PORTUGALIAE MATHEMATICA
Vol. 53 Fasc. 1 – 1996
AN INVERSE PROBLEM FOR A GENERAL DOUBLY-CONNECTED BOUNDED DOMAIN WITH
IMPEDANCE BOUNDARY CONDITIONS
E.M.E. Zayed
P∞
Abstract: The spectral function θ(t) = ν=1 exp(−t λν ), where {λν }∞
ν=1 are the
P2
∂ 2
1
2
eigenvalues of the negative Laplacian −∆ = − i=1 ( ∂xi ) in the (x , x )-plane, is studied
for a general doubly-connected bounded domain Ω in IR2 together with its smooth inner
boundary ∂Ω1 and its smooth outer boundary ∂Ω2 , where piecewise smooth impedance
boundary conditions on the two parts Γ1 , Γ2 of ∂Ω1 and on the two parts Γ3 , Γ4 of ∂Ω2
are considered, such that ∂Ω1 = Γ1 ∪ Γ2 and ∂Ω2 = Γ3 ∪ Γ4 .
1 – Introduction
The underlying inverse problem is to determine some geometric quantities
associated with a bounded domain, from a complete knowledge of the eigenvalues
P2
∂ 2
1 2
{λν }∞
i=1 ( ∂xi ) in the (x , x )-plane.
ν=1 for the negative Laplacian −∆ = −
Let Ω ⊆ IR2 be a simply connected bounded domain with a smooth boundary
∂Ω. Consider the impedance problem
(1.1)
− ∆u = λ u
(1.2)
µ
in Ω ,
∂
+γ u=0
∂n
¶
on ∂Ω ,
∂
where ∂n
denotes differentiation along the inward pointing normal to ∂Ω and γ
is a positive constant, with u ∈ C 2 (Ω) ∩ C(Ω).
Denote its eigenvalues, counted according to multiplicity, by
(1.3)
0 < λ1 ≤ λ2 ≤ ... ≤ λν ≤ ... → ∞
Received : September 10, 1994.
as ν → ∞ .
54
E.M.E. ZAYED
The problem of determining some geometric quantities associated with the bounded domain Ω has been discussed recently by Sleeman and Zayed [5], using the
asymptotic expansion of the spectral function
(1.4)
θ(t) =
∞
X
as t → 0+ .
exp(−t λν )
ν=1
Problem (1.1)–(1.2) has been investigated by many authors (see for example the
articles [1–4, 6, 7]) in the following special cases:
Case 1. γ = 0 (The Neumann problem)
(1.5) θ(t) =
µ ¶1/2 Z
|∂Ω|
t
7
|Ω|
+
+ a0 +
1/2
4π t 8(π t)
256 π
K 2 (σ) dσ + O(t) as t → 0+ .
∂Ω
Case 2. γ → ∞ (The Dirichlet problem)
(1.6) θ(t) =
µ ¶1/2 Z
|∂Ω|
1
t
|Ω|
−
+ a0 +
4π t 8(π t)1/2
256 π
K 2 (σ) dσ + O(t) as t → 0+ .
∂Ω
In these formulae, |Ω| is the area of Ω, |∂Ω| is the total length of its boundary
∂Ω, σ is the arc length of the counter clockwise oriented boundary ∂Ω and K(σ)
is the curvature of ∂Ω. The constant term a0 has geometric significance, e.g., if Ω
is smooth and convex, then a0 = 61 and if Ω is permitted to have a finite number
“H” of smooth convex holes, then a0 = (1 − H)/6.
Case 3. (The mixed problem)
If L1 is the length of a part Γ1 of the boundary ∂Ω with the Neumann boundary condition, and if L2 is the length of the remaining part Γ2 = ∂Ω\Γ1 of ∂Ω with
the Dirichlet boundary condition, such that ∂Ω = Γ1 ∪ Γ2 , then with reference to
[1, 8, 9] we get
θ(t) =
(1.7)
µ ¶1/2 ½ Z
L1 − L 2
|Ω|
1
t
+
+ a0 +
4π t 8(π t)1/2
256 π
+
Z
¾
K 2 (σ) dσ + O(t)
Γ2
K 2 (σ) dσ
7
Γ1
as t → 0+ .
Zayed [8] has recently discussed the equation (1.1) together with the piecewise
smooth impedance boundary conditions:
(1.8)
µ
∂
+ γ1 u = 0 on Γ1 ,
∂n1
¶
µ
∂
+ γ2 u = 0 on Γ2 ,
∂n2
¶
AN INVERSE PROBLEM FOR A GENERAL BOUNDED DOMAIN
55
where ∂n∂ 1 and ∂n∂ 2 denote differentiations along the inward-pointing normals to
Γ1 and Γ2 respectively, in which Γ1 is a part of ∂Ω and Γ2 = ∂Ω\Γ1 is the
remaining part of ∂Ω, while the impedances γ1 and γ2 are positive constants.
The author calculates only the first three terms of the asymptotics of the heat
kernel of this problem, and shows that how the lengths of Γ1 and Γ2 and the
impedances γ1 , γ2 enter into the asymptotic expansions of θ(t) for small positive
t.
Now, let Ω be a general doubly-connected domain in IR2 consisting of a simply
connected bounded inner domain Ω1 with a smooth boundary ∂Ω1 and a simply
connected bounded outer domain Ω2 ⊃ Ω1 with a smooth boundary ∂Ω2 where
Ω1 = Ω1 ∪ ∂Ω1 . Suppose that the eigenvalues (1.3) are given for the eigenvalue
equation
(1.9)
−∆u = λ u
in Ω ,
together with the impedance boundary conditions
(1.10)
µ
∂
+ γ1 u = 0 on ∂Ω1 ,
∂n1
¶
µ
∂
+ γ2 u = 0 on ∂Ω2 ,
∂n2
¶
where γ1 and γ2 are positive constants.
Zayed [10] has recently discussed the problem (1.9), (1.10) and has determined
only the first three terms of the asymptotic expansions of the spectral function
θ(t) for small positive t. The author has determined some geometric quantities
associated with the problem (1.9)–(1.10) by using (1.4).
The object of this paper is to discuss a more general inverse problem consisting
of the eigenvalue equation (1.9) together with the piecewise smooth impedance
boundary conditions:
(1.11)
µ
∂
+ γi u = 0
∂ni
¶
on Γi (i = 1, 2, 3, 4) ,
where Γ1 is a part of the inner boundary ∂Ω1 of Ω and Γ2 = ∂Ω1 \Γ1 is the
remaining part of ∂Ω1 such that ∂Ω1 = Γ1 ∪ Γ2 , while Γ3 is a part of the outer
boundary ∂Ω2 of Ω and Γ4 = ∂Ω2 \Γ3 is the remaining part of ∂Ω2 such that
∂Ω2 = Γ3 ∪ Γ4 , and the impedances γi (i = 1, 2, 3, 4) are positive constants.
The basic problem is to determine some geometric quantities associated with
the general doubly connected domain Ω from the complete knowledge of the
eigenvalues {λν } for the impedance problem (1.9), (1.11) using the asymptotic
expansions of the spectral function θ(t) for small positive t.
Note that our main problem (1.9)–(1.11) can be considered as a generalization
of those obtained by Zayed [8, 9, 10].
56
E.M.E. ZAYED
2 – Statement of results
Suppose that the inner boundary ∂Ω1 of Ω is given locally by the equations
= y i (σ1 ) (i = 1, 2), in which σ1 is the arc length of the counterclock-wise
oriented inner boundary ∂Ω1 and y i (σ1 ) ∈ C ∞ (∂Ω1 ). Suppose also that the
outer boundary ∂Ω2 of Ω is given locally by the equations xi = y i (σ2 ) (i = 1, 2),
in which σ2 is the arc length of the counterclock-wise oriented outer boundary
∂Ω2 and y i (σ2 ) ∈ C ∞ (∂Ω2 ).
Let k1 (σ1 ) and k2 (σ2 ) be the curvatures of ∂Ω1 and ∂Ω2 respectively. Let L1
and L2 be the lengths of the parts Γ1 and Γ2 of ∂Ω1 respectively and let L3 and
L4 be the lengths of the parts Γ3 and Γ4 of ∂Ω2 respectively. Then, the results
of our main problem (1.9), (1.11) can be summarized in the following cases:
xi
Case 1. (0 < γ1 ¿ 1, γ2 À 1, 0 < γ3 ¿ 1, γ4 À 1)
½h
Z
³
´i
|Ω|
1
−1
θ(t) =
+
L
−
L
+
γ
k
(σ
)
dσ
1
2
1
1
1
2
4π t 8(π t)1/2
Γ2
h
³
+ L3 − L4 +
(2.1)
+
+
Z
Γ4
µ ¶1/2 ½ Z ·
1
t
256 π
7
Γ1
Z ·
k12 (σ1 )
2π
−
L2
Z ·
k22 (σ2 )
µ
Γ2
+
γ4−1
Γ4
µ
−
k12 (σ1 ) −
¶3
2π
L4
k2 (σ2 ) dσ2
γ2−1
¶3
¸
γ4−1
´i¾
64 π γ1
− γ12
7 L1
µ
dσ1 + 7
¸
1
(γ1 L1 − γ3 L3 )
2π
+
Z ·
Γ3
¶¸
k22 (σ2 )
¾
dσ2 + O(t)
dσ1
64 π γ3
− γ32
−
7 L3
µ
¶¸
dσ2
as t → 0+ .
Case 2. (0 < γ1 ¿ 1, γ2 À 1, γ3 À 1, 0 < γ4 ¿ 1)
In this case, the asymptotic expansion of θ(t) has the same form (2.1) with
the interchanges γ3 ↔ γ4 , Γ3 ↔ Γ4 and L3 ↔ L4 .
Case 3. (γ1 , γ2 À 1, 0 < γ3 , γ4 ¿ 1)
θ(t) =
(2.2)
½X
Z
4
2 h
i¾
X
1
|Ω|
−1
+
L
−
L
+
γ
k
(σ
)
dσ
i
i
1
1
1
i
4π t 8(π t)1/2 i=3
Γi
i=1
−
¸
µ ¶1/2 ½X
µ ¶3
4
2 Z ·
1
t
2π
1 X
γi−1 dσ1
γ i Li +
k12 (σ1 ) −
2π i=3
256 π
Li
i=1 Γi
+7
4 Z ·
X
i=3 Γi
k22 (σ2 )
64 π γi
−
− γi2
7 Li
µ
¶¸
¾
dσ2 + O(t)
as t → 0+ .
57
AN INVERSE PROBLEM FOR A GENERAL BOUNDED DOMAIN
Case 4. (0 < γ1 , γ2 ¿ 1, γ3 , γ4 À 1)
In this case, the asymptotic expansion of θ(t) has the same form (2.2) with
the interchanges γ1 ↔ γ3 , γ2 ↔ γ4 , L1 ↔ L3 , L2 ↔ L4 , Γ1 ↔ Γ3 , Γ2 ↔ Γ4 and
k1 (σ1 ) ↔ k2 (σ2 ).
Case 5. (γ1 À 1, 0 < γ2 ¿ 1, γ3 À 1, 0 < γ4 ¿ 1)
In this case, the asymptotic expansion of θ(t) has the same form (2.1) with
the interchanges γ1 ↔ γ2 , γ3 ↔ γ4 , L1 ↔ L2 , L3 ↔ L4 , Γ1 ↔ Γ2 and Γ3 ↔ Γ4 .
Case 6. (γ1 À 1, 0 < γ2 ¿ 1, 0 < γ3 ¿ 1, γ4 À 1)
In this case, the asymptotic expansion of θ(t) has the same form (2.1) with
the interchanges γ1 ↔ γ2 , L1 ↔ L2 and Γ1 ↔ Γ2 .
Case 7. (0 < γ1 ¿ 1, γ2 À 1, γ3 , γ4 À 1)
½h
Z
³
´i
|Ω|
1
−1
θ(t) =
+
L
−
L
+
γ
k
(σ
)
dσ
1
2
1
1
1
2
4π t 8(π t)1/2
Γ2
−
4 ³
X
Li +
Z
γi−1
i=3
(2.3)
Γi
k2 (σ2 ) dσ2
µ ¶1/2 ½ Z ·
1
t
+
256 π
+
Z ·
Γ2
7
Γ1
k12 (σ1 )
+ O(t)
−
µ
2π
L2
k12 (σ1 )
¶3
γ2−1
¸
´¾
−
γ 1 L1
2π
64 πγ1
−
− γ12
7 L1
µ
dσ1 +
4 Z ·
X
i=3 Γi
¶¸
dσ1
k22 (σ2 )
−
µ
2π
Li
¶3
γi−1
¸
dσ2
¾
as t → 0+ .
Case 8. (γ1 À 1, 0 < γ2 ¿ 1, γ3 , γ4 À 1)
In this case, the asymptotic expansion of θ(t) has the same form (2.3) with
the interchanges γ1 ↔ γ2 , L1 ↔ L2 and Γ1 ↔ Γ2 .
Case 9. (γ1 , γ2 À 1, 0 < γ3 ¿ 1, γ4 À 1)
In this case, the asymptotic expansion of θ(t) has the same form (2.3) with
the interchanges γ1 ↔ γ3 , γ2 ↔ γ4 , L1 ↔ L3 , L2 ↔ L4 , Γ1 ↔ Γ3 , Γ2 ↔ Γ4 and
k1 (σ1 ) ↔ k2 (σ2 ).
Case 10. (γ1 , γ2 À 1, γ3 À 1, 0 < γ4 ¿ 1)
In this case, the asymptotic expansion of θ(t) has the same form (2.3) with
58
E.M.E. ZAYED
the interchanges γ1 ↔ γ4 , γ2 ↔ γ3 , L1 ↔ L4 , L2 ↔ L3 , Γ1 ↔ Γ4 , Γ2 ↔ Γ3 and
k1 (σ1 ) ↔ k2 (σ2 ).
Case 11. (γ1 À 1, 0 < γ2 ¿ 1, 0 < γ3 , γ4 ¿ 1)
½h
¾
Z
4
´i X
³
|Ω|
1
−1
θ(t) =
Li
L2 − L1 + γ 1
k1 (σ1 ) dσ1 +
+
4π t 8(π t)1/2
Γ1
i=3
4
´
X
1 ³
γ 2 L2 −
+
γ i Li
2π
i=3
(2.4)
+
µ ¶1/2 ½Z ·
t
1
256 π
+7
+7
Z ·
Γ2
4 Z
X
2π
L1
¶3
64 π γ2
− γ22
7 L2
¶¸
dσ1
Γ1
k12 (σ1 )
i=3 Γi
µ
·
K12 (σ1 ) −
µ
−
k22 (σ2 )
64 π γi
− γi2
−
7 Li
µ
¶¸
¸
γ1−1 dσ1
¾
as t → 0+ .
dσ2 + O(t)
Case 12. (0 < γ1 ¿ 1, γ2 À 1, 0 < γ3 , γ4 ¿ 1)
In this case, the asymptotic expansion of θ(t) has the same form (2.4) with
the interchanges γ1 ↔ γ2 , L1 ↔ L2 and Γ1 ↔ Γ2 .
Case 13. (0 < γ1 , γ2 ¿ 1, γ3 À 1, 0 < γ4 ¿ 1)
θ(t) =
½X
Z
2
h
³
´i¾
|Ω|
1
−1
+
L
+
L
−
L
+
γ
(σ
)
dσ
k
i
4
3
2 2
2
3
4π t 8(π t)1/2 i=1
Γ3
+
(2.5)
2
´
1
t
1 ³X
γ i Li − γ 4 L4 +
2π i=1
256 π
µ ¶1/2 ½ X
2 Z ·
64 π γi
− γi2
−
7 Li
µ
+7
Z ·
Γ4
¶¸
k22 (σ2 ) −
dσ1 +
Z ·
Γ3
k22 (σ2 )
64 π γ4
− γ42
7 L4
µ
7
¶¸
−
i=1 Γi
µ ¶3
¾
2π
L3
k12 (σ1 )
γ3−1
dσ2 + O(t)
¸
dσ2
as t → 0+ .
Case 14. (0 < γ1 , γ2 ¿ 1, 0 < γ3 ¿ 1, γ4 À 1)
In this case, the asymptotic expansion of θ(t) has the same form (2.5) with
the interchanges γ3 ↔ γ4 , L3 ↔ L4 and Γ3 ↔ Γ4 .
AN INVERSE PROBLEM FOR A GENERAL BOUNDED DOMAIN
59
Case 15. (0 < γi ¿ 1, i = 1, 2, 3, 4)
4
X
Li
2
4
´
X
1 ³X
|Ω|
i=1
+
+
θ(t) =
γ
L
−
γ
L
i i
i i
4π t 8(π t)1/2 2π i=1
i=3
(2.6)
+
+
µ ¶1/2 ½X
2 Z ·
t
7
256 π
4 Z ·
X
i=3 Γi
i=1 Γi
k22 (σ2 ) −
k12 (σ1 ) −
64 π γi
− γi2
7 Li
µ
64 π γi
− γi2
7 Li
µ
¶¸
¶¸
dσ1
¾
as t → 0+ .
dσ2 + O(t)
Case 16. (γi À 1, i = 1, 2, 3, 4)
½X
Z
2 ³
´
1
|Ω|
−1
L
+
γ
k
(σ
)
dσ
−
θ(t) =
i
1
1
1
i
4π t 8(π t)1/2 i=1
Γi
+
(2.7)
4 ³
X
Li +
i=3
γi−1
Z
Γi
k2 (σ2 ) dσ2
µ ¶1/2 ½X
2 Z ·
1
t
+
256 π
+
4 Z ·
X
i=3 Γi
i=1 Γi
k22 (σ2 )
−
µ
k12 (σ1 )
2π
Li
¶3
γi−1
´¾
−
¸
µ
2π
Li
¾
¶3
γi−1
¸
dσ2 + O(t)
dσ1
as t → 0+ .
With reference to the formulas (1.5)–(1.7) and to the articles [8], [10] [11] the
asymptotic expansions (2.1)–(2.7) may be interpreted as:
i) Ω is a general doubly connected bounded domain in IR2 and we have the
piecewise smooth impedance boundary conditions (1.11) with small/large
impedances γi (i = 1, 2, 3, 4) as indicated in the specifications of the sixteen respective cases, where we notice that γi small approaches Neumann
boundary conditions, while γi large approaches Dirichlet boundary conditions.
ii) For the first four terms, Ω is a general doubly connected bounded domain
in IR2 of area |Ω|.
In (2.1), it has H = 1 − π3 (γ1 L1 − γ3 L3 ) holes, the part Γ1 of ∂Ω1 is of
π γ1
2 1/2 together with the Neulength L1 and of curvature [k12 (σ1 ) − 64
7 ( L1 − γ1 )]
mann boundary condition, while the remaining part Γ2 = ∂Ω1 \Γ1 of ∂Ω1 is of
60
E.M.E. ZAYED
2π 3 −1 1/2
length (L2 + γ2−1 Γ2 k1 (σ1 ) dσ1 ) and of curvature [k12 (σ1 ) − ( L
) γ2 ]
together
2
with the Dirichlet boundary condition. Similarly, the part Γ3 of ∂Ω2 is of length
π γ3
2 1/2 together with the Neumann boundL3 and of curvature [k22 (σ2 )− 64
7 ( L3 −γ3 )]
ary condition, while the remaining part Γ4 = ∂Ω2 \Γ3 of ∂Ω2 is of length
R
2π 3 −1 1/2
) γ4 ]
together with
(L4 + γ4−1 Γ4 k2 (σ2 ) dσ2 ) and of curvature [k22 (σ2 ) − ( L
4
the Dirichlet boundary condition, provided H is an integer.
P
In (2.2), it has H = 1 + π3 4i=3 γi Li holes, the part Γ1 of ∂Ω1 is of length
R
2π 3 −1 1/2
(L1 + γ1−1 Γ1 k1 (σ1 ) dσ1 ) and of curvature [k12 (σ1 ) − ( L
) γ1 ] , while the re1
R
maining part Γ2 = ∂Ω1 \Γ1 of ∂Ω1 is of length (L2 + γ2−1 Γ2 k1 (σ1 ) dσ1 ) and
2π 3 −1 1/2
of curvature [k12 (σ1 ) − ( L
) γ2 ]
together with the Dirichlet boundary condi2
tions on Γ1 and Γ2 . Similarly, the part Γ3 of ∂Ω2 is of length L3 and of curvature
π γ3
2 1/2 while the remaining part Γ = ∂Ω \Γ of ∂Ω is of
[k22 (σ2 ) − 64
4
2
3
2
7 ( L3 − γ3 )]
64 π γ4
2
2
1/2
length L4 and curvature [k2 (σ2 ) − 7 ( L4 − γ4 )]
together with the Neumann
boundary conditions on Γ3 and Γ4 , provided H is an integer.
In (2.3), it has H = 1 + π3 γ1 L1 holes, the part Γ1 of ∂Ω1 is of length
π γ1
2 1/2 together with the Neumann
L1 and of curvature [k12 (σ1 ) − 64
7 ( L1 − γ1 )]
boundary condition, while the remaining part Γ2 = ∂Ω1 \Γ1 of ∂Ω1 is of length
R
2π 3 −1 1/2
(L2 + γ2−1 Γ2 k1 (σ1 ) dσ1 ) and of curvature [k12 (σ1 ) − ( L
) γ2 ]
together with
2
the Dirichlet boundary condition. Similarly, the parts Γ3 and Γ4 of ∂Ω2 are
R
R
respectively of lengths (L3 + γ3−1 Γ3 k2 (σ2 ) dσ2 ), (L4 + γ4−1 Γ4 k2 (σ2 ) dσ2 ) and
2π 3 −1 1/2
2π 3 −1 1/2
) γ3 ] , [k22 (σ2 ) − ( L
) γ4 ]
together with the
of curvatures [k22 (σ2 ) − ( L
3
4
Dirichlet boundary conditions on Γ3 and Γ4 , provided H is an integer.
P
In (2.4), it has H = 1 − π3 (γ2 L2 − 4i=3 γi Li ) holes, the part Γ2 of ∂Ω1
π γ2
2 1/2 together with the
is of length L2 and of curvature [k12 (σ1 ) − 64
7 ( L2 − γ2 )]
Neumann boundary condition, while the remaining part Γ1 = ∂Ω1 \Γ2 of ∂Ω1
R
2π 3 −1 1/2
is of length (L1 + γ1−1 Γ1 k1 (σ1 ) dσ1 ) and of curvatures [k12 (σ1 ) − ( L
) γ1 ]
1
together with the Dirichlet boundary condition. Similarly, the parts Γ3 and Γ4 of
π γ3
2 1/2 ,
∂Ω2 are respectively of lengths L3 , L4 and curvatures [k22 (σ2 ) − 64
7 ( L3 − γ3 )]
π γ4
2 1/2 together with the Neumann boundary conditions on
[k22 (σ2 ) − 64
7 ( L4 − γ4 )]
Γ3 and Γ4 , provided H is an integer.
P
In (2.5), it has H = 1 − π3 ( 2i=1 γi Li − γ4 L4 ) holes, the parts Γ1 and Γ2
π γ1
2 1/2 ,
of ∂Ω1 are of lengths L1 , L2 and of curvatures [k12 (σ1 ) − 64
7 ( L1 − γ1 )]
π γ2
2 1/2 together with the Neumann boundary conditions on
[k12 (σ1 ) − 64
7 ( L2 − γ2 )]
Γ1 and Γ2 . Similarly, the part Γ4 of ∂Ω2 is of length L4 and curvature [k22 (σ2 ) −
64 π γ4
2 1/2 together with the Neumann boundary condition, while the re7 ( L4 − γ4 )]
R
maining part Γ3 = ∂Ω2 \Γ4 of ∂Ω2 is of length (L3 + γ3−1 γ3 k2 (σ2 ) dσ2 ) and of
2π 3 −1 1/2
) γ3 ] together with the Dirichlet boundary condition,
curvature [k22 (σ2 ) − ( L
3
provided H is an integer.
R
AN INVERSE PROBLEM FOR A GENERAL BOUNDED DOMAIN
61
In (2.6), it has H = 1− π3 ( 2i=1 γi Li − 4i=3 γi Li ) holes, the parts Γ1 and Γ2 of
π γ1
2 1/2 ,
∂Ω1 are respectively of lengths L1 , L2 and of curvatures [k12 (σ1 )− 64
7 ( L1 −γ1 )]
64 π γ2
2
2
1/2
[k1 (σ1 ) − 7 ( L2 − γ2 )] together with the Neumann boundary conditions on Γ1
and Γ2 . Similarly, the parts Γ3 and Γ4 of ∂Ω2 are respectively of lengths L3 , L4
π γ3
2 1/2 , [k 2 (σ ) − 64 ( π γ4 − γ 2 )]1/2 together
and of curvatures [k22 (σ2 ) − 64
2 2
4
7 ( L3 − γ3 )]
7 L4
with the Neumann boundary conditions on Γ3 and Γ4 , provided H is an integer.
In (2.7), it has only one hole (i.e., H = 1), the parts Γ1 and Γ2 of ∂Ω1 are
R
R
respectively of lengths (L1 + γ1−1 Γ1 k1 (σ1 ) dσ1 ), (L2 + γ2−1 Γ2 k1 (σ1 ) dσ1 ) and
2π 3 −1 1/2
2π 3 −1 1/2
of curvatures [k12 (σ1 ) − ( L
) γ1 ] , [k12 (σ1 ) − ( L
) γ2 ] , together with the
1
2
Dirichlet boundary conditions on Γ1 and Γ2 . Similarly, the parts Γ3 and Γ4 of
R
R
∂Ω2 are respectively of lengths (L3 +γ3−1 Γ3 k2 (σ2 ) dσ2 ), (L4 +γ4−1 Γ4 k2 (σ2 ) dσ2 )
2π 3 −1 1/2
2π 3 −1 1/2
and of curvatures [k22 (σ2 ) − ( L
) γ3 ] , [k22 (σ2 ) − ( L
) γ4 ]
together with
3
4
the Dirichlet boundary conditions on Γ3 and Γ4 .
P
P
3 – Formulation of the mathematical problem
With reference to [2], [5], [7] one can show that the spectral function θ(t)
associated with the problem (1.9), (1.11) is given by the formula
(3.1)
θ(t) =
ZZ
G(x, x; t) dx ,
Ω
where G(x1 , x2 ; t) is the Green’s function for the heat equation
(3.2)
∆u =
∂u
,
∂t
subject to the piecewise smooth impedance boundary conditions
(3.3)
µ
∂
+ γi G(x1 , x2 ; t) = 0
∂ni
¶
for x1 ∈ Γi (i = 1, 2, 3, 4) ,
and the initial condition
(3.4)
lim G(x1 , x2 ; t) = δ(x1 − x2 ) ,
t→0+
where δ(x1 − x2 ) is the Dirac delta function located at the source point x2 .
Let us write
(3.5)
G(x1 , x2 ; t) = G0 (x1 , x2 ; t) + χ(x1 , x2 ; t) ,
62
E.M.E. ZAYED
where
(3.6)
G0 (x1 , x2 ; t) = (4π t)
−1
|x1 − x2 |2
exp
4t
½
¾
,
is the “fundamental solution” of the heat equation (3.2), while χ(x1 , x2 ; t) is the
“regular solution” chosen in such a way that G(x1 , x2 ; t) satisfies the piecewise
smooth impedance boundary conditions (3.3).
On setting x1 = x2 = x we find that
(3.7)
θ(t) =
|Ω|
+ R(t) ,
4π t
where
(3.8)
R(t) =
ZZ
χ(x; x; t) dx ,
Ω
The problem now is to determine the asymptotic expansion of R(t) as t → 0+ .
In what follows, we shall use Laplace transforms with respect to t, and use s 2 as
the Laplace transform parameter; thus we define
2
(3.9)
G(x1 , x2 ; s ) =
Z
+∞
0
2
e−s t G(x1 , x2 ; t) dt .
An application of the Laplace transform to the heat equation (3.2) shows that
G(x1 , x2 ; s2 ) satisfies the membrane equation
(∆ − s2 ) G(x1 , x2 ; s2 ) = −δ(x1 − x2 )
(3.10)
in Ω ,
together with the piecewise smooth impedance boundary conditions
(3.11)
µ
∂
+ γi G(x1 , x2 ; s2 ) = 0,
∂ni
¶
for x1 ∈ Γi (i = 1, 2, 3, 4) .
The asymptotic expansion of R(t) as t → 0+ may then be deduced directly from
the asymptotic expansion of R(s2 ) as s → ∞, where
(3.12)
R(s2 ) =
ZZ
χ(x, x; s2 ) dx .
Ω
63
AN INVERSE PROBLEM FOR A GENERAL BOUNDED DOMAIN
4 – Construction of the Green’s function
It is well known (see [4, 5, 7]) that the membrane equation (3.10) has the
fundamental solution
G0 (x1 , x2 ; s2 ) =
(4.1)
1
K0 (srx1 x2 ) ,
2π
where rx1 x2 = |x1 − x2 | is the distance between the points x1 = (x11 , x21 ), x2 =
(x12 , x22 ) of the region Ω and K0 is the modified Bessel function of the second kind
and of zero order. The existence of this solution enables us to construct integral
equations for G(x1 , x2 ; s2 ) satisfying the piecewise smooth impedance boundary
conditions (3.11) for small/large impedances γi (i = 1, 2, 3, 4) as indicated in the
specifications of the sixteen respective cases. Therefore, Green’s theorem gives:
Case 1. (0 < γ1 ¿ 1, γ2 À 1, 0 < γ3 ¿ 1, γ4 À 1)
In this case, we have the integral equation
G(x1 , x2 ; s2 ) =
∂
K0 (sryx2 ) + γ1 K0 (sryx2 ) dy
∂n1y
1
π
Z
1
−
π
Z
Γ2
1
−
π
Z
∂
G(x1 , y; s )
K0 (s ryx2 ) + γ3 K0 (sryx2 ) dy
∂n3y
Γ3
1
+
π
Z
+
(4.2)
1
K0 (srx1 x2 ) +
2π
Γ1
G(x1 , y; s2 )
¾
∂
∂
G(x1 , y; s2 ) K0 (s ryx2 ) + γ2−1
K0 (sryx2 ) dy
∂n2y
∂n2y
½
2
Γ4
½
½
¾
¾
∂
∂
K0 (sryx2 ) dy .
G(x1 , y; s2 ) K0 (sryx2 ) + γ4−1
∂n4y
∂n4y
½
¾
Similarly, the integral equations of G(x1 , x2 ; s2 ) for the other fifteen cases can be
found easily.
On applying the iteration methods (see [8], [9]) to the integral equation (4.2),
we obtain the Green’s function G(x1 , x2 ; s2 ) which has a regular part in the
following form:
∂
K0 (s ryx2 ) + γ1 K0 (sryx2 ) dy −
∂n1y
Γ1
½
¾
∂
∂
−1
K0 (srx1 y ) K0 (s ryx2 ) + γ2
K0 (sryx2 ) dy −
∂n2y
∂n2y
(4.3) χ(x1 , x2 ; s2 ) =
1
− 2
2π
Z
Γ2
1
2π 2
Z
K0 (srx1 y )
½
¾
64
E.M.E. ZAYED
1
− 2
2π
Z
+
1
2π 2
Z
+
1
2π 2
Z Z
∂
K0 (sryx2 ) + γ3 K0 (s ryx2 ) dy
K0 (s rx1 y )
∂n3y
Γ3
¾
½
Γ4
∂
∂
K0 (srx1 y ) K0 (s ryx2 ) + γ4−1
K0 (sryx2 ) dy
∂n4y
∂n4y
½
Γ1 Γ1
K0 (srx1 y ) Mγ1 (y, y0 )
¾
½
∂
K0 (s ry0 x2 ) + γ1 K0 (sry0 x2 ) dy dy0
∂n1y0
¾
∂
∂
K0 (srx1 y ) Mγ −1 (y, y0 ) K0 (sry0 x2 ) + γ2−1
K0 (sry0 x2 ) dy dy0
2
0
∂n
∂n
Γ2 Γ2
2y
2y
¾
½
Z Z
1
∂
0
K0 (s ry0 x2 ) + γ3 K0 (sry0 x2 ) dy dy0
K0 (srx1 y ) Lγ3 (y, y )
+ 2
2π Γ3 Γ3
∂n3y0
1
2πi2
Z Z
1
+ 2
2π
Z Z
1
− 2
2π
Z ½Z
+
½
¾
∂
K0 (srx1 y ) Lγ −1 (y, y0 ) K0 (s ry0 x2 )
4
Γ4 Γ4 ∂n4y
¾
∂
+ γ4−1
K0 (sry0 x2 ) dy dy0
∂n4y0
Γ1
½
Γ2
∂
K0 (srx1 y ) Mγ∗−1 (y, y0 ) dy
2
∂n2y
¾½
∂
K0 (s ry0 x2 )
∂n1y0
¾
+ γ1 K0 (sry0 x2 ) dy0
1
− 2
2π
Z ½Z
1
− 2
2π
Z ½Z
−
1
2π 2
Z ½Z
+
1
2π 2
½Z
Γ2
Γ1
Γ3
K0 (srx1 y ) Mγ∗1 (y, y0 ) dy
Γ1
¾½
K0 (sry0 x2 )+γ2−1
Z ½Z
1
+ 2
2π
Z ½Z
1
+ 2
2π
Z ½Z
Γ4
Γ2
Γ3
¾
K0 (srx1 y ) L∗γ3 (y, y0 ) dy
¾½
∂
K0 (sry0 x2 )+γ1 K0 (s ry0 x2 ) dy0
∂n1y0
K0 (srx1 y ) Lγ1 (y, y0 ) dy
¾½
∂
K0 (sry0 x2 )+γ3 K0 (s ry0 x2 ) dy0
∂n3y0
Γ3
Γ1
¾
∂
K0 (srx1 y ) Mγ∗−1 (y, y0 ) dy
4
∂n
Γ4
4y
1
+ 2
2π
∂
K0 (sry0 x2 ) dy0
∂n2y0
K0 (srx1 y ) L∗γ1 (y, y0 ) dy
Γ1
Γ2
¾½
∂
K0 (s ry0 x2 )+γ1 K0 (sry0 x2 ) dy0
∂n1y0
¾½
K0 (srx1 y ) Mγ∗3 (y, y0 ) dy
Γ3
¾
K0 (sr
¾
y0 x
¾½
−1
2 )+γ4
K0 (sry0 x2 )+γ2−1
∂
K0 (srx1 y ) L∗γ −1 (y, y0 ) dy
2
∂n2y
+ γ3 K0 (sry0 x2 ) dy0
¾
¾½
∂
K0 (sry0 x2 ) dy0
∂n4y0
¾
∂
K0 (sry0 x2 ) dy0
∂n2y0
∂
K0 (s ry0 x2 )
∂n3y0
¾
65
AN INVERSE PROBLEM FOR A GENERAL BOUNDED DOMAIN
1
− 2
2π
∂
K0 (srx1 y ) Mγ −1 (y, y0 ) dy K0 (sry0 x2 )
4
Γ4 ∂n4y
Γ2
¾
∂
K0 (s ry0 x2 ) dy0
+ γ2−1
∂n2y0
¾½
Z ½Z
∂
1
0
K0 (srx1 y ) Lγ −1 (y, y ) dy K0 (sry0 x2 )
− 2
2
2π Γ4 Γ2 ∂n2y
¾
∂
+ γ4−1
K0 (s ry0 x2 ) dy0
∂n4y0
¾½
Z ½Z
∂
∂
1
∗
0
K0 (srx1 y ) Lγ −1 (y, y ) dy
K0 (s ry0 x2 )
− 2
4
2π Γ3 Γ4 ∂n4y
∂n3y0
¾½
Z ½Z
¾
+ γ3 K0 (sry0 x2 ) dy0
−
1
2π 2
Z ½Z
Γ4
+ γ4−1
Γ3
0
K0 (srx1 y ) L+
γ3 (y, y ) dy
¾½
K0 (sry0 x2 )
∂
K0 (sry0 x2 ) dy0 ,
∂n4y0
¾
where
(4.4)
0
Mγ1 (y, y ) =
∞
X
Kγ(ν)
(y0 , y) ,
1
ν=0
1
∂
=
K0 (sryy0 ) + γ1 K0 (sryy0 )
π ∂n1y
½
(4.5)
Kγ(0)
(y0 , y)
1
(4.6)
Mγ −1 (y, y0 ) =
(4.7)
2
(0)
(4.8)
ν=0
Kγ −1 (y0 , y) =
2
Lγ3 (y, y0 ) =
∞
X
¾
,
(ν)
(−1)ν Kγ −1 (y0 , y) ,
2
∂2
∂
1
K0 (sryy0 ) + γ2−1
K0 (sryy0 )
π ∂n2y0
∂n2y ∂n2y0
∞
X
½
¾
,
(−1)ν Kγ(ν)
(y0 , y) ,
3
ν=0
(0)
where Kγ3 (y0 , y) has the same form (4.5) with the interchanges γ1 ↔ γ3 and
n1 ↔ n 3 ,
0
L−1
γ4 (y, y ) =
(4.9)
∞
X
ν=0
(0)
(ν)
Kγ −1 (y0 , y) ,
4
where Kγ −1 (y0 , y) has the same form (4.7) with the interchanges γ2 ↔ γ4 and
4
66
E.M.E. ZAYED
n2 ↔ n 4 ,
(4.10)
Mγ∗−1 (y, y0 ) =
2
(4.11)
(0)
Kγ −1 (y0 , y) =
∗
2
(4.12)
Mγ∗1 (y, y0 )
=
∞
X
ν=0
(ν)
Kγ −1 (y0 , y) ,
∗
2
1
∂
K0 (sryy0 ) + γ2−1
K0 (sryy0 )
π
∂n2y
½
∞
X
¾
,
(−1)ν ∗Kγ(ν)
(y0 , y) ,
1
ν=0
1
∂2
∂
=
K0 (sryy0 ) + γ1
K0 (sryy0 )
π ∂n1y ∂n2y0
∂n1y
½
Kγ(0)
(y0 , y)
1
(4.13)
∗
(4.14)
L∗γ3 (y, y0 ) =
∞
X
¾
,
¾
,
Kγ(ν)
(y0 , y) ,
3
∗
ν=0
(4.15)
(4.16)
∂2
∂
1
K0 (sryy0 ) + γ3
K0 (sryy0 )
π ∂n3y ∂n1y0
∂n3y
½
Kγ(0)
(y0 , y) =
3
∗
∞
X
0
Lγ1 (y, y ) =
(−1)ν Kγ(ν)
(y0 , y) ,
1
ν=0
(0)
where Kγ1 (y0 , y) has the same form (4.5),
Mγ∗−1 (y, y0 ) =
(4.17)
4
∞
X
ν=0
∗
(ν)
Kγ −1 (y0 , y) ,
4
(0)
where ∗Kγ −1 (y0 , y) has the same form (4.11) with the interchanges γ2 ↔ γ4 and
4
n2 ↔ n 4 ,
(4.18)
L∗γ1 (y, y0 ) =
∞
X
+
Kγ(ν)
(y0 , y) ,
1
ν=0
(4.19)
+
Kγ(0)
(y0 , y) =
1
(4.20)
Mγ∗3 (y, y0 )
=
1
∂
∂2
K0 (sryy0 ) + γ1
K0 (sryy0 )
π ∂n1y ∂n4y0
∂n1y
∞
X
½
¾
,
(−1)ν ∗∗Kγ(ν)
(y0 , y) ,
3
ν=0
(0)
where ∗∗Kγ3 (y0 , y) has the same form (4.15) with the interchanges n1 ↔ n2 ,
(4.21)
L∗γ −1 (y, y0 ) =
2
∞
X
ν=0
(ν)
(−1)ν ∗Kγ −1 (y0 , y) ,
2
AN INVERSE PROBLEM FOR A GENERAL BOUNDED DOMAIN
67
(0)
where ∗Kγ −1 (y0 , y) has the same form (4.11)
2
Mγ −1 (y, y0 ) =
(4.22)
4
∞
X
ν=0
(ν)
(−1)ν Kγ −1 (y0 , y) ,
4
(0)
where Kγ −1 (y0 , y) has the same form (4.7) with the interchanges γ2 ↔ γ4 ,
4
n2 ↔ n 4 ,
0
(4.23)
Lγ −1 (y, y ) =
2
∞
X
ν=0
(ν)
Kγ −1 (y0 , y) ,
2
(0)
where Kγ −1 (y0 , y) has the same form (4.7),
2
L∗γ −1 (y, y0 ) =
(4.24)
4
∞
X
ν=0
(ν)
(−1)ν ∗Kγ −1 (y0 , y) ,
4
(0)
where ∗Kγ −1 (y0 , y) has the same form (4.11) with the interchanges γ2 ↔ γ4 ,
4
n2 ↔ n 4 ,
0
L+
γ3 (y, y )
(4.25)
=
∞
X
+
Kγ(ν)
(y0 , y) ,
3
ν=0
(0)
where +Kγ3 (y0 , y) has the same form (4.19) with the interchanges γ1 ↔ γ3 ,
n1 ↔ n 3 .
(ν)
In these formulae, we note that Kγi (y0 , y) being the iterates of the kernels
(0)
Kγi (y0 , y) (i = 1, 2, 3, 4) respectively.
Similarly, we can find χ(x1 , x2 ; s2 ) for the other fifteen cases.
On the basis of (4.3), the function χ(x1 , x2 ; s2 ) will be estimated for s → ∞
together with small/large impedances γi (i = 1, 2, 3, 4). The case when x1 and
x2 lie in the neighbourhood of the parts Γ1 , Γ2 of the inner boundary ∂Ω1 of Ω
or in the neighbourhood of the parts Γ3 , Γ4 of the outer boundary ∂Ω2 of Ω is
particularly interesting. In what follows, we shall use coordinates similar to those
obtained in Pleijel [4], Sleeman and Zayed [5] and Zayed [8, 9, 10] to examine
this case.
5 – Coordinates in the neighbourhood of the boundary
Let hi > 0 (i = 1, 2, 3, 4) be sufficiently small. Let ni (i = 1, 2, 3, 4) be the
minimum distances from a point x = (x1 , x2 ) of the region Ω to the parts Γi
68
E.M.E. ZAYED
(i = 1, 2, 3, 4) respectively. Let ni (σ1 ) (i = 1, 2) denote the inward drawn unit
normals to the parts Γi (i = 1, 2) of the inner boundary ∂Ω1 of Ω respectively,
while ni (σ2 ) (i = 3, 4) denote the inward drawn unit normals to the parts Γi
(i = 3, 4) of the outer boundary ∂Ω2 of Ω respectively. Then, we note that the
coordinates in the neighbourhood of the parts Γ1 , Γ2 of ∂Ω1 and its diagrams
(see [9]) are in the same form as in Section 5.2 of Zayed [9] with the interchanges
n1 ↔ ni , h1 ↔ hi , I1 ↔ Ii , D(I1 ) ↔ D(Ii ) and δ1 ↔ δi (i = 1, 2). Thus, we
have the same formulae (5.2.1)–(5.2.5) of Section 5.2 in [9] with the interchanges
n1 ↔ ni , n1 (σ1 ) ↔ ni (σ1 ), t1 (σ1 ) ↔ ti (σ1 ) (i = 1, 2).
Similarly, the coordinates in the neighbourhood of the parts Γ3 , Γ4 of ∂Ω2
and its diagrams (see [9]) are in the same form as in Section 5.1 of Zayed [9]
with the interchanges n2 ↔ ni , h2 ↔ hi , I2 ↔ Ii , D(I2 ) ↔ D(Ii ) and δ2 ↔ δi
(i = 3, 4). Thus, we have the same formulae (5.1.1)–(5.1.5) of Section 5.1 in [9]
with the interchanges n2 ↔ ni , n2 (σ2 ) ↔ ni (σ2 ), t2 (σ2 ) ↔ ti (σ2 ) (i = 3, 4).
6 – Some local expansions
It now follows that the local expansions of the functions
(6.1)
K0 (srxy ) ,
∂
K0 (srxy ) (i = 1, 2, 3, 4) ,
∂niy
when the distance between x and y is small, are very similar to those obtained in
Sections 4 and 5 of [5] (see, also Section 6 in [9]). Consequently, for small/large
impedances γi (i = 1, 2, 3, 4) the local behaviour of the kernels
(6.2)
Kγ(0)
(y0 , y),
i
Kγ(0)
(y0 , y),
i
∗
+
Kγ(0)
(y0 , y)
i
(i = 1, 3) ,
Kγ(0)
(y0 , y) ,
3
∗∗
and
(6.3)
(0)
Kγ −1 (y0 , y),
i
(0)
Kγ −1 (y0 , y)
∗
(i = 2, 4) ,
i
when the distance between y and y0 is small, follows directly from the knowledge
of the local expansions of the functions (6.1).
Definition 1. Let ξ 1 and ξ 2 be points in the upper half-plane ξ 2 > 0 of the
(ξ 1 , ξ 2 )-plane, then we define
ρ12 =
q
(ξ11 − ξ21 )2 + (ξ12 + ξ22 )2 .
AN INVERSE PROBLEM FOR A GENERAL BOUNDED DOMAIN
69
An eλ (ξ 1 , ξ2 ; s)-function is defined for points ξ 1 and ξ 2 belong to sufficiently small
domains D(Ii ) (i = 1, 2, 3, 4) except when ξ 1 = ξ 2 ∈ Ii (i = 1, 2, 3, 4), where λ is
called the degree of this function. For every positive integer Λ, it has the local
expansion (see [4], [5], [8], [9]):
λ
e (ξ 1 , ξ 2 ; s) =
(6.4)
∗
X
f (ξ11 ) (ξ12 )p1
(ξ22 )p2
µ
∂
∂ξ11
¶`1 µ
∂
∂ξ12
¶`2
K0 (s ρ12 )
+ RΛ (ξ 1 , ξ 2 ; s) ,
where ∗ denotes a sum of a finite number of terms in which f (ξ11 ) is an infinitely
differentiable function. In this expansion p1 , p2 , `1 , `2 are integers where p1 ≥ 0,
p2 ≥ 0, `1 ≥ 0, λ = min(p1 + p2 − q), q = `1 + `2 and the minimum is taken
P
over all terms which occur in the summation ∗ . The remainder RΛ (ξ 1 , ξ 2 ; s)
has continuous derivatives of all order d ≤ Λ satisfying
P
(6.5)
D d RΛ (ξ 1 , ξ2 ; s) = O(s−Λ e−Asρ12 )
as s → ∞ ,
where A is a positive constant.
Thus, using methods similar to those obtained in Sections 6–10 of [5], we
can show that the functions (6.1) are eλ -functions with degree λ = 0, −1 respectively. Consequently, for small impedances γi (i = 1, 3) the functions (6.2)
are eλ -functions with degrees λ = 0, −1, −1, −1, respectively while, for large
impedances γi (i = 2, 4) the functions (6.3) are eλ -functions with degrees λ = 0, 1
respectively (see also [8]).
Definition 2. If x1 and x2 are points in large domains Ω + Γi (i = 1, 2, 3, 4),
then we define
r12 = min(rx1 y + rx2 y )
y
R12 = min(rx1 y + rx2 y )
y
∗
r12
= min(rx1 y + rx2 y )
y
if y ∈ Γ1 ,
if y ∈ Γ2 ,
if y ∈ Γ3 ,
and
∗
R12
= min(rx1 y + rx2 y )
y
if y ∈ Γ4 .
An E λ (x1 , x2 ; s)-function is defined and infinitely differentiable with respect
to x1 and x2 when these points belong to large domains Ω + Γi (i = 1, 2, 3, 4)
except when x1 = x2 ∈ Γi (i = 1, 2, 3, 4). Thus, the E λ -function has a similar
70
E.M.E. ZAYED
local expansion of the eλ -function (see [4], [5], [8]). With the help of Sections 8
and 9 in [15], it is easily seen that the formula (4.3) is an E 0 (x1 , x2 ; s)-function
and consequently we get
G(x1 , x2 ; s2 ) = O
nh
+O
(6.6)
+O
+O
i
1 + |log sr12 | e−A1 sr12
nh
nh
nh
i
o
1 + |log sR12 | e−A2 sR12
i
∗
∗
1 + |log sr12
| e−A3 sr12
i
o
o
∗
∗
1 + |log sR12
| e−A4 sR12
o
,
which is valid for s → ∞ and for small/large impedances γi (i = 1, 2, 3, 4)
as indicated in the specification of case 1, where Ai (i = 1, 2, 3, 4) are positive constants. Formula (6.6) shows that G(x1 , x2 ; s2 ) is exponentially small for
s → ∞. Similar statements are true in the other fifteen cases.
With reference to Section 10 in [5], if the eλ -expansions of the functions (6.1)–
(6.3) are introduced into (4.3) and if we use formulae similar to (6.4), (6.9) of
Section 6 in [5], we obtain the following local behaviour of χ(x1 , x2 ; s2 ) when r12 ,
∗ and R∗ are small, which is valid for s → ∞ and for small γ , γ and
R12 , r12
1
3
12
large γ2 , γ4 :
χ(x1 , x2 ; s2 ) =
(6.7)
4
X
χi (x1 , x2 ; s2 ) ,
i=1
where
a) if x1 and x2 belong to a sufficiently small domain D(I1 ), then
(6.8)
1
∂
χ1 (x1 , x2 ; s ) = −
1 − γ1
2π
∂ξ12
2
½
µ
¶−1 ¾
n
K0 (s ρ12 ) + O s−1 e−A1 sρ12
o
;
b) if x1 and x2 belong to a sufficiently small domain D(I2 ), then
(6.9)
χ2 (x1 , x2 ; s2 ) =
∂
1
1 − γ2−1
2π
∂ξ12
½
µ
¶¾
n
K0 (s ρ12 ) + O s−1 e−A2 sρ12
o
;
c) if x1 and x2 belong to a sufficiently small domain D(I3 ), then
(6.10)
χ3 (x1 , x2 ; s2 ) =
∂
1
1 − γ3
2π
∂ξ12
½
µ
¶−1 ¾
n
K0 (s ρ12 ) + O s−1 e−A3 sρ12
o
;
o
.
d) if x1 and x2 belong to a sufficiently small domain D(I4 ), then
(6.11)
1
∂
χ4 (x1 , x2 ; s ) = −
1 − γ4−1
2π
∂ξ12
2
½
µ
¶¾
n
K0 (s ρ12 ) + O s−1 e−A4 sρ12
AN INVERSE PROBLEM FOR A GENERAL BOUNDED DOMAIN
71
∗ ≥ δ > 0 and R∗ ≥ δ > 0, the function
When r12 ≥ δ1 > 0, R12 ≥ δ2 > 0, r12
3
4
12
χ(x1 , x2 ; s2 ) is of order O(e−Bs ) as s → ∞, where B is a positive constant. Thus,
since
∗
R12
r∗
R12
r12
= lim
= ∗lim 12 = lim
=1,
lim
∗ →0
r12 →0 ρ12
R12 →0 ρ12
r12 →0 ρ12
R12
ρ12
(see [8], [9]), then we have the asymptotic formulae (6.9)–(6.11) with ρ 12 in the
∗ and R∗ in the
small domains D(Ii ) (i = 1, 2, 3, 4) being replaced by r12 , R12 , r12
12
large domains Ω + Γi (i = 1, 2, 3, 4) respectively.
Similar formulae for the other fifteen cases can be found.
7 – Construction of our results
Since for ξ 2 ≥ hi (i = 1, 2, 3, 4) the functions χi (x, x; s2 ) are of order O(e−2sAi hi )
(i = 1, 2, 3, 4), the integral over the region Ω of the function χ(x, x; s2 ) can be
approximated in the following way (see (3.12)):
2
R(s ) =
(7.1)
4 Z
X
hi
Z
Li
χi (x, x; s2 ) {1 − k2 (ξ 1 ) ξ 2 } dξ 1 dξ 2
1
3
i=3 ξ =0 ξ =0
Z
2
hi Z L i
X
−
+
χi (x, x; s2 ) {1 + k1 (ξ 1 ) ξ 2 } dξ 1 dξ 2
1
3
i=1 ξ =0 ξ =0
4
X
−2sAi hi
O{e
}
as s → ∞ .
i=1
If the eλ -expansions of χi (x, x; s2 ) (i = 1, 2, 3, 4) are introduced into (7.1), one
obtains an asymptotic series of the form
(7.2)
R(s2 ) =
p
X
an s−n + O(s−p−1 )
as s → ∞ ,
n=1
where the coefficients an , for all sixteen cases, are calculated from the eλ -expansions with the help of formula (10.3) of Section 10 in [9] (see also [4], [5]).
On inverting Laplace transforms and using (3.7), we arrive at our results
(2.1)–(2.7).
REFERENCES
[1] Gottlieb, H.P.W. – Eigenvalue of the Laplacian with Neumann boundary conditions, J. Austral. Math. Soc. Ser. B, 26 (1985), 293–309.
72
E.M.E. ZAYED
[2] Kac, M. – Can one hear the shape of a drum?, Amer. Math. Monthly, 73 (1966),
1–23.
[3] McKean, H.P. and Singer, I.M. – Curvature and the eigenvalues of the Laplacian, J. Diff. Geom., 1 (1967), 43–69.
[4] Pleijel, øA. – A study of certain Green’s functions with applications in the theory
of vibrating membranes, Arkiv für Math., 2 (1954), 553–569.
[5] Sleeman, B.D. and Zayed, E.M.E. – An inverse eigenvalue problem for a general
convex domain, J. Math. Anal. and Appl., 94 (1983), 78–95.
[6] Smith, L. – The asymptotics of the heat equation for a boundary value problem,
Invent. Math., 63 (1981), 467–493.
[7] Stewartson, K. and Waechter, R.T. – On hearing the shape of a drum:
further results, Proc. Camb. Phil. Soc., 69 (1971), 353–363.
[8] Zayed, E.M.E. – Hearing the shape of a general convex domain, J. Math. Anal.
and Appl., 142 (1989), 170–187.
[9] Zayed, E.M.E. – Heat equation for an arbitrary doubly connected region in IR 2
with mixed boundary conditions, ZAMP J. Appl. Math. Phys., 40 (1989), 339–355.
[10] Zayed, E.M.E. – On hearing the shape of an arbitrary doubly connected region
in IR2 , J. Austral. Math. Soc. Ser. B, 31 (1990), 472–483.
[11] Zayed, E.M.E. – Heat equation for an arbitrary multiply connected region in IR 2
with impedance boundary conditions, IMA J. Appl. Math., 45 (1990), 233–241.
E.M.E. Zayed,
Mathematics Department, Faculty of Science,
Zagazig University, Zagazig – EGYPT