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An inverse eigenvalue problem of the wave equation for a multi-connected region in R2 together with three different types of boundary conditions
Applied Mathematics and Computation 154 (2004) 361–388 www.elsevier.com/locate/amc
An inverse eigenvalue problem of the wave equation for a multi-connected region in R2 together with three different types of boundary conditions E.M.E. Zayed Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt
Abstract P 1 1=2 The trace of the wave kernel lðtÞ ¼ 1 x¼1 expðitEx Þ, where fEx gx¼1 are the eiP2 2 o 2 1 2 genvalues of the negative Laplacian r ¼ k¼1 ðoxk Þ in the ðxp;ffiffiffiffiffiffi x ffiÞ-plane, is studied for a variety of bounded domains, where 1 < t < 1 and i ¼ 1. The dependence of lðtÞ on the connectivity of bounded domains and the Dirichlet, Neumann and Robin boundary conditions are analyzed. Particular attention is given for a multi-connected vibrating membrane in R2 surrounded by simply connected bounded domains Xj with smooth boundaries oXj ðj ¼ 1; . . . ; nÞ, where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components Skj Ci ði ¼ 1 þ kj1 ; . . . ; kj Þ of the boundaries oXj is considered, such that oXj ¼ i¼1þk Ci j1 where k0 ¼ 0. The basic problem is to extract information on the geometry of X using the wave equation approach from complete knowledge of its eigenvalues. Some geometrical quantities of X (e.g. the area of X, the total lengths of its boundary, the curvature of its boundary, the number of the holes of X, etc.) are determined from the asymptotic expansion of the trace of the wave kernel lðtÞ for small jtj. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Inverse problem; Wave kernel; Eigenvalues; Multi-connected vibrating membrane; Negative Laplacian; Heat kernel
E-mail address: [email protected] (E.M.E. Zayed). 0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00715-X
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E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
1. Introduction The underlying inverse eigenvalue problem is to deduce some geometrical quantities associated with a bounded domain in R2 from complete knowledge of the eigenvalues of the negative Laplacian by using the wave equation approach. Let X be a simply connected bounded domain in R2 with a smooth boundary oX. Consider the Robin problem r2 / ¼ E/
ð1:1Þ
in X
and
o þc /¼0 on
ð1:2Þ
on oX;
where ono denotes differentiation along the outward normal to the boundary oX Denote its eiand c is positive constant impedance, with u 2 C 2 ðXÞ \ CðXÞ. genvalues, counted according to multiplicity, by 0 < E1 6 E2 6 6 E x 6 ! 1
as x ! 1:
ð1:3Þ
Zayed and Hassan [16] have discussed the problem (1.1) and (1.2) for small/ large impedance c and have determined some geometrical quantities of X using the wave equation approach by analyzing the asymptotic expansion of the trace of the wave kernel lðtÞ ¼
1 X
expðitEx1=2 Þ
as jtj ! 0;
ð1:4Þ
x¼1
pffiffiffiffiffiffiffi which represents a tempered distribution for 1 < t < 1 and i ¼ 1. Note that when c is small, the Robin boundary condition (1.2) looks approximately like the Neumann boundary condition, while when c is large, the Robin boundary condition (1.2) looks approximately like the Dirichlet boundary condition provided ono remains finite. Zayed et al. [16,17,19] and Zayed [20] have discussed the problem (1.1) and (1.2) in the following cases and have obtained the following results: Case 1.1. If c ¼ 0 (the Neumann problem) lðtÞ ¼
jXj joXj H ðjtjÞ þ sign t þ a0 jtj þ Oðt2 sign tÞ 2pt 8
as jtj ! 0:
ð1:5Þ
as jtj ! 0;
ð1:6Þ
Case 1.2. If c ! 1 (the Dirichlet problem) lðtÞ ¼
jXj joXj H ðjtjÞ sign t þ a0 jtj þ Oðt2 sign tÞ 2pt 8
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
where H ðjtjÞ is the Heaviside unit function and 8 t > 0; < 1; sign t ¼ 0; t ¼ 0; : 1; t < 0:
363
ð1:7Þ
In these formulae jXj is the area of X and joXj is the total length of oX while the sign ± of the second term of lðtÞ determines whether we have the Neumann or the Dirichlet problem. 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 1h . 6 Case 1.3 (the mixed problem). If L1 is the length of a component C1 of the boundary oX with the Neumann boundary condition, and if L2 is the length of the remaining component C2 ¼ oX n C1 of oX with the Dirichlet boundary condition, then with reference to the articles [12–17], we get lðtÞ ¼
jXj ðL1 L2 Þ H ðjtjÞ þ sign t þ a0 jtj þ Oðt2 sign tÞ 2pt 8
as jtj ! 0: ð1:8Þ
Note that the order term Oðt2 sign tÞ in (1.6)–(1.8) is yet undetermined. So, in the present paper, we discuss what geometrical quantities are contained in this order term in the case X is a multi-connected vibrating membrane together with the piecewise smooth Dirichlet, Neumann and Robin boundary conditions (1.10) stated below. The object of this paper is to discuss the following more general inverse eigenvalue problem: Let X be a general multi-connected vibrating membrane in R2 surrounded internally by simply connected bounded domains Xj with smooth boundaries oXj ðj ¼ 1; . . . ; n 1Þ and externally by a simply connected bounded domain Xn with a smooth boundary oXn . Suppose that the eigenvalues (1.3) are given exactly for the Helmholtz equation r2 / ¼ E/
ð1:9Þ
in X
together with the following Dirichlet, Neumann and Robin boundary conditions: /¼0 o/ ¼0 on i o þ ci / ¼ 0 oni
on Ci ; i ¼ 1 þ kj1 ; . . . ; k1j ; on Ci ; i ¼ k1j þ 1; . . . ; k2j ; on Ci ; i ¼ k2j þ 1; . . . ; kj ;
ð1:10Þ
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E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
where the boundaries oXj ðj ¼ 1; . . . ; nÞ consist of a finite number Skj of piecewise smooth components Ci ði ¼ 1 þ kj1 ; . . . ; kj Þ such that oXj ¼ i¼1þk Ci where j1 k0 ¼ 0 and ci are positive constants. The basic problem is to determine some geometrical quantities (e.g. the area of X, the total lengths of the boundaries, the curvature of the boundaries, the number of holes of X, etc.) associated with the main problem (1.9) and (1.10) by analyzing the asymptotic expansion of the trace of wave kernel lðtÞ for small jtj. Note that the special cases of the main problem (1.9) and (1.10) have been discussed by Zayed et al. [12–17,19] and Zayed [20,30,31]. Therefore, this problem can be considered as a more general one, which does not seem to have been investigated elsewhere.
2. Statement of the results Suppose that the boundaries oXj ðj ¼ 1; . . . ; nÞ of the multi-connected bounded domain X are given locally by the equations xa ¼ y a ðrj Þ ða ¼ 1; 2Þ in which rj are the arc lengths of the counter clock wise oriented boundaries and y a ðrj Þ 2 C 1 ðoXj Þ. Let Kj ðrj Þ be the curvatures of oXj ðj ¼ 1; . . . ; nÞ respectively and Li be the lengths of the components Ci ði ¼ 1 þ kj1 ; . . . ; kj Þ respectively. Let hi > 0 be sufficiently small numbers. Let ni be the minimum distances from a point x ¼ ðx1 ; x2 Þ of the domain X to the components Ci ði ¼ 1 þ kj1 ; . . . ; kj Þ respectively. Let ni ðrj Þ denote the inward unit normal to the components Ci respectively. Then, we note that the coordinates in the neighborhood of the components Ci ði ¼ 1 þ kj1 ; . . . ; kj ; j ¼ 1; . . . ; N Þ and its diagrams are in the same forms as in Section 5.2 of [22] with the interchanges n1 $ ni , h1 $ hi , I1 $ Ii , DðI1 Þ $ DðIi Þ and d1 $ di . Thus, we have the same formulae (5.2.1)–(5.2.5) of Section 5.2 in [22] with the interchanges n1 $ ni , n ðr1 Þ $ n ðrj Þ and t ðr1 Þ $ t ðrj Þ. Similarly, the coordinates in the neigh1 i 1 i borhoods of the components Ci ði ¼ 1 þ kj1 ; . . . ; kj ; j ¼ N þ 1; . . . ; nÞ and its diagrams are similar to the obtained in Section 5.1. of [22] with the interchanges n2 $ ni , h2 $ hi , I2 $ Ii , DðI2 Þ $ DðIi Þ and d2 $ di . Thus, we have the same formulae (5.1.1)–(5.1.5) of Section 5.1 in [22] with the interchanges n2 $ ni , n ðr2 Þ $ n ðrj Þ and t ðr2 Þ $ t ðrj Þ. 2
i
2
i
Theorem 2.1. With the assumptions stated above, the asymptotic expansion of the trace of the wave kernel lðtÞ for the Dirichlet, Neumann and Robin problem (1.9) and (1.10) can be written in the following form: a1 lðtÞ ¼ H ðjtjÞ þ a2 sign t þ a3 jtj þ a4 t2 sign t þ Oðt3 sign tÞ as jtj ! 0; t ð2:1Þ
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
365
where 0 < ci 1 ði ¼ k2j þ 1; . . . ; mj Þ and ci 1 ði ¼ mj þ 1; . . . ; kj Þ ðj ¼ 1; . . . ; N ; N þ 1; . . . ; nÞ. Here, the coefficients am ðm ¼ 1 4Þ have the forms: a1 ¼ jXj=2p; 82 3 k2j mj n < X X X 1 4 a2 ¼ Li þ Li 5 8 j¼1 : i¼k þ1 i¼k2j þ1 1j 2 39 = Z k1j kj X X 4 Li þ Li þ c1 Kj ðrj Þ drj 5 ; i ; Ci i¼1þk i¼m þ1 j
j1
8 0 1 0 19 mj mj < = N n X X X X 2n 1 @ @ þ ci Li A ci Li A ; a3 ¼ ; 6 2p : j¼1 i¼k þ1 i¼k þ1 j¼N þ1 2j
2j
8 Z Z k1j k2j n < X X X 1 Kj2 ðrj Þ drj þ 7 Kj2 ðrj Þ drj a4 ¼ 512 j¼1 : i¼1þk Ci C i i¼k þ1 1j
j1
Z 64 pci þ7 Kj2 ðrj Þ c2i drj 7 Li i¼k2j þ1 Ci 9 3 # = Z " kj X 2p þ Kj2 ðrj Þ c1 drj : i ; L i C i i¼m þ1 mj X
j
With reference to the formulae (1.5)–(1.8) and to the articles [12– 17,19,20,30–33], the asymptotic expansion (2.1) may be interpreted as follows: i(i) X is a general multi-connected vibrating membrane in R2 and we have the piecewise smooth Dirichlet, Neumann and Robin boundary conditions (1.10) with small/large impedances ci . (ii) For the first four terms, X is a general multi-connected vibrating membrane in R2 of area jXj, it has 8 0 1 0 19 mj mj N n = X X X 3
2j
holes, provided h is a positive integer. Ci ði ¼ 1 þ kj1 ; . . . ; k1j ; j ¼ 1; . . . ; nÞ are of lengths PnThePkcomponents 1j i¼1þkj1 Li Þ and of curvatures Kj ðrj Þ together with the Dirichlet j¼1 ð boundary conditions. P Pk2j The components Ci ði ¼ k1j þ 1; . . . ; k2j ; j ¼ 1; . . . ; nÞ are of lengths nj¼1 ð i¼k Li Þ and curvatures Kj ðrj Þ together with the Neumann 1j þ1
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E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
boundary conditions. . ; mj ; j ¼ 1; . . . ; nÞ are Pn Pmj The components Ci ði ¼ k2j2 þ 1; . . 64 1=2 of lengths j¼1 ð i¼k L Þ and of curvatures ½K ðr Þ ðpci c2i Þ together i j j 7 Li 2j þ1 with the Neumann boundary conditions, Pwhile R Pkj the components Ci ði ¼ mj þ 1; . . . ; kj ; j ¼ 1; . . . ; nÞ are of lengths nj¼1 ½ i¼m ðLi þ c1 K2 i Ci j j þ1 2 2p 3 1 1=2 ðrj Þ drj Þ and of curvatures ½Kj ðrj Þ ð Li Þ ci together with the Dirichlet boundary conditions. Corollary 2.1. If we consider the main problem (1.9) and (1.10) with 0 < ci 1 ði ¼ k2j þ 1; . . . ; kj ; j ¼ 1; . . . ; nÞ, then the P asymptotic expansion of lðtÞ follows kj directly from (2.1) by setting mj ¼ kj with i¼k as zero. j þ1 Corollary 2.2. If we consider the main problem (1.9) and (1.10) with ci 1 ði ¼ k2j þ 1; . . . ; kj ; j ¼ 1; . . . ; nÞ, then the asymptotic expansion of lðtÞ follows Pk2j directly from (2.1) by setting mj ¼ k2j with i¼k as zero. 2j þ1 Remark 2.1. In the case n ¼ 1 (i.e. X is simply connected), then we have the results of Zayed et al. [12,16,17,19]. Remark 2.2. In the case n ¼ 2 (i.e. X is doubly-connected), then we have the results of Zayed et al. [13–15] and Zayed [20]. We close this section with the following question: What is interpretation of X if ‘‘h’’ is not integer? This is an interesting open problem, which has been left for the interested readers.
3. Formulation of the mathematical problem With reference to the articles [12–17], it can be easily seen that the trace of the wave kernel lðtÞ associated with the main problem (1.9) and (1.10) is given by Z Z lðtÞ ¼ ½Gðx ; x ; tÞ x ¼x ¼x dx; ð3:1Þ X
1
2
1
2
where Gðx ; x ; tÞ is GreenÕs function for the wave equation 1 2 2 o 2 r 2 Gðx ; x ; tÞ ¼ 0 in X f1 < t < 1g 1 2 ot
ð3:2Þ
subject to the boundary conditions (1.10) and the initial conditions oGðx ; x ; tÞ 1
limGðx ; x ; tÞ ¼ 0; t!0
1
2
lim t!0
ot
2
¼ dðx x Þ: 1
2
ð3:3Þ
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
367
Let us write Gðx ; x ; tÞ ¼ G0 ðx ; x ; tÞ þ ,ðx ; x ; tÞ; 1
2
1
2
1
ð3:4Þ
2
where
G0 ðx ; x ; tÞ ¼ 1
2
H ðjtj jx x jÞ 1 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2p t2 jx x j 1
ð3:5Þ
2
is the ‘‘fundamental solution’’ of the wave equation (3.2) while ,ðx ; x ; tÞ is the 1 2 ‘‘regular solution’’ chosen in such a way that Gðx ; x ; tÞ satisfies the boundary 1 2 conditions (1.10). By setting x ¼ x ¼ x , we find that 1
2
a1 lðtÞ ¼ H ðjtjÞ þ RðtÞ; t
ð3:6Þ
where a1 ¼ jXj=2p and RðtÞ is given by Z Z ½,ðx ; x ; tÞ x ¼x ¼x dx: RðtÞ ¼ 1
X
2
1
2
ð3:7Þ
The problem now is to determine the asymptotic expansions of RðtÞ for small jtj. In what follows, we shall use Fourier transforms with respect to 1 < t < 1 andp use < g < 1 as the Fourier transform parameter. Thus, ffiffiffiffiffiffi1 ffi we define for i ¼ 1, that b ; x ; gÞ ¼ Gðx 1
2
Z
þ1
1
e2pigt Gðx ; x ; tÞ dt: 1
ð3:8Þ
2
An application of the Fourier transform to the wave equation (3.2) shows b ; x ; gÞ satisfies the reduced wave equation that Gðx 1
2
b ; x ; gÞ ¼ dðx1 x2 Þ ðr2 þ 4p2 g2 Þ Gðx 1
2
in X
ð3:9Þ
together with the boundary conditions (1.10). The asymptotic expansions of RðtÞ for small jtj may then be deduced directly b from the asymptotic expansions of RðgÞ for large jgj, where Z Z b RðgÞ ¼ ½b , ðx ; x ; gÞ x ¼x ¼x dx: ð3:10Þ X
1
2
1
2
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E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
4. Derivation of the results It is well known (see for example [12–17]) that the membrane equation (3.9) ^ has the fundamental solution G0 ðx ; x ; gÞ ¼ 12 Y0 ð2pgrx x Þ, where rx x is the 1 2 1 2 1 2 distance between the points x ¼ ðx11 ; x21 Þ and x ¼ ðx12 ; x22 Þ of the domain X 1 2 while Y0 ð2pgrx x Þ is the Bessel function of the second kind and of zero order. 1 2 The existence of this solution enables us to construct integral equations for ^ Gðx ; x ; gÞ satisfying the boundary conditions (1.10) for small/large impedances 1 2 ci . Therefore, if we consider the main problem (1.9) and (1.10) with 0 < ci 1 ði ¼ k2j þ 1; . . . ; mj Þ and ci 1 ði ¼ mj þ 1; . . . ; kj Þ where ðj ¼ 1; . . . ; N ; N þ 1; . . . ; nÞ then GreenÕs theorem gives the following integral equation: ^ 1 Gðx ; x ; gÞ ¼ Y0 ð2pgrx x Þ 1 2 1 2 2 # Z " k 1j N X 1X o ^ Gðx ; y ; gÞ Y0 ð2pgr y x Þ dy þ 2 2 j¼1 i¼1þk oniy 1 Ci j1
" # Z ^ k2j N X 1X o Gðx ; y ; gÞ Y0 ð2pgr y x Þ dy 2 2 j¼1 i¼k þ1 Ci 1 oniy 1j
"
# Z ^ mj N X 1X o Gðx ; y ; gÞ þ ci Y0 ð2pgr y x Þ dy 2 2 j¼1 i¼k þ1 Ci 1 oniy 2j
# Z " kj N X 1X o ^ þ Gðx ; y ; gÞ 2 j¼1 i¼mj þ1 Ci oniy 1
" 1 þ c1 i
#
o Y0 ð2pgr y x Þ dy 2 oniy
# Z " k1j n X 1 X o ^ Gðx ; y ; gÞ Y0 ð2pgr y x Þ dy 2 2 j¼N þ1 i¼1þk oniy 1 Ci j1
" # Z ^ k2j n X 1 X o þ Gðx ; y ; gÞ Y0 ð2pgr y x Þ dy 2 2 j¼N þ1 i¼k þ1 Ci 1 oniy 1j
"
# Z ^ mj n X 1 X o þ Gðx ; y ; gÞ þ ci Y0 ð2pgr y x Þ dy 2 2 j¼N þ1 i¼k þ1 Ci 1 oniy 2j
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
# Z " kj n X 1 X o ^ Gðx ; y ; gÞ 2 j¼N þ1 i¼mj þ1 Ci oniy 1 " # 1 o 1 þ ci Y0 ð2pgr y x Þ dy : 2 oniy
369
ð4:1Þ
By applying the iteration methods (see for example [12–17,30–33]) to the ^ integral equation (4.1), we obtain GreenÕs function Gðx ; x ; gÞ which has a 1 2 regular part in the following form: b , ðx ; x ; gÞ ¼ 1
2
72 1X Ak ; 4 k¼1
ð4:2Þ
where # Z " k1j N X X o A1 ¼ Y0 ð2pgrx y Þ Y0 ð2pgr y x Þ dy ; 2 1 oniy Ci j¼1 i¼1þk j1
A2 ¼
# o Y0 ð2pgrx y Þ Y0 ð2pgr y x Þ dy ; 2 1 oniy Ci
"
# o Y0 ð2pgrx y Þ þ ci Y0 ð2pgr y x Þ dy ; 2 1 oniy Ci
Z mj N X X j¼1 i¼k2j þ1
A4 ¼
n X j¼N þ1
A6 ¼
Ci
# Z " k1j X o Y0 ð2pgrx y Þ Y0 ð2pgr y x Þ dy ; 2 1 oniy Ci i¼1þk
n X
j1
n X
# o Y0 ð2pgrx y Þ Y0 ð2pgr y x Þ dy ; 2 1 oniy Ci
Z k2j X
j¼N þ1 i¼k1j þ1
A7 ¼
" # o 1 o Y0 ð2pgrx y Þ 1 þ ci Y0 ð2pgr y x Þ dy ; 2 1 oniy oniy
Z kj N X X j¼1 i¼mj þ1
A5 ¼
"
Z k2j N X X j¼1 i¼k1j þ1
A3 ¼
"
# o Y0 ð2pgrx y Þ þ ci Y0 ð2pgr y x Þ dy ; 2 1 oniy Ci
Z mj X
j¼N þ1 i¼k2j þ1
"
370
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388 n X
A8 ¼
Ci
j¼N þ1 i¼mj þ1
A9 ¼
" # o 1 o Y0 ð2pgrx y Þ 1 þ ci Y0 ð2pgr y x Þ dy ; 2 1 oniy oniy
Z kj X
Z "
Z k1j N X X Ci
j¼1 i¼1þkj1
A10 ¼
A11 ¼
Z
Ci
"
# o Y0 ð2pgrx y ÞM2 ðy ; y Þ Y0 ð2pgr y x Þ dy dy 0 ; 2 1 oniy 0 Ci 0
"
# o Y0 ð2pgrx y ÞMci ðy ; y Þ þ ci Y0 ð2pgr y x Þdy dy 0 ; 2 1 0 on iy Ci
Z Z mj N X X Ci
j¼1 i¼k2j þ1
A12 ¼
# o Y0 ð2pgrx y Þ M1 ðy ; y 0 ÞY0 ð2pgr y x Þ dy dy 0 ; 2 1 oniy
Ci
Z k2j N X X j¼1 i¼k1j þ1
Z kj N X X "
Z
Ci
j¼1 i¼mj þ1
0
o Y0 ð2pgrx y ÞMc1 ðy ; y 0 Þ i 1 oniy
Ci
#
1 þ c1 i
o Y0 ð2pgr y x Þ dy dy 0 ; 2 oniy 0
A13 ¼
n X
Ci
j¼N þ1 i¼1þkj1
A14 ¼
n X
Z k2j X
A15 ¼
n X
Z
# o Y0 ð2pgrx y Þ M3 ðy ; y 0 ÞY0 ð2pgr y x Þ dy dy 0 ; 2 1 oniy
"
# o Y0 ð2pgrx y ÞM4 ðy ; y 0 Þ Y0 ð2pgr y x Þ dy dy 0 ; 2 1 oniy 0 Ci
"
# o Y0 ð2pgrx y ÞLci ðy ; y Þ þ ci Y0 ð2pgr y x Þdy dy 0 ; 2 1 oniy 0 Ci
Z Z mj X
j¼N þ1 i¼k2j þ1
A16 ¼
Ci
Ci
j¼N þ1 i¼k1j þ1
n X
Z "
Z k1j X
Ci
Z kj X
j¼N þ1 i¼mj þ1
" 1 þ c1 i
0
Z
Ci
Ci
#
o Y0 ð2pgrx y ÞLc1 ðy ; y 0 Þ i 1 oniy
o Y0 ð2pgr y x Þ dy dy 0 ; 2 oniy 0
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
A17 ¼
8 Z
Ci
:
j¼1 i¼k1j þ1
Ci
371
Y0 ð2pgrx y ÞM5 ðy ; y 0 Þ dy 1
9 =
;
Y0 ð2pgr y x Þ dy ; 2
A18 ¼
N X j¼1
9 8 # " > > Z Z k1j = < N X X X o Y0 ð2pgrx y Þ M6 ðy ; y 0 Þ dy > 1 > oniy Ci ; i¼k þ1 Ci : j¼1 i¼1þk k2j
1j
j1
o Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A19 ¼
8 Z
Ci
:
j¼1 i¼k2j þ1
Ci
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy 1
9 =
;
Y0 ð2pgr y x Þ dy 0 ; 2
A20 ¼
N X
mj X
j¼1 i¼k2j þ1
9 8 " # > Z > Z k1j = 1 > j¼1 i¼1þk oniy Ci : Ci ; j1
!
o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
9 8 # > >X Z < Z " k1j kj = N N X X X o 0 A21 ¼ Y0 ð2pgrx y Þ Mc1 ðy ; y Þ dy i > 1 Ci > ; : j¼1 i¼mj þ1 Ci oniy j¼1 i¼1þk j1
Y0 ð2pgr y x Þ dy 0 ; 2
A22 ¼
N X
kj X
j¼1 i¼mj þ1
9 8 # " > Z > Z k1j = 1 > j¼1 i¼1þk oniy Ci : Ci ;
1 þ c1 i
j1
o oniy 0
! Y0 ð2pgr y x Þ dy 0 ; 2
372
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
A23 ¼
8 Z
j¼1 i¼k1j þ1
:
j¼1 i¼k2j þ1
Ci
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy
1
9 =
;
o Y0 ð2pgr y x Þ dy 0 2 oniy 0
A24 ¼
8 Z
j¼1 i¼k2j þ1
: !
j¼1 i¼k1j þ1
Ci
Y0 ð2pgrx y ÞM5 ðy ; y 0 Þ dy
;
1
9 =
o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A25 ¼
N X j¼1
9 8 # " > > Z Z k j = < N X X X o 0 Y0 ð2pgrx y Þ Mc1 ðy ; y Þ dy i > 1 > oniy ; i¼k þ1 Ci : j¼1 i¼mj þ1 Ci k2j
1j
o Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A26 ¼
8 Z
1 þ c1 i
Ci
o oniy 0
: j¼1 !
i¼k1j þ1
Ci
Y0 ð2pgrx y ÞM5 ðy ; y 0 Þ dy
1
9 =
;
Y0 ð2pgr y x Þ dy 0 ;
2
A27 ¼
N X
mj X
j¼1 i¼k2j þ1
9 8 # " > >X Z < Z k j = N X o 0 Y0 ð2pgrx y Þ Mc1 ðy ; y Þ dy i > 1 > j¼1 i¼m þ1 Ci oniy Ci : ; j
!
o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 on 0 iy
A28 ¼
8 Z
1 þ c1 i
Ci
o oniy 0
: j¼1 !
i¼k2j þ1
Ci
Y0 ð2pgrx y ÞMc1 ðy ; y 0 Þ dy
Y0 ð2pgr y x Þ dy 0 ; 2
1
i
9 =
;
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
A29 ¼
8 Z < X k1j n X
n X
Ci
j¼N þ1 i¼1þkj1
Z k2j X
: j¼N þ1
i¼1þk1j
Ci
373
Y0 ð2pgrx y ÞM7 ðy ; y 0 Þ dy 1
9 =
;
Y0 ð2pgr y x Þ dy 0 ;
2
A30 ¼
9 # " > Z = X o Y0 ð2pgrx y Þ M8 ðy ; y 0 Þdy > 1 Ci > Ci oniy ; : j¼N þ1 i¼1þk
8 > X Z < n X k2j
n X
j¼N þ1 i¼k1j þ1
k1j
j1
o Y0 ð2pgr y x Þdy 0 ; 2 oniy 0
A31 ¼
8 Z < X k1j n X
n X
Ci
j¼N þ1 i¼1þkj1
Z mj X
: j¼N þ1
i¼k2j þ1
Ci
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy 1
9 =
;
Y0 ð2pgr y x Þ dy 0 ;
2
A32 ¼
9 # " > Z = X o Y0 ð2pgrx y Þ M8 ðy ; y 0 Þdy > 1 Ci > Ci oniy ; : j¼N þ1 i¼1þk
8 > X Z < n
mj X
n X
j¼N þ1 i¼k2j þ1
j1
!
k1j
o þ ci Y0 ð2pgr y x Þdy 0 ; 2 oniy 0
9 8 # > Z > Z " k1j kj = < X n X X o 0 A33 ¼ Y0 ð2pgrx y Þ Lc1 ðy ; y Þdy i > 1 Ci > ; : j¼N þ1 i¼mj þ1 Ci oniy j¼N þ1 i¼1þk n X
j1
0
Y0 ð2pgr y x Þdy ; 2
A34 ¼
n X
8 Z > kj n < X X
j¼N þ1 i¼mj þ1
1 þ c1 i
Ci
> : j¼N þ1 !
o oniy 0
9 # > Z " k1j = X o Y0 ð2pgrx y Þ M8 ðy ; y 0 Þ dy > 1 oniy Ci ; i¼1þk j1
Y0 ð2pgr y x Þ dy 0 ; 2
374
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
A35 ¼
8 Z < X k2j n X
n X
Ci
j¼N þ1 i¼k1j þ1
: j¼N þ1
Z mj X Ci
i¼k2j þ1
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy
;
1
9 =
o Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A36 ¼
8 Z < X mj n X
n X
Ci
j¼N þ1 i¼k2j þ1
: j¼N þ1
Z k2j X Ci
i¼k1j þ1
Y0 ð2pgrx y ÞM7 ðy ; y 0 Þ dy
;
1
9 =
! o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A37 ¼
8 > X Z < n X k2j
n X
Ci
j¼N þ1 i¼k1j þ1
> : j¼N þ1
Z "
kj X i¼mj þ1
Ci
9 > =
#
o Y0 ð2pgrx y Þ Lc1 ðy ; y 0 Þ dy i > 1 oniy ;
o Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A38 ¼
8 Z < X kj n X
n X
1 þ c1 i
: j¼N þ1 !
Ci
j¼N þ1 i¼mj þ1
o oniy 0
Z k2j X i¼k1j þ1
Ci
Y0 ð2pgrx y ÞM7 ðy ; y 0 Þ dy 1
9 =
;
Y0 ð2pgr y x Þ dy 0 ; 2
A39 ¼
8 Z > mj n < X X
n X
Ci
j¼N þ1 i¼k2j þ1
!
> : j¼N þ1
9 # > Z " kj = X o 0 Y0 ð2pgrx y Þ Lc1 ðy ; y Þ dy i > 1 oniy ; i¼mj þ1 Ci
o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A40 ¼
n X
8 Z < X kj n X
j¼N þ1 i¼mj þ1
1 þ c1 i
: j¼N þ1 !
Ci
o oniy 0
Z mj X i¼k2j þ1
Ci
Y0 ð2pgr y x Þ dy 0 ; 2
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy 1
9 =
;
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
A41 ¼
8 Z > k1j N n < X X X Ci
j¼1 i¼1þkj1
> : j¼N þ1
375
9 > =
# Z " k1j X o Y0 ð2pgrx y Þ M8 ðy ; y 0 Þ dy > 1 oniy Ci ; i¼1þk j1
0
Y0 ð2pgr y x Þ dy ;
2
A42 ¼
9 8 # > Z > Z " k1j k1j N = 1 oniy Ci > Ci ; : j¼1 i¼1þk i¼1þk
n X j¼N þ1
j1
j1
0
Y0 ð2pgr y x Þ dy ;
2
A43 ¼
8 Z < X k1j N n X X Ci
j¼1 i¼1þkj1
Z k2j X
: j¼N þ1
Ci
i¼k1j þ1
Y0 ð2pgrx y ÞM7 ðy ; y 0 Þ dy 1
9 =
;
Y0 ð2pgr y x Þ dy 0 ;
2
A44 ¼
j¼N þ1
9 8 # " > > Z Z k1j = < N X X X o Y0 ð2pgrx y Þ M6 ðy ; y 0 Þ dy > 1 > oniy Ci ; i¼k þ1 Ci : j¼1 i¼1þk k2j
n X
1j
j1
o Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A45 ¼
8 Z < X k1j N n X X Ci : j¼N þ1
j¼1 i¼1þkj1
Z mj X i¼k2j þ1
Ci
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy 1
9 =
;
Y0 ð2pgr y x Þ dy 0 ;
2
A46 ¼
n X j¼N þ1
"
9 8 # > Z > Z " k1j mj = 1 > oniy Ci ; i¼k þ1 Ci : j¼1 i¼1þk 2j
j1
#
o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A47 ¼
N X
8 > X Z n < X k1j
j¼1 i¼1þkj1
Ci
> : j¼N þ1
Y0 ð2pgr y x Þ dy 0 ; 2
kj X i¼mj þ1
Z " Ci
#
9 > =
o Y0 ð2pgrx y Þ Lc1 ðy ; y 0 Þ dy i > 1 oniy ;
376
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
A48 ¼
9 8 # > Z > Z " k1j kj N = 1 > oniy Ci ; i¼mj þ1 Ci : j¼1 i¼1þk
n X j¼N þ1
j1
"
#
1 þ c1 i
o Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A49 ¼
N X
8 > Z < X n X k2j
j¼1 i¼k1j þ1
k1j
X
Z "
Ci > : j¼N þ1 i¼1þkj1 1
Ci
9 > =
#
o Y0 ð2pgrx y Þ M8 ðy ; y 0 Þ dy > 1 oniy ;
o Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A50 ¼
8 Z
n X
j¼N þ1 i¼1þkj1
Ci :
Ci
j¼1 i¼k1j þ1
Y0 ð2pgrx y ÞM5 ðy ; y 0 Þ dy 1
9 =
;
Y0 ð2pgr y x Þ dy 0 ; 2
A51 ¼
8 Z < X k2j N n X X j¼1 i¼k1j þ1
Z k2j X
Ci : j¼N þ1
i¼k1j þ1
Ci
Y0 ð2pgrx y ÞM7 ðy ; y 0 Þ dy
1
9 =
;
o Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A52 ¼
n X
8 Z
j¼N þ1 i¼k1j þ1
:
j¼1 i¼k1j þ1
Ci
Y0 ð2pgrx y ÞM5 ðy ; y 0 Þ dy 1
9 =
;
o Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A53 ¼
8 Z < X k2j N n X X j¼1 i¼k1j þ1
Ci
: j¼N þ1
o Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
Z mj X i¼k2j þ1
Ci
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy 1
9 =
;
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
A54 ¼
8 Z
n X
Ci
j¼N þ1 i¼k2j þ1
!
:
Ci
j¼1 i¼k1j þ1
377
Y0 ð2pgrx y ÞM5 ðy ; y 0 Þ dy 1
9 =
;
o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A55 ¼
8 > Z n < X X k2j
N X
Ci
j¼1 i¼k1j þ1
> : j¼N þ1
kj X
Z "
i¼mj þ1
Ci
9 > =
#
o Y0 ð2pgrx y Þ Lc1 ðy ; y 0 Þ dy i > 1 oniy ;
o Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A56 ¼
n X
8 Z
j¼N þ1 i¼mj þ1
1 þ c1 i
: j¼1 !
Ci
o oniy 0
i¼k1j þ1
Ci
Y0 ð2pgrx y ÞM5 ðy ; y 0 Þ dy
1
9 =
;
0
Y0 ð2pgr y x Þ dy ;
2
A57 ¼
N X
mj X
j¼1 i¼k2j þ1
8 Z > < X n Ci
> : j¼N þ1 !
9 # " > Z = X o 0 Y0 ð2pgrx y Þ M8 ðy ; y Þ dy > 1 oniy Ci ; i¼1þk k1j
j1
o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A58 ¼
n X
8 Z
j¼N þ1 i¼1þkj1
:
Ci
j¼1 i¼k2j þ1
Ci
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy 1
9 =
;
Y0 ð2pgr y x Þ dy 0 ; 2
A59 ¼
8 Z < X mj N n X X j¼1 i¼k2j þ1
Ci
!
: j¼N þ1
Z k2j X i¼k1j þ1
o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
Ci
Y0 ð2pgrx y ÞM7 ðy ; y 0 Þ dy 1
9 =
;
378
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
A60 ¼
8 Z
n X
Ci
j¼N þ1 i¼k1j þ1
:
j¼1 i¼k2j þ1
Ci
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy 1
9 =
;
o Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A61 ¼
8 Z < X mj N n X X : j¼N þ1
Ci
j¼1 i¼k2j þ1
Z mj X i¼k2j þ1
Ci
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy 1
! o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
9 =
;
A62 ¼
8 Z
n X
Ci
j¼N þ1 i¼k2j þ1
!
:
j¼1 i¼k2j þ1
Ci
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy 1
9 =
;
o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A63 ¼
mj X
N X
8 > X Z < n Ci
j¼1 i¼k2j þ1
> : j¼N þ1 !
kj X
Z "
i¼mj þ1
Ci
9 > =
#
o Y0 ð2pgrx y Þ Lc1 ðy ; y 0 Þ dy i > 1 oniy ;
o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A64 ¼
n X
8 Z
9 =
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy ; 1 : j¼N þ1 i¼mj þ1 Ci j¼1 i¼k2j þ1 Ci ! o 1 þ c1 Y0 ð2pgr y x Þ dy 0 ; i 2 oniy 0
A65 ¼
8 Z > kj N n < X X X j¼1 i¼mj þ1
1 þ c1 i
Ci
o oniy 0
> : j¼N þ1 !
9 # > Z " k1j = X o Y0 ð2pgrx y Þ M8 ðy ; y 0 Þ dy > 1 oniy Ci ; i¼1þk j1
Y0 ð2pgr y x Þ dy 0 ; 2
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
379
9 8 # > Z > Z " k1j kj n N = 1 Ci > ; : j¼1 i¼mj þ1 Ci oniy j¼N þ1 i¼1þk j1
0
Y0 ð2pgr y x Þ dy ; 2
A67 ¼
8 Z < X kj N n X X j¼1
9 =
Z k2j X
Y0 ð2pgrx y ÞM7 ðy ; y 0 Þ dy ; 1 : i¼mj þ1 Ci j¼N þ1 i¼k1j þ1 Ci ! o 1 þ c1 Y0 ð2pgr y x Þ dy 0 ; i 2 oniy 0
A68 ¼
n X j¼N þ1
9 8 # > Z > Z " k2j kj = 1 > oniy ; i¼k þ1 Ci : j¼1 i¼mj þ1 Ci 1j
o 0 Y0 ð2pgr y x Þ dy ; 2 oniy 0
A69 ¼
8 Z < X kj N n X X j¼1
9 =
Z mj X
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy ; 1 : C i i¼mj þ1 j¼N þ1 i¼k2j þ1 ! o 1 þ c1 Y0 ð2pgr y x Þ dy 0 ; i 2 oniy 0 Ci
A70 ¼
n X j¼N þ1
9 8 # > Z > Z " mj kj N = 1 > oniy ; i¼k þ1 Ci : j¼1 i¼mj þ1 Ci 2j
!
o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A71 ¼
8 Z > kj N n < X X X j¼1 i¼mj þ1
1 þ c1 i
Ci
o oniy 0
> : j¼N þ1 !
9 # > Z " kj = X o Y0 ð2pgrx y Þ Mc1 ðy ; y 0 Þ dy i > 1 oniy ; i¼mj þ1 Ci
Y0 ð2pgr y x Þ dy 0 ; 2
380
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388 n X
A72 ¼
j¼N þ1
9 8 # > Z > Z " kj kj N = 1 > oniy ; i¼mj þ1 Ci : j¼1 i¼mj þ1 Ci
1 þ c1 i
o oniy 0
!
Y0 ð2pgr y x Þ dy 0 ; 2
where we deduce also that 1 X ðmÞ ð1Þm K1 ðy ; y 0 Þ;
M1 ðy ; y 0 Þ ¼
m¼0
1 o Y0 ð2pgr y y 0 Þ; 2 oniy 0
ð0Þ
K1 ðy ; y 0 Þ ¼
0
1 X
m¼0
M2 ðy ; y Þ ¼
ðmÞ
K2 ðy ; y 0 Þ;
1 o Y0 ð2pgr y y 0 Þ; 2 oniy
ð0Þ
K2 ðy ; y 0 Þ ¼
Mci ðy ; y 0 Þ ¼
1 X
KcðmÞ ðy ; y 0 Þ; i
m¼0
0
1 1 o B Kcð0Þ þ ci AY0 ð2pgr y y 0 Þ; ðy ; y 0 Þ ¼ @ i 2 oniy 0
Mc1 ðy ; y 0 Þ ¼ i
1 X
ðmÞ
ð1Þm Kc1 ðy ; y 0 Þ;
i
m¼0
0
1 2
1B o o C ð0Þ Kc1 ðy ; y 0 Þ ¼ @ þ c1 AY0 ð2pgr y y 0 Þ; i i 2 oniy 0 oniy oniy 0
M3 ðy ; y 0 Þ ¼
M4 ðy ; y 0 Þ ¼
Lci ðy ; y 0 Þ ¼
1 X m¼0
ðmÞ
K1 ðy ; y 0 Þ;
1 X m ðmÞ ð1Þ K2 ðy ; y 0 Þ;
m¼0 1 X m¼0
m
ð1Þ KcðmÞ ðy ; y 0 Þ; i
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
Lc1 ðy ; y 0 Þ ¼ i
1 X
M5 ðy ; y 0 Þ ¼
ðmÞ
Kc1 ðy ; y 0 Þ;
i
m¼0
1 X ðmÞ ð1Þm K3 ðy ; y 0 Þ;
m¼0
1 o2 Y0 ð2pgr y y 0 Þ; 2 oniy oniy 0
ð0Þ
K3 ðy ; y 0 Þ ¼
M6 ðy ; y 0 Þ ¼
381
1 X
ðmÞ
K4 ðy ; y 0 Þ;
m¼0
1 ð0Þ K4 ðy ; y 0 Þ ¼ Y0 ð2pgr y y 0 Þ; 2 Mci ðy ; y 0 Þ ¼
1 X m ð1Þ KcðmÞ ðy ; y 0 Þ; i
m¼0
0
1
1 B o2 o C þ ci Kcð0Þ ðy ; y 0 Þ ¼ @ AY0 ð2pgr y y 0 Þ; i 2 oniy oniy 0 oniy 0
M7 ðy ; y 0 Þ ¼
M8 ðy ; y 0 Þ ¼
1 X
ðmÞ
1 X m ðmÞ ð1Þ K4 ðy ; y 0 Þ;
m¼0
Mc1 ðy ; y 0 Þ ¼
K3 ðy ; y 0 Þ;
m¼0
ðy ; y 0 Þ ¼ Mc1 i
i
1 X
ðmÞ
Kc1 ðy ; y 0 Þ;
1 X
i
m¼0
m
0
ðmÞ
ð1Þ Kc1 ðy ; y Þ;
m¼0
i
and 1
0
1 o C ð0Þ Kc1 ðy ; y 0 Þ ¼ @1 þ c1 AY0 ð2pgr y y 0 Þ: i i 2 oniy 0
In these formulae, we note for example that KcðmÞ ðy ; y 0 Þ are the iterates of i ^ On the basis of (4.2), the function ,ðx ; x ; gÞ will be estimated for 1 2 jgj ! 1 together with small/large impedances ci . The case when x and x lie in
Kcð0Þ ðy ; y 0 Þ. i
1
2
382
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
the neighborhood of oXj is particularly interesting. For this case we use the local expansions of the functions: Y0 ð2pgrx y Þ;
o Y0 ð2pgrx y Þ; oniy
ð4:3Þ
when the distance between x and y is small, which are very similar to that obtained in [12–17]. Consequently, for small/large impedances ci the local behaviour of the kernels: ð0Þ
ð0Þ
K1 ðy ; y 0 Þ;
K2 ðy ; y 0 Þ;
ðy ; y 0 Þ; Kcð0Þ i
Kcð0Þ ðy ; y 0 Þ; i
ð0Þ
K3 ðy ; y 0 Þ;
ð0Þ
Kc1 ðy ; y 0 Þ; i
ð0Þ
K4 ðy ; y 0 Þ;
ð0Þ
Kc1 ðy ; y 0 Þ; i
ð4:4Þ ð4:5Þ
when the distance between y and y 0 is small, follows directly from knowledge of the local expansions of (4.3). in the upper half-plane n2 > 0 of the Definition 4.1. If n and n are points qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 2 2 2 ðn1 ; n2 Þ-plane, then we define q12 ¼ ðn11 n12 Þ þ ðn21 þ n22 Þ . An ek ðn ; n ; gÞ-function is defined for points n and n belonging to suffi1 2 1 2 ciently small domains DðIi Þ ði ¼ 1 þ kj1 ; . . . ; kj Þ ðj ¼ 1; . . . ; nÞ except when n ¼ n 2 Ii , where k is called the degree of this function. For every positive
1
2
integer ^, it has the following local expansions (see [8,12–17]): !l1 !l2 X o o 1 2 P1 2 P2 k f ðn1 Þðn1 Þ ðn2 Þ Y0 ð2pgq12 Þ e ðn ; n ; gÞ ¼ 1 2 on11 on21 þ R^ ðn ; n ; gÞ;
ð4:6Þ
1 2
P where denotes a sum of a finite number of terms in which f ðn11 Þ is an infinitely differentiable function. In this expansion P1 ; P2 ; l1 ; l2 ; l3 are integers, where P1 P 0, P2 P 0, l1 P 0, k ¼ minðP1 þ P2 qÞ, q ¼ l1 P þ l2 and the mini mum is taken over all terms which occur in the summation . The remainder R^ ðn ; n ; gÞ has continuous derivatives of all orders d 6 ^ satisfying 1 2 d ^
D R ðn ; n ; gÞ ¼ O½g^ expðAgq12 Þ
as jgj ! 1;
ð4:7Þ
1 2
where A is a positive constant. Thus, using methods similar to that obtained in [12–17], we can show that the functions (4.3) are ek -functions with degrees k ¼ 0; 1, respectively. Consequently, the function (4.4) are ek -functions with degrees k ¼ 0; 0; 1; 1 re-
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
383
spectively while for small/large impedances ci the functions (4.5) are ek -functions with degrees k ¼ 0; 1; 0; 1 respectively. Definition 4.2. If x and x are points in large domains X þ Ci , then we define 1
2
ri ¼ minðrx y þ rr x y Þ if y 2 Ci ði ¼ 1 þ kj1 ; . . . ; kj ; j ¼ 1; . . . ; N Þ; y
2
1
and Ri ¼ minðrx y þ rr x y Þ y
if y 2 Ci ði ¼ 1 þ kj1 ; . . . ; kj ; j ¼ N þ 1; . . . ; nÞ:
2
1
An Ek ðx ; x ; gÞ-function is defined and infinitely differentiable with respect 1 2 to x and x when these points belong to large domains X þ Ci except when 1 2 x ¼ x 2 Ci . Thus, the Ek -function has a similar local expansion of the ek 1 2 function (see [8,12–17]). With reference to [8,12–17], it is easily seen that (4.2) is an E0 ðx ; x ; gÞ1 2 function and consequently we get ^
Gðx ; x ; gÞ ¼ 1
2
kj N X X
Of½1 þ j logð2pgri Þj expðAi gri Þg
j¼1 i¼1þkj1
þ
n X
kj X
Of½1 þ j logð2pgRi Þj expðAi gRi Þg;
ð4:8Þ
j¼N þ1 i¼1þkj1
which is valid for jgj ! 1 and for small/large impedances ci , where Ai are ^ positive constants. The estimate (4.8) shows that Gðx ; x ; gÞ is exponentially 1 2 small for jgj ! 1. This proves that the integral (3.8) converges for jgj ! 1. k Following [12–17], if the e -expansions of the functions (4.3)–(4.5) are introduced into (4.2) and if we use formulae similar to (6.4) and (6.9) of Section 6 ^ in [16], we obtain the following local expansion of ,ðx ; x ; gÞ when ri and Ri are 1 2 small, which is valid for jgj ! 1 and for small/large impedances ci : ^
,ðx ; x ; gÞ ¼ 1
2
kj n X X j¼1 i¼1þkj1
^
,i ðx ; x ; gÞ; 1
2
ð4:9Þ
where: (a) If x and x belong to sufficiently small domains DðIi Þ ði ¼ 1 þ kj1 ; . . . ; k1j ; 1 2 j ¼ 1; . . . ; N Þ, then ^ 1 ,i ðx ; x ; gÞ ¼ Y0 ð2pgq12 Þ þ Ofg1 expðAi gq12 Þg: 1 2 4
ð4:10Þ
384
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
(b) If x and x belong to sufficiently small domains DðIi Þ ði ¼ k1j þ 1; . . . ; k2j ; 1 2 j ¼ 1; . . . ; N Þ, then ^ 1 ,i ðx ; x ; gÞ ¼ Y0 ð2pgq12 Þ þ Ofg1 expðAi gq12 Þg: 1 2 4
ð4:11Þ
(c) If x and x belong to sufficiently small domains DðIi Þ ði ¼ k2j þ 1; . . . ; mj ; 1 2 j ¼ 1; . . . ; N Þ, then 8 !1 9 = ^ 1< o 1 ci Y ð2pgq12 Þ þ Ofg1 expðAi gq12 Þg: ,i ðx ; x ; gÞ ¼ 1 2 ; 0 4: on21 ð4:12Þ (d) If x and x belong to sufficiently small domains DðIi Þ ði ¼ mj þ 1; . . . ; kj ; 1 2 j ¼ 1; . . . ; N Þ, then ( !) ^ 1 o 1 1 ci ,i ðx ; x ; gÞ ¼ Y0 ð2pgq12 Þ þ Ofg1 expðAi gq12 Þg: 1 2 4 on21 ð4:13Þ (e) If x and x belong to sufficiently small domains DðIi Þ ði ¼ kj1 þ 1; . . . ; k1j ; 1 2 j ¼ N þ 1; . . . ; nÞ, then ^ 1 ,i ðx ; x ; gÞ ¼ Y0 ð2pgq12 Þ þ Ofg1 expðAi gq12 Þg: 1 2 4
ð4:14Þ
(f) If x and x belong to sufficiently small domains DðIi Þ ði ¼ k1j þ 1; . . . ; k2j ; 1 2 j ¼ N þ 1; . . . ; nÞ, then ^ 1 ,i ðx ; x ; gÞ ¼ Y0 ð2pgq12 Þ þ Ofg1 expðAi gq12 Þg: 1 2 4
ð4:15Þ
(g) If x and x belong to sufficiently small domains DðIi Þ ði ¼ k2j þ 1; . . . ; mj ; 1 2 j ¼ N þ 1; . . . ; nÞ, then 8 !1 9 = ^ 1< o 1 ci Y ð2pgq12 Þ þ Ofg1 expðAi gq12 Þg: ,i ðx ; x ; gÞ ¼ 1 2 ; 0 4: on21 ð4:16Þ (h) If x and x belong to sufficiently small domains DðIi Þ ði ¼ mj þ 1; . . . ; kj ; 1 2 j ¼ N þ 1; . . . ; nÞ, then ( !) ^ 1 o 1 1 ci ,i ðx ; x ; gÞ ¼ Y0 ð2pgq12 Þ þ Ofg1 expðAi gq12 Þg: 1 2 4 on21 ð4:17Þ
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385
When ri P di > 0 ði ¼ 1 þ kj1 ; . . . ; kj ; j ¼ 1; . . . ; N Þ and Ri P di > 0 ^ ði ¼ 1 þ kj1 ; . . . ; kj ; j ¼ N þ 1; . . . ; nÞ the functions ,i ðx ; x ; gÞ are of order 1 2 OfexpðBgÞg as jgj ! 1, where B is a positive constant. Thus, since lim
ri !0
ri Ri ¼ lim ¼1 q12 Ri !0 q12
ð4:18Þ
(see [8,12–17]), then we have the asymptotic formulae (4.10)–(4.17) with q12 in the small domains DðIi Þ are replaced by ri and Ri in the large domains X þ Ci respectively. ^ Since for n2 P hi > 0 the functions ½,i ðx ; x ; gÞ x ¼x ¼x are of orders 1 2 1 2 Ofexpð4gAi hi Þg, the integral over the multi-connected region X of the func^ tion ½,ðx ; x ; gÞ x ¼x ¼x can be approximated in the following way (see (3.10)): 1
^
2
RðgÞ ¼
1
n X
2
kj X
j¼N þ1 i¼1þkj1
N X
kj X
j¼1 i¼1þkj1
þ
n X
kj X
Z
hi 2
n ¼0
Z
hi 2
n ¼0
Z
Li 3
n ¼0
Z
Li 3
^
½,i ðx ; x ; gÞ x
n ¼0
1
2
1
f1 Kj ðn ¼x ¼x
^
½,i ðx ; x ; gÞ x 1
2
1
1
Þn2 gdn2 dn1
2
f1 þ Kj ðn ¼x ¼x
1
Þn2 gdn2 dn1
2
Ofexpð2gAi hi Þg as jgj ! 1:
ð4:19Þ
j¼1 i¼1þkj1 ^
If the ek -expansions of ½,i ðx ; x ; gÞ x ¼x ¼x (see [12–17,30,31]) are introduced 1 2 1 2 into (4.19), one obtains an asymptotic series of the form ^ ia2 a3 2ia4 1 2þ 3 þO 4 RðgÞ ¼ as jgj ! 1; ð4:20Þ g pg pg pg where the coefficients a2 , a3 , a4 are given explicity in Section 2 which have been calculated from the ek -expansions with the help of the formale (11.3) of Section 11 in [16]. On inverting Fourier transforms to both sides of (4.20) we get RðtÞ ¼ a2 sign t þ a3 jtj þ a4 t2 sign t þ Oðt3 sign tÞ as jtj ! 0:
ð4:21Þ
From (3.6) and (4.21) we arrive at our result (2.1) and then the proof of Theorem 2.1 is completed.
5. Discussions and conclusions This paper deals with the very interesting problem of the influence of the boundary conditions on the distribution of the eigenvalues of the negative Laplacian in R2 . Our main result (2.1) has been derived using the method of
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Kac [6], Pleijel [8], Zayed et al., [12–17,19] and Zayed [20,30,31], via the construction of GreenÕs function of the wave equation as an integral equation which can be solved by iteration. So, we have shown that the asymptotic expansion of the trace of the wave kernel lðtÞ for small jtj reflects some geometrical properties of the general multi-connected vibrating membrane X in R2 with the piecewise smooth Dirichlet, Neumann and Robin boundary conditions (1.10). It is well known that the wave equation approach has given very strong result; the definitive one is that of H€ ormander [5] who has studied the distribution tr½expðitP Þ near t ¼ 0 for an elliptic positive semidefinite pseudodifferential operator P of order m in Rn . Recently, the wave equation approach for solving particular inverse problems has been discussed by Zayed et al. [12–17,19] and Zayed [20,30–33] who have studied the spectral distribution lðtÞ for small jtj for some different bounded domains with certain boundary conditions. An alternative method is the heat equation approach, which has been investigated by many authors, see for example P1[2,4,6–11,18,21– 29] who have studied the trace of the heat kernel hðtÞ ¼ x¼1 expðtEx Þ as t ! 0 for some bounded domains with certain boundary conditions. With reference to these articles, one can ask a question, is it possible just by listening with a perfect ear to hear the shape of a bounded drum even if we can not see it? This question has been put nicely by Kac [6], who simply asked, ‘‘Can one hear the shape of a drum? Gordon et al. [3] have shown that the answer of KacÕs question is negative, while Arendt [1] has shown recently that KacÕs question has a positive answer if we restrict ourselves to Dirichlet, Neumann and Robin boundary conditions. Arendt has shown also that the answer of KacÕs question is still negative for arbitrary boundary conditions. This is an interesting open problem, which has been left for the interested readers.
References [1] W. Arendt, Does diffusion determine the body?, J. Reine. Angew. Math. 550 (2002) 97–123. [2] H.P.W. Gottlieb, Eigenvalues of the Laplacian with Neumann boundary condition, J. Aust. Math. Soc. Ser. B 26 (1985) 293–309. [3] C. Gordon, D.L. Webb, S. Wolpert, One can not hear the shape of a drum, Bull. Amer. Math. Soc. 27 (1992) 134–138. [4] P. Greiner, An asymptotic expansion for the heat equation, Arch. Rat. Mech. Anal. 41 (1971) 163–218. [5] L. H€ ormander, The spectral function of an elliptic operator, Acta. Math. 121 (1968) 193–218. [6] M. Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966) 1–23. [7] H.P. McKean, I.M. Singer, Curvature and the eigenvalues of the Laplacian, J. Diff. Geom. 1 (1967) 43–69. [8] A. Pleijel, A study of certain GreenÕs functions with applications in the theory of vibrating membranes, Ark. Mat. 2 (1954) 553–569. [9] B.D. Sleeman, E.M.E. Zayed, An inverse eigenvalue problem for a general convex domain, J. Math. Anal. Appl. 94 (1983) 78–95.
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[10] L. Smith, The asymptotics of the heat equation for a boundary value problem, Invent. Math. 63 (1981) 467–493. [11] K. Stewartson, R.T. Waechter, On hearing the shape of a drum: further results, Proc. Camb. Philos. Soc. 69 (1971) 353–363. [12] 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 (2001) 171–180. [13] 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, Int. J. Math. Math. Sci. 25 (2001) 717–726. [14] 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. [15] E.M.E. Zayed, I.H. Abdel-Halim, 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, Appl. Math. Comput. 132 (2002) 187–204. [16] 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. [17] E.M.E. Zayed, A.A.M. Hassan, The wave equation approach for a region in R2 with two impedance boundary conditions, J Inst. Math.Comput. Sci. (Math. Ser.) 2 (1989) 153–167. [18] 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. [19] E.M.E. Zayed, A.I. Younis, The wave equations approach to an inverse problem for a general convex domain with impedance boundary conditions, Bull. Greek Math. Soc. 37 (1995) 123– 130. [20] E.M.E. Zayed, The wave equation approach to Robin inverse problems for a doublyconnected region, Bull. Calcutta Math. Soc. 80 (1988) 331–342. [21] E.M.E. Zayed, Hearing the shape of a general convex domain, J. Math. Anal. Appl. 142 (1989) 170–187. [22] E.M.E. Zayed, Heat equation for an arbitrary doubly connected region in R2 with mixed boundary conditions, Z. Angew. Math. Phys. 40 (1989) 339–355. [23] E.M.E. Zayed, On hearing the shape of an arbitrary doubly connected region in R2 , J. Aust. Math. Soc. Ser. B 31 (1990) 472–483. [24] 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. [25] E.M.E. Zayed, An inverse problem for a general multiply connected bounded domain, Appl. Anal. 59 (1995) 121–145. [26] E.M.E. Zayed, An inverse problem for a general doubly connected bounded domain with impedance boundary conditions, Portugal Math. 53 (1996) 53–72. [27] 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. [28] E.M.E. Zayed, Short-time asymptotics of the heat kernel of the Laplacian of a general doubly connected domain in R2 with Robin boundary conditions, Soochow J. Math. 25 (1999) 71–89. [29] E.M.E. Zayed, Short-time asymptotics of the heat kernel of the Laplacian for a multiplyconnected domain in R2 with Robin boundary conditions, Appl. Anal. 77 (2001) 177–194. [30] E.M.E. Zayed, The asymptotics of the two-dimensional wave equation for a general multiconnected vibrating membrane with piecewise smooth Robin boundary condititons, Acta. Math. Sinica, English Series, in press. [31] E.M.E. Zayed, The wave equation approach to an inverse problem for a general multiconnected domain in R2 with mixed boundary conditions, Appl. Math. Comput. 144 (2003) 457–474.
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[32] E.M.E. Zayed, An inverse problem of the three-dimensional wave equation for a general annular vibrating membrane with piecewise smooth boundary condititons, J. Appl. Math. Comput. 12 (2003) 81–105. [33] E.M.E. Zayed, The wave equation approach for solving inverse eigenvalue problems of a multi-connected region in R3 with Robin boundary conditions, Appl. Math. Comput., in press.
An inverse eigenvalue problem of the wave equation for a multi-connected region in R2 together with three different types of boundary conditions E.M.E. Zayed Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt
Abstract P 1 1=2 The trace of the wave kernel lðtÞ ¼ 1 x¼1 expðitEx Þ, where fEx gx¼1 are the eiP2 2 o 2 1 2 genvalues of the negative Laplacian r ¼ k¼1 ðoxk Þ in the ðxp;ffiffiffiffiffiffi x ffiÞ-plane, is studied for a variety of bounded domains, where 1 < t < 1 and i ¼ 1. The dependence of lðtÞ on the connectivity of bounded domains and the Dirichlet, Neumann and Robin boundary conditions are analyzed. Particular attention is given for a multi-connected vibrating membrane in R2 surrounded by simply connected bounded domains Xj with smooth boundaries oXj ðj ¼ 1; . . . ; nÞ, where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components Skj Ci ði ¼ 1 þ kj1 ; . . . ; kj Þ of the boundaries oXj is considered, such that oXj ¼ i¼1þk Ci j1 where k0 ¼ 0. The basic problem is to extract information on the geometry of X using the wave equation approach from complete knowledge of its eigenvalues. Some geometrical quantities of X (e.g. the area of X, the total lengths of its boundary, the curvature of its boundary, the number of the holes of X, etc.) are determined from the asymptotic expansion of the trace of the wave kernel lðtÞ for small jtj. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Inverse problem; Wave kernel; Eigenvalues; Multi-connected vibrating membrane; Negative Laplacian; Heat kernel
E-mail address: [email protected] (E.M.E. Zayed). 0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00715-X
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1. Introduction The underlying inverse eigenvalue problem is to deduce some geometrical quantities associated with a bounded domain in R2 from complete knowledge of the eigenvalues of the negative Laplacian by using the wave equation approach. Let X be a simply connected bounded domain in R2 with a smooth boundary oX. Consider the Robin problem r2 / ¼ E/
ð1:1Þ
in X
and
o þc /¼0 on
ð1:2Þ
on oX;
where ono denotes differentiation along the outward normal to the boundary oX Denote its eiand c is positive constant impedance, with u 2 C 2 ðXÞ \ CðXÞ. genvalues, counted according to multiplicity, by 0 < E1 6 E2 6 6 E x 6 ! 1
as x ! 1:
ð1:3Þ
Zayed and Hassan [16] have discussed the problem (1.1) and (1.2) for small/ large impedance c and have determined some geometrical quantities of X using the wave equation approach by analyzing the asymptotic expansion of the trace of the wave kernel lðtÞ ¼
1 X
expðitEx1=2 Þ
as jtj ! 0;
ð1:4Þ
x¼1
pffiffiffiffiffiffiffi which represents a tempered distribution for 1 < t < 1 and i ¼ 1. Note that when c is small, the Robin boundary condition (1.2) looks approximately like the Neumann boundary condition, while when c is large, the Robin boundary condition (1.2) looks approximately like the Dirichlet boundary condition provided ono remains finite. Zayed et al. [16,17,19] and Zayed [20] have discussed the problem (1.1) and (1.2) in the following cases and have obtained the following results: Case 1.1. If c ¼ 0 (the Neumann problem) lðtÞ ¼
jXj joXj H ðjtjÞ þ sign t þ a0 jtj þ Oðt2 sign tÞ 2pt 8
as jtj ! 0:
ð1:5Þ
as jtj ! 0;
ð1:6Þ
Case 1.2. If c ! 1 (the Dirichlet problem) lðtÞ ¼
jXj joXj H ðjtjÞ sign t þ a0 jtj þ Oðt2 sign tÞ 2pt 8
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
where H ðjtjÞ is the Heaviside unit function and 8 t > 0; < 1; sign t ¼ 0; t ¼ 0; : 1; t < 0:
363
ð1:7Þ
In these formulae jXj is the area of X and joXj is the total length of oX while the sign ± of the second term of lðtÞ determines whether we have the Neumann or the Dirichlet problem. 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 1h . 6 Case 1.3 (the mixed problem). If L1 is the length of a component C1 of the boundary oX with the Neumann boundary condition, and if L2 is the length of the remaining component C2 ¼ oX n C1 of oX with the Dirichlet boundary condition, then with reference to the articles [12–17], we get lðtÞ ¼
jXj ðL1 L2 Þ H ðjtjÞ þ sign t þ a0 jtj þ Oðt2 sign tÞ 2pt 8
as jtj ! 0: ð1:8Þ
Note that the order term Oðt2 sign tÞ in (1.6)–(1.8) is yet undetermined. So, in the present paper, we discuss what geometrical quantities are contained in this order term in the case X is a multi-connected vibrating membrane together with the piecewise smooth Dirichlet, Neumann and Robin boundary conditions (1.10) stated below. The object of this paper is to discuss the following more general inverse eigenvalue problem: Let X be a general multi-connected vibrating membrane in R2 surrounded internally by simply connected bounded domains Xj with smooth boundaries oXj ðj ¼ 1; . . . ; n 1Þ and externally by a simply connected bounded domain Xn with a smooth boundary oXn . Suppose that the eigenvalues (1.3) are given exactly for the Helmholtz equation r2 / ¼ E/
ð1:9Þ
in X
together with the following Dirichlet, Neumann and Robin boundary conditions: /¼0 o/ ¼0 on i o þ ci / ¼ 0 oni
on Ci ; i ¼ 1 þ kj1 ; . . . ; k1j ; on Ci ; i ¼ k1j þ 1; . . . ; k2j ; on Ci ; i ¼ k2j þ 1; . . . ; kj ;
ð1:10Þ
364
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where the boundaries oXj ðj ¼ 1; . . . ; nÞ consist of a finite number Skj of piecewise smooth components Ci ði ¼ 1 þ kj1 ; . . . ; kj Þ such that oXj ¼ i¼1þk Ci where j1 k0 ¼ 0 and ci are positive constants. The basic problem is to determine some geometrical quantities (e.g. the area of X, the total lengths of the boundaries, the curvature of the boundaries, the number of holes of X, etc.) associated with the main problem (1.9) and (1.10) by analyzing the asymptotic expansion of the trace of wave kernel lðtÞ for small jtj. Note that the special cases of the main problem (1.9) and (1.10) have been discussed by Zayed et al. [12–17,19] and Zayed [20,30,31]. Therefore, this problem can be considered as a more general one, which does not seem to have been investigated elsewhere.
2. Statement of the results Suppose that the boundaries oXj ðj ¼ 1; . . . ; nÞ of the multi-connected bounded domain X are given locally by the equations xa ¼ y a ðrj Þ ða ¼ 1; 2Þ in which rj are the arc lengths of the counter clock wise oriented boundaries and y a ðrj Þ 2 C 1 ðoXj Þ. Let Kj ðrj Þ be the curvatures of oXj ðj ¼ 1; . . . ; nÞ respectively and Li be the lengths of the components Ci ði ¼ 1 þ kj1 ; . . . ; kj Þ respectively. Let hi > 0 be sufficiently small numbers. Let ni be the minimum distances from a point x ¼ ðx1 ; x2 Þ of the domain X to the components Ci ði ¼ 1 þ kj1 ; . . . ; kj Þ respectively. Let ni ðrj Þ denote the inward unit normal to the components Ci respectively. Then, we note that the coordinates in the neighborhood of the components Ci ði ¼ 1 þ kj1 ; . . . ; kj ; j ¼ 1; . . . ; N Þ and its diagrams are in the same forms as in Section 5.2 of [22] with the interchanges n1 $ ni , h1 $ hi , I1 $ Ii , DðI1 Þ $ DðIi Þ and d1 $ di . Thus, we have the same formulae (5.2.1)–(5.2.5) of Section 5.2 in [22] with the interchanges n1 $ ni , n ðr1 Þ $ n ðrj Þ and t ðr1 Þ $ t ðrj Þ. Similarly, the coordinates in the neigh1 i 1 i borhoods of the components Ci ði ¼ 1 þ kj1 ; . . . ; kj ; j ¼ N þ 1; . . . ; nÞ and its diagrams are similar to the obtained in Section 5.1. of [22] with the interchanges n2 $ ni , h2 $ hi , I2 $ Ii , DðI2 Þ $ DðIi Þ and d2 $ di . Thus, we have the same formulae (5.1.1)–(5.1.5) of Section 5.1 in [22] with the interchanges n2 $ ni , n ðr2 Þ $ n ðrj Þ and t ðr2 Þ $ t ðrj Þ. 2
i
2
i
Theorem 2.1. With the assumptions stated above, the asymptotic expansion of the trace of the wave kernel lðtÞ for the Dirichlet, Neumann and Robin problem (1.9) and (1.10) can be written in the following form: a1 lðtÞ ¼ H ðjtjÞ þ a2 sign t þ a3 jtj þ a4 t2 sign t þ Oðt3 sign tÞ as jtj ! 0; t ð2:1Þ
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365
where 0 < ci 1 ði ¼ k2j þ 1; . . . ; mj Þ and ci 1 ði ¼ mj þ 1; . . . ; kj Þ ðj ¼ 1; . . . ; N ; N þ 1; . . . ; nÞ. Here, the coefficients am ðm ¼ 1 4Þ have the forms: a1 ¼ jXj=2p; 82 3 k2j mj n < X X X 1 4 a2 ¼ Li þ Li 5 8 j¼1 : i¼k þ1 i¼k2j þ1 1j 2 39 = Z k1j kj X X 4 Li þ Li þ c1 Kj ðrj Þ drj 5 ; i ; Ci i¼1þk i¼m þ1 j
j1
8 0 1 0 19 mj mj < = N n X X X X 2n 1 @ @ þ ci Li A ci Li A ; a3 ¼ ; 6 2p : j¼1 i¼k þ1 i¼k þ1 j¼N þ1 2j
2j
8 Z Z k1j k2j n < X X X 1 Kj2 ðrj Þ drj þ 7 Kj2 ðrj Þ drj a4 ¼ 512 j¼1 : i¼1þk Ci C i i¼k þ1 1j
j1
Z 64 pci þ7 Kj2 ðrj Þ c2i drj 7 Li i¼k2j þ1 Ci 9 3 # = Z " kj X 2p þ Kj2 ðrj Þ c1 drj : i ; L i C i i¼m þ1 mj X
j
With reference to the formulae (1.5)–(1.8) and to the articles [12– 17,19,20,30–33], the asymptotic expansion (2.1) may be interpreted as follows: i(i) X is a general multi-connected vibrating membrane in R2 and we have the piecewise smooth Dirichlet, Neumann and Robin boundary conditions (1.10) with small/large impedances ci . (ii) For the first four terms, X is a general multi-connected vibrating membrane in R2 of area jXj, it has 8 0 1 0 19 mj mj N n = X X X 3
2j
holes, provided h is a positive integer. Ci ði ¼ 1 þ kj1 ; . . . ; k1j ; j ¼ 1; . . . ; nÞ are of lengths PnThePkcomponents 1j i¼1þkj1 Li Þ and of curvatures Kj ðrj Þ together with the Dirichlet j¼1 ð boundary conditions. P Pk2j The components Ci ði ¼ k1j þ 1; . . . ; k2j ; j ¼ 1; . . . ; nÞ are of lengths nj¼1 ð i¼k Li Þ and curvatures Kj ðrj Þ together with the Neumann 1j þ1
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boundary conditions. . ; mj ; j ¼ 1; . . . ; nÞ are Pn Pmj The components Ci ði ¼ k2j2 þ 1; . . 64 1=2 of lengths j¼1 ð i¼k L Þ and of curvatures ½K ðr Þ ðpci c2i Þ together i j j 7 Li 2j þ1 with the Neumann boundary conditions, Pwhile R Pkj the components Ci ði ¼ mj þ 1; . . . ; kj ; j ¼ 1; . . . ; nÞ are of lengths nj¼1 ½ i¼m ðLi þ c1 K2 i Ci j j þ1 2 2p 3 1 1=2 ðrj Þ drj Þ and of curvatures ½Kj ðrj Þ ð Li Þ ci together with the Dirichlet boundary conditions. Corollary 2.1. If we consider the main problem (1.9) and (1.10) with 0 < ci 1 ði ¼ k2j þ 1; . . . ; kj ; j ¼ 1; . . . ; nÞ, then the P asymptotic expansion of lðtÞ follows kj directly from (2.1) by setting mj ¼ kj with i¼k as zero. j þ1 Corollary 2.2. If we consider the main problem (1.9) and (1.10) with ci 1 ði ¼ k2j þ 1; . . . ; kj ; j ¼ 1; . . . ; nÞ, then the asymptotic expansion of lðtÞ follows Pk2j directly from (2.1) by setting mj ¼ k2j with i¼k as zero. 2j þ1 Remark 2.1. In the case n ¼ 1 (i.e. X is simply connected), then we have the results of Zayed et al. [12,16,17,19]. Remark 2.2. In the case n ¼ 2 (i.e. X is doubly-connected), then we have the results of Zayed et al. [13–15] and Zayed [20]. We close this section with the following question: What is interpretation of X if ‘‘h’’ is not integer? This is an interesting open problem, which has been left for the interested readers.
3. Formulation of the mathematical problem With reference to the articles [12–17], it can be easily seen that the trace of the wave kernel lðtÞ associated with the main problem (1.9) and (1.10) is given by Z Z lðtÞ ¼ ½Gðx ; x ; tÞ x ¼x ¼x dx; ð3:1Þ X
1
2
1
2
where Gðx ; x ; tÞ is GreenÕs function for the wave equation 1 2 2 o 2 r 2 Gðx ; x ; tÞ ¼ 0 in X f1 < t < 1g 1 2 ot
ð3:2Þ
subject to the boundary conditions (1.10) and the initial conditions oGðx ; x ; tÞ 1
limGðx ; x ; tÞ ¼ 0; t!0
1
2
lim t!0
ot
2
¼ dðx x Þ: 1
2
ð3:3Þ
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367
Let us write Gðx ; x ; tÞ ¼ G0 ðx ; x ; tÞ þ ,ðx ; x ; tÞ; 1
2
1
2
1
ð3:4Þ
2
where
G0 ðx ; x ; tÞ ¼ 1
2
H ðjtj jx x jÞ 1 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2p t2 jx x j 1
ð3:5Þ
2
is the ‘‘fundamental solution’’ of the wave equation (3.2) while ,ðx ; x ; tÞ is the 1 2 ‘‘regular solution’’ chosen in such a way that Gðx ; x ; tÞ satisfies the boundary 1 2 conditions (1.10). By setting x ¼ x ¼ x , we find that 1
2
a1 lðtÞ ¼ H ðjtjÞ þ RðtÞ; t
ð3:6Þ
where a1 ¼ jXj=2p and RðtÞ is given by Z Z ½,ðx ; x ; tÞ x ¼x ¼x dx: RðtÞ ¼ 1
X
2
1
2
ð3:7Þ
The problem now is to determine the asymptotic expansions of RðtÞ for small jtj. In what follows, we shall use Fourier transforms with respect to 1 < t < 1 andp use < g < 1 as the Fourier transform parameter. Thus, ffiffiffiffiffiffi1 ffi we define for i ¼ 1, that b ; x ; gÞ ¼ Gðx 1
2
Z
þ1
1
e2pigt Gðx ; x ; tÞ dt: 1
ð3:8Þ
2
An application of the Fourier transform to the wave equation (3.2) shows b ; x ; gÞ satisfies the reduced wave equation that Gðx 1
2
b ; x ; gÞ ¼ dðx1 x2 Þ ðr2 þ 4p2 g2 Þ Gðx 1
2
in X
ð3:9Þ
together with the boundary conditions (1.10). The asymptotic expansions of RðtÞ for small jtj may then be deduced directly b from the asymptotic expansions of RðgÞ for large jgj, where Z Z b RðgÞ ¼ ½b , ðx ; x ; gÞ x ¼x ¼x dx: ð3:10Þ X
1
2
1
2
368
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
4. Derivation of the results It is well known (see for example [12–17]) that the membrane equation (3.9) ^ has the fundamental solution G0 ðx ; x ; gÞ ¼ 12 Y0 ð2pgrx x Þ, where rx x is the 1 2 1 2 1 2 distance between the points x ¼ ðx11 ; x21 Þ and x ¼ ðx12 ; x22 Þ of the domain X 1 2 while Y0 ð2pgrx x Þ is the Bessel function of the second kind and of zero order. 1 2 The existence of this solution enables us to construct integral equations for ^ Gðx ; x ; gÞ satisfying the boundary conditions (1.10) for small/large impedances 1 2 ci . Therefore, if we consider the main problem (1.9) and (1.10) with 0 < ci 1 ði ¼ k2j þ 1; . . . ; mj Þ and ci 1 ði ¼ mj þ 1; . . . ; kj Þ where ðj ¼ 1; . . . ; N ; N þ 1; . . . ; nÞ then GreenÕs theorem gives the following integral equation: ^ 1 Gðx ; x ; gÞ ¼ Y0 ð2pgrx x Þ 1 2 1 2 2 # Z " k 1j N X 1X o ^ Gðx ; y ; gÞ Y0 ð2pgr y x Þ dy þ 2 2 j¼1 i¼1þk oniy 1 Ci j1
" # Z ^ k2j N X 1X o Gðx ; y ; gÞ Y0 ð2pgr y x Þ dy 2 2 j¼1 i¼k þ1 Ci 1 oniy 1j
"
# Z ^ mj N X 1X o Gðx ; y ; gÞ þ ci Y0 ð2pgr y x Þ dy 2 2 j¼1 i¼k þ1 Ci 1 oniy 2j
# Z " kj N X 1X o ^ þ Gðx ; y ; gÞ 2 j¼1 i¼mj þ1 Ci oniy 1
" 1 þ c1 i
#
o Y0 ð2pgr y x Þ dy 2 oniy
# Z " k1j n X 1 X o ^ Gðx ; y ; gÞ Y0 ð2pgr y x Þ dy 2 2 j¼N þ1 i¼1þk oniy 1 Ci j1
" # Z ^ k2j n X 1 X o þ Gðx ; y ; gÞ Y0 ð2pgr y x Þ dy 2 2 j¼N þ1 i¼k þ1 Ci 1 oniy 1j
"
# Z ^ mj n X 1 X o þ Gðx ; y ; gÞ þ ci Y0 ð2pgr y x Þ dy 2 2 j¼N þ1 i¼k þ1 Ci 1 oniy 2j
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
# Z " kj n X 1 X o ^ Gðx ; y ; gÞ 2 j¼N þ1 i¼mj þ1 Ci oniy 1 " # 1 o 1 þ ci Y0 ð2pgr y x Þ dy : 2 oniy
369
ð4:1Þ
By applying the iteration methods (see for example [12–17,30–33]) to the ^ integral equation (4.1), we obtain GreenÕs function Gðx ; x ; gÞ which has a 1 2 regular part in the following form: b , ðx ; x ; gÞ ¼ 1
2
72 1X Ak ; 4 k¼1
ð4:2Þ
where # Z " k1j N X X o A1 ¼ Y0 ð2pgrx y Þ Y0 ð2pgr y x Þ dy ; 2 1 oniy Ci j¼1 i¼1þk j1
A2 ¼
# o Y0 ð2pgrx y Þ Y0 ð2pgr y x Þ dy ; 2 1 oniy Ci
"
# o Y0 ð2pgrx y Þ þ ci Y0 ð2pgr y x Þ dy ; 2 1 oniy Ci
Z mj N X X j¼1 i¼k2j þ1
A4 ¼
n X j¼N þ1
A6 ¼
Ci
# Z " k1j X o Y0 ð2pgrx y Þ Y0 ð2pgr y x Þ dy ; 2 1 oniy Ci i¼1þk
n X
j1
n X
# o Y0 ð2pgrx y Þ Y0 ð2pgr y x Þ dy ; 2 1 oniy Ci
Z k2j X
j¼N þ1 i¼k1j þ1
A7 ¼
" # o 1 o Y0 ð2pgrx y Þ 1 þ ci Y0 ð2pgr y x Þ dy ; 2 1 oniy oniy
Z kj N X X j¼1 i¼mj þ1
A5 ¼
"
Z k2j N X X j¼1 i¼k1j þ1
A3 ¼
"
# o Y0 ð2pgrx y Þ þ ci Y0 ð2pgr y x Þ dy ; 2 1 oniy Ci
Z mj X
j¼N þ1 i¼k2j þ1
"
370
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388 n X
A8 ¼
Ci
j¼N þ1 i¼mj þ1
A9 ¼
" # o 1 o Y0 ð2pgrx y Þ 1 þ ci Y0 ð2pgr y x Þ dy ; 2 1 oniy oniy
Z kj X
Z "
Z k1j N X X Ci
j¼1 i¼1þkj1
A10 ¼
A11 ¼
Z
Ci
"
# o Y0 ð2pgrx y ÞM2 ðy ; y Þ Y0 ð2pgr y x Þ dy dy 0 ; 2 1 oniy 0 Ci 0
"
# o Y0 ð2pgrx y ÞMci ðy ; y Þ þ ci Y0 ð2pgr y x Þdy dy 0 ; 2 1 0 on iy Ci
Z Z mj N X X Ci
j¼1 i¼k2j þ1
A12 ¼
# o Y0 ð2pgrx y Þ M1 ðy ; y 0 ÞY0 ð2pgr y x Þ dy dy 0 ; 2 1 oniy
Ci
Z k2j N X X j¼1 i¼k1j þ1
Z kj N X X "
Z
Ci
j¼1 i¼mj þ1
0
o Y0 ð2pgrx y ÞMc1 ðy ; y 0 Þ i 1 oniy
Ci
#
1 þ c1 i
o Y0 ð2pgr y x Þ dy dy 0 ; 2 oniy 0
A13 ¼
n X
Ci
j¼N þ1 i¼1þkj1
A14 ¼
n X
Z k2j X
A15 ¼
n X
Z
# o Y0 ð2pgrx y Þ M3 ðy ; y 0 ÞY0 ð2pgr y x Þ dy dy 0 ; 2 1 oniy
"
# o Y0 ð2pgrx y ÞM4 ðy ; y 0 Þ Y0 ð2pgr y x Þ dy dy 0 ; 2 1 oniy 0 Ci
"
# o Y0 ð2pgrx y ÞLci ðy ; y Þ þ ci Y0 ð2pgr y x Þdy dy 0 ; 2 1 oniy 0 Ci
Z Z mj X
j¼N þ1 i¼k2j þ1
A16 ¼
Ci
Ci
j¼N þ1 i¼k1j þ1
n X
Z "
Z k1j X
Ci
Z kj X
j¼N þ1 i¼mj þ1
" 1 þ c1 i
0
Z
Ci
Ci
#
o Y0 ð2pgrx y ÞLc1 ðy ; y 0 Þ i 1 oniy
o Y0 ð2pgr y x Þ dy dy 0 ; 2 oniy 0
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
A17 ¼
8 Z
Ci
:
j¼1 i¼k1j þ1
Ci
371
Y0 ð2pgrx y ÞM5 ðy ; y 0 Þ dy 1
9 =
;
Y0 ð2pgr y x Þ dy ; 2
A18 ¼
N X j¼1
9 8 # " > > Z Z k1j = < N X X X o Y0 ð2pgrx y Þ M6 ðy ; y 0 Þ dy > 1 > oniy Ci ; i¼k þ1 Ci : j¼1 i¼1þk k2j
1j
j1
o Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A19 ¼
8 Z
Ci
:
j¼1 i¼k2j þ1
Ci
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy 1
9 =
;
Y0 ð2pgr y x Þ dy 0 ; 2
A20 ¼
N X
mj X
j¼1 i¼k2j þ1
9 8 " # > Z > Z k1j =
!
o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
9 8 # > >X Z < Z " k1j kj = N N X X X o 0 A21 ¼ Y0 ð2pgrx y Þ Mc1 ðy ; y Þ dy i > 1 Ci > ; : j¼1 i¼mj þ1 Ci oniy j¼1 i¼1þk j1
Y0 ð2pgr y x Þ dy 0 ; 2
A22 ¼
N X
kj X
j¼1 i¼mj þ1
9 8 # " > Z > Z k1j =
1 þ c1 i
j1
o oniy 0
! Y0 ð2pgr y x Þ dy 0 ; 2
372
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
A23 ¼
8 Z
j¼1 i¼k1j þ1
:
j¼1 i¼k2j þ1
Ci
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy
1
9 =
;
o Y0 ð2pgr y x Þ dy 0 2 oniy 0
A24 ¼
8 Z
j¼1 i¼k2j þ1
: !
j¼1 i¼k1j þ1
Ci
Y0 ð2pgrx y ÞM5 ðy ; y 0 Þ dy
;
1
9 =
o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A25 ¼
N X j¼1
9 8 # " > > Z Z k j = < N X X X o 0 Y0 ð2pgrx y Þ Mc1 ðy ; y Þ dy i > 1 > oniy ; i¼k þ1 Ci : j¼1 i¼mj þ1 Ci k2j
1j
o Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A26 ¼
8 Z
1 þ c1 i
Ci
o oniy 0
: j¼1 !
i¼k1j þ1
Ci
Y0 ð2pgrx y ÞM5 ðy ; y 0 Þ dy
1
9 =
;
Y0 ð2pgr y x Þ dy 0 ;
2
A27 ¼
N X
mj X
j¼1 i¼k2j þ1
9 8 # " > >X Z < Z k j = N X o 0 Y0 ð2pgrx y Þ Mc1 ðy ; y Þ dy i > 1 > j¼1 i¼m þ1 Ci oniy Ci : ; j
!
o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 on 0 iy
A28 ¼
8 Z
1 þ c1 i
Ci
o oniy 0
: j¼1 !
i¼k2j þ1
Ci
Y0 ð2pgrx y ÞMc1 ðy ; y 0 Þ dy
Y0 ð2pgr y x Þ dy 0 ; 2
1
i
9 =
;
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
A29 ¼
8 Z < X k1j n X
n X
Ci
j¼N þ1 i¼1þkj1
Z k2j X
: j¼N þ1
i¼1þk1j
Ci
373
Y0 ð2pgrx y ÞM7 ðy ; y 0 Þ dy 1
9 =
;
Y0 ð2pgr y x Þ dy 0 ;
2
A30 ¼
9 # " > Z = X o Y0 ð2pgrx y Þ M8 ðy ; y 0 Þdy > 1 Ci > Ci oniy ; : j¼N þ1 i¼1þk
8 > X Z < n X k2j
n X
j¼N þ1 i¼k1j þ1
k1j
j1
o Y0 ð2pgr y x Þdy 0 ; 2 oniy 0
A31 ¼
8 Z < X k1j n X
n X
Ci
j¼N þ1 i¼1þkj1
Z mj X
: j¼N þ1
i¼k2j þ1
Ci
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy 1
9 =
;
Y0 ð2pgr y x Þ dy 0 ;
2
A32 ¼
9 # " > Z = X o Y0 ð2pgrx y Þ M8 ðy ; y 0 Þdy > 1 Ci > Ci oniy ; : j¼N þ1 i¼1þk
8 > X Z < n
mj X
n X
j¼N þ1 i¼k2j þ1
j1
!
k1j
o þ ci Y0 ð2pgr y x Þdy 0 ; 2 oniy 0
9 8 # > Z > Z " k1j kj = < X n X X o 0 A33 ¼ Y0 ð2pgrx y Þ Lc1 ðy ; y Þdy i > 1 Ci > ; : j¼N þ1 i¼mj þ1 Ci oniy j¼N þ1 i¼1þk n X
j1
0
Y0 ð2pgr y x Þdy ; 2
A34 ¼
n X
8 Z > kj n < X X
j¼N þ1 i¼mj þ1
1 þ c1 i
Ci
> : j¼N þ1 !
o oniy 0
9 # > Z " k1j = X o Y0 ð2pgrx y Þ M8 ðy ; y 0 Þ dy > 1 oniy Ci ; i¼1þk j1
Y0 ð2pgr y x Þ dy 0 ; 2
374
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
A35 ¼
8 Z < X k2j n X
n X
Ci
j¼N þ1 i¼k1j þ1
: j¼N þ1
Z mj X Ci
i¼k2j þ1
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy
;
1
9 =
o Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A36 ¼
8 Z < X mj n X
n X
Ci
j¼N þ1 i¼k2j þ1
: j¼N þ1
Z k2j X Ci
i¼k1j þ1
Y0 ð2pgrx y ÞM7 ðy ; y 0 Þ dy
;
1
9 =
! o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A37 ¼
8 > X Z < n X k2j
n X
Ci
j¼N þ1 i¼k1j þ1
> : j¼N þ1
Z "
kj X i¼mj þ1
Ci
9 > =
#
o Y0 ð2pgrx y Þ Lc1 ðy ; y 0 Þ dy i > 1 oniy ;
o Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A38 ¼
8 Z < X kj n X
n X
1 þ c1 i
: j¼N þ1 !
Ci
j¼N þ1 i¼mj þ1
o oniy 0
Z k2j X i¼k1j þ1
Ci
Y0 ð2pgrx y ÞM7 ðy ; y 0 Þ dy 1
9 =
;
Y0 ð2pgr y x Þ dy 0 ; 2
A39 ¼
8 Z > mj n < X X
n X
Ci
j¼N þ1 i¼k2j þ1
!
> : j¼N þ1
9 # > Z " kj = X o 0 Y0 ð2pgrx y Þ Lc1 ðy ; y Þ dy i > 1 oniy ; i¼mj þ1 Ci
o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A40 ¼
n X
8 Z < X kj n X
j¼N þ1 i¼mj þ1
1 þ c1 i
: j¼N þ1 !
Ci
o oniy 0
Z mj X i¼k2j þ1
Ci
Y0 ð2pgr y x Þ dy 0 ; 2
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy 1
9 =
;
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
A41 ¼
8 Z > k1j N n < X X X Ci
j¼1 i¼1þkj1
> : j¼N þ1
375
9 > =
# Z " k1j X o Y0 ð2pgrx y Þ M8 ðy ; y 0 Þ dy > 1 oniy Ci ; i¼1þk j1
0
Y0 ð2pgr y x Þ dy ;
2
A42 ¼
9 8 # > Z > Z " k1j k1j N =
n X j¼N þ1
j1
j1
0
Y0 ð2pgr y x Þ dy ;
2
A43 ¼
8 Z < X k1j N n X X Ci
j¼1 i¼1þkj1
Z k2j X
: j¼N þ1
Ci
i¼k1j þ1
Y0 ð2pgrx y ÞM7 ðy ; y 0 Þ dy 1
9 =
;
Y0 ð2pgr y x Þ dy 0 ;
2
A44 ¼
j¼N þ1
9 8 # " > > Z Z k1j = < N X X X o Y0 ð2pgrx y Þ M6 ðy ; y 0 Þ dy > 1 > oniy Ci ; i¼k þ1 Ci : j¼1 i¼1þk k2j
n X
1j
j1
o Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A45 ¼
8 Z < X k1j N n X X Ci : j¼N þ1
j¼1 i¼1þkj1
Z mj X i¼k2j þ1
Ci
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy 1
9 =
;
Y0 ð2pgr y x Þ dy 0 ;
2
A46 ¼
n X j¼N þ1
"
9 8 # > Z > Z " k1j mj =
j1
#
o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A47 ¼
N X
8 > X Z n < X k1j
j¼1 i¼1þkj1
Ci
> : j¼N þ1
Y0 ð2pgr y x Þ dy 0 ; 2
kj X i¼mj þ1
Z " Ci
#
9 > =
o Y0 ð2pgrx y Þ Lc1 ðy ; y 0 Þ dy i > 1 oniy ;
376
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
A48 ¼
9 8 # > Z > Z " k1j kj N =
n X j¼N þ1
j1
"
#
1 þ c1 i
o Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A49 ¼
N X
8 > Z < X n X k2j
j¼1 i¼k1j þ1
k1j
X
Z "
Ci > : j¼N þ1 i¼1þkj1 1
Ci
9 > =
#
o Y0 ð2pgrx y Þ M8 ðy ; y 0 Þ dy > 1 oniy ;
o Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A50 ¼
8 Z
n X
j¼N þ1 i¼1þkj1
Ci :
Ci
j¼1 i¼k1j þ1
Y0 ð2pgrx y ÞM5 ðy ; y 0 Þ dy 1
9 =
;
Y0 ð2pgr y x Þ dy 0 ; 2
A51 ¼
8 Z < X k2j N n X X j¼1 i¼k1j þ1
Z k2j X
Ci : j¼N þ1
i¼k1j þ1
Ci
Y0 ð2pgrx y ÞM7 ðy ; y 0 Þ dy
1
9 =
;
o Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A52 ¼
n X
8 Z
j¼N þ1 i¼k1j þ1
:
j¼1 i¼k1j þ1
Ci
Y0 ð2pgrx y ÞM5 ðy ; y 0 Þ dy 1
9 =
;
o Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A53 ¼
8 Z < X k2j N n X X j¼1 i¼k1j þ1
Ci
: j¼N þ1
o Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
Z mj X i¼k2j þ1
Ci
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy 1
9 =
;
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
A54 ¼
8 Z
n X
Ci
j¼N þ1 i¼k2j þ1
!
:
Ci
j¼1 i¼k1j þ1
377
Y0 ð2pgrx y ÞM5 ðy ; y 0 Þ dy 1
9 =
;
o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A55 ¼
8 > Z n < X X k2j
N X
Ci
j¼1 i¼k1j þ1
> : j¼N þ1
kj X
Z "
i¼mj þ1
Ci
9 > =
#
o Y0 ð2pgrx y Þ Lc1 ðy ; y 0 Þ dy i > 1 oniy ;
o Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A56 ¼
n X
8 Z
j¼N þ1 i¼mj þ1
1 þ c1 i
: j¼1 !
Ci
o oniy 0
i¼k1j þ1
Ci
Y0 ð2pgrx y ÞM5 ðy ; y 0 Þ dy
1
9 =
;
0
Y0 ð2pgr y x Þ dy ;
2
A57 ¼
N X
mj X
j¼1 i¼k2j þ1
8 Z > < X n Ci
> : j¼N þ1 !
9 # " > Z = X o 0 Y0 ð2pgrx y Þ M8 ðy ; y Þ dy > 1 oniy Ci ; i¼1þk k1j
j1
o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A58 ¼
n X
8 Z
j¼N þ1 i¼1þkj1
:
Ci
j¼1 i¼k2j þ1
Ci
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy 1
9 =
;
Y0 ð2pgr y x Þ dy 0 ; 2
A59 ¼
8 Z < X mj N n X X j¼1 i¼k2j þ1
Ci
!
: j¼N þ1
Z k2j X i¼k1j þ1
o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
Ci
Y0 ð2pgrx y ÞM7 ðy ; y 0 Þ dy 1
9 =
;
378
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
A60 ¼
8 Z
n X
Ci
j¼N þ1 i¼k1j þ1
:
j¼1 i¼k2j þ1
Ci
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy 1
9 =
;
o Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A61 ¼
8 Z < X mj N n X X : j¼N þ1
Ci
j¼1 i¼k2j þ1
Z mj X i¼k2j þ1
Ci
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy 1
! o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
9 =
;
A62 ¼
8 Z
n X
Ci
j¼N þ1 i¼k2j þ1
!
:
j¼1 i¼k2j þ1
Ci
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy 1
9 =
;
o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A63 ¼
mj X
N X
8 > X Z < n Ci
j¼1 i¼k2j þ1
> : j¼N þ1 !
kj X
Z "
i¼mj þ1
Ci
9 > =
#
o Y0 ð2pgrx y Þ Lc1 ðy ; y 0 Þ dy i > 1 oniy ;
o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A64 ¼
n X
8 Z
9 =
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy ; 1 : j¼N þ1 i¼mj þ1 Ci j¼1 i¼k2j þ1 Ci ! o 1 þ c1 Y0 ð2pgr y x Þ dy 0 ; i 2 oniy 0
A65 ¼
8 Z > kj N n < X X X j¼1 i¼mj þ1
1 þ c1 i
Ci
o oniy 0
> : j¼N þ1 !
9 # > Z " k1j = X o Y0 ð2pgrx y Þ M8 ðy ; y 0 Þ dy > 1 oniy Ci ; i¼1þk j1
Y0 ð2pgr y x Þ dy 0 ; 2
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
379
9 8 # > Z > Z " k1j kj n N =
0
Y0 ð2pgr y x Þ dy ; 2
A67 ¼
8 Z < X kj N n X X j¼1
9 =
Z k2j X
Y0 ð2pgrx y ÞM7 ðy ; y 0 Þ dy ; 1 : i¼mj þ1 Ci j¼N þ1 i¼k1j þ1 Ci ! o 1 þ c1 Y0 ð2pgr y x Þ dy 0 ; i 2 oniy 0
A68 ¼
n X j¼N þ1
9 8 # > Z > Z " k2j kj =
o 0 Y0 ð2pgr y x Þ dy ; 2 oniy 0
A69 ¼
8 Z < X kj N n X X j¼1
9 =
Z mj X
Y0 ð2pgrx y ÞMci ðy ; y 0 Þ dy ; 1 : C i i¼mj þ1 j¼N þ1 i¼k2j þ1 ! o 1 þ c1 Y0 ð2pgr y x Þ dy 0 ; i 2 oniy 0 Ci
A70 ¼
n X j¼N þ1
9 8 # > Z > Z " mj kj N =
!
o þ ci Y0 ð2pgr y x Þ dy 0 ; 2 oniy 0
A71 ¼
8 Z > kj N n < X X X j¼1 i¼mj þ1
1 þ c1 i
Ci
o oniy 0
> : j¼N þ1 !
9 # > Z " kj = X o Y0 ð2pgrx y Þ Mc1 ðy ; y 0 Þ dy i > 1 oniy ; i¼mj þ1 Ci
Y0 ð2pgr y x Þ dy 0 ; 2
380
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388 n X
A72 ¼
j¼N þ1
9 8 # > Z > Z " kj kj N =
1 þ c1 i
o oniy 0
!
Y0 ð2pgr y x Þ dy 0 ; 2
where we deduce also that 1 X ðmÞ ð1Þm K1 ðy ; y 0 Þ;
M1 ðy ; y 0 Þ ¼
m¼0
1 o Y0 ð2pgr y y 0 Þ; 2 oniy 0
ð0Þ
K1 ðy ; y 0 Þ ¼
0
1 X
m¼0
M2 ðy ; y Þ ¼
ðmÞ
K2 ðy ; y 0 Þ;
1 o Y0 ð2pgr y y 0 Þ; 2 oniy
ð0Þ
K2 ðy ; y 0 Þ ¼
Mci ðy ; y 0 Þ ¼
1 X
KcðmÞ ðy ; y 0 Þ; i
m¼0
0
1 1 o B Kcð0Þ þ ci AY0 ð2pgr y y 0 Þ; ðy ; y 0 Þ ¼ @ i 2 oniy 0
Mc1 ðy ; y 0 Þ ¼ i
1 X
ðmÞ
ð1Þm Kc1 ðy ; y 0 Þ;
i
m¼0
0
1 2
1B o o C ð0Þ Kc1 ðy ; y 0 Þ ¼ @ þ c1 AY0 ð2pgr y y 0 Þ; i i 2 oniy 0 oniy oniy 0
M3 ðy ; y 0 Þ ¼
M4 ðy ; y 0 Þ ¼
Lci ðy ; y 0 Þ ¼
1 X m¼0
ðmÞ
K1 ðy ; y 0 Þ;
1 X m ðmÞ ð1Þ K2 ðy ; y 0 Þ;
m¼0 1 X m¼0
m
ð1Þ KcðmÞ ðy ; y 0 Þ; i
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
Lc1 ðy ; y 0 Þ ¼ i
1 X
M5 ðy ; y 0 Þ ¼
ðmÞ
Kc1 ðy ; y 0 Þ;
i
m¼0
1 X ðmÞ ð1Þm K3 ðy ; y 0 Þ;
m¼0
1 o2 Y0 ð2pgr y y 0 Þ; 2 oniy oniy 0
ð0Þ
K3 ðy ; y 0 Þ ¼
M6 ðy ; y 0 Þ ¼
381
1 X
ðmÞ
K4 ðy ; y 0 Þ;
m¼0
1 ð0Þ K4 ðy ; y 0 Þ ¼ Y0 ð2pgr y y 0 Þ; 2 Mci ðy ; y 0 Þ ¼
1 X m ð1Þ KcðmÞ ðy ; y 0 Þ; i
m¼0
0
1
1 B o2 o C þ ci Kcð0Þ ðy ; y 0 Þ ¼ @ AY0 ð2pgr y y 0 Þ; i 2 oniy oniy 0 oniy 0
M7 ðy ; y 0 Þ ¼
M8 ðy ; y 0 Þ ¼
1 X
ðmÞ
1 X m ðmÞ ð1Þ K4 ðy ; y 0 Þ;
m¼0
Mc1 ðy ; y 0 Þ ¼
K3 ðy ; y 0 Þ;
m¼0
ðy ; y 0 Þ ¼ Mc1 i
i
1 X
ðmÞ
Kc1 ðy ; y 0 Þ;
1 X
i
m¼0
m
0
ðmÞ
ð1Þ Kc1 ðy ; y Þ;
m¼0
i
and 1
0
1 o C ð0Þ Kc1 ðy ; y 0 Þ ¼ @1 þ c1 AY0 ð2pgr y y 0 Þ: i i 2 oniy 0
In these formulae, we note for example that KcðmÞ ðy ; y 0 Þ are the iterates of i ^ On the basis of (4.2), the function ,ðx ; x ; gÞ will be estimated for 1 2 jgj ! 1 together with small/large impedances ci . The case when x and x lie in
Kcð0Þ ðy ; y 0 Þ. i
1
2
382
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
the neighborhood of oXj is particularly interesting. For this case we use the local expansions of the functions: Y0 ð2pgrx y Þ;
o Y0 ð2pgrx y Þ; oniy
ð4:3Þ
when the distance between x and y is small, which are very similar to that obtained in [12–17]. Consequently, for small/large impedances ci the local behaviour of the kernels: ð0Þ
ð0Þ
K1 ðy ; y 0 Þ;
K2 ðy ; y 0 Þ;
ðy ; y 0 Þ; Kcð0Þ i
Kcð0Þ ðy ; y 0 Þ; i
ð0Þ
K3 ðy ; y 0 Þ;
ð0Þ
Kc1 ðy ; y 0 Þ; i
ð0Þ
K4 ðy ; y 0 Þ;
ð0Þ
Kc1 ðy ; y 0 Þ; i
ð4:4Þ ð4:5Þ
when the distance between y and y 0 is small, follows directly from knowledge of the local expansions of (4.3). in the upper half-plane n2 > 0 of the Definition 4.1. If n and n are points qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 2 2 2 ðn1 ; n2 Þ-plane, then we define q12 ¼ ðn11 n12 Þ þ ðn21 þ n22 Þ . An ek ðn ; n ; gÞ-function is defined for points n and n belonging to suffi1 2 1 2 ciently small domains DðIi Þ ði ¼ 1 þ kj1 ; . . . ; kj Þ ðj ¼ 1; . . . ; nÞ except when n ¼ n 2 Ii , where k is called the degree of this function. For every positive
1
2
integer ^, it has the following local expansions (see [8,12–17]): !l1 !l2 X o o 1 2 P1 2 P2 k f ðn1 Þðn1 Þ ðn2 Þ Y0 ð2pgq12 Þ e ðn ; n ; gÞ ¼ 1 2 on11 on21 þ R^ ðn ; n ; gÞ;
ð4:6Þ
1 2
P where denotes a sum of a finite number of terms in which f ðn11 Þ is an infinitely differentiable function. In this expansion P1 ; P2 ; l1 ; l2 ; l3 are integers, where P1 P 0, P2 P 0, l1 P 0, k ¼ minðP1 þ P2 qÞ, q ¼ l1 P þ l2 and the mini mum is taken over all terms which occur in the summation . The remainder R^ ðn ; n ; gÞ has continuous derivatives of all orders d 6 ^ satisfying 1 2 d ^
D R ðn ; n ; gÞ ¼ O½g^ expðAgq12 Þ
as jgj ! 1;
ð4:7Þ
1 2
where A is a positive constant. Thus, using methods similar to that obtained in [12–17], we can show that the functions (4.3) are ek -functions with degrees k ¼ 0; 1, respectively. Consequently, the function (4.4) are ek -functions with degrees k ¼ 0; 0; 1; 1 re-
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
383
spectively while for small/large impedances ci the functions (4.5) are ek -functions with degrees k ¼ 0; 1; 0; 1 respectively. Definition 4.2. If x and x are points in large domains X þ Ci , then we define 1
2
ri ¼ minðrx y þ rr x y Þ if y 2 Ci ði ¼ 1 þ kj1 ; . . . ; kj ; j ¼ 1; . . . ; N Þ; y
2
1
and Ri ¼ minðrx y þ rr x y Þ y
if y 2 Ci ði ¼ 1 þ kj1 ; . . . ; kj ; j ¼ N þ 1; . . . ; nÞ:
2
1
An Ek ðx ; x ; gÞ-function is defined and infinitely differentiable with respect 1 2 to x and x when these points belong to large domains X þ Ci except when 1 2 x ¼ x 2 Ci . Thus, the Ek -function has a similar local expansion of the ek 1 2 function (see [8,12–17]). With reference to [8,12–17], it is easily seen that (4.2) is an E0 ðx ; x ; gÞ1 2 function and consequently we get ^
Gðx ; x ; gÞ ¼ 1
2
kj N X X
Of½1 þ j logð2pgri Þj expðAi gri Þg
j¼1 i¼1þkj1
þ
n X
kj X
Of½1 þ j logð2pgRi Þj expðAi gRi Þg;
ð4:8Þ
j¼N þ1 i¼1þkj1
which is valid for jgj ! 1 and for small/large impedances ci , where Ai are ^ positive constants. The estimate (4.8) shows that Gðx ; x ; gÞ is exponentially 1 2 small for jgj ! 1. This proves that the integral (3.8) converges for jgj ! 1. k Following [12–17], if the e -expansions of the functions (4.3)–(4.5) are introduced into (4.2) and if we use formulae similar to (6.4) and (6.9) of Section 6 ^ in [16], we obtain the following local expansion of ,ðx ; x ; gÞ when ri and Ri are 1 2 small, which is valid for jgj ! 1 and for small/large impedances ci : ^
,ðx ; x ; gÞ ¼ 1
2
kj n X X j¼1 i¼1þkj1
^
,i ðx ; x ; gÞ; 1
2
ð4:9Þ
where: (a) If x and x belong to sufficiently small domains DðIi Þ ði ¼ 1 þ kj1 ; . . . ; k1j ; 1 2 j ¼ 1; . . . ; N Þ, then ^ 1 ,i ðx ; x ; gÞ ¼ Y0 ð2pgq12 Þ þ Ofg1 expðAi gq12 Þg: 1 2 4
ð4:10Þ
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E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
(b) If x and x belong to sufficiently small domains DðIi Þ ði ¼ k1j þ 1; . . . ; k2j ; 1 2 j ¼ 1; . . . ; N Þ, then ^ 1 ,i ðx ; x ; gÞ ¼ Y0 ð2pgq12 Þ þ Ofg1 expðAi gq12 Þg: 1 2 4
ð4:11Þ
(c) If x and x belong to sufficiently small domains DðIi Þ ði ¼ k2j þ 1; . . . ; mj ; 1 2 j ¼ 1; . . . ; N Þ, then 8 !1 9 = ^ 1< o 1 ci Y ð2pgq12 Þ þ Ofg1 expðAi gq12 Þg: ,i ðx ; x ; gÞ ¼ 1 2 ; 0 4: on21 ð4:12Þ (d) If x and x belong to sufficiently small domains DðIi Þ ði ¼ mj þ 1; . . . ; kj ; 1 2 j ¼ 1; . . . ; N Þ, then ( !) ^ 1 o 1 1 ci ,i ðx ; x ; gÞ ¼ Y0 ð2pgq12 Þ þ Ofg1 expðAi gq12 Þg: 1 2 4 on21 ð4:13Þ (e) If x and x belong to sufficiently small domains DðIi Þ ði ¼ kj1 þ 1; . . . ; k1j ; 1 2 j ¼ N þ 1; . . . ; nÞ, then ^ 1 ,i ðx ; x ; gÞ ¼ Y0 ð2pgq12 Þ þ Ofg1 expðAi gq12 Þg: 1 2 4
ð4:14Þ
(f) If x and x belong to sufficiently small domains DðIi Þ ði ¼ k1j þ 1; . . . ; k2j ; 1 2 j ¼ N þ 1; . . . ; nÞ, then ^ 1 ,i ðx ; x ; gÞ ¼ Y0 ð2pgq12 Þ þ Ofg1 expðAi gq12 Þg: 1 2 4
ð4:15Þ
(g) If x and x belong to sufficiently small domains DðIi Þ ði ¼ k2j þ 1; . . . ; mj ; 1 2 j ¼ N þ 1; . . . ; nÞ, then 8 !1 9 = ^ 1< o 1 ci Y ð2pgq12 Þ þ Ofg1 expðAi gq12 Þg: ,i ðx ; x ; gÞ ¼ 1 2 ; 0 4: on21 ð4:16Þ (h) If x and x belong to sufficiently small domains DðIi Þ ði ¼ mj þ 1; . . . ; kj ; 1 2 j ¼ N þ 1; . . . ; nÞ, then ( !) ^ 1 o 1 1 ci ,i ðx ; x ; gÞ ¼ Y0 ð2pgq12 Þ þ Ofg1 expðAi gq12 Þg: 1 2 4 on21 ð4:17Þ
E.M.E. Zayed / Appl. Math. Comput. 154 (2004) 361–388
385
When ri P di > 0 ði ¼ 1 þ kj1 ; . . . ; kj ; j ¼ 1; . . . ; N Þ and Ri P di > 0 ^ ði ¼ 1 þ kj1 ; . . . ; kj ; j ¼ N þ 1; . . . ; nÞ the functions ,i ðx ; x ; gÞ are of order 1 2 OfexpðBgÞg as jgj ! 1, where B is a positive constant. Thus, since lim
ri !0
ri Ri ¼ lim ¼1 q12 Ri !0 q12
ð4:18Þ
(see [8,12–17]), then we have the asymptotic formulae (4.10)–(4.17) with q12 in the small domains DðIi Þ are replaced by ri and Ri in the large domains X þ Ci respectively. ^ Since for n2 P hi > 0 the functions ½,i ðx ; x ; gÞ x ¼x ¼x are of orders 1 2 1 2 Ofexpð4gAi hi Þg, the integral over the multi-connected region X of the func^ tion ½,ðx ; x ; gÞ x ¼x ¼x can be approximated in the following way (see (3.10)): 1
^
2
RðgÞ ¼
1
n X
2
kj X
j¼N þ1 i¼1þkj1
N X
kj X
j¼1 i¼1þkj1
þ
n X
kj X
Z
hi 2
n ¼0
Z
hi 2
n ¼0
Z
Li 3
n ¼0
Z
Li 3
^
½,i ðx ; x ; gÞ x
n ¼0
1
2
1
f1 Kj ðn ¼x ¼x
^
½,i ðx ; x ; gÞ x 1
2
1
1
Þn2 gdn2 dn1
2
f1 þ Kj ðn ¼x ¼x
1
Þn2 gdn2 dn1
2
Ofexpð2gAi hi Þg as jgj ! 1:
ð4:19Þ
j¼1 i¼1þkj1 ^
If the ek -expansions of ½,i ðx ; x ; gÞ x ¼x ¼x (see [12–17,30,31]) are introduced 1 2 1 2 into (4.19), one obtains an asymptotic series of the form ^ ia2 a3 2ia4 1 2þ 3 þO 4 RðgÞ ¼ as jgj ! 1; ð4:20Þ g pg pg pg where the coefficients a2 , a3 , a4 are given explicity in Section 2 which have been calculated from the ek -expansions with the help of the formale (11.3) of Section 11 in [16]. On inverting Fourier transforms to both sides of (4.20) we get RðtÞ ¼ a2 sign t þ a3 jtj þ a4 t2 sign t þ Oðt3 sign tÞ as jtj ! 0:
ð4:21Þ
From (3.6) and (4.21) we arrive at our result (2.1) and then the proof of Theorem 2.1 is completed.
5. Discussions and conclusions This paper deals with the very interesting problem of the influence of the boundary conditions on the distribution of the eigenvalues of the negative Laplacian in R2 . Our main result (2.1) has been derived using the method of
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Kac [6], Pleijel [8], Zayed et al., [12–17,19] and Zayed [20,30,31], via the construction of GreenÕs function of the wave equation as an integral equation which can be solved by iteration. So, we have shown that the asymptotic expansion of the trace of the wave kernel lðtÞ for small jtj reflects some geometrical properties of the general multi-connected vibrating membrane X in R2 with the piecewise smooth Dirichlet, Neumann and Robin boundary conditions (1.10). It is well known that the wave equation approach has given very strong result; the definitive one is that of H€ ormander [5] who has studied the distribution tr½expðitP Þ near t ¼ 0 for an elliptic positive semidefinite pseudodifferential operator P of order m in Rn . Recently, the wave equation approach for solving particular inverse problems has been discussed by Zayed et al. [12–17,19] and Zayed [20,30–33] who have studied the spectral distribution lðtÞ for small jtj for some different bounded domains with certain boundary conditions. An alternative method is the heat equation approach, which has been investigated by many authors, see for example P1[2,4,6–11,18,21– 29] who have studied the trace of the heat kernel hðtÞ ¼ x¼1 expðtEx Þ as t ! 0 for some bounded domains with certain boundary conditions. With reference to these articles, one can ask a question, is it possible just by listening with a perfect ear to hear the shape of a bounded drum even if we can not see it? This question has been put nicely by Kac [6], who simply asked, ‘‘Can one hear the shape of a drum? Gordon et al. [3] have shown that the answer of KacÕs question is negative, while Arendt [1] has shown recently that KacÕs question has a positive answer if we restrict ourselves to Dirichlet, Neumann and Robin boundary conditions. Arendt has shown also that the answer of KacÕs question is still negative for arbitrary boundary conditions. This is an interesting open problem, which has been left for the interested readers.
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