The wave equation approach to the two-dimensional inverse problem for a general bounded domain with piecewise smooth mixed boundary conditions

2001,21B( 2): 171-181

THE WAVE EQUATION APPROACH TO THE TWO-DIMENSIONAL INVERSE PROBLEM FOR A GENERAL BOUNDED DOMAIN WITH PIECEWISE SMOOTH MIXED BOUNDARY CONDITIONS 1 E.M.E.Zayed

I.H.Abdel-Halim

Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt E-mail: [email protected][email protected]

Abstract

The spectral distribution 'j'i(t)

1

= 2:::=1 exp(-itEJ),

where {K'}:=I are the

eigenvalues of the negative Laplacian -~ = - 2::~=1 (8~v)2 in the (Xl, x 2)-plane, is studied for a variety of domains, where

-00

< t < 00

and i

= R.

The dependence of 'j'i( t)on the

connectivity of a domain and the boundary conditions are analyzed. Particular attention is given to a general bounded domain

n in R 2

with a smooth boundary

an,

where a finite

number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth parts fj(j

=

Some geometrical properties of

1,···,n) of an are considered such that an

n

(e.g., the area of

n, the

=

uj=lfj.

total lengths of the boundary,

the curvature of its boundary, etc.) are determined, from the asymptotic expansions of

'j'i( t) for ItI -+ Key words

o. Inverse problem, spectral distribution, wave equation, eigenvalues

1991 MR Subject Classification

1

35K99, 35P05, 35P99

Introduction

Let n be a simply connected bounded domain in R 2 with a smooth boundary the two-dimensional Robin problem

{

an.

Consider

-~u = EU~ in n, (an + -y)u - 0, on an,

where :.. denotes differentiation along the inward pointing normal to constant. Denote its eigenvalues, counted according to multiplicity, by

(1.1 ) (1.2)

an and -y is a

positive

(1.3) I

Received November 3,1999; revised February 5,2001

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Zayed and Hassan[15,16] have discussed the problem (1.1 )-( 1.2) for small / large impedance 1 and have determined some geometric quantities of the domain 0, using the asymptotic expansion of the spectral distribution

= L exp(-itEJ) 00

ji(t)

as

Itl-t 0, i

= yCT.

(1.4)

w=l

Zayed[21,23] has discussed the problem (1.1)-(1.2) in the following cases: (i) Case 1.11

=

°

(Neumann problem)

101H(ltl) +

ji(t) = 2

1ft

(ii) Case 1.2 1 -t

ji(t)

IX>

lasOlsignt + ao ItI + O(t 2signt) as Itl-t 0.

(1.5)

(Dirichlet problem)

= ElH(ltl) - 1001 signt + ao ItI + O(esignt) as ItI -t 0, 21ft S

(1.6)

where H(ltl)is the Heaviside's unit function and

signt

=

{

I,

if t > 0,

0,

if t

= 0,

-1,

if t

< 0.

In these formula, 101 is the area of 0, 1001 is the total length of 00. The constant term ao has geometric significance, e.g., if 0 is smooth and convex, then ao = ~ and if 0 is permitted to have a finite number of smooth convex holes"h", then ao = (1 - h)/6. (iii) Case 1.3 (The mixed problem) If L 1 is the length of a part f 1 of the boundary 00 with the Neumann boundary conditions, and if L 2 is the length of the remaining part f 2 = aO\f 1 of 00 with the Dirichlet boundary conditions, then with reference to the articles [19, 21, 23], we get

ji(t)

1 2 = 2101 H(ltl) + (L ~ L ) signt + ao ItI + O(t 2signt) as Itl-t 0. 1ft

(1.7)

The object of this paper is to discuss the following more general inverse problem: Suppose that the eigenvalues (1.3) are known exactly for the Helmholtz equation (1.1) together with the following piecewise smooth Dirichlet, Neumann and Robin boundary conditions

u = 0,

on fj (j=I,···,k),

au on f j (j=k+l,···,m), anj a (8 +1j)U = 0, on fj (j=m+l,···,n), nj -=0,

(1.S)

where the boundary 00 of the domain 0 consists of a finite number of piecewise smooth parts

f j, (j = 1,···, n) such that 00 = Uj'=lf j, {)~j denotes differentiations along the inward pointing normals to the parts I' j respectively and the parameters 1j are positive constants. The basic problem is to determine some geometrical properties of 0 (e.g., the area, the total length of the boundary, the curvature of the boundary, etc) associated with the main

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problem (1.1) and (1.8) by using the asymptotic expansions of the spectral distribution (1.4) for small Itl. We close this section with the remark that the special cases of the problem (1.1) and (1.8) have been discussed by many authors, see for example Zayed and Hassan[15,16] , Zayed and Younis[19] and Zayed[21,23]. Thus, this problem can be considered as a more general one which does not seem to have been investigated elsewhere.

2

Statement of Results

Suppose that the boundary an of the domain n is given locally by the equations XO yO(cr), (0: = 1,2) in which a is the arc-length ofthe counterclockwise oriented boundary, yO(cr) E Coo (an) and let h j > 0 (j = 1, ... , n) be sufficiently small. Let L j (j = 1" .. , n) be the length of the parts f j of an respectively, and K(cr) be the curvature of an. Let nj (j = 1,"" n) be the minimum distances from a point ~ = (xl, x 2 ) of n to the parts f j (j = 1,"" n) respectively. Let nj (cr) denote the inward drawn unit normals to the parts I' j respectively. We note that ~

the coordinates in the neighborhood of fj and its diagram are in the same form as in Sec.3 of [15] (see also [24, 25]) with the interchanges n f-> nj, h f-> hj, I f-> I j. C(I) f-> V(Ij) and o f-> OJ (j = 1"", n). Thus, we have the same formulae (3.1)- (3.4) of Sec.3 in [15] with the interchanges c( cr) f-> K( cr), n f-> nj and ~(cr) f-> nj (o ) (j = 1, ... , n). Theorem 2.1 With the assumptions stated above, the asymptotic expansion of the spectral distribution ji(t) for small It I of the main problem (1.1) and (1.8) can be written in the form:'

ji(t) = al H(ltJ) + a2signt + a31tl + a4t2signt + O(t 3signt) as t

where the coefficients A. Case 2.1

a,AII =

Itl----+ 0,

(2.1)

1,2,3,4) are in the following forms:

IfO<,j«

1(j=m+1,"',c) and,j»

1(j=c+1,···,n)

B. Case 2.2 If,j»1(j=m+1,"',n) In this case, the coefficients a ll (1I = 1,2,3,4) follow immediately from the case 2.1, by setting c = m with I:j'~m+l as zero. C. Case 2.3 If 0 <,j« 1(j = m+ 1,···,n)

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In this case, the coefficients av(v = 1,2,3,4) follow immediately from the case 2.1, by setting c

3

= n with I:J=n+1 as

zero.

The proof of Theorem 2.1

With reference to the articles [15, 16, 19, 23], it is easy to show that ji(t) associated with the main problem (Ll) and (1.8) is given by

ji(t)

=

JJ

(3.1 )

G(:, :;t)d:,

n

where G(XI' Xz; t) is the Green's function for the wave equation.

(~-

()z ji2)G(XI,XZ; t) ot ~ ~

=0

in

n x {-oo < t < oo},

(3.2)

subject to the piecewise smooth Dirichlet, Neumann and Robin boundary conditions (1.8) and the initial conditions

aG(XI, Xz; t)

lim G(XI, Xz; t) = 0,

t--+O

~

at

lim

~

t--+'O

= b(XI - xz),

- -

(3.3)

where b(Xl - xz) is the Dirac delta function located at the source point Xz. Let

~

(3.4)

where (3.5) is the "fundamental solution" of the wave equation (3.2) while N( Xl, Xz; t) is the "regular solution" chosen in such a way that G (Xl, xi; t) satisfies the piecewise smooth Dirichlet, Neumann and Robin boundary conditions (1.8). On setting

- = Xz- = -X, we find that

Xl

ji(t)

= ElH(ltl) + K(t), 27rt

where

K(t) =

(3.6)

JJ

(3.7)

N(:, :;t)d:.

n

In what follows we shall use Fourier transforms with respect to -00

-00


00

and use

< 7] < 00 as the Fourier transform parameter. Thus, we define

J +00

8(:'1, :,z; 7])

=

e-Z,,-i'lt G(:,l' :,z; t)dt.

-00

(3.8)

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An application of the Fourier transform to the wave equation (3.2) shows that G(Xl, X2; TJ) satisfies the reduced wave equation (3.9) together with the piecewise smooth Dirichlet, Neumann and Robin boundary conditions (1.8). The asymptotic expansion of K(t), for small ItI, may then be deduced directly from the asymptotic expansion of K(TJ), for large ITJI, where

K (TJ)

=

JJ (~, ~; Af

n

TJ)d ~ .

(3.10)

It is well known (see for example [15], [16], [19]) that the equation (3.9) has the fundamental

solution

Go(:.;, :::;TJ)

=

-~Yo(21rTJr::::) n, while

where r:::: =

1:

1 -

:'21 is the distance between the

order. The existence of this solution enables us to construct integral equations for G(Xl, X2; TJ) -satisfying the piecewise smooth Dirichlet, Neumann and Robin boundary conditions (1.8) for points Xl and X2 of the region

Yois the Bessel function of the second kind and of zero

small / large impedances 'Yj, where 0 < 'Yj << 1, (j = m+l,· .. , c) and 'Yj Therefore, Green's theorem gives the following integral equation:

>> l(j = c+l,· .. , n).

--

Similarly, we can find the integral equations for G(Xl, X2'; TJ) for the other two cases. On applying the iteration methods (see, for example [15, 16]) to the integral equation (3.11), we obtain the Green's function G(X1' X2; TJ), which has a regular part of the form 20

Af (Xl, X2; TJ)

=(L

rn e-L

where

Am) /4,

(3.11)

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Where 00

Mi(y, y') = """,.....

, K 3 (y , y) ~

~

2) _1)V K;V)(y', y)

v=O

"'"

(i = 1,2,···,8),

,..",

a + "Ij )Y (27l"'T/r y)], = -21 [( -",y__ o unj y

,

Kg ( ~,~) =

t

,21[(1 + "Ij-1 anja y )Yo (27r" 1r:: .:) ].

In these formulae, we note for example, that K;v)(y', Y) being the iterates of K;O)(y', Y) (i

= 1, 2,···,8). On

together with small

"""" ""

the basis of (3.12), the function.if

I

(Xl, X2;

large impedances Ti- The case when

of the piecewise smooth parts

rj

Xl ~

""

......

71) will be estimated for large 1"11 and

X2

lie in the neighborhood

~

is particularly interesting. For this case, we use the local

expansions of the functions:

Y O(27l"'T/r _x _y), ~Yo(27r71r;r unj y __y) (j

= 1, .. ·,n),

(3.13)

when the distance between z and y is small, which are very similar to those obtained in Sections ~

~

4, 5 of [15]. Consequently, the local behaviour of the kernels Ki(y', y) (i

= 1,2,···,8)

when

the distance between Yand y' is small, follows directly from knowledge of the local expansions of the functions (3.13).

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Definition 1

Let

6

Vo1.21 Ser.B

and6be points in the upper half-plane

e > 0. Then, we define

An e A ( ~l, ~2; 7] )-function is defined for points ~1l6 belonging to sufficiently small domains "-'

""

1)(1j) except when

6 =

-

"V

""

6 E Ij (j = 1"", n) and A is called the degree of this function. For

~

every positive integer A, it has the expansion:

where 2:* denotes a sum of a finite number of terms in which f(~}) is an infinitely differentiable functions ( see [8, 10, 15, 16] ). In this expansion Pll P2, £, m are integers, Pl :2: 0, P2 :2: 0, £ :2: 0, A = min(Pl + P2 - q) and q = £ + m. The remainder R A(6 ,6 ; 7]) has continuous derivatives of order k

:S A satisfying

where Aj (j = 1, ... , n) are positive constants. Thus, using methods similar to those obtained in Sections 6-10 of [15, 16], we can show that the functions (3.13) are e A _ functions with degrees A = 0, -1 respectively. Consequently the kernels K(y',y) (i = 1,"',8) are eA-functions with degrees A = 0,0,0,0,-1,1,-1,1 re-

--

spectively.

Definition 2

Let xlandx2 be points in large domains

n + fj

(j

= 1"", n),

then we

define

r2 = min(r x 1 Y +rX 2Y) if¥ E f j (j Y

- -

- -

.-

ra = min(rx 1 Y

+ r __y)

r 4 = min( rXl

+ r y) if ~

Y

- -

Y

Y

X2

- -

X2

= k + 1, .. ·,m),

if ~ E fj (j = m E I'j (j = c

+ 1"", c),

+ 1, ... , n).

- -

An E A(Xll X2; 7])-function is defined and infinitely differentiable with respect to Xl and X2 ""

""

""

when these points belong to large domains n + fj(j = 1"", n).Thus the EA-function has a similar local expansion of the e A _ function ( see [8], [15] ). "-'

With the help of Sections 8,9 in [15] it is easily seen that formula (3.12) is an EO(Xl, X2; 7])function and consequently G(Xll X2; 7]) can be estimated as follows k

G(Xl' X2; 7]) "'"

f'"..,;

=L

0 {[I + Ilog(21r7]rdlJe-Ajl)Tl}

j=l

+

m

L j=k+l

0 ([I

+ Ilog(21r7]r2) IJ e -Ajl)T2}

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No.2

c

L:

+

O{[1+llog(27r7]T3)lJe-Aj'lrs}

j=m+1 n

L: 0 {[I + Ilog(27r7]T4)1J e-

+

Aj

'l r

4

}

(3.14)

,

j=c+1

which is valid for 17]1 --+ 00 and for small/large impedances 1'j(j = m + 1"", n) where Aj(j = 1"", n) are positive constants. Similar statements are true in the other two cases. The estimate

--

(3.14) shows that G(X1' X2; 7]) is exponentially small for large 17]1 . This proves that G(X1' X2; 7]) converges for 17]1 --+ 00. With reference to Section 10 in [15], if the eA-expansions of the functions K;(y', y) (i =

--

1,2"",8) are introduced into (3.12) and if we use formulae similar to (6.4), (6.9) of Section 6 in [15], we obtain the following local behaviour of the regular part N (Xll X2; 7]) when T1, T2, T3 and

T4 are small, which is valid for 17]1 --+

00

--

and for small / large impedances 1'j (j = m + 1,···, n): n

N(X1,X2;7]) - ~

-

-

= L:iJj (Xll X2;7]), j=l

(3.15)

--

where (a): if Xl and x2belong to sufficiently small domains D(Ij ) (j = 1,· .. , k), then, ~

1

Nj (:1, :2; 7]) = 4" Yo(27r7]P12 ) + O{7]

-1

exp( - Aj7]P12)}, as 17]1 --+

(3.16)

00.

- -

(b): if x1andx2belong to sufficiently small domains D(Ij ) (j = k + 1, ... , m), then, (3.17)

- -

(c): if x1andx2belong to sufficiently small domains D(Ij) (j = m + 1, ... , e), then,

a

1

(f

JVj(X1, X2; 7]) = --{I -1'j(a C2) - ~ 4 ~1

-

-1

}Yo(27r7]pd + O{7]

-1

exp( -A j7]pd}, as /7]1--+

00.

(3.18)

(d): if x1andx2belong to sufficiently small domains D(Ij) (j = e + 1, ... , n), then, ~

o 1 -1 a -1 JVj(:l,:;; 7]) = 4"{1 -1'j (a~i )}Yo(27r7]P12) + O{7] exp( -Aj7]P12)}, as 17]1 --+

00.

(3.19)

When T1 2: bj > O(j = 1, .. ·,k), T2 2: s, > O(j = k + 1,· .. ,m), T3 2: bj > O(j = m+ 1,,, .,e), and T4 2: bj > O(j = e+ 1, ···,n) the function N(XllX2;7]) is of order O{e- B'1 } ~

as 17]1 --+

00,

B =constant

~

> O. Thus, since rO'--+O lim ~ = 1 (0: = 1,2,3,4) (see [15], [16]) then we P12

have the asymptotic formulae (3.16)-(3.19) with P12 in the small domain D(Ij) (j = 1,···, n) are replaced by To: (0: = 1 - 4) in the large domains n + f j (j = 1,···, n) respectively. Since for

e 2: h j > 0 (j = 1, .. ·, n) the functions N j (;::::.; 7]) are of order O{ -27]A j h j},

the integral over the domain

n of the function N(::., ::.; 7]) can be approximated in the following

way (see (3.10)):

L: J JiJj (::.'::.; 7]){1 n

K(TJ) =

hj

i.,

J=le=o€,=o

K(e)e}dede

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ACTA MATHEMATICA SCIENTIA

"

+L

O{ exp( -21]A j hj

)} ,

as

71--+ 00.

17

Vol.21 Ser.B

(3.20)

j=1

If the e.\-expansion of

Nj (?!. ?!. i1] ) (j = 1,"',n)

(see [15,16]) are introduced into (3.20).

and with the help of the formula (11.3) of Sec.ll in [15], we deduce after inverting Fourier transforms and using (3.6) that the result (2.1) has been constructed, and the proof of the theorem 2.1 follows.

4

Discussions and Conclusions

The problem of determining some geometric quantities of a general bounded domain in R 2 with a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions, from complete know ledge of its eigenvalues has been discussed in the present paper by using the wave equation approach. It is well known that the wave equation approach has given very strong result; the definite one is that of Hormander[5] who has studied the distribution tr[exp( -itP)]

near to t = 0 for an elliptic positive semi-definite Pseduodifferential operator P of order m in H", Recently, the wave equation approach in solving particular problems has been discussed by Zayed and Hassan[15,16], Zayed and Younis[19j and Zayed[21,23] who have studied the spectral

distribution ji(t) = 2:::=1 exp( -itE3) as ItI --+ 0 for some different bounded domains in R 2 with certain boundary conditions. An alternative method is the heat equation approach, which has been investigated by many authors, see for example, Pleijel[8], Kac[6], McKean and Singer['], Stewartson and Waechter[13], Smith[12], Gottlieb[2-4], Protter[9], Gordon,Webb and Wolpert[1], Sleeman and Zayed[lO,ll], Zayed and Younis[17,18] and Zayed[20,22,24-30]. They have studied the 1

asymptotic expansion ofthe trace of the heat kernel 0(t) = 2:::=1 exp( -tEw ) as t --+ 0, for some bounded domains in R 2 with certain boundary conditions. With reference to the above articles one can ask a question, is it possible just by listening with a perfect ear to hear the shape of the domain n? This question has been put rather nicely by Kac[6], who simply asked, "Can one hear the shape of a drum?" Recently Gordon, Webb and Wolpert[1] have shown that the answer of Kac's question is negative. They have shown explicitly two domains that although having different shapes, have the same eigenvalues (i.e., isospectral domains). This theoretical result was experimentally verified in [14] by employing thin microwave cavities shaped in the form of two different domains known to be isospectral. If we look at the two methods mentioned above, we can see the differences between them in solving the inverse problems. In particu- . lar, the present paper provides useful techniques to inverse problem methods via the spectral distribution of the Laplacian. References Gordon C, Webb D L, Wolpert S. One can not hear the shape of a drum, Bull Amer Math Soc, 1992, 27: 134-138 2 Gottlieb H P W. Hearing the shape of an annular drum. J Austral Math Soc Ser B, 1983, 24: 435-438 3 Gottlieb H P W. Eigenvalues of the Laplacian with Neumann boundary conditions. J Austral Math Soc Ser B, 1985, 26: 293-309 4 Gottlieb H P W. Eigenvalues of the Laplacian for rectilinear regions. J Austral Math Soc Ser B, 1988, 29: 270-281

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5 Hormander L. The spectral function of an elliptic operator. Acta Math, 1968,121: 193-218 6 Kac M. Can one hear the shape of a drum? Amer Math Monthly, part 11,1966,73(4): 1-23. 7 McKean H P, Singer I M. Curvature and the eigenvalues of the Laplacian. J Diff Geom, 1967,1:43-69

8 Pleijel A. A study of certain Green's funct.ions with applications in the theory of vibrating membranes. Ark Mat, 1954,2: 553-569 9 Protter M H. Can one hear the shape of a drum?revisited. SIAM, Rev, 1987,29: 185-197 10 Sleeman B D, Zayed E M E. An inverse eigenvalue problem for a general convex domain. J Math Anal Appl, 1983,94: 78-95 11 Sleeman B D, Zayed E M E. Trace formulae for the eigenvalues of the Laplacian. Z Angew Math Phys, 1984,35: 106-115 12 Smith L. The asyrnptotics of the heat equation for a boundary value problem. Invent Math, 1981, 63: 467-493 13 Stewartson K, Waechter R T. On hearing the shape of a drum: further results. Proc Carnb Phil Soc, 1971,69: 353-363 14 Sridhar S, Kudrolli A. Experiments on Not hearing the shape of drums. Phys Rev Lett, 1994,72: 2175-2178 15 Zayed E M E, Hassan A A M. The wave equation approach to an inverse problem for a general convex domain. Bull Calcutta Math Soc, 1989,81: 139-156 16 Zayed E M E, Hassan A A M. The wave equation approach to an inverse problem for a region in R 2 with two impedance boundary conditions. J Institute Math Comput Sci (Math Ser), 1989,2: 153-167 17 Zayed E M E, Younis A I. An inverse problem for a general convex domain with impedance boundary conditions. Quart Appl Math, 1990,48(1): 181-188 18 Zayed E M E, Younis A I. On hearing the shape of rectilinear regions. J Math Phys, 1994,35: 3490-3496 19 Zayed E M E, Younis A I. The wave equation approach to an inverse problem for a general convex domain with impedance boundary conditions. Bull Greek Math Soc, 1995,37: 123-130 20 Zayed E M E. An inverse eigenvalue problem for the Laplace operator. In: Lecture Notes in Mathematics, 964. Berlin: Springer-Verlag, 1982. 718-726 21 Zayed E M E. The wave equation approach to inverse problems: An extension to higher dimensions. Bull Calcutta Math Soc, 1986,78: 281-291 22 Zayed E M E. Eigenvalues of the Laplacian for the third boundary value problem. J Austral Math Soc Ser B, 1987,29: 79-87 23 Zayed E M E. The wave equation approach to Robin inverse problem for a doubly connected region. Bull Calcutta Math Soc, 1988,80: 331-342 24 Zayed E M E. Hearing the shape of a general convex domain. J Math Anal Appl, 1989,142: 170-187 25 Zayed E M E. Heat equation for an arbitrary doubly-connected region in R 2 with mixed boundary conditions. J Applied Math Phys (ZAMP), 1989,40: 339-355 26 Zayed E M E. On hearing the shape of an arbitrary doubly-connected region in R2. J Austral Math Soc, Ser B, 1990,31: 472-483 27 Zayed E M E. Heat equation for an arbitrary multiply connected region in R 2 with impedance boundary conditions. IMA J Appl Math, 1990,45: 233-241 28 Zayed E M E. An inverse problem for a general multiply connected bounded domain. Appl Anal, 1995, 59: 121-145 29 Zayed E M E. Short-time asymptotics of the heat kernel of the Laplacian of a bounded domain with Robin boundary conditions. Houston J Math, 1998,24: 377-385 30 Zayed E M E. The asyrnptotics of the heat semigroup for a general bounded domain with mixed boundary conditions. Acta Math Sinica , English Series, 2000,16: 627-636