The spectrum of irregular domains

Nonlimlrr

Andy&,

Tbemy, bfethds

Pergamon

& Applhtitwu, Vol. 30, No. 1, pp. 541-547.1997 Proc. 2nd Worki Congress of Nonlitwar Analysts 0 1997 Elseviu

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Ltd

Rimed inGreuBritain. Auri*ts reauved 0362-546x197

PII: s0362-s45x(

THE SPECTRUM

OF IRREGULAR

$17.00

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DOMAINS

N. MANDOUVALOS Department of Mathematics, Aristotle University of Thessaloniki, 54006Thessaloniki,Greece Key wordsandphrases:Hyperbolicgeometry,Spictrum,Bottom of thespectrumandof the essential spectrum,Minkowski dimension,Kleiniangroups. 1. INTRODUCTION The purpose of this paper is to study certain aspects of a hyperbolic-type

geometry

and to obtain uppierand lower bounds on the bottom of the spectrum E and essentialspectrum E,, for a region Q c lRN which is provided with a hyperbolic-type conformal metric. The methods we have usedare designedto provide quantitative information in the casewhere the boundaty L of Q is a fractal set, and make fundamental useof the Minkowski dimension.If L is the limit set of a Kleinian group r acting on the hyperbolic space RN+’ , we are able to obtain sharper information when the conformal metric on the region of discontinuity a(T) E RN = lRN u {-> is chosento be r invariant. This isjoint work with Professor E.B. Davies of King’s College London and details of the work have beenpublishedin [ 11. 2. ANALYTICAL

AND GEOMETRIC CONSIDERATIONS

Throughout this paper we let Q denote a possibly unbounded region in IRN whoseboundary L = &J is closed. We let cr be a non-negative continuous function on 6 suchthat: (0

o(x) = 0 if and only if x E L.

(ii)

If L is compact and R is unboundedthen o(x) 3 0 as ] x] + 00 in 0.

(2.1) If, however, L is only

closedthen we require that o(x) + - as d(x) = d(x,L) + 00

(iii)

where d is the Euclidean distance.

(2.2)

] Vo(x)] is locally bounded on 2.

(2.3)

We study the conformal metric ds2= o(x)-2()dxH2 541

(2.4)

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on R, which we call a hyperbolic-type

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metric even though one need not be able to compute its curvature

tensor because o need not be twice differentiable. The fundamental metric tensor go satisfies 112 = o-N

WgJ and the L2 space for this metric is

= {f: jn[f12 CT-~< -}.

L2(&YNdx)

The Laplace operator is given formally by -AOf = -oNV.(02-NVf)

= (N-2)oVo.Vf

- o2 Af,

where V, A denote the Euclidean gradient and Laplacian, and is associated with the quadratic form

Q,(f) = 1jVf1202-N. 51 Rigorously

we define -Ah, to be the non-negative self-adjoint operator associated with the closure

of the quadratic form Q,, initialty defined on We use quadratic

c:w

form techniques to obtain upper and lower bounds on the bottom

spectrum E and the bottom of the essential spectrum

of the

E,,, in terms of the geometry of L = XI and the

properties of cs. We are primarily interested in two particular cases. In the first L is an arbitrary compact subset of IRN with empty interior, and o = d where

d(x) = min{l x-y1 : y E L} is the Euclidean distance function.

In the second case L is the limit set of a Kleinian group acting on RN+‘, and CI = n(r) region of discontinuity

for the action of I

function on 52 which transforms

appropriately

c 6’

is the

on IRN. We then choose o to be a positive continuous under the action of I.

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3. UPPER BOUNDS

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ON E AND

Eess

Throughout this section we assume that L is compact and that xl$-tLsup(Va(x)(

The function

f(y) =

j orTNdx O(X)Sl

is a monotonically

I 1.

decreasing function of y which is finite for

y = N since {x E Sz : o(x) I 1) is a bounded region in lRN. Under very weak conditions on o there is a critical exponent

y, 2 0 such that f(y) < 00 if y > y, and f(y) = - if y < y,. We do not make any

assumptions about what happens when y = y,, although in many examples one has f(yo) = 0. We shall see later that the critical exponent is closely related to the Minkowski

dimension of the set L.

The following Theorem holds: THEOREM

3.1. Let us assume that limsuplVcr(x)(

Il.

x+L

Then

12

OIE
l

y. 5qN2.

The proof of this Theorem is in [I]. In

order to relate the critical exponent

weighted Minkowski

D,(L)

dimension Do(L)

y,

to the Minkowski

dimension of L, we define the

of L by

= inf{n: 1L”,] I ~.a~-~ for some c C 00 and all 0 < a < 1} ,

where IL”, I is the Euclidean volume of L”, = {x: o(x) < E}. The ordinary

Minkowski

D(L) 1 H(L), where H(L)

dimension

is the Hausdorff

D(L) then arises from the choice o = d. One always has dimension of L.

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3.2. If there exists a constant K > 0 suchthat

LEMMA

K-Id(x) I (T(X)I Kd(x)

(3.2)

for all x closeenoughto L then Do(L) = D(L). THEOREM

3.3. If o satisfiesconditions (2.1) - (2.3) then y, I D,(L).

Moreover, if there exist positive constants c,, c2 and u such that c,aN-l’ I IL”, I I c2.sNP for all O<&
3.4. Under the above conditions one has

Y, = DO4 = D,(L) = W-h provided o satisfies(3.2). We conjecture that the result of Theorem 3.4 isvalid for all geometrically finite Kleinian groups.

4. LOWER BOUNDS ON E The problem of obtaining lower bounds on E is much harder, and we present a number of different approaches. Our emphasisis upon obtaining explicit bounds, rather than simply proving that E is strictly positive. Ideally theselower bounds should be of the sameorder of magnitude as the upper boundsof the last section. We start by describing a known approach to the problem, which will be applied in Theorem 4.2. The idea is to find a constant c > 0 suchthat

jlVf12 2 C IN2

02 ’

for all f E C”(Q) c 7hzE IR2.

(4.1)

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We assume, without loss of generality, that L = 80 is closed and unbounded. We also assume that o satisfies (3.2) for x E Sz. Then (4.1) is equivalent to (IVfl2 2 c’ ‘!J2

(4.2)

with a different constant. Here we shall investigate a stronger condition, which is often more convenient for applications. If z E Sz and /WI = 1 then we define d,Jz)=inf{ltl:tE Clearly d(z) I dJz)

IR and z+twC

a}.

for all w such that 1d = 1. We say that XJ is uniformly visible from 62 if

1{w: dJz) 5 cd(z)}1 2 a

(4.3)

for some c > 0, a > 0 and all z E 52, where 1.I denotes the Lebesgue measure on the unit circle normalized so that the circle has total length unity.

THEOREM

4.1. If o satisfies condition

(3.2) for x E Q then under the hypothesis (4.3) we have:

It is well known that (4.3) is implied by a uniform exterior cone condition or by a uniform Lipschitz condition on dS& but we are interested here in much more irregular regions Cl. For an application of Theorem 4.1 to the case where L is the limit set of a convex cocompact Kleinian group, see theorem 4.2. THEOREM 4.2. Let I be a convex cocompact Kleinian group having an ordinary set n = G4(IJ E C which is not connected. Then E2-

a 2K2c2 ’

where K, a, c are determined explicitly. Theorem 4.1 can be extended to higher dimensions, but the next theorem is specifically two dimensional.

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4.3. If Sz E IR2 is simply connected and cssatisfies condition (3.2) for all x E Q then E2-

1 16K2 *

Our next method is based upon the validity of a certain differential curvature of the metric (2.4) is given by K. = (N-1){2oAo

inequality.

Since the scalar

- Nl Vol 2},

the hypothesis of our main theorem can be recast as an upper bound on the scalar curvature.

THEOREM

4.4. If 6 > 0 and OAO I (N-1-6j Vd 2

on a, then -A0 2 +i[(N-

l)lVd2 - oAo] 2 +a21Vo/2

as a quadratic form inequality. The proof is preceded by two lemmata. LEMMA Then

4.5. Let M be a Riemannian manifold and X a vector field on M such that divX 2 /Xj2.

-A, 2 a divX as a quadratic form inequality on CT(M). LEMMA4.6.

If Q> 0 and A,@<0

on M then

-A, 1 f (W21Vq2 as a quadratic form inequality on CT(M).

-@-‘A,@)

2 +2[V@j2

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4.7. If o = d and A(a*) I 2(N-6)

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1 2 for some 6 > 0, then E 2 46 .

4.8. If Sz c IRN and suppose o = d satisfies lim sup A(o*) I 2(N - 6) X-+&t

for some 0 < d C N. Then Eess>q6’ The following

2

.

theorem shows how the upper and lower bounds may be combined in one simple

case. THEOREM o = 6, then

4.9. If R = IRN \ L where

L is a compact smooth manifold of dimension

Eess=qS’

2

6, and if

.

REFERENCES I.

DAVIES, E.B. & MANDOUVALOS, N., The hyperbolic domains, Nonlinearity 3,913-945 (1990).

geometry and spectrum

of irregular