THE NEGATIVE LAPLACIAN FOR GENERAL DOUBLY-CONNECTED BOUNDED DOMAINS

Internat. J. Math. & Math. Sci.

VOL. 18 NO. 2 (1995) 245-254

ON THE SPECTRUM

OF

THE NEGATIVE LAPLACIAN FOR GENERAL DOUBLY-CONNECTED BOUNDED DOMAINS

245

E.M. E.ZAYEDandA. I. YOUNIS Mathematics

Department,

Facultyof Science

ZagazigUniversity, Zagazig,

Egypt

(Received June 17,1993and inrevisedformJune6,

1994)

ABSTRACT.

Thispaperisdevotedtoasymptotic formulas forfunctionsrelatedwith thespectrum of thestandardLaplace operatorintwoand three dimensionalboundeddoubly connecteddomains withimpedance boundaryconditions, where theimpedancesareassumedtobepositivefunctions.

Moreover,

asymptotic expressions forthe differenceof eigenvaluesrelatedtoimpedance boundary valueproblemswithdifferentimpedancesare derived. Further resultsmaybe obtained.

KEY

WORDS

AND PHRASES:

Eigenvalues of ^the negative Laplacian, doubly connected domains,impedance eigenvalue problem, asymptotic expansions ofthe heatkernel.

1991

AMS SUBJECT CLASSIFICATION CODES:

35K,35P.

1.

INTRODUCTION

Theunderlying problemistodeduce thegeometrical properties ofamembranefrom acomplete knowledgeof theeigenvalues

{,(o)}’.

^{for the}negative Laplacian-&’’- inR",n 2or 3.

Letflbeasimplyconnected bounded domain in

R"

with asmoothboundary0flin the case n 2, orasmoothbounding surface^$in the casen 3. Considertheimpedance problem

(A,,+.)u-0

in ff,

(1.1)

n

^{+ u-O on aF(or$),}

^(1.2)

where

;-

denotes differentiationalongtheinward-pointing normaltoOf

(or S),

^{and o is apositive}

function. Denoteitseigenvalues,countedaccordingtomultiplicity, by

0<l.h(o)su2(o)<...<l.t,(o)s....-,o

as ^k--,oo.

(1.3) At

thebeginning of^thiscenturytheprincipal

problem

^wasthatof investigating the asymptotic distributionof theeigenvalues

(1.3).

^{Itiswell known}

[1]

that inthecasen 2

i.t,(o)~( 4-)k

^{as k--*oo,}

^(1.4)

while in the casen 3

q(o)~ -v-k

^{as k oo,}

^(1.5)

where f andVarerespectivelythe areaand the volume of the domainf. The problem of determiningfurther information about thegeometry offl has been discussed

by

many authors,^see

246 E.M. E. ZAYED AND A. t. YOUNIS

forexamplePleijel

[2,3],

Kac

[4],

McKeanand Singer

[5],

Stewartsonand Waechter

[6],

^Waechter

[7],

^Greiner

[8],

^Smith

[9],

^Gottlieb

[10-12],

^Hsu

[13],

^Sleeman and Zayed

[14,15]

^and Zayed

[16-23],

usingtheasymptoticbehaviorofthespectralfunction

(R)(t)=

X

exp[-t,(o)] ^{as t-,0.} (1.6)

-1

Thus,if o 0

(Neumann

problem),it iswellknownthat in the casen 2

K(Q)dQ

+O(t) as 0, (1.7)

I9(t) +

8(

t)

⁺

a

⁺

a while in the case n 3

V +

12e/1/lfsf

^s

O(t)=(4t)3+

^H(QQ+

[H(Q)-N(Q)Q

+o(t

m)

^{as 0.} (1.8) Ifo

(Dirichlet

problem),it iswell known that in thecase n 2

la ^(taf

O(t) ^4nt 8(t)

a+a+ ) ^j

ⁿ

^{K2(QQ +o(t)}

^{as 0 (1.9)}

while inthecase n 3

IS[

_{+ H(QQ+}

[H2(Q)-N(Q)Q +O(t )

as 0

(1

10) O(t)

(4nt)3

^16n/

ltl

Anexamination ofthe results

(1.7)

and

(1.9)

shows that in the case n -2 the first term ofO(t) determinesthe area

lof ,

^thesecondtermdeterminesthe total

length]

^{0fl oftheboundary0fl}

and thefourthtermdeterminesthecuatureK(Q)of Ofl^atthe point

Q

^r0flwhilethesign ^{of the} secondtermdetermineswhetherwehaveaNeumannor aDirichletproblem. ^{The third}term a0in

(1.7)

^and

(1.9)

has geometric significance,e.g.,if issmooth and convex, thena0

;

^{whileif flis}

permittedtohaveafinitenumber of smoothconvexholes"h",then_a0 (1

h).

Similarly,anexaminationof the results

(1.8)

^and

(1.10)

^{shows thatinthecase n 3thefirst}

termof

O(t)

determinesthe volumeVof

,

^{the secondterm}determines thesurfacearea

IS

^ofS,

thethirdtermdeterminesthemean cuatureH(Q) ? ^{+ andthefourthtermdetermines} of the surfaceSatthepoint

Q,

^where

R

^and

R

^arethe

theGaussian cuature

N(Q)-

principalradiiof cuature,whilethesign of the secondtermof

(t)

^determineswhetherwehave aNeumannoraDirichletproblem.

We merely^notethat aspects of thequestion of

Kac,

namely,

"n

onehear the shapeofa drum?" have beendiscussedbySleeman andZayed

[14]

whenn 2 andby Zayed

[16]

^whenn 3 for problem

(1.1)-(1.2)

in the case isapositiveconstant.

Suppose

^that is ageneral doublyconnectedboundeddomainin

R ,

n 2or3consistingof asimplyconnected bounded inner domain with asmoothboundary

0

^{in thecase n 2}

(or

^a

smoothboundingsurface

S

^{in thecasen}

3)

^andasimplyconnected boundedouterdomain D with asmoothboundary

0

^{in thecase n 2}

(or

^asmoothbounding surface

S

^{in thecase n}

^3).

Considertheimpedance problem

(,+)u-0

in

, (1.11)

+ot

^{u0 on}

^{0 (or S),}

^(1.12)

and

SPECTRUM OF THE NEGATIVE LAPLACIAN FOR BOUNDED DOMAINS 247

n2+02

^{u ::0 on Og22 (or $2), (1.13)}

and

where denote differentiationsalongthe inward-pointingnormalsto Og2

(or S)

^and

0922 (or S)

respectively, inwhich theimpedances

o

^and

o

^{arepositivefunctions.}

Denoteitseigenvalues,counted accordingtomultiplicity, by

0<I(OI, O2) N(OI, O2) N...N(OI, O2)

^{N... as k}

.

^(1.14)

Theproblemofdeterminingthegeometryof aswellastheimpedances

o

^{ando2 fromacomplete}

knowledgeof the eigenvalues

(1.14)

has been discussedby Zayed

[21]

in thecase n-2 andby Zayed

[22]

in the casen 3 where

o

^and

o=

^{arepositiveconstants,usingthe}asymptotic expansion of thespectralfunction

O(t)- exp[-t(o,o2)

as t0. (1.15)

Thus in the case n 2,Zayed

[21,23]

hasshown that 4nt+

8(n/)

- ^{I I1}

7

’"

g:(o)-T i0=,1-

^{dQ+O(t) as t0}

^(1.16)

where

0

^andK(Q

), (QeO)

^arerespectively the total

length

andthe cuature of

0

^while

#=1

^andK=(Q),

^(QeO:)

^arerespectivelythe totallengthand thecuwatureof

O.

In

^{the case}n 3,Zayed

[22]

has shown that o()- V

Is, l+lsl+

(4t)

^{3n+ 16t 1}

where

S1 I,x(Q)

^andNI(Q),

( l)

^arerespeotivelythe

suac

area,meanuaiureand Oaussian eatreof thesurface

S,

while

S I,(Q)

^ad

(Q), (0 )

^arrespectivelythesurface area, manuatureand 8aussiancuatureof the

suTace S.

^Furihr

interetations

^offoulae

^(1.16)

and

(1.17)

^anbe found inZayed

[21-23].

In

Theorem 1, wegeneralize th resul

(1.16)

^and

(1.17)

^tothesewhen and

=

^arepositive

functionssatisfyingtheLipshitz ondition, byusingihexprssion

{(,, )

⁺

)-, (1.18)

where

P

isapositiveonstant.

In

Theorem 2, we showthatthisgeneralization

plays

animpontrole inestablishingamethod tostudytheasymptoticbehaviorofthe difference

(( ^{) l(a,, ,)), (1.19)}

forlargevalesof wherethe threepairsoffunctions

(, ), (al, )

^and

(, )

are distinct and satisfying the Lipschitz oondition and th summation is taken over all values of k for which

(, )

The method uses aninteresiingandimpoaaniTaubriantheorem dueioHardyand Litilewood anddeveloped byTitchmarsh

[24].

248 E.N.E. ZAYEI)AND A. I. YOUNIS

Theorems3, 4and Corollaries1-5 contain furtherresults which can be considered as agen- eralization of theresults ofTheorem 2.

2. STATEMENTAND PROOFS OF RESULTS

THEOREM 1. If the functions%(Q),QeOf

(or S)

^and%(Q),Qeo2

(or S.)

satisfy the LipschitzconditionandifPis apositiveconstant, then in the casen 2

16P³ 2

P "

+

2 ^{K?(Q [o,(Q )K,(Q

1024P,- o,

-2(Q)]}dQ+O(]

^as

P,

(2.1) while in the case n 3

8ytP^t,.2. 16t

P

⁺

24P ^’3,

^-1

128;tP -1

o,(Q)H,(Q) +--(Q)]}dQ

+O as P

.

^(2.2)

Notethat the expression

(1.18)

isjust theLaplacetransform of thefunctiontO(t)withrespectto andP>0 istheLaplacetransform parameter. With thisconnectionwededuce that formulae

(2.1)

and

(2.2)

canbeconsidered as ageneralization offormulae

(1.16)

^and

(1.17)

respectively.

THEOREM2. If the threepairsof functions(ol(Q),

o:(Q

)), (%(Q),

II(Q ))

^and((Q),

(Q

)) aredistinct andsatisfying^theLipschitzcondition, thenwededucefor that

[+ ^o(K)

in the case n 2, (2.3)

{,(., ⁾

^,(p

^)}

,(0,,09.

+o(

3a)

in the case n 3,

(2.4)

where

and

al=

Ioen[lz(Q)- ^II(Q)]dQ- Ioal[c(Q)- ^q(Q)]dQ,

bl Is2[fSz(Q ^51(Q

^)]dQ+

Is,[Ctz(Q ^ct(Q

^)]dQ

Formulae

(2.3)

^and

(2.4)

^canbeconsidered as ageneralizationof the familiar formulae of Gel’landand Levitan

[25]

for the difference oftracesoftwoSturm-Liouville operators.

Letus nowgive^theproofsof Theorems 1,2. To proveTheorem 1,weshall use theLaplace transform ofGreen’sfunctionforthe heatequation

(A, )u

^{0,n 2or3 withrespecttothe time}

t,anduseS astheLaplacetransform parameter.

PROOFOF

THEOREM

1. Withreference^to

[26,

^Sec.

2],

let 1

SPECTRUH OF THE ^NEGATI_VE I_.APIACIAN FOR B()UNI)ED DOI’.UINS 249

betheGreen’sfunction of theexpression(A

s2)u

in the domain^Q_CR togetherwith theboundary conditions

(1.12)

^and

(1.13)

^on

0f2

respectively,wheresis asufficientlylarge positiveconstant while x and

x

^{arepoints belongto}

.

^In

^{(2.5), K0}

^{is themodified Besselfunctionofthesecond}

kind andof zeroorder,while

-

^isaregular partof theGreen’sfunction.

Withreferenceto

[2],

^wededuce that as

x

^{x theequality}

(2.6)

,x ^;- s ,x

^s

;- _{o,

_o)+

_{?} {o, o}

+s

}

where

{,({)}

^{arenormalized}eigenfunctionsand^s,s,, implies

(2.7)

og +V _5,x.-s - ^5,x.-

2= s -x

{(o,,

o)+

Thus wegetthe formula

2 ^{(o, o9

⁺

}-

+

,., _4=s=

^g,

,{,-s’

^dx.

^(2.8)

Using methods similartothoseobtainedin

[14], [21], [23]

^{wecan showthat}

ff s’({.{" ^$2)

^dx

^{=1 o"l} J ^{0"2 ’-h{ fo.Oz}

^(Q

^{Q fen..}

^(Q

^Q

512S4i-I

+ 0 as s oo. (2.9)

Oninserting

(2.9)

into

(2.8)

andletting^S2=

P

we arrive at

(2.1).

Similarly,^let

- ^{x_,x;-s 4n[x_-x_ (1 ’-(x_, x_ ,;-sZ), (2.10)}

betheGreen’sfunctionof theexpression(5

sZ)u

in thedomain fa

__.

R togetherwiththeboundary conditions

(1.12)

^and

(1.13)

^on

S

^and

Sz

respectively.

Withreferenceto

[3],

wededuce that as

x

^xtheequality

(2.6)

implies

+(x_,x;-s -

^x,x;-s

=(s2-sZ),Yq {ix(o,o:)+sZ}{i.t(o,oz)+sZ} ^(2.11)

Thus weget the formula

5: ^{(..(o,. o=,}

⁺

^{s._}_._ v} lf.f(

,-

--ss+ -’ ^{x_,x_;-s dx_. (2.12)}

Using methods^{similar to}thoseobtained in

[16], [22],

we canshowthat

250 E. H. E. ZAYED AND A. I. YOUN1S

fs

[H,(Q )- 3o,(Q )]dQ

12ns

^,-

7

fs{[H,(Q)_3c,(Q)],__[N,(Q)_2,(Q)H,(Q)

64rrs

,-

On inserting

(2.13)

^into

(2.12)

^andletting

s"=

Pwearriveat

(2.2).

Finally,^{we notethat the}proofofeither

(2.9)

or

(2.13)

is omittedhere since it isverysimilar tothose obtained in

[21]

^or

[22]

respectively.

PROOFOF THEOREM2.Withreferenceto

[26],

^letusassumethat

a2(Q

^:a(Q), (Q

0)

andO2(Q)

(Q),

(Q

02)

and introducethe non-negative andnon-decreasingfunction

()

{, 2)- (a, )},

^(2.14)

(.)

moreover welet

{a(,

^z)-

a(R,)} {,)

+

2(a,)

+

3P}

(P)= .Z { ,(a, )

+

} {,,(a, )

⁺

}

^{(2. s)}

Using formula

(2.1)

^firstfor the functions

(a(Q), O(Q)),

then forthe functions((Q),

Oz(Q))

^and subtracting the secondonefromthefirst,wefindaftersomereduction that

2 + +

+0)

^{as P (2.16)}

where

az I[z(Q) (Q)][ ^Kz(

^{Q (Q z(Q}

)dQ

⁺

^f,[Rz(Q)-

^a(Q

)][ ^Kx(Q)

^a(Q)-(Q

^)]dQ

Formula

(2.16)

^{canbewrittenfor}anyc<

(, 2)

^intheequivalentform

2

(+p)+(P)+.16P

^{e+0 as P. (2.17)}

Further, notingthat

()

^o (

) (x ,

weget

d(X) a

as

P

(.la)

(

+

P)

^4p

ApplyingaTauberianTheorem ofHardyandLittlewood

(see,

forexample

[24]),

wefindthat

e(l-x

^as

^{x. (.9}

Analogously,oneestablishes theasymptoticformula

Further, notingthat

SPFCTRUH OF THE NEGATIVtz LAPI.\CIAN FOR BOUNDED ^I)OHAINS 251

where

ot(Q) min{o(Q), %(Q)},

(2.21)

[(Q min{

(’2(Q),

[, (Q)}, (Q) min{o,(Q),

et2(Q)

and thefact thatask the functions

IB(Q min{o2(Q), 32(Q)},

.k(4,e4)

X ^{[a,(c, 3;) bt,(c2, [2)}, (2.22)}

and likewise for

(a,13)

^are asymptotically equal to

gZ.,

^{we obtain}

^(2.3)

^{for the special case}

%(0 %(0), (O

⁽of)^and

132(0) 13(a), (O 02).

Similarly, we derive

(2.4)

for the special case

%(Q)aa(Q),

(Q

S)

^and (Q)^a(Q), (Q $2)asfollows: Usingformula

(2.2)

firstfor thefunctions(a(Q),

(Q)),

^then for the functions (%(Q), 2(Q

))

^and subtracting ^{the second} onefrom the first, we find forany

c<(%,

)

^that

+ +0 as P

(2.23)

2 (+

p)3

⁺

^(P)

^8np3a

where

Onusing thesame natureof

W(P),

we write theintegralin

(2.23)

^{in the}asymptotic^form

as P^m (2.24)

j, 16=e

Consequently,^wededucethat

bl

3/2

(.) -Z.

^as

^Z.

^oo.

^(2.25)

Analogously,one establishestheasymptotic formula

bx

3/2

X {k((3/,,2,12)--l,k((/,l,l)} ~--Z.

^as

^L

^o.

^(3.26)

On using

(2.21)

^andthefact thatas

.

^{oothefunctions}

^(2.22)

^for

^(,)

^{andlikewisefor}

^(a,,)

b

3

areasymptoticallyequalto weobtain

(2.4)

for thespecialcase(Q) a(Q), (Q G

Sa)

and

(Q) ,(Q),

(Q

u sz).

In

ordertoprovethe theorem in thegeneralcase itissufficient toapplytheequality

{,(m,)-,(a,,,)}- E {,,(o0,o)-,(a,,)}

X ^{{I-q(o0, o)} la(ct2,12)}, (2.27)

.(,02) X

252 E. N. E. ZAYEI) AND A. 1_. YOUNIS

%(Q)=

max{ctt(Q),(Q)}, o;(Q)-- max{[,(Q),[2(Q)},

andapplythe specialcaseof the theoremwhich wejustproved.

3. FURTHER RESULTS

COROLLARY1. Onusing formulae

(1.4)

^and

(1.5)

^wededuceasm that

,Yl {g’(ct’’[52)-la’(czl’[3)}= -

^{m +m +o(m)o(m) inin thecasethecase nn 3.2, (3.1)(3.2)}

UsingTheorem 2 weeasilyprove^thefollowingTheorems:

THEOREM3. Let the threepairs offunctions(o(Q), o2(Q)), (oh(Q),

13(Q)),

(ck2(Q),13.(Q)) andthequantity

a ,,

0be thesame asin

(2.3).

Furthermore,onthe half-axis

[c,

+)letafunction

](.)

^ofconstantsignbegivenwhich isabsolutelycontinuous oneachinterval

[c,d],d

^<o; further

xf(x)

weassumethattheexpression isbounded almosteverywhere

andf*(R)f(?,.)dZ.

^{oo.Then as}

L

^c

weget

-,,(o,.og,x

+o(1) ^{f(t)dt. (3.3)}

THEOREM 4. Letthe threepairs of functions(Ol(Q), %(Q)), (Ctl(Q),

131(Q)),

((Q), 13:(Q)) and thequantity

b ,,

0bethesame asin

(2.4).

Furthermore,onthehalf-axis

[c,

+oo)letafunction

f(Z.)

^ofconstantsign^begivenwhichisabsolutelycontinuous oneachinterval

[c,d],d

<0%further

xf(x)

we assume thatthe expression isboundedalmosteverywhereand

f[ lf(:L)d?,.

^{x,. Thenas}

X weget

-.,(,,,,,,9-

+o(1) ^It It/2f(t)dt.

^(3.4)

PROOF. Onsetting

,(o,,o2) k

where the summation is taken over all values ofk, forwhich

t,(x, o) ,

wededuce forany

<

a(ch, %)

^that

Y /’[I.t,(ox, o2)] {ta(ch, 13) t.t,(ax, Ix)} Ak(k) (3.5)

(o,o) k

On inserting

(2.3)

and

(2.4)

^into

(3.5)

^weget easily

(3.3)

^and

(3.4)

respectively.

COROLRY2. Onusing themeanvaluetheorem,^wededuceforanyc<g(%,

o)

^that

k-1

2 ^/’[g(o, o)] {g(, )

g(ax,

x)} [’(k(k), (3.6)

g(o,o) k

where

(al, )

g(o,

oz)

g(,

)

and the summation is takenoverall values ofk,for which Consequently,if

f() ,

^{>0 wededuce form} that in thecase n 2

SPECTRUM OF THE NEGATIVE LAPLACIAN FOR BOUNDED DObbINS 253

.{p(,2)-lx(cq,[5)---

^m

^+o(m’),

whileif

f(Z.)

^Z.’,i

-

^{wededuce form} that in thecase n 3

(3.7)

ib

62

^{( a}

--if-m)

^+o(mz’

^v3)

^{if >--}

(2i

+1) ²

--:_In --m +o

In

if i---.

v 2

(3.8)

COROLLARY3. Assumingthat the function

f(Z.)

ofTheorem3has the form:

f(Z.)- ZJ,

-1 then wededuceas that

al -,i+1

2(i+1) ^{" +o} 1)

^{if >-1,}

,k((]l, 02) {lLI,k((Y.2,12)

ILI,

k(C{I, I1)}

(3.9)

(o,o2) k

[

^{if =-1.}

COROLLARY4.Assumingthatthefunctionf(,)of Theorem 4 has the form

[(Z.) .i,

-3/2 wededuceforZ. that

bt _..(2//3a

+

o((2 +3)

if >-3/2

(3.10)

la,(ol, oz){la,(h,l2)_,(ctl, l) } J(2i

⁺

³⁾

0,,,(o,.o2),x

l_.ln.+o(ln.

^{if i--3/2.}

COROLLARY 5. If

t.t,(ctl, I) "

^{0wededuce form} that in the casen 2

,.q(tl,[5)=m

⁺

-

ⁿ

^-[

^{m +o In}

^-[

^m

^(3.11)

while inthecase n 3

,1,((1, I)

^"m+

b

^m1/+o(m

1/3). (3.12) REFERENCES

[1] COURANT,

^R. and

HILBERT, D., Methods.of

Mathematical Physics, Vol. 1, Wiley- Interscience,New York,1953.

[2] PLEIJEL,/1,,., A

studyofcertainGreen’sfunctionswithapplicationsinthetheoryofvibrating membranes,

Ar.k. ^Mat,

²

(1954),

553-569.

[3] ,

^{OnGreen’sfunctionsandthe}eigenvaluedistributionofthe three-dimensional membrane equation,Skand.

Mat. Konger. X.I!, (1954),

^222-240.

[4] KAC, M.,

Canonehear theshapeofa

drum?..Arner. Math.

^Monthly.,73

(1966),

^1-23.

[5] ^McKEAN,

^H.

^P.,

^Jr.and

SINGER,

I.

M.,

Curvatureandtheeigenvaluesof theLaplacian,

J.

Diff.Geom,1

(1967),

43-69.

[6] STEWARTSON,

K.and

WAECHTER,

R.

T.,

On hearingthe shapeof a drum: further results,

Proc.

^Camb.

Philos. Sot,,

⁶⁹

(1971),

^353-363.

[7] WAECHTER,

^R.

T.,

On hearing^theshapeofadrum:

An

extensiontohigherdimensions, Proc.Camb. Philos.Soc.,72

(1972),

^439-447.

[8] GREINER, P., An

asymptotic expansionfor the heatequation, Arch. Rat. Mech.Anal.,41

(1971),

163-218.

[9] SMITH, L.,

^Theasymptoticsof the heatequation for aboundaryvalueproblem,

Invent.

Math. 63

(1981),

467-493.

254 E. ,1. E. ZAYEI)ANI),\. I. YOUNIS

1()]

GOTTLIEB, H.P., Hearingtheshape of^anannulardrum, J. Austral.Math. Soc. Ser.B24 (1983),^435-438.

[11]

Eigenvalues^{of the} Laplacian withNeumann boundary conditions,J. Austral.

Math.Soc. Ser. B26

(1985),

^293-309.

12]

Eigenvalues of the Laplacian forrectilinear regions,J.Austral.Math. Soc.Set.

B29

(1988),

270-281.

13]

HSU, P.,Onthe O-functionofacompactRiemannian manifold withboundary, C. R.Acad.

Sci.Paris,309

(1989),

507-510.

[14]

^{SLEEMAN, B. D. and}

ZAYED,

E. M. E., An inverse eigenvalue problem^fora general convexdomain, J. Math. Anal.Appl.,94

(1983),

78-95.

Trace formulae for the eigenvalues of the Laplacian, J. Appl.

Math.

Ph/s.

[15]

(ZAMP), 5 (1984),

^106-115.

16] ZAYED,

E.M.E., Aninverseeigenvalueproblem^forageneralconvex domain:Anextension tohigher dimensions,J.Math.Anal.Appl. 112

(1985),

455-470.

17]

Eigenvalues^oftheLaplacian:

An

extensiontohigherdimensions,

IMA

J. Appl.

Math.,33

(1984),

^83-99.

[18] ,

Eigenvalues oftheLaplacian forthe thirdboundaryvalueproblem, J. Austral.

Math.Soc. Ser. B29

(1987),

79-87.

19]

Eigenvalues of the Laplacian forthe thirdboundaryvalueproblem: Anextension tohigher dimensions, J. Math. Anal.Appl., 130

(1988),

78-96.

[20]

^Hearingtheshape ofageneralconvexdomain, J. Math.Anal.Appl:142

(1989),

170-187.

[21]

On hearingtheshapeofanarbitrary doubly-connectedregioninR

:,

J.Austral.

Math.

^{Soc.Set. B31}

^(1990),

^472-483.

[22]

Hearing theshapeofageneral doubly-connecteddomain in

R

withimpedance boundaryconditions,J.Math.Phys.,31

(1990),

2361-2365.

[23]

^Heatequation foranarbitrary multiply-connectedregionin

R:’

withimpedance boundaryconditions,

IMA

J. Appl. Math.,45

(1990),

233-241.

[24] TITCHAMRSH,

E. C., Eigenfunction Expansions Associated withSecond Order Differ- entialEquations,Vol. 2, ClarendonPress, Oxford,1958.

[25]

GEL’FAND,I.M.and

LEVITAN,

B.

M.,

Onasimpleidentityforeigenvalues^ofasecond order differential operator, Dokl. Akad. Nauk.SSSR88

(1953),

^593-596.

[26] ZAYED,

E. M.

E.,

Someasymptoticspectralformulaefortheeigenvaluesof the Laplacian, J.Austral. Math.Soc. Set.

B

30

(1988),

220-229.