Eigenvalues of Elliptic Boundary Value Problems With an Indefinite Weight Function
Author(s): Jacqueline Fleckinger and Michel L. Lapidus
Source: Transactions of the American Mathematical Society, Vol. 295, No. 1 (May, 1986), pp.
305-324
Published by: American Mathematical Society
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TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 295, Number 1, May 1986

EIGENVALUESOF ELLIPTICBOUNDARY VALUEPROBLEMSWITH AN INDEFINITE WEIGHT FUNCTION
JACQUELINEFLECKINGER1AND MICHELL. LAPIDUS2
ABSTRACT.We consider generalselfadjointelliptic eigenvalueproblems(P)= Ar(x)u,;Atu

is positive and of order 2min an open set Q C Rk. Here, the operator ;A1
and the "weight"r is a function which changes sign in Q and is allowedto bediscontinuous. A scalar A is said to be an eigenvalueof (P) if vqu= Aru-in
the variational sensc for some nonzero u satisfying the appropriategrowthand boundaryconditions. We determinethe asymptotic behaviorof the eigenvalues of (P), under suitable assumptions. In the case when Q is bounded, we
assumed Dirichlet or Neumann boundary conditions. When Q is unbounded,
we work with operators of "Schrodingertype"; if we set ri = mgc(ir,0),
two cases appear naturally: First, if Q is of "weightedfinite measure" (i.e.,< +oo or AQ(r_)k/2m < +OO),we obtain an extension of theJ^n(r+)k/2m
well-known Weyl asymptotic formula; secondly, if Q is of "weightedinfinite
measure" (i.e., rQ(r+)k/2m = +oo or CQ(r_)k/2m = +oo), our results extend the de Wet-Mandlformula (which is classical for Schrodingeroperatorswith weight r--1). When Q is bounded, we also give lower bounds for the
eigenvaluesof the Dirichlet problemfor the Laplacian.

t. Introduction. The purposeof this paperis to provideestimateson theellipticboundaryvalueproblemsSu = Aruinof ;;rightnondefinite"eigenvalues
whentheweightfunction
conditions)boundaryQ C Rk (withDirichletorNeumann(i.e.,changessignin Q).
r is "indeSnite"
elliptic
forselfadjoint
oftheeigenvalues
Themainresultsconcerntheasymptotics
operatorsof order2m. Foran openset Q whichis not necessarilybounded,weextendthe classicalWeylformulaas wellas its analogue,alsocalledthe de Wettype. Wealsoobtainlowerboundsfor
Mandlformulaforoperatorsof Schrodingerof the Dirichletproblemfor the Laplacian.Theseresultsextend
the eigenvaluesby Lapidus[La1, La 2] andFleckinger-El
the earlieronesobtainedindependently
Fetnassi[F1F1, F1F 2].
problems
to ellipticeigenvalue
lead,afterlinearization,
problems
Manynonlinearwith an indefiniteweightfunction.(See,e.g., the surveypaperby de Figueiredo
Received by the editors June 26, 1985.
1980 MathematicsSubject Classification.Primary 35P20, 35J2(), 35J25, 35J35, 35J40,
35P15; Secondary34B25, 35J10, 35P30 47A70, 49G05.
Key wordsandphrases.Elliptic boundaryvalue problems,indefiniteweight function, asymptotic behavior of eigenvalues,lower bounds of eigenvalues,spectral theory, variationalmethods,
operatorsof Schrodingertype.
1Researchpartiallysupportedby the CentreNational de la RechercheScientifiqueand Argonne
National Laboratory.
2Supportedin part by a Grant from the Faculty Researchand InnovationFund at the University of Southern Californiaand by Argonne National Laboratory. The second author would liketo dedicate his researchin this work to Odile.
(g)1986 American Mathematical Society
0002-9947/86 $1.00 + $.25 per page
305

306

JACQUELINE FLECKINGER AND M. L. LAPIDUS[dF]andthe workof Hessand Kato [HK, H].) A vast literaturein engineering,
physicsand appliedmathematics
dealswith suchproblemsarising,for instance,in the studyof transporttheory,reaction-diffusion
equations,fluiddynamics,etc.
(See,e.g., [DGHa, GN, LuR, LuW andKKLeZ].)
Mostof the literatureon this subject studiedsincethe endof the last century(Richardson[Ri],Bocher[Bo],Hilbert[Hi],etc.) is concernedwith the one dimensionalcase,whichis stillthe objectof activeresearch.(See,forinstance,[KZ,
Be].) To our knowledge,the firstresultin the multidimensionalcase is due toHolmgren(1904)[Ho];he considers
the Dirichletproblem/\u + Ar(x,y)u = 0, on aboundedopenset set Q c R2, whenr is continuousandchangessign;he provesin
this casethe existenceof an infinitenumberof positive(andnegative)eigenvalues
whichcanbe characterized
by the "min-max
principle".The asymptoticdistributionof theseeigenvalues
hasbeenestablishedby Pleijel(1942)[Pl]forthe Dirichlet
andNeumannproblems.
Thespectraltheoryof similarabstractproblemshasbeenstudiedmorerecently
by Weinberger[Wn],and its applicationto the Dirichletproblemfor an elliptic
operatorof order2 by ManesandMicheletti[MM].
Pleijel'sresultsonthe asymptoticdistributionof eigenvalues
havebeenextended
by manySovietmathematicians
duringthe 1970s.(See,forinstance,the surveypaperby BirmanandSolomjak[BS 3], theirbook[BS 4] andthe relevantreferences
therein.) In most of these papers,the weightfunctionneednot be continuous.
Birmanand Solomjak[BS 1, BS 2], for example,obtainasymptoticestimatesfor boundedopen sets and (possiblydegenerate)selfadjointellipticoperatorsof
order2m. Rozenbljum[Ro 1] extendstheseresulsto someproblemsdefinedon
unbounded
opensets suchthat AQ[rlk/2m< +x.
Recently,unawareof Pleijel'swork,Lapidus[La 1] determined
the asymptotic
behaviorof the eigenvaluesof the Laplacian
withDirichletor Neumannboundary
conditionson a boundedopenset of Rk in the casewhenthe weightfunctionis
not necessarily
continuous;
he alsogiveslowerboundshavingthe correct"coupling
constantbehavior"forthe eigenvaluesof the Dirichletproblem.In somecases,heobtainsestimateson the remainder
term[La 2].
At aboutthe sametime, Fleckingerand El Fetnassi[FlF 1, FlF 2] proved
an analogousasymptoticestimatefor the Dirichletproblemcorresponding
to ageneralselfadjoint
ellipticoperatorof order2mwitha continuous
weightfunction;
asymptotics
werealsoobtainedforoperatorsof "Schrodinger-type"on unboundeddomainsfora sufficiently
smoothweightfunction.
Thepresentpaperextendstheseresults:forgeneralellipticproblems,
we do notassumethat the openset Q is boundedor that the weightfunctionr is continuous;
the latteris of interestin viewof possibleapplications
to nonlinearproblems.
We now indicatehow this workis organized.Afterhavingstated our main
resultsin thenextsection,wepresentin §3the prototypal
exampleof the Laplacian
on an openset of finitevolumeand givelowerboundsfor the eigenvaluesof thecorresponding
Dirichletproblem.Wethenextendthisstudyin §4to generalelliptic
boundaryvalueproblemsof order2m on an openset Q (whichis boundedforthe
Neumannproblem).Finally,in §5,wedetermine
the asymptoticdistributionof theeigenvaluesof certainSchrodinger
operatorson Q c Rk with DirichletboundaryELLIPTIC PROBLEMS WITH INDEFINITE WEIGHTS

307

andwe dealwiththe fourdifferentcases:in this case,Q is unbounded
conditions;< +x.
= +x or gQ(r+)k/2m
g (r+)k/2m
Sincethe firstversionof thispaperwassubmitted,the workof Birman-Solomjak
on boundaryvalueproblemswithindefiniteweightswasbrought
andRozenbljum
thereis someoverlapbetweenTheorem4.1 herein
to ourattention.In particular,
andTheorem2 in [BS 1 or BS 2], as wellas Theorem5.2 hereinandTheorem2
5.4,
ourresultsgo muchfurther;Theorem5.1 andProposition
in [Ro 1]. However,opensets, appearto be entirelynewas wellas the possible
statedforunbounded
resultingfromTheorems5.1 and5.2.
combinations
2. Notation and main results. Weintroducesomenotationwhichwill be
the aricle:
usedthroughout
Letm andk be integers> 1.
Let Q be an openset in Rk withboundaryAQ.
The interior(resp.,the closure)of Q C Rk is denotedby Q° (resp.,Q); if,[QIstandsfor its Lebesguek-dimensionalin addition,Q is Lebesguemeasurable,
in Rk.
measureor "volume"of R in Q.
If R c Q, the notationQ\R indicatesthe complement
functionsin
differentiable
Forj E N, Cj(Q)is the spaceof j timescontinuouslyQ.

oforderloll= ol1+ +ak: Da =ForOl = (ol1, . . ., °lk) E Nk, Da is thederivative
Alal/0xl1 aSkak; if, in addition,( = (41,..., (k) E Rk, then ( = 411 (kak.
the Sobolevspaceof functionsu E L2(Q)suchthat the distriHm(Q)odenotes
Dau are in L2(Q)for loll < m (see, e.g., [Ad]). Recallthat
butionalderivatives
Hm(Q)is a Hilbertspaceforthe norm
A/

| U| Hm(X)= tt E [Dau(x)l2dx)

1/2

CO(Q) withrespectto
is thesubspaceof Hm(Q)obtainedbycompleting
Hom(Q)
differentiable
the normof Hm(Q);here,CO(Q) denotesthe spaceof continuously
functionswithcompactsupportcontainedin Q.
functiondefinedon Q, we set f+ = max(f,O)and
Finally,if f is a real-valued
f_ = maxt-f, O)
Wecannowstateourhypothesesandpresentourmainresults:(2.1) Let A be a formallyselfadjointuniformlyellipticoperatorof order2mdefinedon Q:
A=E

Da (as (x)D ))1oel<m

1p1 <mwhereaOs = a;3,,for IOll < m and 1d1 < m; we assumethat ao > Oand when
Q is bounded(resp.,unbounded)ao E Lk/2m(Q)if k > 2m and ao E L1(Q)ifaOs E L°°(Q) if O< loll + 1d1 < 2m andk < 2m [resp.,aO E L°°(Q)]; moreover,aOs E C°(Q) if lol| + 1d1= 2mfunctionon Q;we supposethat Q+ =real-valued(2.2) Let r be a measurable{x E Q:r(x) > O}andQ_ = {x C Q:r(x) < O}areof positiveLebesguemeasure.In
withp > k/2m§§3-4(resp.,§5),wealsoassumethatr E LP(Q)[resp.,r E LlPoc(Q)]ofif k > 2m andp = 1 if k < 2m. Further,in §§3-4,Q is suchthat the imbedding(2.3)
(2.4)
(2.5)

N+(>)
N-(>)
1 Ckn2/kjX (>r(x))k/2mpA(x)dx,
llrill
L l/2(Q),1Ai

as n as> oo,A +x,-w.

JACQUELINE FLECKINGER AND M. L. LAPIDUS

308

Hm(Q)into L2(Q)is compact;this is alwaysthe caseif Q is bounded,k < 2m or
an is smoothenough(see [Ad, Theorem
6.13,p. 151]).
Westudythespectrum(i.e.,theset of eigenvaluesA)of thehomogeneous
Dirichlet or Neumannboundaryvalueproblem(in the variational
sense)
Su = Aru, in 0.

(P)

WhenQ hasfiniteLebesguemeasure,this spectrumis discreteandconsistsof adoublesequenceof realeigenvaluesof finitemultiplicity(onenonnegative
andonenegative)*- < An+l < )n <* < >2 < >1 < (°) < )21 < )\2 << An < An+l <Xwith 1Ai1 tendingto +oo as n tendsto oo (eacheigenvalue
is repeatedaccording
to multiplicity).The sameresultholds,underappropriate
assumptions,for an
operatorof "Schrodinger
type"on X withDirichletboundaryconditions.
Let N+(A)[resp.,N-(A)]be the numberof nonnegative[resp.,negative]eigenvaluesAn+lessthanA > O[resp.,;An greaterthanA < O].
Assumethat Qhas weightedfinitevolume.(Ofcourse,this assumption
is satisfiedif Q is bounded.) If IQ+\Q+I= o (thisis the case, for example,if r is
continuous),
we showforthe Dirichletproblemthat (seeTheorem4.1)
n+whereiiA(x)is the usualBrowder-Garding
densityon X,tbv(z)= (21r)-k|dt,

x E 0,{(ERk:A'(x,()<l}andJq'(x,() is the symbolof the principalpartof A:
J4/(x,() =E

aa!s(x)(a+pXfor (x, () E Qx Rk.

1oel=1(31=m

Similarly,
if IQ-\Q°-I= o,thenn_
In particular,
whenA = -1\, we have*bA(x) = (2fr)k Bk,

whereBk = 7rk/2/r(l + k/2) denotesthe volumeof the unit ball in Rk; since
N+(>n+) = n, we thenobtainan extension
of Hermann
Weyl'sasymptoticformula(seeTheorems3.1 and3.2):

whereCk= (27r)2(Bk)-2/k
denotesWeyl'sconstant.
REMARKS.1. Actually,(2.5) containstwoequations:one involvingthe minus
signsaloneanda secondone for the plussigns;we shalluse this convenientsign
convention
throughout
the paper.

(2.7)>IfOgQ(r+)k/2m
asN+(A)[xl +x.
| < (>r(x)+X, thenestimate
q(x))k/2m,uS(2.3) or (2.4)(x) holds
dx, respectively
as A ix, as A ix.

ELLIPTIC PROBLEMS WITH INDEFINITE WEIGHTS

309

2. Note that the resultsfornegativeeigenvalues
can be deduceddirectlyfromthoseforpositiveeigenvalues
by changingr into -r.
3. Forsimplicity,
we writeQAinsteadof (Q)°.
If, in addition,Q is a boundedopenset satisfyingthe segmentproperty(forexample,if AQis locallyLipschitz)andif l0QI= O,the formulae(2.3) and (2.4)
still holdforthe Neumannproblem(seeTheorem4.2).
Whenr(x) _ 1, N-(A) = O and (2.3) is the usualformula(see, e.g., [CH,
ChapterVI, §4,pp. 429-445;Br; Ga; Kc; FlMe; Me; RS, TheoremXIII.78,p.
271;etc.]).
WhenQ is unbounded,
we assumethat g is an operatorof "Schrodinger
type",
i.e., Jq= £ + q, where£ satisfies(2.1).
(2.6)Here,the "potential"
q is a nonnegative
functionon Qsuchthatq-1(x)r(x)Underothersuitablehypotheses,we showthat the spectrumof (P) is discrete.
Moreover,we provethat (seeTheorem5.1 andProposition5.4)
winnX
if JgQ(rJr)k/2m = +X, respectively;
here,Qx= {x E Q:Ar(x)> q(x)}and,uL= 11A,
by definition.
Thisestimateis well-known
whenr(x) _ 1. (See,e.g., [T, Chapter17, §§8-14,
pp. 174-185;Ro 2; Me; Fl; RS, TheoremXIII.81,p. 275;etc.].)(SeeTheorem5.2 andRemark1 followingit as wellas Proposition5.4.)
Wealsoestablishlowerboundsforthe eigenvaluesof the Dirichletproblemon
a boundeddomainQ;for instance,forthe Laplacianandfor k > 3, we have(seeTheorem3.3)(2.8)|>n | > okn2/||rA||Lk/2(Q))

where69k is a constantdependingonlyon k whichis explicitlyknown.In viewof
the extendedWeylformula(2.5),theselowerboundsexhibitthe correct"coupling
constantbehavior".
We referthe readerto Courant-Hilbert[CH, ChapterVI], Reed-Simon[RS,ChapterXIII,§§1,15]orWeinberger[Wn,Chapter3]forthe Courant-Weyl
methodof proofof Weyl'sformula.Forthe variational
theoryof ellipticboundaryvalue
problems,we pointout Lions-Magenes[LM]andAgmon[Ag]. Finally,the basic
propertiesof Sobolevspacesusedin this papercanall be foundin Adams[Ad].
3. An example: the Laplacianon an open set with finite volume. We
nowillustrateourmethodsforfindingthe asymptoticbehaviorof the eigenvalues,in the case of the Laplacianwith Dirichletor Neumannboundaryconditionson
an openset with finitevolumeor on a boundedopenset, respectively(partsAandB). In partC, we obtainlowerboundsforthe eigenvaluesof the corresponding
Dirichletproblemon a boundeddomain.
A. TheDirichletproblem.Weconsiderthe eigenvalue
problem(E)-Au = Aru, in Q;

u = O on AQ.

nx. Theprecise asymptoticbehavioris given by

310

JACQUELINE FLECKINGER AND M. L.

LAPIDUS(3.1) Q is an opensubsetof Rk with finite
Lebesguemeasure,satisfying(2.-2)
andsuchthat IQ+\Q+I= oand IQ-\Q°
I=°
Theeigenvaluesof the variational
problemwith (E) arethe complex
numbersA suchthat thereexistsa nonzerouassociatedE Ho(Q) satisfying-Au = Aru(inthe distributional
sense).
It is wellknown(see,e.g., [Ho;Pl; Wn,
Chapter3;MM; dF, ChapterI, §§1-2;etc.]) thatthe eigenvalues
AA (whentheyexist)arecharacterized
by the "max-min
principle"(3.2)\+ FnGi;wuEF7X
{1Q
( )
}
wherein denotesthe set of n-dimensional
subspacesof Ho(Q);an analogousformulaholdsforAn.
ThehypothesisIQ+I> o impliesthat AA existsfor
all n > 1 andAA ) ix as
THEOREM3.1.

Underassumptions(2.2) and(3.1), estimates(2.3)-(2.5)hold.
If r is positivein Q,we haveN-(A) = 0
this happensif andonly if
IQ-I= o). Formulae(2.3) and (2.5) which(actually,areclearlyequivalent are
when
r is positiveandcontinuous(see,e.g., [CH,Theorem14,p. 435;RS, classical
Theorem
XIII.78,
p. 271;Me, Theorem5.12,p. 188;etc.]).
Weshallneedtwolemmasin orderto derive
Theorem3.1. Thefollowingresult
isessentiallyknownandfork > 3, forinstance,canbe obtainedby combining[RS,
Theorem
XIII.80,p. 274]and [LiY,Step (iv), pp.
317-318].(Forthe casek = 1,
see[GoK,Theorem8.1, p. 309].)
LEMMA3. 1. Whenr is positivein
LP(Q) withp > k/2 if k > 3 andp
= 1 ifk< 3, (2.3) and(2.5) hold.
In the following,we shalldenoteby An+(r,Q) the nth nonnegative
eigenvalueof(E)to indicatethe dependence
on r andQ.
LEMMA3.2 (MONOTONICITY
PRINCIPLES). An+(r,0) doesnot increasewhen
ror Q increases.
REMARKS.1. Hereandthereafter,
weadoptthefollowing
convention:
wheneverwemakeuseof an eigenvalue
An+, we implicitly
assumethatitexists(equivalently,
that
the right-handsideof (3.2) or of its counterpartis finite).
2. The monotonicity
withrespectto Q is onlyvalidforDirichletproblems.
Lemma3.2 is a simpleconsequenceof (3.2)andwhenr is positive(see,e.g., [CH, Theorem3, p. 409 andTheoremis well-known
270
andWn, Theorem7.1, p. 58, Example7.2, 7, p. 411;PbS,Proposition4, p.
p. 60 andTheorem8.1, p. 62].
PROOF OF THEOREM3.1. Wefollowthe
methodconsists
in findinglowerandupperboundsfor An+ usedin [La1, La 2]: the idea
With
this aimandthe helpof the monotonicity havingthe sameasymptotics.
principles,
we reducethe problem
tothestudyof positiveweights.
It clearlysufficesto establish(2.5) for
An+ since An(r,Q)=-A+(-r,Q) andr_= (-r)+
Fix6 > 0. By Lemma3.2, we have
(3.3)

An
(r+ + 6, Q)< An(r, Q)< An+(r, Q+);ELLIPTIC PROBLEMS WITH INDEF'INITE WEIGHTS

311

this holdssincer < r+ + 6 andQ+ is an opensubsetof Q of positivemeasure,by(2.2)and (3.1).
By Lemma3.1 appliedto the positiveweightsr+ + 6 in Q andr in Q+, we see
that
(3.4)

lim n-2/kA+(r++ 6,Q) = Ck||r++ b||

and
(3.5)

lim n-2/kA+(r,Q+)=ckilRllL 1/2 (QoWeinferfromequations(3.3)to (3.5)that(3.6)

Ck ||r+ + 6^lLk/2(Q) <<lim n 2/k>+ (r,Q)

noo

lim n-2/kA+(r,Q)< CkllrllLl/2(Qo).

Now, it followsfrom the Lebesguedominatedconvergencetheoremthatr+ + b||Lk/2(Q)tendsto llrllLk/2(QO) as 6 tendsto 0; moreover,by (3.1),
|

rk/2

Q+

= |

rk/2

Q+

= |(r+)k/2

Q[Notethat by omittingthe zeroset of r, one doesnot alterthe valueof the latter
integral.]Consequently,
we obtain(2.5)by letting6 tendto zeroin (3.6). 0
REMARKS.
1. Whenr is continuous,
Q+ is openandTheorem3.1 holdsforan
arbitraryopensubsetQ of Rk withfiniteLebesguemeasure(in particular,
foranyboundedopenset).
2. WhenQ+ is a "Jordancontendedset" (roughlyspeaking,if Q+ is bounded
andwell approximatedfromwithinand withoutby a finiteunionof cubes see,e.g., [LoS, Chapter8, §§6-7;RS, p. 271; Pe, Chapter2, §§11-12]),we have
IQ+\Q+I=Osincethen,by [LoS,Proposition6.1, p. 332],g0(0+)g=O;this was
the assumptionof [La 1, Theorem1, p. 266].
B. TheNeumann
problem.
Wesupposethat(3.7)Q isa boundedopenset satisfyingthesegmentproperty(see[Ag,Definition
2.1, p. 11])suchthat t00I
=O,IQ+\Q+I=0 and IQ-\Q°
I=0.
Weconsiderthe eigenvalue
problem(Et)-Au = Aruin 0;

au/0n=O on aQ;

where0/0n denotesthe normalderivative.
The scalarA is said to be an eigenvalueof the variationalproblemassociatedwith (E') if -/\u = Aru
forsomenonzerou E H1(Q).
The corresponding
nonnegative
eigenvalues+(r,Q)are,in this case,givenby(3.2),whereHo(Q)is replacedby H1(Q).
SinceHo(Q) c H1(Q), itfollowsfromthe "max-minprinciple"
asin [CH,
Theorem5, p. 410;RS, Proposition4, p. 270 or Wn, Theorem7.1, p. 58 andExample7.4, pp. 61-62]that(3.8)+(r, Q) < A+(r,Q);

JACQUELINE FLECKINGER AND M. L. LAPIDUS

312

and,as above,the eigenvalues
aremonotonewithrespectto the weight;therefore,
for6 > O,we have
(3.9)

+ (r+ + 6,Q) < + (r,Q) < >+(r,Q).

As fortheDirichletproblem,usinganextensionof (2.3)or (2.5)whenQsatisfies(3.7)andthe weightis positive[FlMe;Me, Theorem5.12,p. 188],we obtain
THEOREM3.2.

Under
assumptions(2.1), (2.2)and(3.7), estimates(2.3)-(2.5)

hold.
Next,we commenton possibleextensionsor interpretationsof the aboveresults:
REMARKS.1. Thehypothesisconcerning
the segmentpropertyin Theorem3.2
canbe weakenedas in [Me,Condition(C'),p. 156].
2. It is noteworthy
thatnoassumption
hasbeenmadeaboutQo= {x E Q:r(x) =O}in orderto obtainTheorems3.1 and3.2.
3. It is easyto derivefromTheorems
3.1and3.2a corresponding
resultformixed
Dirichlet-Neumannboundaryconditions;naturally,the asymptoticdistributionofthe eigenvalues
remainsunchangedin this case.
4. Theconclusionsof Theorems3.1or3.2canbe interpreted
as follows:the positive(resp.,negative)eigenvaluesof (E) or (E')havethe sameasymptoticbehavior
as the eigenvaluesin the usualsense of the ellipticoperator-A/r(x) [resp.,+A/r(x)], definedin Q+ (resp.,Q°) for the corresponding
boundaryconditions.
Here,we use implicitlythe factthat ||r+||Lk/2(X)= ||r||Lk/2(X+).
[Theprecedingremarksapply,with obviouschanges,to the generalelliptic
boundaryvalueproblemsstudiedin §4.B.]
5. Physically,whenr is positive,the eigenvaluesof (E) representthe natural
frequencies
of a vibratingmembranefixedalongits boundaryaQand of massdensityequalto r as wellas of tensionunity.(See,e.g., [CH,ChaptersV andVI
or Kc].) Therefore,
in the casewhenr changessign,wemayinterpretTheorem3.1
by sayingthat the largevaluesOfAn+ (resp.,An ) aredeterminedby the "positivemass"(resp.,"negativemass") distributionof the "membrane".
EXAMPLE3.1. Wenowconsidera simpleexampleof discontinuous
weightfunction. Naturally,our assumptionswouldallowus to treat muchmoresingularweights.
Let R be an opensubsetof Rk withfinitemeasureI0ILet 0+andR be two
disjointmeasurable
subsetsof R of positivemeasuresuchthat In+\R+I= o and|Q_\Q°|=O.Wedefine(-1
if xe0,r(x) = < °if x E 0\(R_u0+),
t +1 if xe0+.
WethendeducefromTheorem3.1 that, forthe Dirichletproblem,we have
COROLLARY
3 . 1 . AX

Ck(n/lQA 1)2/k, as n > oo.

ThesameestimateholdsfortheNeumann
problemundertheadditional
assumptionsthat janI
= o andR isboundedandsatisfiesthe segmentproperty.
REMARK. The geometricfeaturesof the aboveexamplesuggestthe following
naturalextensionof the well-known
isospectralproblem(see,e.g., [Kc andY, pp.

ELLIPTIC PROBLEMS WITH INDEFINITE WEIGHTS

313

23-24]):let Q1,Q+,r1 andQ2,Q+, r2, be definedas in Example3.1. Assumethat
incountingmultiplicities,
the problem(E) [or(E')]hasthe sameset of eigenvalues,
Q1andQ2. Underwhich(generic)conditionsdoesthereexist an isometrywhich
Q2,Q+ andQ2? In otherwords,to what
sendsQ1,Q+ andQ1 onto,respectively,
extentdoesthe spectrumdeterminethe signof the weight?
Weobtainherelowerbounds(resp.,upper)
C. Lowerboundsof theeigenvalues.
boundsfor the eigenvaluesA+ (resp.,An ) of the Dirichletproblem(E) studiedin partA. Theseboundsare validfor all n and compatiblewith the asymptotic
behaviorobtainedin Theorem3.1.
domainof Rk withC2 boundary
THEOREM3.3. AssumethatQis a bounded
we havefor the> o, respectively,
AQand supposethat k > 3. Then,if IQ+I
Dirichletproblem(E) in Q

1Ai1> Skn / ||ri||Lk/2(Q),(3.10)

whereSk = (k(k - 2)/4e)(Sk_l)2/kn
of theunitspherein Rk.

Vn> 1,e = exp(1) andUk-l = kBk denotesthe areao. FixPROOF. It clearlysufficesto finda lowerboundfor A+ when IQ+I,
n > 1 and6 > 0. Sincer < r+ + 6, we derivefromLemma3.2 that
+ 6,Q.
>+(r,Q)> A+(r+(3.11)
By [LiY,Theorem2, p. 314]appliedto the positiveweightr+ + 6, we have+611-1
A+(r + 6 0)> skn2/kilr(3.12)

Combining(3.11)and (3.12),we nowobtain(3.10)by letting6 ) 0 andapplying
theorem. O
the dominatedconvergence
SinceN+(A+)= n, we deduce
COROLLARY
3.2.

(k(k-2))(A2k1N+(X)< 0\k/2 (|(r+) ) )resultof Li andYau[LiY,Theorem2, p. 314]
Wehavejust useda remarkablethe existingupperboundsforthe numberof "boundwhichimproved
significantly
operatorin R3. (See [LiY,p. 311 andLb, p. 243]forstates"of the Schrodinger
references
to previousworkson this subject.)Note, however,that the methodof
Li andYaudoesnot seema priorito applyto indefiniteweights.
=oisnotneededforTheorem3.3
that IQ+\Q+I
REMARKS.1. Theassumption
to hold.
2. We note that upperboundsfor 1A+1 can be deducedsimilarlyfromthe
withpositiveweights.
of Dirichletproblems
knownupperboundsfortheeigenvalues(r,Q)| < 1AA(r,Qa)1
Indeed,by (3.5), 1AA
3. Asis thecasewiththeresultof LiandYau,Theorem3.3extendsto a compact
by a constant
Sk mustthen be replaced
Riemannian
manifoldwith c2 boundary;dependingon the Sobolevconstantof the manifold.
ellipticoperatorsof order2;4. Theorem3.3 alsoextendsto generalselfadjoint
in this case,the newconstantoccurringin (3.10)dependson Sk andthe constant

314

JACQUELINE FLECKINGER AND M. L. LAPIDUS

of uniformellipticityof the operator.Thisis easilyseenby usinga monotonicity
argument(basedon Lemma4.2) andTheorem3.3.
5. It maybe worthpointingout that, fork > 3, Theorem3.3 providesus withI, o.
a newproofof the factthat Ai existsforall n > 1 andAi > ioo whenI0+
with [MM,Proofof Proposition3, p. 290or dF, Proofof Proposition(Compare1.11,p. 43].)
4. Elliptic boundary value problems with bounded coefficients. Vt7eellipticoperatorsof order2m.
extendthe resultsof §§3.Aand3.B to selfadjoint
moregeneralboundaryvalue
onresults
obtaintoorder
Intheory.
A. Abstracte.g., [Wn, Chapter3 and(see,theoryabstracttheexplainbriefly
we
problems,
bounded(§4.B)as wellwithoperators
toapplied
be
will
later
It
I]).
ChapterdF,
operators).
Schrodinger
case,
latter
the
in
is,
(that
coefficients(§5)
unboundedas(4.1)Let H be a realor complexHilbertspace.
(4.2) Let (V,H,a) be a variationaltriple: V is a densesubspaceof H withanda is a hermitian,boundedandcoerciveformdefinedon
imbeddingcontinuous
V (see [LM])(4.3)b is a hermitianformon V suchthat, forsomepositiveconstant6,
Ib(u,u)l< ba(u,u), Fu E V.
problemeigenvalue
Weconsiderthe followingvariationala(u, v) = Ab(u,v), Vv E V;of (Q) [oris in the spectrumof (Q)] if thereexistsa nonzero
here,A is an eigenvalue(Q)

u E V suchthat the last equationholds.
REMARK. In the exampleof §3, we considerH = L2(Q), a(u,u) = ||U||H1(Q),b(u,u) = AQrlul2and V = Ho(Q) [resp.,V = H1(Q)]for the Dirichlet[resp.,
Neumann]boundaryvalueproblemon an open set Q with finitevolume(resp.,
on a boundedopen set Q). In this manner,Theorems3.1 and 3.2 will becomeof Theorems4.1 and4.2, respectively.
corollaries
By (4.2), a inducesin V an innerproductequivalentto the originalone; by
operatorT fromtheorem,we definea boundedselfadjoint
the Rieszrepresentation(V,a(, )) to itselfbyb(u,v) = a(Tu, v), F(u, v) E V x V.
(4.4)
of (Q) if andonly if 1/S is an eigenvalueof
It followsthat A is an eigenvalue
T:Tu = (1/A)u,forsomenonzerou E V.
(4.5) Wenowassumethat T is compact.This is the case in all the problems
studiedhere;it holdsin particularif V is compactlyembeddedin H and b ison H.
continuous
It followsfromthe classicalspectraltheoryof compactselfadjointoperatorsin
Hilbertspacesthat the spectrumof (Q) consistsof a doublesequence< An+l< An<< >1 < (°) < >1 << An< An+l<

andis repeatedaccordingly).
has finitemultisplicity(eacheigenvalue(b7V) if necessary is given
whichwe writeAn+(a, V) or Ani
Whenit exists,A2nr
principle"
by the "max-min(4.6){b(u,u): a(u, u) = 1} );x,UEiFn.X+= FmE

(4.8) 1 \+ l-n2m/k t /& rk/2m(x),tsA (x) dx) , as n x.

ELLIPTIC PROBLEMS WITH INDEFINITE WEIGHTS

315

wherein iS the set of n-dimensional
subspacesof V. A similarformulaholdsfor
An sinceAn (v)-b) =-An+(V)b).
Justas before,we deducefrom(4.6)the followingabstractmonotonicity
principles:
LEMMA 4.1. If b1,b2are two hermitian
formssatisfying(4.3) and suchthatb1(u,u) < b2(u,u) for all u E V, thenAn+ (b1,V) > An+ (b2,V) .
LEMMA4.2. Letal,a2
becoercive
formssatisfying(4.2) andsuchthata1(u,u)> a2(uvu)
for all u E V; thenAn+ (a1,V) > An+ (a2) V)LEMMA4.3. If (V1,H, a) and(V2) H, a) aretwo"variational
triples"suchthat
Vl C V2 n thenA+(b,V1) > An+ (b,V2 ) B. Operatorsof order2m. In the following,we supposethat Q is a nonempty
open subsetof Rk and that r and Q satisfy (2.2); furtherthe operatorvq=
EIaI<m,lal<mD(aaD:) fulfills(2.1).
Wefirstconsiderthe Dirichletboundaryvalueproblem:vqu= Aru,u E Hom(Q).
(4.7)Weassumethat Qhasfinitevolumeandthat IQ+\Q+I= oandIQ-\Q°
I=o.

Wecannowapplythe aboveabstracttheorywithH = L2(Q),V = Hom(Q)anda(u,v) = ,/Q

E

aO:DauD:v.

lxxl<m1d1<m

It followsfromstandardresultson Sobolevspacesandfromthe uniformellipticityof A (i.e., '(x, () > clfl2m,for all ( E Rk andsomec > O)that a satisfies(4.2).
By (2.2)andthe Sobolevimbedding
theorem[Ad, Theorem5.4, p. 97],we seethat b(u,v) = AQruv satisfies(4.3).
Whenk > 2m, we alsohave,by usingthe Sobolevandthe Holderinequalities
that,foru,v E V| ruv < ||r||LP(X)||U||L9(Q)IIVIIL2*(Q))
Q
where2* = 2k/(k - 2m) and 1/p + 1/s + 1/2* = 1.
Therefore,(4.5) followsfromthe Rellich-Kondrachov
theorem[Ad, Theorem
6.2, p. 144]. Similarresultsareobtainedwhenk < 2m by use of [Ad, Theorem
5.4, CasesB andC, p. 97 andTheorem6.2, equations(4) and (6), p. 144].Muchlikein [dF, Proofof Proposition1.11,p. 43],onethenshowsthat the assumption
In+I,oimpliesthat 1Ai1 >+w as n >w.
Wenowstate
THEOREM4.1. Underhvypothveses(2.1), (2.2) and (4.7), estimates(2.3) and(2.4) holdand/\ -2m/k

PROOF. Estimate(2.3)holdswhenr is continuousandboundedawayfromzero(see [Me, Theorem5.12,p. 188]);hence,for r nonnegative,
we obtainthe same

JACQUELINE FLECKINGER AND M. L.

316

LAPIDUS

resultby replacingr by r + 6 andletting6
we thenobtain(4.8)and (2.3)exactlyas in > O. In lightof Lemmas4.1 and4.3,
the proofof Theorem3.1. Notethat|

QI

rk/2mpA=

Likewise,
we have|QI

rk/2mpA=|

Q(r+)k/2mpg

THEOREM4.2. For the Neumann
problem,if r,Q and Jqsatisfy(2.1), (2.2)
and(3.7), then(2.3), (2.4) and(4.8) hold.
PROOF. WenowconsiderV = Hm(Q);we use, in particular,
Lemma4.3 with
V1= Hom(Q)andV2= Hm(Q);andthe proofgoesthroughasin
Theorems
3.2 and4.1. [15. The Schrodingeroperator. In thissection,we givesufficientconditions
underwhicha positiveoperatorof "Schrodinger
type"andwithanindefinite
weighthasdiscrete"spectrum"in an unbounded
openset
Q c Rk andwe thendetermine
theasymptotic
distributionof its eigenvalues.
theresults
andmethodsof j4 and previousworks[Ro Wecombine,in particular,
2; Ro 3; Fl; etc.] on Schrodinger
operators
withpositiveweights.
DEFINITION5.1. Letf bea continuousfunctiondefinedon an opensubset£ ofRk.Wesay thatf satisfies(E)if thereexists
a positivenumbersobecontinuously
extendedto S = {x E Rk:dist(x,) < so} and, suchthatf can
for all E E (Oso)thereexistsr1> O suchthat,for all (x,y) E
S x S with Ix - Yl < r1,wehave
If(x)- f(Y)l < sIf(z)I
REMARKS.1. Forinstance,f(x) = (1withff > 1, satisfies(E).
2. If a positivefunctionf satisfies(E),so
does
its
inverseIn the following,
we set QP = {X E Q |X| > P} forp > O. 1/f.
(5.1) Let q be a nonnegative
functionin Llko/c2m(Q),tendingto +oo at infinity
andsatisfying(E)on QR1 n Q+, forsomeR1 > °(5.2) Let r be a functiondefinedon Q suchthat q-lr tendsto Oat infinity.
Moreover,we supposethat(a) Assumption(2.2)holdsand IQ+\Q°+I =°(b) Therestrictionof r to QR1 n Q+
satisfies(E).
(r+)k/2m = +0O(Notethat r is allowedto changesignanywhere
in Q.)REMARK. (5.2) holds,for example,if Q =Rkwith r(x) =-1 if |x| < 1 andr(x)= +1 if Ixl> 1.
(5.3) Let £ be a formallyselfadjoint,
uniformlyellipticoperatorof order2mdefinedon Q andverifying(2.1):+IXI2)a,

(C)
JQ

£=

E
Dt(a:D:);
a1<m
,'S1
<m

we
supposethat for all 1a1 = 1A1= m, aO: satisfies
assumethat thereexistsd > Osuchthat forall open (E)on Q/R1; in addition,we
subsets£ ofQ,|,
w Ial=ldl=m

aO:DauD:u> dilullHm(w))
btuE Hm(S)ELLIPTIC PROBLEMS WITH INDERINITE WEIGHTS

317

DEFINITION5.2. If £ is an opensubsetof Rk, V°(S) denotesthe completionofCO(£) withrespectto theHilbertnorm/
llullvq(w)=t||U||Hm(w)

\ 1/2
+

|

q|u|2)

ThespaceV1(S) is theset of restrictions
to S of elementsof V°(Q).
Westudythe variational
eigenvalue
problem(S)

J4U= (£ + q)u= Aru, in Q, withDirichletboundaryconditions.

In otherwords,A is an eigenvalueof (S) if thereexistsa nonzerou in V°(Q)suchthat the aboveequationholdsin the distributional
sense.
We now applythe abstracttheorydevelopedin j4.A; here, H = L2(Q)isequippedwith its usualinnerproduct(, ), V = V°(Q), b(u,u) = (ru,u) anda(u,u) = (u, u) + (qu,u), foru E V.
It followsfrom(5.1)and(5.3)that a is coerciveon V°(Q);further,the factthat
q tendsto infinityat infinityimpliesthat the imbeddingof V°(Q) into L2(Q)iscompact.Moreover,(4.5)holdsbecause,forp > O,) <I I - (zEPq ) (XQ
qlUl
6(P)II?£IIVq(QP)7
whereb(p)tendsto zeroas p > +oo;this followssince,by (5.2),r/q tendsto zeroat infinity.
Consequently,
the spectrumof (S) is discrete;moreover,the eigenvaluesof (S)
arecharacterized
by the "max-min"principle"(4.6).
Let A+(r,) [resp.,A+(r,)] denotethe nonnegative
eigenvaluesof the variationalDirichlet[resp.,Neumann]boundaryvalueproblem
JQP:4U= Aru, in S c Q;

u E V°(S) [resp.,u E V1(S)].

Set yO+()i;r,,JJ,
Jq)= EA,+(rW)<.A
1 andNi+(A;r,s,:4)= EA+(row)<S 1; oneindicatesin this waythe dependence
on the weightfunctionr, the openset S andthe
operator:4.
Notethat
No+(A;r,Q,:4)= N+ (>))
whereN+(A) standsfor the numberof nonnegative
eigenvaluesof (S) less thanA > O.

Weshalluse the subsequentlemma,whichis derivedfromLemma4.3.
LEMMA 5.1. If W1and W2 are two disjointopensets in £ c Q such thatwl UW2 = X, then

No+(A;
r,wl, A)+ No+(A;r,W2, ) < No+(>;r)) ) < N+(>;r)) )< N+ (A;r,wl, A)+ N+ (A;r,W2, )Thisis duein particular
to the inclusionsVq°(w1) d3Vq°(w2) c Vq°(w) and V1(w) c V1(w1) d3V1(£2 ) £318

JACQUELINE FLECKINGER AND M. L.

LAPIDUSLet us considera coveringof Rk by acountablefamilyof disjointopencubes
withcenterxf andsidesof length:
Rk= U Q¢.

{Q¢}¢eZk¢Zk

Let R > R1. Set B = {x E Q:Ixl < R} andG

= QR= {X E Q:Ixl > R}; write
G+ = {x E G:r(x) > O} and G_ = {X E G:
r(x) < O}.
Fix A > O;let Ga = {x E G:Ar(x)- q(x) > O}.
Wenotethat G+ andGa areopen;moreover,
we have
Ga c G+ and dist(G>,G\G+)> O;
thisfollowssincer > ey/Son Gaandr,q areon G+;hereandthereafter,
wechooseR1 (andR > R1) SOlargethat q >continuous
eyon QR1forsomepositiveconstantey;this is possiblein viewof (5.1). Since,by (5.2),
GA,is bounded,the sets I andJ
thatwe nowintroducearefinite.
Put I = {¢ E zk Qf c G>} and J = {¢ E zk
Qf n Ga 7&0}. Wechoose71so
smallthat (U/;E
JQf) c G+.
Wefurthermakesometechnicalassumptions
on r, q and GA,(recallthat r is
positive
on G>):
z

(5.5)

lim

'
E

z

(r¢)k/2m) E

%¢E
J\I

' -1

(r )k/2m)

k¢EI

= O

where
rf = r(x¢)for¢ E zk.
(5.6) Finally,we supposethat thereexist >'> Oand c > Osuchthat Ga is
Jordancontended(see [LoS, Chapter8])and [GA]< c[G>/2],
VA > >') where[GA]= tGArk/2m.Intuitively,(5.5)meansthatdGais of zeromeasurewith
respect
tor(x)dx;moreover,hypothesis(5.6)is of Tauberian
type.
Wecannowstate
THEOREM5.1. Underhypotheses(5.1)-(5.3)and (5.5)-(5.6),the spectrumof(S)
is discreteand
estimate(2.7) holds:N+(>)/ [(Ar-q)+]k/2mU)
asA+oo,n+where
Q+ = {x E Q:r(x) > O}.
The ideaof the proofconsistsin breakingRk
intotheestimatesso obtainedwith the helpof Lemma5.1.disjointpiecesand linking
Insidethe boundedopen
setB, we applythe resultsof j4.B, wherer
is allowedto be "singular";outsideB)
wereducethe problemto Ga on whichr is"smooth"andpositiveanda methodsimilarto that of [Fl]canbe used.
ToproveTheorem5.1,we shallneed
the-following
results:
PROPOSITION5.1. Thereexist two positive
constantsc' andall
A sufWciently
large,

C/Ak/2m[G>]< W(#A)< CAk/2m [G>] X
where(A) = AQ+[(Ar- q)+]k/2m,US .

c" such that, forThe
Thisresult
follows
noweasilyfollows
fromhypothesis
by letting(5.5);£ 0; here
observe
andthat
thereafter,
E Oimplies£ denotesthat 71 )the°*319

ELLIPTIC PROBLEMS WITH INDEFINITE WEIGHTS

PROOF OF PROPOSITION5.1. It followsfrom(5.1)-(5.3)that(p()i)= |(>r_ q)k/2m8,£ < c"|G>(,\r)k/2m
G>,

andfrom(5.6)that(Ar _ q)k/2mp: > c' |(>r_ q)k/2mp,/ > ||G,(Ar)k/2m;G,GA/2

notethat r-r+ and)r - q = ()ir-q)+ on G) 2 Ga/2. 0
5.1 holdsfor No+(A;r,Q,A1),thenit holdsfor
PROPOSITION5 .2 . If Theorem£' is theprincipalpartof £.
and
+
q
£'
'
=
where
Q,
J4),
r,
No+(A;
of interpolation
PROOF OF PROPOSITION5.2. This is a simpleconsequence
theoremsandof Lemma4.2 if we notice-as in [Ro 2, Lemma1.1,p. 353] that
+

(p((l

<

s)A)

(1

+

(p(A))

Cl06)

forsomepositiveconstantc1.
variableusedin Definition5.1. 0
PROPOSITION5. 3.

9-1 (A)E W+(>,f) = 1
limO
J
p 1(A)E (R (>,f)- lim limO

> lim

I

where
f

IQ|
[(1i: s)Arm-q¢]k/2m,u
and
Qf = q(x)( , f)U -(21r)-kJd(.
{&,ER Lial-ldl=maCtR(l:f)((>+d<l}

PROOF OF THEOREM5.1. In light of Proposition5.2, it sufficesto establish

Theorem5.1 for JA'.
Now,we makeuse of the covering(Q¢)¢zk
(5.7)

E

of Rk andof Lemma5.1:

No+(A;r,Q¢,/) < No+(>;r,Q,A )I< N1+(A;r,B, At)+ E N1 (A;rXQf) A )J+N1+(A;r,D,A')+N1+(A;r,G\G+,/),
Q)
whereD = G+\(U¢EJ
Next,weremarkthatN1+(A;r,G\G+, ') = Obecauseon G\G+, r is nonpositive
N+(A;r,D, ') = O;theMoreover,and,by (4.6),thereareno positiveeigenvalues.

(5.8) N+ (>; r,N+(>)
B, A')| (>r)k/2m,u/:)| [(Ar _ q)+]k/2m,uz,
as A +oo.as A +oo,

JACQUELINE FLECKINGER AND M. L.

320

LAPIDUSlatterequalityholdssinceD c Ua,whereU) = {xE G+:Ar(x)- q(x)(AUz
qlul2/
JUA
rlul2)> > and,by (4.6),N+(A;r,U>,') = O.
By Theorems4.1 and4.2, we have

< 0};

hence

B+

whereB+ = {x E B:r(x) > O}andi = 0 or 1
accordingto theboundary
conditions;in fact, by (2.2), we may chooseR big
enoughso that IB+] > 0; moreover,weobservethat, becauseB is bounded,
X

B+

X

(>r)k/2mp,g,,

B+[(Ar _ q)+]k/2m,tb

asS

> +oo

Itfollows,in particular,that p-l(A)Ni(A;r,B, ') > Oas A > +oo.
Thanksto (5.7),weonlyhaveto workon U¢EJ
QF,
wherer is positivelybounded
frombelowby A-leyand satisfies(g). On eachcube QF,
wheref E J, we cancompareJk'witha homogeneous
operatorwithconstantcoefficients;
precisely,for
i= 0 or 1, it followsfromcondition(E)and
fromLemmas4.1 and4.2 that, fornsmallenough,(5.9)

Ni ([(1-6)>-q¢/r¢];r¢,Q¢, ¢) < Ni+(>;rQsQ
24)< Ni+([(l+ 6)>-q¢/r¢];r¢,Q¢,:¢)
where£.¢= EI<xl=ldl=ma^,:(x¢)D+: and,as above,r<;= r(x,¢);furthermore,
weknow
by a specialcaseof [Me, Theorem5.1, p. 175
and Theorem5.12,p. 188]
that
thereexistsC independentof ¢ suchthat(5.10) INi+([(1i s)S - q¢/r¢];r¢,Q¢,£¢)-H¢IQ¢l[(l+ 6)>rf-q¢]
I<

C (Arf) (k-1) /2m

In light of Lemma5.1, Propositions5.1-5.3
andconclude
the proofof Theorem5.1 by choosing£ = equations(5.7)-(5.10),we>-1/2(2m-1) and by letting
A > +oo.
O
Thenextstatementis reallya corollaryof theisan extensionof Weyl'sformula(Theorem4.1).proofof Theorem5.1 althoughit
THEOREM5 . 2. If in assumption(5.2)(c)we suprpose
insteadthatJQ|r|k/2m isfinite,we mustreplacethe conclusionof Theorem
5.1 bythatof Theorem
4.1:
n+
REMARKS.
1. Of course,analogousestimatesholdprovidedthat hypotheses(5.1), (5.2)and (5.5),(5.6)are for negativeeigenvalues
changedIt is noteworthy
thatthe valueof JQ(r_)k/2m doesnot interfereaccordingly.
withthe asymptotics
of N+(A;S,r,Q) and vice versa. Wegive an
examplebelow
whereN+(A)and
N-(A) do not havethe sameestimate(oneis given
byTheorem
5.1 andthe
otherby Theorem5.2). Wenotethat,in the case
whenJQ(r+)k/2m arebothfinite,
Theorem
5.2 (andits analoguefor N-) has beenproved
underslightlydifferent
hypotheses
by Rozenbljum[Ro 1, Theorem2, p. 246].

N (>)(21r) (Bk) |>|/X as A -oo,

ELLIPTIC PROBLEMS WITH INDEFINITE WEIGHTS

321

2. A resultsimilarto Theorems5.1 and 5.2 couldalso be obtainedby the
problems(undersuitablecompactness
samemethodsformixedDirichlet-Neumannhypotheses).
of the eigen3. Recently,Gurarie[Gu]has studiedthe asymptoticdistribution
type in Rk havinga smoothpositiveweightvaluesof operatorsof Schrodinger
operators,
function;in his work,basedin parton the theoryof pseudodifferential
speed"condition.
the analogueof condition(E)is the so-called"finitepropagation
Thefollowingresultis of interestforcertainapplications:
5.1 and5.2 remainvalidif weof Theorems
PROPOSITION
5.4. Theconclusions
thatq tendsto +oo at infinityandreplaceit bythe assumption
dropthehypothesis
thatr is positiveoutsideB.
of V intoH is continuous
Inthiscase,as in theabstractpartof 04,theimbedding(butnot compact);howeverT is compactby (5.4)sincewe still assumethat q-lr
tendsto 0 at infinity.
the followingeigenEXAMPLE
5.1. Wenowillustrateourresultsby consideringvalueproblem:aRu= (-/\ + q)u= Aru, in Rk,

(ES)
where

if Ixl<1,(-1
if Ixl= 1,
r(x) = < 0
t + 1 if Ixl> 1,
andwhereq satisfies(5.1) [e.g.,q(x)= (1 + 1Sl2)a withff > 1].
Accordingto Theorem5.1, Theorem5.2 andthe precedingRemark1, we thenhave
COROLLARY
5 . 1.

and
N+(A) (2X) kBk|

Ixl>l[(>-q(x))+]k/2dx, as A

> +oo,

whereBk denotesthe volumeof the unitballin Rk.
numerical
choiceof the sharp
NOTEADDEDIN PROOF.Becauseof anerroneous
Sobolevconstant[LY,equation(19), p. 316],the claimin [LiY,Theorem2 andCorollary2]thattheconstantfoundthereis betterthanin [Lb]is notjustified.The
boundseemsstilldueto Liebin [Lb],bestconstantforthe Cwikel-Lieb-Rozenbljum
exponentsin
as is clearlydiscussedon page475 of E. H. Lieb, On characteristic
turbulence,Comm.Math.Phys.92 (1984),473-480.In j3.C of the presentpaper,
one shouldthereforeuse [Lb, Theorem2, p. 243]ratherthan [LiY, Theorem2,
if Lk and Ck are definedas in [Lb, p. 243]and [LiY,p. 314];morespecifically,
one mustreplaceok by Ck= (Lk)-2/kin equation(3.10)of
p. 310],respectively,
Theorem3.3 andsubstituteLk-1Ni(A)for the left-handside of the inequalityin
Corollary
3.2.
Wewishto thankProfessorElliottH. Liebforbringingthesefactsto ourattention.

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JACQUELINE FLECKINGER AND M. L. LAPIDUSACKNOWLEDGEMENT. We wish to thank Argonne National Laboratory which
made this collaboration possible. Jacqueline Fleckinger is grateful to the Universityof Southern California for its hospitality and to the organizers of the International
Conference at Pan Am University (Texas). Michel L. Lapidus is indebted to the
Mathematical Sciences Research Institute at Berkeley of which he was a member
while part of this paper was completed and to the Institute for Mathematics andits Applications at Minneapolis for its hospitality in November 1984.
Both authors are thankful to E. Brian Davies for his remarks and to Didier
Robert and Barry Simon for some references.
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JACQUELINE FLECKINGER AND M. L. LAPIDUS[We] H. Weyl, Das asymptotischeVerteilungsgesetz
der EigenwertelinearerpartiellerDifferentialgleichungen,Math. Ann. 71 (1911), 441-469.
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DEPARTEMENT DE MATHEMATIQuES 1R2, UNIVERSITE PAUL SABATIER, 118 ROUTE DE
NARBONNE, 31062 TOULOUSE, CEDEX, FRANCE (Current address of Jacqueline Fleckinger)
MATHEMATICALSCIENCES RESEARCH INSTITUTE, BERKELEY, CALIFORNIA 94720

Currentaddress(M. L. Lapidus): Department of Mathematics, DRB 306, University ofSouthern California, Los Angeles, California 90089- 1113