An inverse eigenvalue problem for an arbitrary multiply connected bounded region in R2

AN INVERSE EIGENVALUE PROBLEM FOR AN

ARBITRARY

MULTIPLY CONNECTED BOUNDED REGION IN R

2

E. M. E.ZAYED

Mathematics

Department,

Facultyof Science Zagazig University

Zagazig,

Egypt

(Received June

26,1990 and inrevisedformJuly 26,

1990)

ABSTRACT.

The basicproblemis to determinethegeometry ofanarbitrary multiplyconnected bounded regioninR togetherwiththemixedboundaryconditions, from thecomplete knowledgeof theeigenvalues

{’

}’-1

for the

Laplace

operator, usingtheasymptoticexpansion of thespectralfunction

0(t)

exp(-t)

ast--0.

KEY

WORDS

AND PHRASES. Inverse

problem,

Laplace’s

operator, eigenvalue problem, spectral function.

1980

AMS

SUBJECT

CLASSIFICATION CODE. 35K,

35P 1.

INTRODUCTION.

The underlyingproblemis todeduce the preciseshapeofamembrane fromthecomplete knowledge of theeigenvalues

{j}’.

for theLaplace operator

A2-i-i(

.)2

in the

xx2-plane.

Let

__.R be asimplyconnected bounded domain withasmoothboundary 0. Consider the Neumann/Dirichlet problem

(A

/)u

0 in

,

(1.1)

Ou

---0 or u-0 on 0,

(1.2)

On

where denotes differentiationalongtheinwardpointingnormalto0V2andu

_

C2(Q)t’)C(-).

Denote

itseigenvalues,countedaccordingtomultiplicity, by

0<XlX2<...<.j...-,oo

as j-,oo.

(1.3)

Theproblemofdeterminingthegeometryoff2has beeninvestigated by Pleijel

],

Kac

[2],

McKean

and

Singer

[3],

Stewartson

andWaechter

[4],

Smith

[5],

Sleemanand

Zayed

[6,7],

Gottlieb

[8],

Greiner

[9],

Zayed

[10-13]

and the referencesgiven there, usingtheasymptotic expansionof thetracefunction

O(t)-tr[exp(-tA2)]=

,

exp(-tX,)

as t--0.

(1.4)

0(t)

L___[+

101

+a0+

k2(o)do+0(t)

as t-0,

(1.5)

41r/

8(m)

while, in thecaseof Dirichletboundaryconditions

(D.b.c.):

0(t)

=4m-8(m)m

101

+a0+

k2(oo+t)

as t0,

(1.6)

n

In

theseformulae,

I1

isthe areaof

,

lis

thetotallengthof0and

k(o)

isthe cuwature of #.

e

constantte_{a0 has}geometric significance,

e.g.,

if is_{smooth and convex, then a0}

;

and if

ispermittedtohave a finite numberof smooth convex holes

"H",

_{then a0}

(1

H).

Theobject of thispaperis todiscuss thefollowingmoregeneralinverse

problem: t

be an arbitrary multiplyconnectedboundedregionin

R

which issurrounded internallyby simplyconnected bounded domains

,

withsmoothboundaries

0,

m andexternally byasimplyconnected boundeddomain

=

with asmooth

bounda

#=. Suppose

that theeigenvalues

(1.3)

aregivenfor the eigenvalue equation

(A+X)u=0

in

,

(1.7)

togetherwith oneof thefollowingmixedboundaryconditions: 0u

=0 on 0i, i=l,...,k and u=0 on

0i,

i=k+l,...,m,

(1.8)

0n

u-0 on

0,,

i=l k and

0=0

on

0i,

i=k+l m,

(1.9)

0n

where

on

denote differentiationsalongthe inwardintingnormalstothe boundaries

0,

m, respectively.

Thebasicproblemis to deteinethegeometof

om

theasymptotic expansionof

e

spectral function

(1.4)

for small

sitive

t.

Note

thatproblems

(1.-(1.9)

have beeninvestigated recentlyby yed

[11]

in the specialca where is an

arbitra

doubly

connected boundedregion

(i.e.,

m=2).

2.

STAMENT OF OUR ULTS.

Suppose

that the boundaries

0,

1,...,maregiven locally

by

theequations

x"

y’(o),

n 1, 2 in which

o,

1,...,m are the arc-lengths of the counterclocise oriented boundaries

₀

and

y"(o)

C=(0).

tL

and

k(o)

be thelengthsand thecuwaturesof

0,

mrespectively.

en,

the results of our mainproblem

(1.-(1.9)

canbe summarized in thefollowingcases:

CASE

1.

.b.c.

on0i, 1 ,kandD.b.c. on

0,,

k + 1

,m)

0(t)

+

L,-

L,

+

(2-m)

7i

+

+

0(t)

as 0.

CASE

2.

.b.c.

on

a,

1,..., kandN.b.c.on

a

+1,...,

m)

In

this case theasymptotic expansionof

0(t)

as 0 has the me form

(2.1)

withtheinterchanges 1,...,k

,

k+ 1 ,m.

With referencetoformulae

(1.4), (1.5)and

toarticles

[6], [11],

[12]

theasymptotic expansion

(2.1)

maybe interpretedasfollows:

(i)

fis anarbitrary multiply connected boundedregionin

R

and wehave themixedboundary conditions

(1.8)

or

(1.9)

asindicated inthespecificationsof thetworespectivecases.

(ii) For

the first four terms,fis anarbitrarymultiplyconnectedboundedregionin

R:’

of area f2

I.

In

case 1,it has

H

(m

1)

holes, the boundaries

O,,

k areoflengths

L,

andofcurvatures

i-I

k,(o,),

1, ...,k togetherwith

Neumann

boundaryconditions, while the boundaries

Of2,,

k+ m

areoflengths

,

L,

and ofcurvatures

k,(oi),

k + m togetherwith Dirichletboundaryconditions,

i-k+l

provided

H

is aninteger.

Weclosethis section withthefollowingremarks:

REMARK

2.1.

On

settingk 0 in formula

(2.1)

withthe usual definition that iszero,weobtain

i-I

the resultsof Dirichletboundaryconditions onOf2i, 1, ...,m.

REMARK

2.2.

On

setting k m in formula

(2.1)

withthe usual definition that iszero,we

i-m+l

obtainthe resultsof

Neumann

boundaryconditions on

OQ,,

1, ...,m.

3.

FORMULATION OF

THE

MATHEMATICAL PROBLEM

Itiseasytoshowthat thespectralfunction

(1.4)

associated withproblems

(1.7)-(1.9)

isgiven by

(3.1)

where

G(x_,,x_:;/)is

Green’s

functionfor the heatequation

A2--

u-0,

(3.2)

subjecttothe mixedboundaryconditions

(1.8)

or

(1.9)

and the initialcondition

-,o

\-where

&(x_,,

-_x2//

isthe Dirac delta function locatedatthe sourcepointx,-x2.

Let

us write

where

G(x_,_;t)- Go(X_,

_x.z;t

(3.4)

sothat

G(x_l,_x2;t)

satisfiesthe mixedboundaryconditions

(1.8)or (1.9).

O(t)

+

K(t),

(3.6)

where

Theproblemnowis todetermine theasymptotic expansionof

K(t)

for smallpositivet.

In

whatfollows we shall useLaplacetransforms withrespecttot, and uses asthe

Laplace

transformparameter;thus we define

(x_,,x_:;s:’-

I(R)e-’2’G(x_,,x_2;t)dt.

(3.8,

An

applicationof the

Laplace

transformtothe heat equation

(3.2)

shows that

-(x_t,xz;s

\

satisfies the membraneequation

(A:,-S-)-(X_,X_:;S:’)---6(x_-_x:,)

in

,,

(3.9)

togetherwiththemixedboundaryconditions

(1.8)

or

(1.9).

The asymptotic expansion of

K(t)

for small positivet, may then be deduced directly from the asymptotic expansionof

(s

)

forlarge positives, where

-(sz,

ff(x_,x_;sm)dx_.

(3.10,

4.

CONSTRUCTION OF GREEN’S

FUNCTION.

It

iswell known

[6]

that themembraneequation

(3.9)

has thefundamentalsolution

)

(s%)

(4.

)

where

rx,

x2 isthedistancebetween thepoints

x

(x,x)

and

x

(x,x)

of theregion fwhile

K0

isthe modified Bessel functionof the secondkindandof zero order. The existence of this solution enables ustoconstructintegral equationsfor

(xj,x_:;s

2)

satisfyingthe mixedboundaryconditions

(1.8)

or

(1.9).

Therefore,

Green’s

theoremgives:

CASE

1.

(N.b.c.

on0ffi, 1,...,kand D.b.c. on

Of,,

k+

1,...,rn)

0

sr

lfO (x_

+_

-

y;s s

t,-,/

"_

K

-

ay_.

(4.2)

CASE

2.

(D.b.c.

on(gQ, 1,...,kandN.b.c. on

0,,

k+1

m)

In

this case

Green’s

function

"(x_,x_z;s

has the same form

(4.2)

with the interchanges

0,,

On

applyingthe iteration method

(see [11

],

[12])

totheintegralequation

(4.2),

weobtain

Green’s

function

x_l,x_..;s2/,

whichhastheregularpart:

K

sr.:

K

sr,

d

y

+

--

.,/,

+

",.,

o,,,,"-o,

M,(

y_,

y

’)

ani,.

O--Ko(.Vr,ld

__)

ydy

f f

--CKo(%)mfY

y’

Kosr.,_.._dydy’

i- _..niy

where

+,-1

/,

.f{fKo(sr,_,,._)L:(yy,)ay,.,

,

_"

(4.3)

(4.4)

(4.5)

(4.6)

(4.7)

(4.8)

(4.9)

(4.10)

and

(4.11)

[

On

thebasisof

(4.3)the

function

Xl,X,_;s’-)

will beestimatedforlargevalues ofs. The case when

x

and

x,.

liein theneighborhoodsofOg2,, m isparticularly interesting.

For

thiscase,weneedto

usethefollowingcoordinates.

5. COORDINATFINTItENEIGItBORItOODSOFOf2,,i=a m.

Let

ni, i-1 rn be the minimum distances from a point

x---(x_i,x

-]\/

ofthe region g2 to the

boundariesOf2i, 1,...,m respectively.

Let

n,(o,),

1, ...,mdenote the inward drawn unit normalsto

Off2,,

,m respectively. Wenotethat the coordinates in theneighborhoodofOg2,, -k+1, ...,m anditsdiagrams

(see

[11

])

arein the same formas in section5.1of

[11

with theinterchanges o

o,,

n2

n,,

h

h,,

I2

I,, D(I)

D(li)

and 5

&,,

k+ ,m. Thus,wehave the same formulae

(5.1.1)-(5.1.5)

of section 5.1 in

[11]

with the interchanges n2"-

n,,

nz(o,.)-

n,(oi),

t2(o2)’,-

t,(o,),

k.(o2)

k,(o,),

i--k + m.

Similarly,the coordinates inthe neighborhoodof Of

2,,

1, ...,k anditsdiagrams

(see [11])

are similartothose obtained in section5.2of

[11

withtheinterchangesoi

o;,

n

n,,

h

h,,

I

li,

D(li)

D(/,)

and b hi, k. Thus,wehave the same formulae

(5.2.1)-(5.2.5)

ofsection5.2 in

[11

withthe interchanges

n

n,,

n(ol)

ni(o,), tl(oi)

t,(;,)

and

ki(o)

k,(oi),

k.

6.

SOME LOCAL EXPANSIONS.

It

nowfollows that the localexpansionsof the functions

when the distance between x and

y

issmall, areverysimilar

o

those obtained in section 6 of

[11].

Con-sequently,for k,k+1, ...,m,thelocal behavior of hefollowingkernels:

when the distance betweenyand

y’

issmall,follows directly from theknowledgeof the localepansions

of(a..

DEFINITION

1.

Let

l

and

2

bepointsinthe

upper

half-plane

j-

0,hen we define

)12 V(l

2l)

4-

(21

4-

)2.

(6.4)

where

"

denotes a sumof a finite number of terms in which

f(l)

isaninfinitelydifferentiable function.

In

thisexpansion,

P,

P2,

l,tnareintegers,where

Pl

0,

P2

0, a:0,

g

min(P

+

P:,

q),

q + m and the minimum is taken overalltermswhichoccur in the summation

Y.’.

The

remainderR^(t,2;s)

has continuous derivativesoforder ds

A

satisfying

,_2;s

-0(s-Ae

s/’’2)

as s o,

(6.6)

where

A

is apositiveconstant.

Thus, usingmethods similar to those obtained in section7of

[11],

we can show that the functions

(6.1)

aree

X-functions

withdegrees k 0, -1 respectively.

Consequently,

the functions

(6.2)

are

e-functions

withdegrees k- 0,-1,whilethe functions

(6.3)

are

eX-functions

withdegrees

g

0,1 respectively.

DEFINITION

2.

IfXl andx2

arepointsinlargedomainsf+0fi, k,k+1,...,rn, then we define

and

f2-min(rxly+ry)

if

yE Ofi,

i-1 ,k,

J

2 min

rx,

+

r

if

y

_

Of2i, k+ m.

An

Ek(X_l,

_x2;s)ofunction

isdefinedandinfinitelydifferentiable withrespecttox

1_

andx

2_

when these

points belong

m

large

domains +

0fi

except

whenx

x

0i, ,m. Thus,the

E-function

has a similarlocalexpansionof the

e-function

(see [6], [11 ]).

By

thehelpof section8in

[11],

it iseasilyseenthatformula

(4.3)is

an

E(x_,x_2;s)-function

and consequently

(x_’_x2;s2)

"i-,

0

{[1+

logsfl2,

+

Y.

0

[1

₊

logs/2l

(6.7)

i-k+l

which is validfors

,

where

A,

1, ...,m arepositiveconstants. Formula

(6.7)shows

(x_,,_x,;s2)

isexponentiallysmallfors

7.

THE

ASYMPTOTIC BEHAVIOR OF

X_1,_X2;S2).

small-where,

ifx

_{and x2}belongtosufficientlysmall domains

D(li),

k,k+1,...,m,then

-Ko(Sl,)

+

O

(s

exp(-A,slt)}.

(7.2)

k

andl2>),>O,i--k+l,...,m

the function

(x_,x_2;s21

isof order When

fla

>0,

O{exp(-cs)}ass

--,oo,c>0. Thus,since lim

=---12

lim 1,

thenifxl

_{and x2}belongtolargedomains 12---0P2 /--0P

ff+

02,

,k,

wededuce for s oo_that

i(X_I,X_2;S2)

--Ko(sf2)+O{s-’exp(-Aisf,2)}

(7.3)

while, if

x

and_{x2 belong}tolargedomains if2 +

0,

k + m, we deduce for s that

$. CONSTRI/CTION

OF OUR

RF_NULTS.

Sincefor

h,

0,i 1, ...,k,k+1,...,,the

functionsx

,x_;s

areoforderO

{exp(-2hi)},

theintegralof the function

.(x_,

x;s overtheregion f2canbeapproximatedin thefollowingway

(see

(3.10)):

h, L,

’(s2)-,./

,

x_,x_;s

{1-k,(x)2Idd2

2-o

-o

+

E

O{exp(-2sA,h,)}

as s oo.

(8.1)

i-1

Ifthe

e’-expansions

of

,(x_,x;s21,

i-a\/_

k,k+a,...,m, are introduced

in.to

(8.1),

one obtains an asymptoticseries of theform:

(s2)

X

a,

s-

+0(s-i-x)

as s--oo,

(8.2)

where the coefficients

a,,

arecalculated fromthe

eX-expansions

by

the

help

offormula

(10.3)

ofsection10 in

[11].

a,

L,

+

L,

o\i-1

a2-(2-m),

(8.3)

k

(o,)doi

+

.,.

kCoi)do

On

inserting

(8.3)

int

(8.2)

andinverting Laplacetransforms andusing

(3.6)

wearriveatourresult

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PLEIJEL,

A.

A

study ofcertain

Green’s

functionswith applications in the theory of vibrating membranes,Ark.

Mat.

2

(1954),

553-569.

[2]

KAC, M. Can

one hear theshapeof adrum?

Amer.

Math.Month, 73,

No.

4,

Part

II

(1966),

1-23.

[3]

McKEAN,

H. P.

and

SINGER, I. M. Curvature

and the eigenvaluesof the Laplacian,

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Diff.

Geometry,

1

(1967),

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STEWARTSON, K.

and

WAECHTER,

R.

T. On

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(1971),

353-363.

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SMITH,

L.

Theasymptoticsof the heatequationfor aboundaryvalueproblem,

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(1981),

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and

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E. M. E.

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and

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E. M. E. Trace

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Applied.Math.Phys., 3_5

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H. P.

W.

Eigenvaluesof theLaplacianforrectilinearregions,

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Austral, Math.

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E. M. E.

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E.

M. E.

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Applied

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hearingthe

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Math. Soc,

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472-483. *Presentaddress: Mathematics

Department

Faculty

of Science Universityof Emirates

P.O. Box

15551