Indefinite Elliptic Boundary Value Problems on Irregular Domains
Author(s): Jacqueline Fleckinger and Michel L. Lapidus
Source: Proceedings of the American Mathematical Society, Vol. 123, No. 2 (Feb., 1995), pp.
513-526
Published by: American Mathematical Society
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PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 123, Number 2, February 1995

INDEFINITE ELLIPTIC BOUNDARY VALUE PROBLEMSON IRREGULARDOMAINS
JACQUELINEFLECKINGERAND MICHELL. LAPIDUS(Communicatedby Palle E. T. Jorgensen)
ABSTRACT. We establish estimates for the remainderterm of the asymptoticsof the Dirichlet or Neumann eigenvalueproblem-Au(x) = Ar(x)u(x), x e Q2c Rndefinedon the bounded open set Q2C Rn; here, the "weight" r is a real-valued
function on Q2which is allowed to changesign in Q2and the boundary A2 isirregular.We even obtain errorestimates when the boundaryis "fractal".
These results-which extend earlier work of the authors [particularly,
J. Fleckinger& M. L. Lapidus, Arch. Rational Mech. Anal. 98 (1987), 329356; M. L. Lapidus, Trans. Amer. Math. Soc. 325 (1991), 465-529]-arealreadyof interest in the special case of positive weights.

1. INTRODUCTION

In this paper, we study the influence of the irregularityof the weight r andof the boundary aQ on the asymptotics of the eigenvalues for the following
boundary value problem (in its variational formulation):(P)

in Q c R,-Au =Aruwith Dirichlet or Neumann homogeneous boundary conditions.
Here, Q is a bounded open set and r is a real-valued function on Q thatis allowed to change sign in Q (in which case it is called an "indefiniteweightfunction") and may be discontinuous. Under suitable assumptions on r and
Q ([P1], [BS], [FF], [LI], [FL1], ... ), there exists a countable sequence of
positive [resp., negative] eigenvaluestending to +oo [resp., -o ]; furthermore,
Received by the editors May 19, 1993.
1991 Mathematics Subject Classification.Primary 35J05, 35J20, 35P20; Secondary 28A75,
31B20, 31B35, 41A46, 47A70.
Key wordsand phrases. Dirichlet and Neumann Laplacians,indefinite elliptic problems,variational boundaryvalue problems on irregulardomains, indefinite weight functions, asymptoticsof
eigenvalues,remainderestimate, interplaybetween the irregularityof the weight and of the boundary, Whitney-typecoverings,"fractal"boundaries,vibrations of drums with fractal boundaryand
with variablemass density.
The second author'sresearchwas partiallysupportedby the National Science FoundationunderGrants DMS-8904389 (transferredto the University of California as NSF Grant DMS-9196085)
and DMS-9207098, as well as by the M. G. Michael Award for Excellence in Researchfrom theFranklinCollege of Arts and Sciences of the University of Georgia, and by Yale University.
(? 1994 American Mathematical Society
0002-9939/94 $1.00 + $.25 per page
513

514

JACQUELINEFLECKINGERAND M. L. LAPIDUS

the number of positive eigenvaluesless than A for the above problem, denotedby Nj?(i; r, n) (where i = 0 [resp., i = 1 ] corresponds to the Dirichlet [resp.,

Neumann] problem) satisfies(1.1) N(A;r,

Q)

W(;

r+,Q) :=(27r) ni7 n,/2

r'nl2

asA.

+oo ,

where r+ = max (r, 0) and Rn is the volume of the unit ball in Rn.
Of course, we also introduce N7 (i; r, Q), the number of negative eigenvalues largerthan A < 0.
We study here the asymptotics of the "remainderterm":(1.2)
R-(A; r, Q) := NL(A; r, Q)- W(A; r?, Q)
and extend (in the special case of the Dirichlet and Neumann Laplacians)earlier
results of the authors. In particular, we obtain estimates for the "remainder
term" of (P) valid when aQ is very irregularand even "fractal". We thereby
obtain-as was suggested in [L2, Remark 3.5(c), p. 490]-the counterpartof[FL2, Theorem 2, p. 339] and [L2, Theorems 2.3 and 4.1, pp. 482-483 andpp. 510-511] for the Dirichlet and Neumann Laplacians with an "indefinite
weightfunction" and on an open set with "roughboundary".In the former case[FL2], the weight function r is allowed to be indefinite and discontinuous but
the boundary aQ is assumed to be (relatively) regular, whereas in the lattercase [L2], the boundary is allowed to be "fractal"but r _1 .
Our results show, in particular,that there is an interesting interplaybetween
the singularities (or the oscillations) of the weight function r and the irregularities of the boundary aQ. (See esp. Theorem 2 and Remarks 3 and 4 below.)
They are new and of (mathematical or physical) interest even when the weightfunction r is positive. They enable one, for example, to study the vibrationsof "drums with fractal boundary" ([L2-4], [LF1,2]) and variablemass density.
Other significantphysical applications include the study of flows throughporous
media and of the vibrations of crackedbodies.
"Indefinite elliptic problems" (i.e., involving an indefinite weight function)
occur naturally by linearization of many semilinear elliptic equations and areof broad interest in applied mathematics, engineering, physics, and biology;
for example, transport theory, hydrodynamics,crystal coloration, laser theory,
reaction diffusion equations, ... . (See, e.g., [FL1,2] and the relevantreferences
therein.) Recent mathematical works on these problems include [Be], [BS],[Fal-3], [FF], [FL1-3], [Hel-2], [HeKa], [KKZ], [KZ], [L1], [We].
The rest of this paper is organized as follows:
In Section 2, we consider the Dirichlet problem studied in [FL2, Section 3].
There we assume only that Q4, the interior of Q? := {x E Q/ r(x) ; 0}, is"Jordancontented" (without the hypothesis of "segmentproperty")and recoverand strengthenthe asymptotic estimate of [FL2, Theorem 2, p. 339].
In Section 3, by use of "Whitney-typecoverings" (with dyadic cubes whichbecome smaller near the boundary),we obtain a result for the Dirichlet problem
without any condition on Q. For the Neumann problem, we recover Theorem
2 of [FL2] (where an hypothesis was not made precise enough, as mentioned in[L2, Remark 3.5(d), p. 490] and [FL3], under some of the same conditions on
Q as in [FM, Mtl,2] where r 1.
The case when aQ is "fractal"is also studied in Section 3 and hence, inthe special case of the Dirichlet and Neumann Laplacians, this paper extends

INDEFINITEELLIPTICPROBLEMSON IRREGULARDOMAINS

515

to indefinite weights some of the main results of [L2,3], esp. [L2, Theorems
2.1 and 4.1, pp. 479 and 510-511], where r =1. (See also [LF1,2] where a
special case of this result is announced for the Dirichlet Laplacian.) The proofof our main result (Theorem 2 from Section 3) combines techniques from both[FL2] and [L2] (as well as [Mt2]) in order to deal with the oscillations and/or
discontinuities of the (possibly) indefinite weight function, as well as with theroughness of the boundary.
2. THE DIRICHLETPROBLEM
ON A "JORDANCONTENTED
SET"
2.A. Hypotheses and results. (H1) Q is a nonempty bounded open set in R'
which is "Jordancontented";the weight function r belongs to L??(Q) and isallowed to change sign; furthermore IQ+I > 0, where I 1 denotes the Lebesgue
measure in R' and Q+ is the interior of Q+ = {x E Rn/ r(x) > O}.
Remark 1. Recall that co c JR is said to be "Jordancontented" (in French,"quarrableau sens de Jordan")if it can be well approximatedfrom within andwithout by a finite union of dyadic cubes (see, e.g., [LoSt, Chapter 6, ??6-7] or[RS, p. 271]). In that case, its boundary aow must have zero n-dimensionalLebesgue measure.
We now consider a covering of Rn by disjoint open cubes (QW)EZnwith
side t7 (so that Rn = UEZQC)and we set(2.1a)=( U

I EZn/ QC no

I(Q?)

)
C

CEI(Q+)(2.1b)IC E Zn/ QCn Q :Ao}

J(Q+)

and

(2.1c)w=(

U
CEJ(!QO)\I(!iO)

We next introduce two hypotheses (which were made in [FL2]):(H2) Q+ is "Jordancontented" and satisfies the "fl-condition": given fi >0, there exist two positive constants co and 'i such that, for all 7 E (0, i0],#(J(n+)\I(ni+)) <- co?7lA
where #A denotes the number of elements in the finite set A.
(H3) r+ satisfies the "y-condition on Q+ ": given y > n, there exists a
positive constant cl, which does not depend on
enough and for all E I(Q+),',

such that, for all 7 small'pC(r) :=J Jr+ -(2.2a)

r11lIn/2(Q)< ci Yf,

with rC> 0 defined by(2.2b)

r,'/2

:=

1 jn

rn/2

-

F-nIIrlIn/2

Moreover, r+ can be extended to a neighborhood of Q+ to a positive bounded

function, still denoted by r+ . Then

516

JACQUELINEFLECKINGERAND M. L. LAPIDUS

Theorem 1. Under hypotheses (H1) to (H3), for all ( E[2(VE1)v := min(/J, y - n), we have

]

where

as - +oo.
Rt+(A;r,Q)=OYi+),
This result is nothing else than Theorem 2, p. 339, in [FL2] for the Dirichletproblem, but with weaker hypotheses. An analogous result could of course
be obtained for negative eigenvalues under analogous hypotheses on Q_ andr_ . The exponent n2-1 + ( depends on the regularityof the weight function r(the larger y, the "smoother" r) and of the boundary aQ (the larger fi, the"smoother" &Q9)so that for smooth data, this exponent is as close as we wantfrom I , the best possible exponent.
2.B. Proof of Theorem 1. We first note that for all p > 0,NO+(A;r, Q?+)< NO+(A;r, n) < No+(A;r+ + p, Q).
Hence, by letting p tend to zero, exactly as in [LI], [FLi], ... , Theorem 1 whenr changes sign can be derived from the case when r is positive.
Thus from now on in this section, we assume that(H4) r is positive, so that Q = Q? and No-(A;r, Q) =O;
in that case, we write in short(2.3)

Ni(A; r, Q)

Ni+(A;r, Q)

I:= I(Q+,

J:=

and
(2.4)

J(Q.+)The monotonicity of No(A;r, Q) with respect to Q allows us to consider
the problem on (Q' U Ki)Owhich is larger than Q; moreover, by use of themethod of the Dirichlet-Neumann bracketing [CH, RS, Mtl,... ], we have...

, FLI-2, L1-2,

No(A; r, Q') < No(A; r, Q) < No(A; r, (n'UK0)?) < NI(A; r, n')+N,(A;r, c()).

Hence, by subtractingthe "Weyl term" W(A; r, Q), we get(2.5)

Ao < Ro(A; r, Q) < Al + A2

where for i = 0 or 1,(2.6)

Ri(A; rC, QC)

Ai
CEI

and
A2 := N1,(; r, c) - W(A; r, Q\Q') < Ni(A; r, co).
The "interior terms" Ao and AI can be handled exactly as the term A
defined by Eq. (19.0) in [FL2], p. 342. (Note that in [FL2], the dimension isdenoted by k instead of n .) Hence, for a given A > 0 large enough, we choose(as in [FL2, Eq. (24), p. 343]):q = -a with a E (O,, ]
(2.8)
(2.7)

and we deduce that there exists c > 0 such that the counterpartof Eq. (36) in[FL2, p. 345] holds:

INDEFINITEELLIPTICPROBLEMSON IRREGULARDOMAINS

517

In the same manner, we obtain an upper bound for AI, which follows from
Eqs. (38) to (40) in [FL2, p. 346]:{AIu< c)A +'.

(2.10)For the "boundaryterm" N1 (A; r, c) which appears in (2.7), we again use
the Dirichlet-Neumann bracketing and the monotonicity of N1(i; r, c) withrespect to r:
A2 ? NiQ(; r, c) < Ni(A; M, c) = Ni(AM; 1, c)(2.11)~<E

N, (AM; I1,QC) < C(#(jJI));n,An12

CEJ\I

where M < xo is an upper bound for r+; note that M exists by hypothesis(H3). The last inequality is well known since the problem corresponding to
N1 (i; 1, QC) is the Laplacian on a cube "without weight", i.e., with r 1_ .
By use of (HI), (H2), (2.4), and (2.8), there exists c > 0 such that
(2.12)#(J\I) ? cia(nfl) with a E (0, (].

Hence, combining (2.11) with (2.12), we obtain as in [FL2, Eq. (28), p. 344]:(2.13)

A2 ? C) -afl < c)Y,provided that we choose a positive number a such that
(2.14)

I11-( --()<a?<.

Note that this choice of a is possible since by hypothesis of Theorem 1,2(v+1)--2

Theorem 1 for r > 0 follows from (2.5), (2.9), (2.10), and (2.13).
3. IRREGULAR BOUNDARIES
3.A. Introduction. In the previous section, the upper bound for No(A;r, Q)
was very simple to establish since by use of the monotonicity of No(A;r, Q)
with respect to Q, we could include Q in a (finite) union of cubes. But thisis not possible for the Neumann problem. Hence for obtaining more precise
estimates at the boundary, we consider a covering of Rn by dyadic cubes. Thispartition is used in particularin [CH] and by many authors (e.g., [Mt2], [L2],...

); it is sometimes referred to as a "Whitney-type covering".

Let 0 be an open set in Rn . For a given t7c,> 0, we consider a covering of
Rnby disjoint open cubes (QCq)CqEZn with side

(3.1)

?1q= ;702q 5, q e N.

Set
(3.2)

Io(O) = {4o E Zn/ QC0C O};?= ( U
CoEIo(O)Qco)

JACQUELINEFLECKINGERAND M. L. LAPIDUS

518

and
00' = ?\?o;(3.3)

Iq(O) = {Cq e Zn/ QUq C ?a-l};

0s = (0q'- u ( U

Qq))

CTqEIq(O)andOq'= O\Oq, for q > 1.
We now weaken the ",f-condition"which implies that Q? is a Jordancontented
set; this enables us to deal with more irregularboundaries and even fractal
boundaries, as in [L2-3].
3.B. Hypotheses and results. (H,) Q is a (nonempty) bounded open set in
Rn with n > 1; the weight function r belongs to L? (Q) and is allowed tochange sign; furthermore, IQ?I > 0(H2) Q+Ois such that there exists d E [n - 1; n) satisfying
X0*(d; aQn):= lim sup<-(n-d)

<

x0,

with
FE:=FFn no and FE :={xERn/dist(x,a(QnO)) < };here, dist ( , .) denotes the Euclidean distance in Rn.
(H3) r+ satisfies the "y-condition"on Q+ (adapted to the covering Qcq)
with y > d; i.e., there exists C2 > 0 which does not depend on CqE Iq(Qno)and there exists i, > 0 such that for all ? E (0, t1j, for all q E N,
pC(r)< C2i1
where pC is defined as in (2.2a).
(H4) For the Neumann problem, we assume that (H2) holds with fE replaced by rF; we assume in addition that Q+ satisfies either the "extension
property"(there exists a continuous linear extension of H1(Q) onto HI (in),e.g., [L2, Definition 4.3, p. 510]) or the "(C') condition" ([Mt2, pp. 154-156]:
there exists a finite covering of aQ by open sets (0j)j=I..,jo , with Ojc iRn;there are open sets Uj' in Rn- , positive numbers hi, upper semi-continuousfunctions ojo: (-hj, 2h1) - Uj, and C1 diffeomorphisms 6j: O n Q -* V1
with
1 E Rn 4 E Uj; ipj(4l) < 4l < 2hjl.
Vj 4=(1Furthermore,for all x and y in O n Q, there exists s > 0 such that the pathYs(x, y) associated to the reunion of segments[6j(x),

Oj(x) + se,] U [6j(x) + se,,Oj(y) + se,] U [6j(y) + se,,Oj(y)]

lies within O n Q (here, el denotes the first vector of the standard basis in
Rn). Moreover, if we denote by pj(x, y) the infimum of the lengths of all such$s (x, y) in Oj n Q, there exists ko such that for all j E {1, 2,..., Jo},pj(x, y) < ko dist (x, y).

Remark 2. (a) If the open set co satisfies the "segment property"[Ag, p. 11],then it satisfies the "(C') condition"; this is the case, for example, if aco is

INDEFINITEELLIPTICPROBLEMSON IRREGULARDOMAINS

519

Lipschitz. The "(C') condition" was first introduced in [FM, p. 914, ?3], andthen used in [Mtl-2], [FL1-2], [L1-4]. Roughly, it means that the boundary &Q
is not "too long": the Euclidean distance dist (x, y) is equivalent to do(x, y),
the minimal length of continuous paths from x to y within Q. Note that if
the "(C') condition" is not satisfied, then the usual Weyl asymptotic formula(1.1) may fail for the Neumann problem. (See, e.g., the counterexampleswhen
i= 1 and r 1 in [FM] and [Mt2, ?VII.1, pp. 200-204].)(b) When n = 2, the simply connected domain co satisfies the "extension
property"if and only if it is a quasidisk, i.e., the image of the unit disk undera quasiconformal mapping. (See, e.g., [L2, Example 4.2, p. 510] and the references therein.) The boundary &co is then a quasicircle(roughly, a distorted
circle) and may be extremely irregular[L2, p. 510].
(c) Intuitively, the more "regular"the boundary OQn', the smaller we may
choose d in hypothesis (H2) . Similarly,the more "regular"the weight functionr, the larger we may choose y in hypothesis (H3) . For example, if r = 1, asin [L2], then clearly, any y > 0 is suitable. Further, it follows from (theargumentprovided in) [FL2, Example 2, p. 333] that if r is Holder continuousof order G E (0, 1) and bounded away from zero on Q+O,then it satisfies the"y-condition"on Qn with y = n + (n0/2); in particular, y > n in that case.
Theorem 2. Underthe above hypotheses,we havefor i = 0 or 1:(i) If d E (n - 1, n) (the "fractalcase")
as)A
-i
+0,
RtQ(; r, Q) := N+(A; r, Q) - W(A; r+, Q) = O(Qt/2),
where t:= max (d, d+n-y).
[Hence t=d if yin and t=d+n-yify < n.](ii) If d = n - 1 (the "nonfractalcase"),{Rt(i; r,1={

I-lg~ )OQ.(Y22

wheny >n,when y < n.

Theorem 2 extends to problems with indefinite weights earlier results obtained for r 1_ ([Mt1-2], [L2-3]). It also makes more precise and extends toirregularboundaries results of [FL2] where an hypothesis needed to be completed (see [L2, Remark 3.5(d), p. 490]).
Remark 3. (a) Hypothesis (H2) which replacesin this section the "fl-condition"says that the (interior Bouligand-) Minkowski dimension D of &Q? is < d.
(See, e.g., [L2, Definition 2.1 (b), pp. 474-475 and ?3]. For relationshipsbetween
the ",8-condition"and the Minkowski dimension, see [L2, Corollary 3.3 andRemark 3.5, pp. 489-490].) When the weight r is "smooth enough" (comparedwith &Q), viz y > n, then t = d and we recover for d > n - 1 results of [L2-3]
in the special case of the (Dirichlet or Neumann) Laplacian. (See Remark 2(c)above and [L2, Theorem 2.3, pp. 482-483].) Indeed, the irregularity(fractality)of the boundary of Q+ can be due to that of Q (oQ n 0(Q+)) or that of r
(O(Q0)\OQ) : d < y < n.
(b) The use of the Minkowskidimension D for roughboundaries(in a related
context) was first made by Brossard and Carmona in [BrCa]. In [BrCa], theauthorsalso obtained (pre-Tauberian)errorestimates (expressedin terms of D )for the short time asymptotics of the "partition function"Z(t) = Trace (etA),
when A is the Dirichlet Laplacianand r-I .

520

JACQUELINEFLECKINGERAND M. L. LAPIDUS(c) Strictly speaking and to be in keeping with the terminology used in [L24], the "fractal"(resp., "nonfractal")case is that when D 7 n - 1 (resp., D =n - 1). Recall from [L2] that we always have D E [n - 1, n] and that if
O< A*(d; &Q.) < oo, then d = D, the (interior) Minkowski dimension ofQ+.
Of course, an analogous theorem holds for Ni-(A; r, Q) under analogous
hypotheses.
As in the previous section, we will prove Theorem 2 for r > 0; the casewhere r changes sign can be derived as indicated above. Therefore, from now
on, we assume that(Hg) r is positive, hence Q = Q+j, and as above we setNi(A; r, Q) = Ni+(A;r, Q)(3.4)

We also let, in view of (3.3),
Iq(3.5a)

I(Q)

and
q :=

(3.5b)

Q"

3.C. A lower bound. We establish here a lower bound for Ri(A; r, Q) =i = 0 or 1, when r is positive. The numbersy and d are given, with y > d. Choose A > 1 and then i0 = A-a where

N1(A; r, Q) - W(A; r, Q),
a >

n7)

with qi small enough so that (H2) or (H4) for the Dirichlet or

Neumann problem, respectively, implies that there exists c3 > 0 such that for
all qeN,(3.6)(#(Iq)) In < C3 n-d

where 11q is defined by (3.1) and Iq by (3.3) and (3.5a).
Since r is bounded, there exists M > 0 such that for almost all x E
Q, Ir(x)l < M, and hence, A being given, there exists P E N such that,for all q > P,
(3.7)

No(Q; r, QCq) = 0.

We define(3.8)

P:= max{q E N/ No(A;M, QCq)$ O}.

Note that the integer P depends on A. By means of (3.3) and the usual inequalities on N1(A; r, Q), we can write:
NI (A; r, Ql)> No(A; r, Ql)> No(A; r, Up)
(3.9)

P

>=
1:1 CqEIq N A;r.
q=0

C)

Then, by subtractingthe "Weyl term", we obtain(3.10)

Ri(A; r, Q) = N1(A;r, Q) - W(A; r, Q) > Al +A2+A3+A4,

INDEFINITEELLIPTICPROBLEMSON IRREGULARDOMAINS

521

with
p

A1 :=

(3.1Oa)1J (No(A; r, QCq)- No (; rCq,QCq)),

q=O qEIqp

A2 :=E(3.1Ob)R(A; rCq,Qq),E

q=O qEIqP.

A3 :=E
E(W(A;
q=O qEIq(3.10Oc)r, QCq)),rCq QCq)-W(A;and(3.O1d)A4-W(A; r, cop).

We first note that, with pC defined as in (2.2a),(3.11)

1Ni (A; r, QC)- Ni (A; rC. QC)I < Ni (A; Ir - rCI QC) < pC(r)An,2.

Therefore, by (H3), (3.1), (3.6), and (3.7), we havePP

-A1 < c Z
(3.12)(#(Iq)) 71y,An/2< Cn/2 E

1y-d

q=O

q=O

< C_nI2 y-d

2-q < C1A(n12)-a(y-dq=O> 1 and thus our choice of a
We note that, when d < y < n, we have n-d
y-d
preceding Eq. (3.6) implies a >Thus, for d E [n - 1, n),(3.13)

d + n - y).

with t = max(d,-A1 < CA/2,

We now consider A2 (which is negative); it can be handled as in [FL2, Eq. (21),p. 342]: we know from [CH, Section 6.4] (or [RS, Proposition 2, pp. 266-267])
that there exists co > 0 such that, for i = 0 or 1, for all A > 1 and all q EN,(3.14)

jRj(A; 1, QCq)l< co[1 + (Al 2)(n-0/2i

Consequently,p(3.15)

1A21< CoE(#(Iq))[ 1 +q

q=O

We deduce from (3.8) that there exists c4 > 0 satisfying(3.16)

2c4< 2P < c4V

Hence, by combining (3.1), (3.6), (3.15), and (3.16), and since r is boundedby M, we obtain
P

P

1A21
< c' E Q-d + CZA(n-1)12
E ,n-l-dq=O

q=O

JACQUELINEFLECKINGERAND M. L. LAPIDUS

522

Therefore,
IA2I< cAd/2 for d>n-1,(3.17i)(3.17ii)
IA21< cA(n-)12 In) for d=n-1.
By definition of rCqin (2.2), it is obvious in view of (3.1Oc)that
A3 = 0.
(3.18)
Finally we deduce from (3.16), (H;), and (H2) or (H4) that

= W(A;r, cp) = cf
(3.19) 1A41

< CAn/2 n-d < C.d/2.
(Ar)nl2 ? CAnP212wpj
ZP~~~~~~iHence it follows from (3.10), (3.13), and (3.17) to (3.19) that, for r regularenough, i.e., for y > d, we have:(i) When d = n - 1,
(3.20a)Ri(A; r, Q) > -cA(3.20b)

>-,aY

lnA
2

if y>nif d < y < n.

(ii) When d E (n- 1, n),Ri(A; r, Q) > _CA,12(3.21a)wheret = max(d, d + n - y).
(3.21b)Remark 4. (a) In particular,when r = 1, we have y > n [see Remark 2(c)];if d E (n - 1, n), we then recover, in the special case of the Laplacian, theestimate Ri(A; r, n) > -cAd/2 which has been established in [L2, Theorem
2.3, p. 482] and [L3]. As was shown in [L2], [L4], this estimate is sharp forevery d e (n - 1, n) and for i = 0 or 1.
(b) Note that when r is Holder continuous on n and bounded away fromzero, then y > n (by Remark 2(c) above) and hence y > d is automaticallyfulfilled.
(c) When n = 1 and 0 < d < 1 , the lower bound is of the form -CAd/2 ln A
[resp. -CAd/2 ] when y > 1 [resp. < 1].
3.D. Upper bound. For the Dirichlet problem, we can simply include n in a
finite union of cubes; up to a set with Lebesgue measure zero, we have
p

Op:=U

nc

U

u U QIP)
QCq

q=O CqEIq

CpEI'pwhere
If ={pEZn/ Qcp n9&0}.

We note that as for Iq in (3.6), we have
(3.22)
Therefore, we can write(3.23)

No(A;r, n) < No(A; r, Op)<#(Ip) < c3q1,d.

p

EZ
q=O CqEIq

Ni(A; r, QCq)+ Z
pNi(; r, QCP).

INDEFINITEELLIPTICPROBLEMSON IRREGULARDOMAINS

523

In (3.23), in order to extend r, we must assume A (and hence P) large
enough.
By subtractingthe Weyl term, we obtain(3.24)p

Ro(i; r, Q) < E

RiQ(; r, QCq)+ E NiQ(; r, QCp)< Bi +B2+B3+B4,E

q=O CqEIq

pEIsp

where
p

(3.25a)(NI(A; r QCq)- Nl (; rCq,QCq))

1

B1 :=

q=O CqEIq
p(3.25b)B2 :=E(NI(A; rq, QCq)- W(A; rCq QCq)),

E

q=O CqEIq
p(3.25c)E (W(A; rCq QCq)
q=O CqEIq

B3 :=: E

Wo.; r,. QCq)),

and(3.25d)B4 :=N(A;r, QCP)
N,

:
CpEIp

The first three terms B1, B2, B3 can be handled exactly as A1, A2, A3 in theprevious section (?3.C), and we obtain:
B1 <CAC/2 with t = max(d, d + n - y),(3.26)(3.27a)B2 < cAd12 for d > n - 1,
(3.27b)B2 < cA(n- )/2 ln A when d = n - 1,
(3.28)
B3 = 0.
Since r is bounded, we haveB4 <>j N1(AM; 1, QC ) < CAnI2In(#(I' ))CpEIp

so that, by (3.1), (3.16), and (3.22),
B4 < c)d/2(3.29)
Hence, by combining (3.26) to (3.29) with (3.24), we have for y > d,
(3.30)y>n,
RO(A;r,Q)<cA 2 lnA when d=n-land< cA 2 + 2<cAt/2

when d=n-I

with t=max(d,and d<y<n,d+n-'y),when dE(n-l,n).

For the Neumann problem, we cannot write (3.23); we simply have by meansof definition (3.5b):(3.31)
RI(A; r, Q) < Bi +B2+B3 +B',
where B1, B2, B3 are given by (3.25a-c) andJACQUELINEFLECKINGERAND M. L. LAPIDUS

524

Since r is bounded,
B4 < N(iQ(M; 1, cop)(3.33)

We now have to study a Neumann problem (without weight function) on theboundary strip cop. We deal as in [FM], [Mtl-2] or [L2, pp. 496-497]. We
denote by u- the extension of u to Ri with u E HI (Rn) . On each cube QCq
we approximate ui by its mean value Uq . Setv:=(3.34)Z

uypIQcQp

CpEIp

where
(X)
lQCP

0 when x E QCp,

By use of inequality (1) in [FM, p. 915] or inequality 5.2 in [Mtl], we obtain

1 ||UIH(Q)
L2(W,p)<(See also [L2].) We note that v defined by (3.34) lies in a (#(Ip))-dimensional
subspace of H1(Ri) . Hence in view of (3.22) and (3.35):(3.35)|uv(3.36)

B4 < C)dA2 .

It follows from (3.26) to (3.28) combined with (3.31) and (3.36) that the counterpartof (3.30) also holds for R1(i; r, Q) [instead of Ro(A; r, Q) ], and hence
Theorem 2 is proved.
Remark 5. (a) Our results (Theorems 1 and 2) could be extended to more general elliptic operators,much as in [Mt1-2], [FL2], and [L2-3], althoughwe chose
not do so in order to keep our argumentsreasonablyshort and simple.
(b) It would be interesting to investigate whether, under suitable hypotheses,
the results of [LP1,2]-that establish when n = 1 (and r- 1) the "modified Weyl-Berryconjecture"of [L2] concerning the existence of a (monotonic)
asymptotic second term for N(A)-can be extended to "fractal strings" ([L2],[L4], [LP1-2]) with variablemass density.
ACKNOWLEDGMENTS

The authors wish to thank the University of Montreal (Canada) and the
University College in Cardiff, Wales (G.B.). J. F. thanks the Universities ofAthens (Ohio) and Rutgers, USA, and M. L. L. is grateful to the Departmentof Mathematics of Yale University, where he was Visiting Professor while partof this paper was completed.
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UNIVERSITA TOULOUSE 1, PLACE ANATOLE FRANCE, 31042 TOULOUSE CEDEX, FRANCEDEPARTMENT OF MATHEMATICS, THE UNIVERSITY OF CALIFORNIA, SPROUL HALL, RIVERSIDE,CALIFORNIA
92521-0135

E-mail address: lapidusDucr . math. edu