591
Internat. J. Math. & Math. Sci.
VOL. 13 NO. 3 (1990) 591-598
HEARING THE SHAPE OF MEMBRANES: FURTHER RESULTS
E.M.E. ZAYED
Department of Mathematics
Zagazig University
Faculty of Science
Zagazig, Egypt
(Received January 18, 1989 and in revised form May 5, 1989)
exp(-t
The spectral function 0(t)
ABSTRACT.
m--I
elgenvalues of the Laplacian in R
n,
m
), t
>
0 where
{m}m=l
are the
2 or 3, is studied for a variety of domains.
n
Particular attention is given to circular and spherical domains with the impedance
and Sj,
J where
j
0 on
u
+
boundary
(or Sj),
conditions-
J are parts of the boundaries of these domains respectively, while
j
,
J
l,...,J are positive constants.
1.
INTRODUCTION.
The underlying problems are to deduce the precise shape of membranes from the
complete knowledge of the eigenvalues
0
<
<
h
<
k2
k3
<
<’’"
n,
for the Laplace operator An in R
km
<
,
as m
2 or 3.
n
{(r,0): 0 < r < a, 0 < 0 < 27} be a circular domain of radius a and
Suppose that the eigenvalues (I. I) are given for the eigenvalue
boundary r.
0 in R together with the impedance boundary conditions:
equation (A + %) u
2
(PI):
Let R
(-=-- + y.)u
0
on rj,
3
where
(1.2)
J,
J
J are positive constants and the boundary Y consists of parts
yj, J
Y.I’
J such that
j
rj
(P2):
Let R
radius
a
and
{(r, 0): r
{(r,0,):
surface
a,
0
S.
(A3+%)u
( + yj) u
o
aj
<
r
<
a, 0
Suppose
J,
aj+ I, J
< e <
that
7,
0
<
ctl=
< 2} be a spherical
eigenvalues
the
0, aj+1= 2}.
domain of
(I. I) are given for the
0 in R together with the impedance boundary conditons:
elgenvalue equation
0
where the surface S consists of parts
on S
Sj,
j
j,
J
J
1,...,J such that
(1.3)
592
E. M. E. ZAYED
{(r,0,): r
Sj
a, 0
,
<
0
aj <
<
J:a1-
].
a.+l,
0,aj+l--
2}.
,
The object of this paper is to determine the geometry of the domains in (PI) and (P2)
as well as the impedances
,J from the asymptotic expansion of the
J
spectral function
}. exp(-tm )’
(t)
for small positive t.
Zayed
when J
[I] has recently investigated probems (PI) and (P2) in the special case
2, that is, when the boundary
surface S consists of two parts
r
consists of two parts
rI, r2
and when the
S 2.
Finally, we close this introduction with the
$I,
remark that the author [2,3] has recently generalized the results of [I] to the case
n
when
c R
2 or 3 is a simply connected bounded domain with a smooth boundary.
n
2.
CONSTRUCTION OF 0(t) FOR PROBLEM (PI).
Following the method of Kac [4] and following closely the procedure of section 2
in Zayed [I], it is easy to show that the spectral function (1.4) associated with
problem (PI) is given by:
e(t)
where
G(x,x’;t)
ff G(x, x;t)dx,
(2.1)
is the Green’s function for the heat equation
(A2
--)
O,
u
(2.2)
subject to the impedance boundary conditions (1.2) and the initial condition
llm
G(x,x’;t)
5(x- x’),
(2.3)
twhere 5(x
x’) is the Dlrac delta function located at the source point x
x’.
Let
us write
C(x.,x.’;t)
C0(x;x’. ;t)
+ x(x,x’.. ;t),
(2.4)
where
Go(,’;t)
is
the
"regular
(4t)
exp{-
"fundamental solution" of
solution"
chosen
so
tl)e
that
conditions (1.2).
On setting x
O(t)
where
K(t)
4t
l:eat
,lua+/-o
G(,’;t)
(2.5)
}’
(2.2), while
satisfies
the
impedance
is
boundary
x’ we find that
area_____fl + K(t),
4t
]/ x(x,x
12
;t)dx.
the
(2.6)
(2.7)
HEARING THE SHAPE OF MEMBRANES:
FURTHER RESULTS
593
The problem now is to determine the asymptotic expansion of K(t) for small positive
In what follows we shall use Laplace transform with respect to "t" and use "s 2’’ as
t.
the Laplace transform parameter; thus
An
that
application
2
G(x,x’
;s
of
2
+(R)
2
(x,,s
x
f e-S
0
t
G(x,x
Laplace
the
(2.8)
;tldt.
transform
to
heat
the
(A2
s
2) G(x,x’;s 2
(x
x’)
in
(2.2)
,
shows
(2.9)
together with the impedance boundary conditions (1.2).
K(t) as t
equation
satisfies the two-dimenslonal membrane equation
The asymptotic expansion of
2)
0, may then be deduced directly from the asymptotic expansion of (s
for
% where
s
2
K(s )=
ff (x,x
s
2
(2.[01
)axe.
to section 3 in Stewartson and Waechter
With reference
[5], it can readily be
shown after some reduction that the impedance boundary conditions (1.2) give
K(s
2)
J
2
[
[
where
(a
j =I
m=-(R)
aj) fj
j+
I (sa)
2
fj(m;s)
(I +
--2
s
a
in which
convergent
m
{Im(sa)Km(sa)
I’ (sa)K’ (sa)
m
(2.11)
(m;s)},
sa[s
m
I
a[sl’(sa) +
m
I (sa)]
m
I’ (sa)
m
I(sa)
+ j
Im and Km are modified Bessel functions.
(2.12)
Im(sa)]
The
therefore,
for large positive s and it is
7.
3
series
expedient
(2.11) is slowly
apply a Watson
to
transformation [5] to obtain
(s 2)
2
a__
J
+(R)
11
aj)
2j= (aj+1
It now follows that the functions f.(v;s),
O fj
(v;s)dv
J
as s
=
(2.13)
J may be expressed in terms of
.
the asymptotic expansions of the modified Bessel functions and their derivatives due
to Olver [6]. These expansions for s
are uniformly valid in u for arg u
<
Now, the following cases can be considered:
CASE 1. (0 <
j)
rj << I, j
In this case, it can be shown for s
}+s2a 2
fj (u;s)
where
T
v2+s 2a2
1/2"
2 2
s a
For n
I/2
/
that
=V A ,nn (T)
L
n=0
0,1,2,3 we deduce that
(2. I4)
594
E. M. E. ZAYED
0
Aj,0
and
-T3(3
Aj ,3
-
(r- 3 ), Aj, 2
A..1,1
ayj+a yj )- z5 (-
23
2 2
+
2(a yj -)
3ayj
-a
2 2
TT(41
yj)
3
4(ayj
-
-)+
2aY.i)
6
21
9
(2.15)
z
On inserting (2.14) into (2.13) we deduce after some simplification that
K(s
r
8s
2
J
3a
--j=!Z
{I
6s
2
(2.16)
s
On inverting Laplace transforms and using (2.6) we have the formula:
+ let_h r + {I
area
o(t
J
3a
8(t)I/2
4t
(a,+
l-J
J=l
a’)Y’ }+ O(t
l/2
as t
J
O.
(2.17)
(0 <
k and
J)
Yi << l, J
Yi >> 1, j k+l
In this case f.(v;s), j
k have the same forms (2.14) and (2.15) while
J have the form (2.14)where
k+l
f.(v;s), j
CASE 2.
5
O,
Aj, 0
Aj,
+
2
4 8ayj
T
Aj ,2
8ayj
Aj,3
z3
)
ayj
+
ayj’
T6
19
5
)
+ 25
13
7
107
4a .j
15
a
43
(8ayj
8
s-yj ’
and
)
(4ayj
+
z5
27
)
(4ayj
+ ? 141
(4aj
15
g)
T
27
8
II
(2.18)
Consequently, we deduce after some reduction that
O(t)
_area4t
fl
3
k
+ {I
J
k
+
1/21 {a ,j-l
8(at)
J=k+l aJ+l-aJ
aJ+l-J
l (aj+l-aj) Yj
o(t
I/2)
a8
-I
(a+yj)
(2.19)
0.
t
j=l
(Yi >>
CASE 3.
I, j
k and 0
< yj <<
I, J
J)
k+1
This case can be deduced from the previous one and ylelds:
area
O(t)
4t
11+
J
{a
8(t)I/2
k
Z (aj+1-3
Jfk+1
I (aj+l-a3) (a+yjl)}
Jffil
J
+ {x -3a
(j+Jffik+1
Z
CASE 4.
In
this
(yj >>
case
1/2
as t
(2.20)
O.
J)
I, J
fj(v;s),
j) yj}g+o(t
J
J
have
the
same
forms
(2.14)
and
(2.18).
Consequently we have the formula:
O(t)
J
area R_
4
8(t)i/2.
Z (j+l-=j)
J=l
With reference to section
v ++ O(t I/2
(a+,]l)}
in Zayed
as t
0.
(2.21)
[I] and the articles by Kac [4], Gottlleb
HEARING THE SHAPE OF MEMBRANES:
FURTHER RESULTS
595
[7], Pleijel [8], and Steeman and Zayed [9], the asymptotic expansions (2.17), (2.19),
(2.20) and (2.21) may be interpreted as:
(1)
is a circular domain of radius a and we have the impedance boundary conditions
(1.2) with small/large impedances
of
the
four
respective
cases,
2
domain in R 2 of area a
In case
is
a bounded
be the number of smooth convex holes in
J
3a
it has n
(li) for the first three terms,
<
Let h
J as indicated in the specifications
j
yj,
or
>.
(a
j) 7j
i+
j=l
holes and a boundary length of
2 together with Neumann boundary conditions, provided h is an integer.
k
(al+l- .I)71
In case 2, it has h
-
holes, the parts
r],
k of the
j
(’+13
’) together with Neumann boundary conditions
j=l
j
k+l
J have lengths
boundary 1" have lengths a
J
while the other parts
rj,
j=k+l
together with Dirlch]et boundary conditions,
In case 4,
(a +
-1
7_.
it has no holes
(h
(j+l- j)
J
0) and a boundary length of
j =I
together with Dirlchlet boundary conditions.
We close this section with the remark that when J
(a*Y I)
(’+I
-’)
2 the results (2.17), (2.19),
(2.20) and (2.21) are in agreement with the results of [I].
3.
CONSTRUCTION OF (t) FOR PROBLEM (P2).
In analogy with the two dimensional membrane problem, it
is clear that
t)
associated with problem (P2) is given by:
G(,;t)d,
(t)
where
G(,’;t)
is the
(A3
(3.1)
Green’s function for the heat equation
-)
u
0,
(3.2)
subject to the impedance boundary conditions (1.3) and the initial condition of the
As we have done in section 2, we can write G(x,x’;t) for problem (P2) in
form (2.3).
a form similar to
(2.4), where
2
G0(,’;t)
(4t)
exp(-
4t
}"
(3.3)
From (2.4), (3.1) and (3.3) we find that
n+
O(t)
volume
K(t)
(3.4)
K(t)
fff (x,x;t)dx.
(3.5)
where
(4t)3/2
An appllcatlon of the Laplace transform to the heat equation (3.2) shows that
G(x,x’;t) satisfies the three-Plmenslonal membrane equation
2-2
(A3
s )G(x,x ;s
(3.6)
(x x’) in
,
E. M. E. ZAYED
596
together with the impedance boundary conditions (1.3), where
fff (x,x;s2)dx.
(s 2)
(3.7)
[I0], it can readily be shown after some
reduction that the impedance boundary conditions (1.3) give
With reference to section 2 in Waechter
where
J
2
(s 2)
a2 m--0
[
(m +
[ (a.j+l- aj) f.j(m;s)},
)
have the same form (2.12) with m replaced by m
fj(m;s)
The series (3.8) if fact diverges since K(t)
+
for small positive t; however,
difficulty may be easily removed by considering the asymptotic expansion for
this
large positive s of
(s2)
a
J
N
2
(m +
i
m=0
)
Inversion of the Laplace transform gives
K(t)
(3.9)
(aj+ -aj) fj(m;s)}.
j--I
KN(t)
and we may then write
(3.1o)
KN(t).
lim
N
On applying a Watson transformation [I0] to (3.9), we find that
2
J
N
(3.11)
Now,
the
four
respective
cases
considered
in
section
2,
can
be
applied
as
follows
CASE I.
(0
< ,{j <<
I, j
J)
On inserting (2.14) and (2.15) into (3.11) and integrating and letting N
(R),
deduce after some simplification that
surface area S
+
16t
K(t)
+
3/2t I/2
0(t I/2) as
[2a2
%) (-
aj+l
12
(3.12)
0.
t
From (3.4) and (3.12) we have the formula
O(t)
volume R surface area S
3/2 +
(4t)
16t
+
123/2 tl2
+ 0(t /2
CASZ 2.
N
O(t
(0
< yj <<
I, j
k and
{2a2
J
(3.13)
0.
as t
yj >> I, J
k+l
J)
On inserting (2.14), (2.15) and (2.18) into (3.11) and integrating and lettlng
we have the formula
volume R
+
(4t)3/2
t2a
2
[ (aj+
j=l
.
J
k
)
2a
j=k+l
(aj+
aj)(a
-I )}
2yo
we
HEARING THE SHAPE OF MEMBRANES:
+
+
1/2
12 3/2 t
0(t I/2)
CASE 3.
{2a
2
597
J
j= (a
+ 2a
aj)(- 3yj)
j+
(a
aj)}
j+
j=k+l
0.
as t
(yj >>
.
FURTHER RESULTS
k
(3.14)
I, j
k and 0
< yj <<
I, j
J)
k+l
.
This case can be deduced from the previous one and yields
volume
8(t)
+
(4t)3/2
123-2/t
J
2
j=k+l
2
I/2 {2a
+ 0(t 1/2)
CASE 4.
{2a
2a
aj)(a-2yjl)}
(aj+ I-
J =I
J
k
).’
+ 2a
(aj+ I- aj)( 3Tj)
J=k+l
j=l
(aj+ I- aj))
O.
as t
(yj >>
k
(aj+ I- aj)
(3.15)
I, j
J)
On inserting (2.14) and (2.18) into (3.11) and integrating and letting N
have the formula
.
J
volume
{2a
(4t) 3/2
16c
j=l
-I
(aj+
aj)(a- 2yj
)} +
+ 0(t 1/2
a
3(rt) 1/2
(3.16)
0.
as t
(R)we
[7], Waechter
[I0], P1eiJel [11], and Zayed [12] the asymptotic expansions (3.13)
(3.16) may be
is a spherical domain of radius a and we have the impedance
interpreted as (1)
J as indicated in
boundary conditions (1.3) with small/large impedances
j
With reference
to section
in [1] and the articles by Gottlleb
specifications
of
respective
,
the
terms,
the
four
is a bounded domain in R 3 of volume
In case I,
it has a surface S of area
surface S have areas 2a
2
cases,
4
3
or
(ii)
for
the
three
first
a
4a
2
the parts
I,
Sj, J
J of the
J
).’
j=l
(aj+
aj)
and mean curvatures
(;- 3yj),
jffil
J
together with Neumann boundary conditions.
In case 2, the parts
2a
2
Sj, J
I,
k of the surface S have areas
k
" (aJ+l-
aj)
and mean curvatures
(- 3Vj)
boundary conditions, while the other parts
J
-I
J
Sj, J
and mean curvature
j---k+1
k together with Neumann
k+1
a
J have areas
together with Dirichlet boundary
conditions.
J
In case 4, it has a surface of area 2a
-I
(0+
I- Ja4)(a- 2yj
j=l
with Dirichlet boundary conditions.
--together
a
and mean curvature
598
E. M. E. ZAYED
Finally, we note that when J
2 the results (3.13)
(3.16) are in agreement
with the results of [I].
4.
DISCUSSIONS.
This paper represents a sensible extension of the author’s previous publication
[I] when the boundary
r
in R 2 or the surface S in R3 consists of two parts (J
2) to
the case when F oc S consists of J parts, where J is a finite positive integer, in
which a great deal of technical analysis has gone into obtaining the results.
[2,3] has recently generalized the results of [I] to the case when
R
n,
n
Zayed
2 or 3
is a simply connected bounded domain, where a considerable amount of mathematical work
has gone into obtaining the results.
With reference to the previous work (See [2],
[3],
[II],
[12]),
we
conclude
that,
there
are
technical
difficulties
and
a
considerable amount of mathematical work in extending the results of the present paper
to the type of domains considered in [2] and [3].
This extension is still an open
problem which rlll be discussed in a forthcoming paper.
ACKNOWLEDGEMENTS.
suggestions
and
The author would like to thank the referee
comments.
for this interesting
He also would like to thank Professor M.S.P. Eastham
(University of London) for his suggestions and comments.
grateful to the Editor Dr. L. Debnath for his co-operation.
Finally,
he is deeply
REFERENCES.
I.
2.
ZAYED, E.M.E., Eigenvalues of the Laplaclan rlth Impedance Boundary Conditions,
Bull. Calcutta Math. Soc., to appear in Vol. 81 (1989).
ZAYED, E.M.E., Hearing the Shape of a General Convex Domain, J. Math. Anal.
Appl. to appear.
ZAYED, E.M.E., Hearing the Shape of a General Convex Domain: An Extension to
Hgher Dimensions, J. Math. Anal. Appl., to appear.
KAC,
4.
M., Can one Hear the Shape of a Drum? Amer. Math. Monthl 73(4), part II,
(1966), 1-23.
5. STEWARTSON, K. and WAECHTER, R.T., On Hearing the Shape of a Drum: Further
Results, Proc. Camb. Phil. Soc. 69 (1971), 353-363.
6. OLVER, F.W.J., The Asymptotic Expansion of Bessel function of Large Order,
Phil. Trans. R. Soc. London Set. A247 (1954), 328-368.
7. GOTTLIEB, H.P.W., Eigenvalues of the Laplaclan rlth Neumann Boundary Conditions,
J. Austral. Math. Soc. Set. B26 (1985), 293-309.
8. PLEIJEL, A., A Study of Certain Green’s Functions rlth Applications in the Theory
of Vibrating Membranes, Arklv. Math. 2 (1953), 553-569.
9. SLEEMAN, B.D. and ZAYED, E.M.E., An Inverse Eigenvalue Problem for a General
Convex Domain, J. Math. Anal. Appl. 94(I) (1983), 78-95.
I0. WAECHTER, R.T., On Hearing the Shape of a Drum: An Extension to Hgher
Dimensions, Proc. Camb. Phil. Soc. 72 (1972), 439-447.
II. PLEIJEL, A., On Green’s Functions and the Eigenvalue l>Istrlbutlon of the Threedimensional Membrane Equations, Skandlnav. Math. Koner Xll (1954), 222-240.
12. ZAYED, E.M.E., An Inverse Eigenvalue Problem for a General Convex Domain: An
Extension to Higher Dimensions, J. Math. Anal. Appl 112(2) (1985), 445-470.
3.