 is the length of the remaining component C2 ¼ oX n C1 of oX with the Dirichlet boundary conditions, then with reference to the articles [20,25,27], we get b l ðtÞ ¼
jXj ðL1 L2 Þ H ðjtjÞ þ sign t þ a0 jtj þ Oðt2 sign tÞ 2pt 8
as jtj ! 0: ð1:8Þ
Further, the order term Oðt2 sign tÞ in (1.6)–(1.8) is yet undetermined. So, in the present paper, we discuss what geometric quantities are contained in this order term. The object of this paper is to discuss the following more general inverse problem: Let X be a general annular bounded domain in R2 consisting of a simply connected bounded inner domain X1 with a smooth boundary oX1 and a simply connected bounded outer domain X2 X1 with a smooth boundary oX2 where X1 ¼ X1 [ oX1 . Suppose that the eigenvalues (1.3) are known exactly for the Helmholtz equation
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D/ ¼ E/
ð1:9Þ
in X;
together with the piecewise smooth Robin boundary conditions o ðJ ¼ 1; . . . ; m; m þ 1; . . . ; nÞ; þ cJ / ¼ 0 on ocJ onJ
ð1:10Þ
where the inner boundary oX1 of X consists of a finite number of piecewise Sm smooth components CJ ðJ ¼ 1; . . . ; mÞ such that oX1 ¼ J ¼1 CJ , while the outer boundary oX2 of X consists of a finite number of piecewise smooth S components CJ ðJ ¼ m þ 1; . . . nÞ such that oX2 ¼ nJ ¼mþ1 CJ and cJ are positive constant impedances. The basic problem is to determine some geometrical quantities of X associated with the main problem (1.9), (1.10) from the asymptotic expansions of b l ðtÞ for small jtj, which does not seem to have been investigated elsewhere. We close this section with the remark that the special cases of the main problem (1.9), (1.10) have been discussed by many authors, see for example [16–20,23,25,27]. Thus, this problem can be considered as a more general one which has not been discussed before. 2. Statement of results Suppose that the inner boundary oX1 of the annular region X is given locally by the equations xa ¼ y a ðr1 Þ ða ¼ 1; 2Þ in which r1 is the arc-length of the counterclockwise oriented inner boundary oX1 , where y a ðr1 Þ 2 C 1 ðoX1 Þ. Suppose that the outer boundary oX2 of the annular region X is given locally by the equations xa ¼ y a ðr2 Þ ða ¼ 1; 2Þ in which r2 is the arc-length of the counterclockwise oriented outer boundary oX2 , where y a ðr2 Þ 2 C 1 ðoX2 Þ. Let LJ ðJ ¼ 1; . . . ; mÞ be the lengths of the components CJ ðJ ¼ 1; . . . ; mÞ, of the inner boundary oX1 and let K1 ðr1 Þ be the curvature of oX1 . Let LJ ðJ ¼ m þ 1; . . . ; nÞ be the lengths of the components CJ ðJ ¼ m þ 1; . . . ; nÞ, of the outer boundary oX2 and let K2 ðr2 Þ be the curvature of oX2 . Let hJ > 0 ðJ ¼ 1; . . . ; nÞ be sufficiently small numbers. Let nJ ðJ ¼ 1; . . . ; nÞ be the minimum distances from a point x ¼ ðx1 ; x2 Þ of the region X to the components CJ ðJ ¼ 1; . . . ; nÞ, respectively. Let n ðr1 Þ denote the inward drawn unit normals to the components CJ ðJ ¼ 1; . J. . ; mÞ of the inner boundary oX1 and let n ðr2 Þ denote the inward drawn unit normals to the components CJ ðJ ¼ J m þ 1; . . . ; nÞ of the outer boundary oX2 , respectively. Then we note that the coordinates in the neighborhood of the components CJ ðJ ¼ 1; . . . ; mÞ and its diagrams are in the same form as in Section 5.2 of [29] with the interchanges n1 $ nJ ; h1 $ hJ ; I1 $ IJ . DðI1 Þ $ DðIJ Þ and d $ dJ ðJ ¼ 1; . . . ; mÞ. Thus, we have the same formulae (5.2.1)–(5.2.5) of Section 5.2 in [29] with the interchanges n1 $ nJ , n ðr1 Þ $ n ðr1 Þ and tðr1 Þ $ tJ ðr1 Þ ðJ ¼ 1; . . . ; mÞ. 1
J
Similarly, the coordinates in the neighborhood of the components oXJ ðJ ¼
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191
m þ 1; . . . ; nÞ and its diagrams are in the same form as in Section 5.1 of [29] with the interchanges n2 $ nJ ; h2 $ hJ ; I2 $ IJ . DðI2 Þ $ DðIJ Þ and d2 $ dJ ðJ ¼ m þ 1; . . . ; nÞ. Thus, we have the same formulae (5.1.1)–(5.1.5) of Section 5.1 in [29] with the interchanges n2 $ nJ , n ðr2 Þ $ n ðr2 Þ and tðr2 Þ $ 2 J tJ ðr2 Þ ðJ ¼ m þ 1; . . . ; nÞ. Theorem 2.1. With the assumptions stated above, the asymptotic expansion of the spectral distribution b l ðtÞ for small jtj of the main problem (1.9), (1.10) can be written in the form: a1 b l ðtÞ ¼ H ðjtjÞ þ a2 sign t þ a3 jtj þ a4 t2 sign t þ Oðt3 sign tÞ t as jtj ! 0; ð2:1Þ where if 0 < cJ 1 ðJ ¼ 1; . . . ; bÞ; cJ 1 ðJ ¼ b þ 1; . . . ; mÞ; 0 < cJ 1 ðJ ¼ m þ 1; . . . ; cÞ, and cJ 1 ðJ ¼ c þ 1; . . . ; nÞ, then the coefficients at ðt ¼ 1; . . . ; 4Þ have the forms: a1 ¼ jXj=2p; (" # b m X X
1 1 a2 ¼ LJ LJ þ 2pcJ 8 J ¼1 J ¼bþ1 " #) c n X X
1 LJ LJ þ 2pcJ þ ; J ¼mþ1
a3 ¼
b X J ¼1
cJ L J
J ¼cþ1 c X
cJ L J
ð2:2Þ
ð2:3Þ
!, 2p;
ð2:4Þ
J ¼mþ1
( b Z X 1 64 pcJ 7 K12 ðr1 Þ c2J dr1 a4 ¼ 512 7 LJ CJ J ¼1 3 # Z " m X 2p 2 dr1 þ K1 ðr1 Þ c1 J L J C J J ¼bþ1 Z c X 64 pcJ 2 2 þ7 K2 ðr2 Þ cJ dr2 7 LJ J ¼mþ1 CJ " # ) 3 Z n X 2p 2 þ K2 ðr2 Þ c1 dr2 : J L J J ¼cþ1 CJ
ð2:5Þ
With reference to formulae (1.6)–(1.8) and to the articles [19,23,27], the asymptotic expansions (2.1) may be interpreted as follows: 1. X is a general annular bounded domain in R2 and we have the piecewise smooth Robin boundary conditions (1.10) with small/large impedances cJ ðJ ¼ 1; . . . ; nÞ.
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2. For the first four terms, X is a general annular bounded domain in R2 of area jXj, it has " !# c b X X 3 h¼ 1þ c LJ cJ LJ p J ¼mþ1 J J ¼1 holes, provided ‘‘h’’ is a positive Pb integer, where the components CJ ðJ ¼ 1; . . . ; bÞ of oX1 are of lengths J ¼1 LJ and of curvatures 1=2 64 pcJ 2 2 K1 ðr1 Þ cJ 7 LJ together with the Neumann boundary conditions, and P the remaining components CJ ðJ ¼ b þ 1; . . . ; mÞ of oX1 are of lengths mJ¼bþ1 ðLJ þ 2pc1 J Þ and of curvatures " 3 #1=2 2p 2 K1 ðr1 Þ c1 J LJ together with the Dirichlet boundary conditions, while the components Pc CJ ðJ ¼ m þ 1; . . . ; cÞ of oX2 are of lengths J ¼mþ1 LJ and of curvatures 1=2 64 pcJ 2 2 K2 ðr2 Þ cJ 7 LJ together with the Neumann boundary conditions, and P the remaining components CJ ðJ ¼ c þ 1; . . . ; nÞ of oX2 are of lengths nJ ¼cþ1 ðLJ þ 2pc1 J Þ and of curvatures " 3 #1=2 2p 2 K2 ðr2 Þ c1 J LJ together with the Dirichlet boundary conditions.
3. The Proof of Theorem 2.1 b ðtÞ asWith reference to the articles [16,19,23,27], it is easy to show that l sociated with the main problem (1.9), (1.10) is given by Z Z b l ðt Þ ¼ Gðx; x ; tÞ dx; ð3:1Þ X
where Gðx1 ; x2 ; tÞ is the Green function for the wave equation
E.M.E. Zayed, I.H. Abdel-Halim / Appl. Math. Comput. 132 (2002) 187–204
o2 D 2 Gðx1 ; x2 ; tÞ ¼ 0 ot
in X f 1 < t < 1g;
193
ð3:2Þ
subject to the piecewise smooth Robin boundary conditions (1.9) and the initial conditions oGðx1 ; x2 ; tÞ lim Gðx1 ; x2 ; tÞ ¼ 0;
t!0
lim
t!0
¼ dðx1 x2 Þ;
ot
ð3:3Þ
where dðx1 x2 Þ is the Dirac delta function located at the source point x2 . Let us write Gðx1 ; x2 ; tÞ ¼ G0 ðx1 ; x2 ; tÞ þ ,ðx1 ; x2 ; tÞ;
ð3:4Þ
where
G0 ðx1 ; x2 ; tÞ ¼
H ðjtj jx1 x2 jÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2p t2 jx1 x2 j
ð3:5Þ
is the ‘‘fundamental solution’’ of the wave equation (3.2) while ,ðx1 ; x2 ; tÞ is the ‘‘regular solution’’ chosen in such a way that Gðx1 ; x2 ; tÞ satisfies the piecewise smooth Robin boundary conditions (1.9). On setting x1 ¼ x2 ¼ x , we find that
jXj b H ðjtjÞ þ KðtÞ; l ðtÞ ¼ 2pt
ð3:6Þ
where KðtÞ ¼
Z Z X
ð3:7Þ
,ðx; x ; tÞ dx:
In what follows we shall use Fourier transforms with respect to 1 < t < 1 and use 1 < g < 1 as the Fourier transform parameter. Thus, we define Z Z þ1 b e2pigt Gðx1 ; x2 ; tÞ dt: ð3:8Þ G ðx1 ; x2 ; gÞ ¼
1
An application of the Fourier transform to the wave equation (3.2) shows that b ðx1 ; x2 ; gÞ satisfies the reduced wave equation G
b ðx1 ; x2 ; gÞ ¼ dðx1 x2 Þ in X; ðD þ 4p2 g2 Þ G
ð3:9Þ
together with the piecewise smooth Robin boundary conditions (1.9). The asymptotic expansion of KðtÞ, for small jtj, may then be deduced dib ðgÞ, for large jgj, where rectly from the asymptotic expansion of K
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b ðgÞ ¼ K
Z Z X
b , ðx; x ; gÞ dx:
ð3:10Þ
It is well known (see for example [19,20,23]) that Eq. (3.9) has the fundamental b 0 ðx1 ; x2 ; gÞ ¼ 1 Y0 ð2pgrx x Þ where rx x ¼ jx1 x2 j is the distance solution G 1 2 1 2 2 between the points x1 and x2 of the region X, while Y0 is the Bessel function of the second kind and of zero order. The existence of this solution enables us to b ðx1 ; x2 ; gÞ satisfying the piecewise smooth construct integral equations for G Robin boundary conditions (1.9) for small/large impedances cJ , where 0 < cJ 1 ðJ ¼ 1; . . . ; bÞ, cJ 1 ðJ ¼ b þ 1; . . . ; mÞ, 0 < cJ 1 ðJ ¼ m þ 1; . . . ; cÞ, and cJ 1 ðJ ¼ c þ 1; . . . ; nÞ. Therefore, the Green theorem gives the following integral equation: b ðx1 ; x2 ; gÞ G
2 3 b Z X 1 1 b ðx1 ; y ; gÞ4 o þ cJ 5Y0 ð2pgr y x Þ dy ¼ Y0 ð2pgrx1 x2 Þ G 2 2 2 J ¼1 CJ onJy 3 2 32 Z m 1 X o o b ðx1 ; y ; gÞ541 þ c1 5Y0 ð2pgr y x Þ dy 4 þ G 2 J 2 J ¼bþ1 CJ onJy onJy 2 3 Z c 1 X o b ðx1 ; y ; gÞ4 þ cJ 5Y0 ð2pgr y x2 Þ dy þ G 2 J ¼mþ1 CJ onJy 3 2 32 Z n 1 X o o b ðx1 ; y ; gÞ541 þ c1 5Y0 ð2pgr y x Þ dy : 4 G 2 J 2 J ¼cþ1 CJ onJy onJy
ð3:11Þ On applying the iteration methods (see, for example [18–20,23]) to the inb ðx1 ; x2 ; gÞ, which has a tegral equation (3.11), we obtain the Green function G regular part of the form !, 20 X b , ðx1 ; x2 ; gÞ ¼ Am 4; ð3:12Þ
m¼1
where A1 ¼
b Z X J ¼1
CJ
2
3 o þ cJ 5Y0 ð2pgr y x2 Þ dy ; Y0 ð2pgrx1 y Þ4 onJy
E.M.E. Zayed, I.H. Abdel-Halim / Appl. Math. Comput. 132 (2002) 187–204
A2 ¼
Z m X
A4 ¼
3 o þ cJ 5Y0 ð2pgr y x2 Þ dy ; Y0 ð2pgrx1 y Þ4 onJy
J ¼cþ1
CJ
2
Z n X CJ
2
Z c X J ¼mþ1
195
3
32
4 o Y0 ð2pgrx y Þ541 þ c1 o 5Y0 ð2pgr y x Þ dy ; 1 2 J onJy onJy
CJ
J ¼bþ1
A3 ¼
2
3
32
4 o Y0 ð2pgrx y Þ541 þ c1 o 5Y0 ð2pgr y x Þ dy ; 1 2 J onJy onJy
82 9 3 > > Z Z < b = X 6 o 7 0 þ cJ 5Y0 ð2pgr 0 Þ dy dy 0 ; Y0 ð2pgrx1 y ÞM1 ðy; y Þ 4 A5 ¼ y x2 > > CJ CJ : onJy0 ; J ¼1 3 o 4 A6 ¼ Y0 ð2pgrx1 y Þ5M2 ðy; y 0 Þ on Jy C C J J J ¼bþ1 9 82 3 > > = < 6 1 o 7 4 1 þ cJ 5Y0 ð2pgr y0 x Þ dy dy 0 ; > 2 > onJy 0 : ;
A7 ¼
2
Z m X
Z
Z
Z
c X J ¼mþ1
CJ
CJ
82 9 3 > > < = 6 o 7 0 þ cJ 5Y0 ð2pgr 0 Þ dy dy 0 ; Y0 ð2pgrx1 y ÞM3 ðy ; yÞ 4 y x2 > > : onJy0 ;
3 o 4 A8 ¼ Y0 ð2pgrx1 y Þ5M4 ðy; y 0 Þ on Jy C C J J J ¼cþ1 82 9 3 > > < = 6 1 o 7 4 1 þ cJ 5Y0 ð2pgr y0 x Þ dy dy 0 ; > 2 > onJy 0 : ; Z n X
Z
2
8 9 2 3 Z b Z m < X = X o 4 A9 ¼ Y0 ð2pgrx1 y Þ5M5 ðy; y 0 Þ dy ; onJy CJ : J ¼bþ1 CJ J ¼1 82 9 3 > > < = 6 o 7 þ cJ 5Y0 ð2pgr 0 Þ dy 0 ; 4 y x2 > > : onJy0 ;
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A10 ¼
( Z b Z c X X CJ
J ¼1
) 0
Y0 ð2pgrx1 y ÞM1 ðy; y Þ dy CJ
J ¼mþ1
82 9 3 > > < = 6 o 7 þ cJ 5Y0 ð2pgr y 0 x2 Þ dy 0 ; 4 > > : onJy0 ; 8 9 2 3 Z b Z n < X = X o 4 Y0 ð2pgrx1 y Þ5M5 ðy; y 0 Þ dy A11 ¼ ; onJy CJ : J ¼cþ1 CJ J ¼1 82 9 3 > > < = 6 o 7 þ cJ 5Y0 ð2pgr 0 Þ dy 0 ; 4 y x2 > > : onJy0 ; A12 ¼
Z (X m b Z X J ¼bþ1
CJ
) 0
Y0 ð2pgrx1 y ÞM6 ðy ; yÞ dy CJ
J ¼1
82 9 3 > > < = o 7 6 0 41 þ c1 Y ð2pgr Þ dy 0 ; 5 0 y x2 J > > onJy 0 : ; A13 ¼
Z ( X Z m c X J ¼bþ1
CJ
J ¼mþ1
) 0
Y0 ð2pgrx1 y ÞM6 ðy; y Þ dy
CJ
82 9 3 > > < = o 7 6 0 41 þ c1 dy 0 ; Y ð2pgr Þ 5 0 J y x2 > > 0 on Jy : ; 8 9 2 3 Z < X Z = m n X o 4 A14 ¼ Y0 ð2pgrx1 y Þ5M2 ðy; y 0 Þ dy : J ¼cþ1 CJ onJy ; J ¼bþ1 CJ 82 9 3 > > < = 6 1 o 7 4 1 þ cJ 5Y0 ð2pgr y0 x Þ dy 0 ; > 2 > onJy 0 : ; A15 ¼
Z (X c b Z X J ¼mþ1
CJ
J ¼1
) 0
Y0 ð2pgrx1 y ÞM3 ðy; y Þ dy CJ
82 9 3 > > < = 6 o 7 þ cJ 5Y0 ð2pgr y 0 x2 Þ dy 0 ; 4 > > : onJy0 ;
E.M.E. Zayed, I.H. Abdel-Halim / Appl. Math. Comput. 132 (2002) 187–204
8 9 2 3 Z < X Z = c m X o 4 A16 ¼ Y0 ðx1 ; y ; gÞ5M7 ðy; y 0 Þ dy : J ¼bþ1 CJ onJy ; J ¼mþ1 CJ 82 9 3 > > < = 6 o 7 þ cJ 5Y0 ð2pgr 0 Þ dy 0 ; 4 y x2 > > : onJy0 ; 8 9 2 3 Z < X Z c n = X o 4 A17 ¼ Y0 ðx1 ; y ; gÞ5M7 ðy; y 0 Þ dy : J ¼cþ1 CJ onJy ; J ¼mþ1 CJ 82 9 3 > > < = 6 o 7 4 þ cJ 5Y0 ð2pgr 0 Þ dy 0 ; y x2 > > : onJy0 ; A18 ¼
Z (X n b Z X
) 0
Y0 ðx1 ; y ; gÞM8 ðy; y Þ dy 9 2 3 > > < = o 7 6 0 dy 0 ; 41 þ c1 Y ð2pgr Þ 5 0 J y x2 > > 0 on Jy : ; J ¼cþ1 8
CJ
J ¼1
CJ
8 9 2 3 Z < X Z = n m X o 4 A19 ¼ Y0 ðx1 ; y ; gÞ5M4 ðy; y 0 Þ dy : J ¼bþ1 CJ onJy ; J ¼cþ1 CJ 82 9 3 > > < = 6 1 o 7 4 1 þ cJ 5Y0 ð2pgr y0 x Þ dy 0 ; > 2 > onJy 0 : ; A20 ¼
Z ( X Z n c X
) 0
Y0 ðx1 ; y ; gÞM4 ðy; y Þ dy 9 2 3 > > < = o 7 6 0 41 þ c1 Y ð2pgr Þ dy 0 ; 5 0 y x2 J > > 0 on Jy : ; J ¼cþ1 8
where M1 ðy ; y 0 Þ ¼
M2 ðy ; y 0 Þ ¼
M3 ðy ; y 0 Þ ¼
CJ
CJ
J ¼mþ1
1 X t ðtÞ ð1Þ K1 ðy 0 ; y Þ;
t¼0 1 X
ðtÞ
K2 ðy 0 ; y Þ;
t¼0 1 X t¼0
ðtÞ
K1 ðy 0 ; y Þ;
197
198
E.M.E. Zayed, I.H. Abdel-Halim / Appl. Math. Comput. 132 (2002) 187–204
M4 ðy ; y 0 Þ ¼
1 X t ðtÞ ð1Þ K2 ðy 0 ; y Þ;
t¼0
M5 ðy ; y 0 Þ ¼
1 X t ðtÞ ð1Þ K3 ðy 0 ; y Þ;
t¼0
M6 ðy ; y 0 Þ ¼
1 X
1 X
ðtÞ
ðtÞ
K3 ðy 0 ; y Þ;
t¼0
M8 ðy ; y 0 Þ ¼
1 X t ðtÞ ð1Þ K4 ðy 0 ; y Þ;
t¼0
and
K4 ðy 0 ; y Þ;
t¼0
M7 ðy ; y 0 Þ ¼
2
3
16 o 7 ð0Þ K1 ðy 0 ; y Þ ¼ 4 þ cJ 5Y0 ð2pgr 0 Þ; yy 2 onJy 0
2
3 2
16 o o 7 ð0Þ þ c1 K2 ðy 0 ; y Þ ¼ 4 5Y0 ð2pgr y 0 y Þ; J 2 onJy onJy 0 onJy
2
3
16 o 7 ð0Þ K3 ðy 0 ; y Þ ¼ 41 þ c1 5Y0 ð2pgr y 0 y Þ; J 2 onJy 0
2
3
16 o2 o 7 ð0 K4 ðy 0 ; y Þ ¼ 4 þ c1 5Y0 ð2pgr y 0 y Þ: J 2 onJy 0 onJy onJy 0
ðtÞ
In these formulae, we note for example, that Ki ðy 0 ; y Þ ði ¼ 1; . . . ; 4Þ are the
ð0Þ
iterates of Ki ðy 0 ; y Þ ði ¼ 1; . . . ; 4Þ, respectively. On the basis of (3.12), the function b , ðx1 ; x2 ; gÞ will be estimated for large jgj together with small/large impedances cJ . The case when x1 and x2 lie in the neighborhood of the com ponents CJ ðJ ¼ 1; . . . ; mÞ of the inner boundary oX1 or in the neighborhood of the components CJ ðJ ¼ m þ 1; . . . ; nÞ of the outer boundary oX2 of the annular region X is particularly interesting. For this case, we use the local expansions of the functions Y0 ð2pgrx y Þ;
o Y0 ð2pgrx y Þ onJy
ðJ ¼ 1; . . . ; nÞ;
ð3:13Þ
E.M.E. Zayed, I.H. Abdel-Halim / Appl. Math. Comput. 132 (2002) 187–204
199
when the distance between x and y is small, which are very similar to that obtained in Sections 4, 5 of [19]. Consequently, the local behavior of the kernels ð0Þ
ð0Þ
ð0Þ
ð0Þ
K1 ðy 0 ; y Þ; K2 ðy 0 ; y Þ; K3 ðy 0 ; y Þ; K4 ðy 0 ; y Þ;
when the distance between y and y 0 is small, and for small/large impedances cJ follows directly from knowledge of the local expansions of (3.13). Definition 1. Let n1 and n2 be points in the upper half-plane n2 > 0. Then, we define qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q12 ¼ ðn11 n12 Þ2 þ ðn21 þ n22 Þ2 : An ek ðn1 ; n2 ; gÞ-function is defined for points n1 n2 belonging to sufficiently small domains DðIJ Þ except when n1 ¼ n2 2 IJ ðJ ¼ 1; . . . ; nÞ and k is called the degree of this function. For every positive integer K, it has the expansion: !‘ !m X o o 1 2 p1 2 p2 k e ðn1 ; n2 ; gÞ ¼ f ðn1 Þðn1 Þ ðn2 Þ Y0 ð2pgq12 Þ on11 on21 þ RK ðn1 ; n2 ; gÞ;
P
where denotes a sum of a finite number of terms in which f ðn11 Þ is an infinitely differentiable function (see [9,11,19,20]). In this expansion p1 ; p2 ; ‘; m are integers, p1 P 0; p2 P 0; ‘ P 0; k ¼ minðp1 þ p2 qÞ and q ¼ ‘ þ m. The remainder RK ðn1 ; n2 ; gÞ has continuous derivatives of order k 6 K satisfying
Dk RK ðn1 ; n2 ; gÞ ¼ OðgK eAJ gq12 Þ as jgj ! 1;
where AJ ðJ ¼ 1; . . . ; nÞ are positive constants. Thus, using methods similar to that obtained in Sections 6–10 of [19], we can show that the functions (3.13) are ek -functions with degrees k ¼ 0; 1, respectively. Consequently the kernels Ki ðy 0 ; y Þ ði ¼ 1; . . . ; 4Þ are ek -functions with degrees k ¼ 0; 0; 1; 1, respectively. Definition 2. Let x1 and x2 be points in large domains X þ CJ ðJ ¼ 1; . . . ; nÞ. Then we define r1 ¼ minðrx1 y þ rx2 y Þ y
r2 ¼ minðrx1 y þ rx2 y Þ y
if y 2 CJ ðJ ¼ 1; . . . ; bÞ;
if y 2 CJ ðJ ¼ b þ 1; . . . ; mÞ;
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E.M.E. Zayed, I.H. Abdel-Halim / Appl. Math. Comput. 132 (2002) 187–204
r3 ¼ minðrx1 y þ rx2 y Þ y
if y 2 CJ ðJ ¼ m þ 1; . . . ; cÞ;
r4 ¼ minðrx1 y þ rx2 y Þ y
if y 2 CJ ðJ ¼ c þ 1; . . . ; nÞ:
An Ek ðx1 ; x2 ; gÞ-function is defined and infinitely differentiable with respect to x1 and x2 when these points belong to large domains X þ CJ ðJ ¼ 1; . . . ; nÞ. Thus the Ek -function has a similar local expansion of the ek -function (see [9,19]). With the help of Sections 8, 9 in [19] it is easily seen that formula (3.12) is an b ðx1 ; x2 ; gÞ can be estimated as follows: E0 ðx1 ; x2 ; gÞ-function and consequently G
b X # $ b ðx1 ; x2 ; gÞ ¼ G O ½1 þ j logð2pgr1 jeAJ gr1
þ þ þ
J ¼1 m X
$ # O ½1 þ j logð2pgr2 jeAJ gr2
J ¼bþ1 c X
$ # O ½1 þ j logð2pgr3 jeAJ gr3
J ¼mþ1 n X
$ # O ½1 þ j logð2pgr4 jeAJ gr4 ;
ð3:14Þ
J ¼cþ1
which is valid for jgj ! 1 and for small/large impedances cJ ðJ ¼ 1; . . . ; nÞ where AJ ðJ ¼ 1; . . . ; nÞ are positive constants. The estimate (3.14) shows that b ðx1 ; x2 ; gÞ is exponentially small for large jgj. This prove that G b ðx1 ; x2 ; gÞ G converges for jgj ! 1. With reference to Section 10 in [19], if the ek -expansions of the functions Ki ðy 0 ; y Þ ði ¼ 1; . . . ; 4Þ is introduced into (3.12) and if we use formulae similar to (6.4) and (6.9) of Section 6 in [19], we obtain the following local behavior of the regular part b , ðx1 ; x2 ; gÞ when r1 ; r2 ; r3 and r4 are small, which is valid for jgj ! 1 and for small/large impedances cJ ðJ ¼ 1; . . . ; nÞ: b , ðx1 ; x2 ; gÞ ¼
n X J ¼1
b , J ðx1 ; x2 ; gÞ;
ð3:15Þ
where (a) if x1 and x2 belong to sufficiently small domains DðIJ Þ ðJ ¼ 1; . . . ; bÞ, then 8 !1 9 < = 1 o b , J ðx1 ; x2 ; gÞ ¼ 1 cJ Y ð2pgq1 2Þ 2 ; 0 4: on1 þ Ofg1 expðAJ gq1 2Þg as jgj ! 1:
ð3:16Þ
E.M.E. Zayed, I.H. Abdel-Halim / Appl. Math. Comput. 132 (2002) 187–204
201
(b) if x1 and x2 belong to sufficiently small domains DðIJ Þ ðJ ¼ b þ 1; . . . ; mÞ, then ( !) 1 o 1 b 1 cJ , j ðx1 ; x2 ; gÞ ¼ Y0 ð2pgq1 2Þ 4 on21 þ Ofg1 expðAJ gq1 2Þg
as jgj ! 1:
ð3:17Þ
(c) if x1 and x2 belong to sufficiently small domains DðIJ Þ ðJ ¼ m þ 1; . . . ; cÞ, then 8 !1 9 < = 1 o b 1 cJ Y ð2pgq1 2Þ , J ðx1 ; x2 ; gÞ ¼ ; 0 4: on21 þ Ofg1 expðAj gq1 2Þg
as jgj ! 1:
ð3:18Þ
(d) if x1 and x2 belong to sufficiently small domains DðIJ Þ ðJ ¼ c þ 1; . . . ; nÞ, then ( !) 1 o 1 b 1 cJ , J ðx1 ; x2 ; gÞ ¼ Y0 ð2pgq1 2Þ 4 on21 þ Ofg1 expðAJ gq1 2Þg as jgj ! 1:
ð3:19Þ
When r1 P dJ > 0 ðJ ¼ 1; . . . ; bÞ, r2 P dJ > 0 ðJ ¼ b þ 1; . . . ; mÞ, r3 P dJ > 0 ðJ ¼ m þ 1; . . . ; cÞ, and r4 P dJ > 0 ðJ ¼ c þ 1; . . . ; nÞ the function b , ðx1 ; x2 ; gÞ is of order OfeBg g as jgj ! 1, B ¼ constant > 0. Thus, since lim ra =q12 ¼ 1 ða ¼ 1; . . . ; 4Þ (see [19,20]) then we have the asymptotic forra !0 mulae (3.16)–(3.19) with q12 in the small domain DðIJ Þ ðJ ¼ 1; . . . ; nÞ are replaced by ra ða ¼ 1; . . . ; 4Þ in the large domains X þ oXJ ðJ ¼ 1; . . . ; nÞ, respectively. Since for n2 P hJ > 0 ðJ ¼ 1; . . . ; nÞ the functions b , J ðx; x ; gÞ are of order Ofexpð2gAJ hJ Þg, the integral over the general annular bounded domain X of the function b , ðx; x ; gÞ can be approximated in the following way (see (3.10)): Z hJ Z LJ n X b b , J ðx; x; gÞf1 K2 ðn1 Þn2 g dn1 dn2 K ðgÞ ¼ J ¼mþ1 m X
þ
J ¼1 n X
n2 ¼0
Z
hJ n2 ¼0
n1 ¼0
Z
LJ
n1 ¼0
b , J ðx; x; gÞf1 þ K1 ðn1 Þn2 g dn1 dn2
Ofexpð2gAJ hJ Þg as jgj ! 1:
ð3:20Þ
J ¼1
b J ðx; x; gÞ ðJ ¼ 1; . . . ; nÞ (see [19,20]) are introduced into If the ek -expansion of , (3.20), and with the help of formula (11.3) of Section 11 in [19], we deduce after
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inverting Fourier transforms and using (3.6) that the result (2.1) has been constructed, and the proof of Theorem 2.1 is completed.
4. Discussions and conclusions The problem of determining some geometric quantities of the general annular bounded domain X in R2 with a finite number of piecewise smooth Robin boundary conditions (1.10), from complete knowledge of its eigenvalues has been discussed in the present paper by using the wave equation approach. It is well known that the wave equation approach has given very strong result; the definite one is that of Hormander [6] who has studied the distribution tr½expðitP Þ near to t ¼ 0 for an elliptic positive semi-definite Pseduodifferential operator P of order m in Rn . Recently, the wave equation approach in solving particular problems has been discussed by Zayed and Abdel-Halim [16–18], Zayed and Hassan [19,20], Zayed and Younis [23] and Zayed [25,27] who have studied the spectral distribution b l ðtÞ ¼
1 X
expðitEx1=2 Þ as jtj ! 0;
x¼1
for some different bounded domains in R2 with certain boundary conditions. An alternative method is the heat equation approach, which has been investigated by many authors, see for example, [2–5,7–14,21,22,24,26,28– 34]. They have studied the asymptotic expansion of the trace of the heat kernel HðtÞ ¼
1 X
expðtEx Þ
as t ! 0;
x¼1
for some bounded domains in R2 with certain boundary conditions. With reference to the above articles one can ask a question, is it possible just by listening with a perfect ear to hear the shape of the bounded domain X even if we can not see it? This question has been put rather nicely by Kac [7], who simply asked, ‘‘Can one hear the shape of a drum?’’ Recently Gordon et al. [2] have shown that the answer of Kac’s question is negative. They have shown explicitly two domains that although having different shapes, have the same eigenvalues (i.e., isospectral domains). This theoretical result was experimentally verified in [15] by employing thin microwave cavities shaped in the form of two different domains known to be isospectral. In particular, the present paper provides useful techniques to inverse problem methods via the spectral distribution of the Laplacian.
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