ARBITRARY MULTIPLY CONNECTED DOMAINS

Internat. J. Math. & Math. Sci.

VOL. 19 NO. 3 (1996) 581-586

581

EIGENVALUES OF THE NEGATIVE LAPLACIAN

FOR

ARBITRARY MULTIPLY CONNECTED DOMAINS

E. M. E.ZAYED

Mathematics

Department Facultyof Science Zagazig University Zagazig,EGYPT

(Received

^{October 6, 1994}and in revisedform

January

17,

1995)

ABSTRACT. The purposeof thispaperis to derive someinterestingasymptotic formulae for spectra of arbitrary multiply connected bounded ^{domains in two or} three dimensions, linked with

’variation

ofpositive distinct functions entering the boundary conditions, using the spectral function

’ ^{{/k (G1, oz,) + p}-2}

^as

^P -

^oo. Further results may be obtained.

k=l

KEY

WORDS AND PHRASES. Inverse problem, negative Laplacian,eigenvalues,spectralfunctions.

1991AMSSUBJECT CLASSIFICATION CODES. 35K,35P.

1. INTRODUCTION.

Theunderlying inverse eigenvalue problem

(1.1)-(1.2)

has been discussed recently byZayed

[1]

and has shown thatsomegeometricquantitiesassociated withabounded domain can be found from a complete knowledge oftheeigenvalues

{#k()}__

^forthenegative Laplacian

A, (b-,)

²ⁱⁿ

i=1

R n(n=2or3).

Letft beasimply connected bounded domainⁱⁿ

R"

with asmoothboundaryOft in thecase n 2

(or

^{a smooth}boundingsurfaceSinthecasen

3).

Consider theimpedance problem

-AnU=AU

^{in ft}

^(1.1)

+

^{a u 0 on}

^{On (or S), (1.2)}

where

-

denotes differentiation along the inward pointing normal to Oft or

S,

and cr is a positive function.

Denoteitseigenvalues, counted accordingtomultiplicity,by

0

< #l(U) _</(u) _< _< #(u) _<

--*oo as k oo.

(1.3)

Itiswellknown

[2]

that in thecase

as k_{--* oo}

(1.4)

while in thecaser 3

lzk(a

^{k as k}

,

(1.5) where

Ill[

^andVarerespectively theareaandthe volume of f.

Thepurposeofthispaperis todiscussthefollowingmoregeneral inverseproblem: Let flbe

bitr

^{multiplycoectedboundeddomain in}

^R"(n

^{2 or}

³⁾

^{wch is}suounded internallyby simply coected bound domns

,

^with smooth undes

O,

inthe casen 2

(or

smooth bounding surfaces

S,

inthecase n 3)where 1,2, m- 1, deemly bya simply^co,cotedbound domn

fl

^thasmooth

bounda 0

^{inthe casen 2}

(or

^asmooth bounding surface

S

^inthe

case n 3).

Suppose

that the eigenvues

0

< (a,...,a) (a,...,a) (a,...,a)

^{as k}

(1.6)

c

o

exactlyfor the impedanceproblem

u

^{Au in}

, (1.7)

( ⁺

^{a, u 0 on}

^{0fi (or}

^{Si) (1.$)}

where

,

^denotedifferentiationsalongthe inwardpointing normalstoOff,or

Si

resptively md positivenetions

(i

1,

m).

InTheorem 2.1, we detene some geometricqutitiesasiatedth themultiply

roman

flfromthe complete owlge oftheeigenvues

(1.6)

fortheproblem

(1.7)-(1.8)

usingthe asptotie expsionof the

setr

^netion

1

k(a], am) + p]2

^{as P}

, (1.9)

where

P

isapositivenstt,wlei epositivenionsdefin on

Ofli

^or

Si (i

1,

m)

^d tisng the Lipsctzndition.

InThrem 2.2, weshow that theasptotic expsion of

(1.9)

as

P

plays impot roleinestablisgamethodtostudythe asptotic behaor of the difference

(,..., m) ( ,flm)] A , (I.lO)

0<(a a)SA

where

a(Q), a(Q), B,(Q)

Q Off, (or

Q S), (i

¹

m)

e, generly

spg,

sfinct nions d tisng the Lipm condition d thesuation is

ten

over

vues

of k forwch

k

(a, a) ^A.

Notethatthrems d

roes

of

ts

par

cont

her results

s

to thorn obtn rntly by

Zay

d

Yours [2].

2. STAMENT OF SULTS

Usingmethodssiltothorn obtnedin

[1

],

[2],

wecsily prove the follong theorems:

EOM2.1. If the nions

a,(Q), Q

Off,

(or Q S,), (i

1,

m) tis

^theLipm

ndition, theninthe casen 2

EIGENVALUESOF THE NEGATIVE LAPLACIAN 583

+

^m)+ a,(Q)dQ 2 a,(QldQ

21

K"(Q)- T

+

_{1024P.._,}

.

-2aT(Q)]

dQ+O as

P-oe,

(21)

where L, and

K(Q) (z

1,...,m) ^arerespectively thetotal lengths and thecurvatures ofOft, atthe pointQ,while nthe casen 3 wededuce that

(22) where

S I, H

^{(Q) and}

N ^(Q)

^are respectively the surface areas, mean curvatures and Gaussian curvaturesofthebounding surfaces

,.q _(i

1,..,

m)

Formulae (2 1) and (2 2) canbeconsidered as ageneralization of the formula

(2

3) obtained by Zayed[3]and theformula(2 3)obtainedbyZayed [4]respectively

THEOREM 2.2. If the functions

a(Q), a(Q), /3(Q),

QEOf

2

(or Q E

S)

(i 1,...,m) are distinct andsatisfytheLipschitzcondition,then wededuce for

A

^cxzthat

E ^[#k(Ol’ OGn)- k(fll

,m)]--[ j/I[ ^{3- /’’ + (1)]}

^{inthe case n 2, (23)}

,,(o

....

^)<_

^{+ o(1) A}

^{3/2 inthe case n 3,}

⁽²

⁴⁾

where

and

bl E [a,(q)-13,(q)]dQ

3. FURTHER RESULTS

COROLLARY3.1. Using formulae(14)and(1.5),wededuceasM cothat

Using Theorem 2.2,weeasilyprovethe following theorems:

inthe case n 2, (3.1) inthecase n 3,

(3.2)

THEOREM 3.1. Let the functions

a,(Q), a,(Q),/,(Q), Q

E

Of (i=

1

,m)

and the quantitya 0 be thesame asin

(2.3)

Furthermore, onthe half-axis

[c, + ^co)

^letafunction

f(A)

^of

constantsign be givenwhich isabsolutelycontinuouson each interval

[c, d],

^d

<

co; furtherwe assume thattheexpression

Af’(A)/f(A)

^isbounded almosteverywhere. Then

(i) If

f+oo ^f(A)dA _E

^co,we

f[l’Zk(tTl’""O’m)]{#k(Otl’""Otm)

^deducefor

^A -

^cothat

k([l’ [m)}

0<k(o

Iflrn ^{f (A)}

^{co, wededuce for}

^A

^cothat

(3.3)

[a,

{f[#k(al a,)] f[Pk(l,’",/,)]} + 0(1) f(A).

(3.4)

0<#,(o

TIIEOREM3.2. Letthefunctions

a,(Q), a,(Q), fl,(Q), Q

E

S, (i 1,...,m)

^{and the}quantity

bl :

^{0bethesame asin}

^(2.4).

Furthermore,onthe half-axis

[c, + co)

^letafunction

f(A)

^ofconstant sign be givenwhich is

absolutely

continuous oneachinterval

[c, d],

^d

<

co; furtherwe assumethatthe expression

Af’(A)/f(A)

^isboundedalmosteverywhere. Then

(i) If

fe

⁺

^A1/2f(A)dA

^{co, wededuce for}

^A --

^oothat

O<,(o

f[]’Zk(O’l,’",O’m)]{gk(Otl,’",Om)

_--/k

(11, m)}

+0(1) Itll/zf(t)dt. (3.5)

If

f+oo ^AV2f,(A)dA

^{co, wededuce for}

^A -

^oothat

O<#k(o

{ftk(Otl, Otm) fzk(/l, im)])

[/71 ^{+ o(1) Itl (t)dt. (3.6)}

COROLLARY3.2. Assuming^that

f(A)

^ofTheorem3.1(i)has theform

f(A) M,

(j

> 1)

(ii)

thenwededuceas

A

cothat

27r

( ^[I 1) + o(1)

/

if j

>

1,

(3.7)

if j= 1.

(3.8)

EIGENVALUESOF THE NEGATIVE LAPLACIAN 585

COROLLARY3.3. Assuming that

f(A)

ofTheorem32(i)has theform

f(A)

^M,

(3 _>

7,) then wededuce asA 00that

b +o

³

.e(27+ 3) (1)

^A

’

^{if 3}

^> ,

+o(1)

lnA if ₃

(3 9) (3 10) COROLLARY3.4. Assumingthat

f(A)

ofTheorem3 l(ii) has the form

f(X)

^M,

(3 >

0)then wededuce asA 00that

E [/-/’(Otl,-..,tm) P:(/I, ...,/m)] [a,

^q-

0(1) ^A’

(3 11) COROLLARY3.5. Assuming that

f(A)

of Theorem 32(ii)has the form

f(X)

^M,

(3 >

:) then wededuceasA 00that

O<m(o:,

[#(tl, tm)- ]:(1,

...,/Sin)

ab,

7r2 (2j

1) + o(1)

if j> ¹

2’

if j= 1 2

(3 12) (3.13) COROLLARY 3.6. If

/k(/51, ...,/,):/:

0 we deduce from Corollaries 3 and 3 3 that as

o(1) In M

if n 2,

+o(1) M1/a

if

n=a.

(3 14) (3 5) THEOREM 3.3. Let the functions a,

(Q),

a,(Q),/,

(Q)

QEOQ,,

(i

^1,

rn)

and the quantity_al

#

⁰be the sameasin

(3

1). Furthermore, onthe half-axis

[c + 00)

letafunction

f(A)

of constantsign be givenwhich isabsolutelycontinuous oneach interval

[c, d],

^d

<

00,furtherwe assume that the expression A

f’(A)/f(A)

is bounded almost everywhere and

foo

^,V_

^{f(A)d 00(.7 > O)}

Then asA 00

+ o(1)] Itla-f(t)dt.

(3 16) THEOREM3.4. Letthefunctions

a,(Q), a,(Q), /3(Q) Q

S,,

(i

^1,

m)

and the quantity

b #

^{0be thesame asin}(3.12) Furthermore,on thehalf-axis

[c, + 00)

let a function

f(A)

ofconstant sign be givenwhich isabsolutelycontinuous oneachinterval

[c, d],

d

<

00,furtherwe assumethat the

expression

Af’(A)/f(A)

^{isboundedalmost}everywhereand

f+oo

^Aj_

^/(A)dA

^oo(jE

^[).

^Thenas

r

^j+

^o(1) Itl’-1/2f(t)dt.

COROLLARY

3.7. Assumingthat

f(X)

ofTheorem 3.3 hasthe form

f(A) At,

where ris a realnumber. Thenas

A -

^ooweget _I

O<(o o)

[2r(r +

^j)

+ o(1)

^{if r}

+

^j

^>

^O,

-j+o(1)

^{lnA if r+j=O.}

(3.18) (3.19)

COROLLARY 3.8. Assuming^that

f(A)

^{of Theorem 3.4}has the form

f(A) ^A r,

whererisa realnumber. Thenas

A -- _v.

^ooweget

#((Yl,"’,(Tra) _{ ftc(l,"’,ra) 1(1 ,Jm)

O<p(o

1 jbl

+ ^{o(1) A}

^{+a+/2 if r}

+

^j

>

r(1 + 2’7 +

^2j)

^2’

b ]

¹

j+o(1)

^{ln, if r+j--}

_2"

(3.20) (3.21)

COROLLARY 3.9.

M

^--,oo

If

#k(l,.-.,fl,)#

0 we deduce from Corollaries 3.7 and 3.8 that as

(3.22) (3.23)

ACKNOWLEDGMENT. The authorexpresseshisgratefulthankstothe refereeforsomeinteresting suggestionsandcomments.

[11

[21 [3]

[4]

REFERENCES

ZAYED, EM.E,

Some asymptotic spectral formulae for the eigenvalues of the Laplacian,

J.

Austral. Math.^Soy. Set.

B,

30(1988),^220-229.

ZAYED, E.M.E.

and

YOUNIS, A.I.,

On thespectrumofthenegative Laplacian for general doubly connectedboundeddomains,Internat. J.Math.&Math. Sci. 18

(1995),

^245-254.

ZAYED, E.M.E.,

Heat equation foranarbitrary multiply connected regionin

R

² withimpedance boundaryconditions,/M

J. AppL

Math.,45

(1990),

^233-241.

ZAYED, E.M.E., An

inverse eigenvalue problem for an arbitrary multiply connected bounded domain in

R

³withimpedance boundaryconditions,

SIAMJ.

Appl. Math.,52

(1992),

^725-729.