Asymptotic behavior of the eigenvalues of the one-dimensional weighted p-Laplace operator

(1)

Ark. M a t . , 4 1 (2003), 267 280

Asymptotic behavior of the eigenvalues of the one-dimensional weighted p-Laplace operator

Julian Fernandez Bonder and J u a n Pablo Pinasco(1)

A b s t r a c t . In t h i s p a p e r we s t u d y t h e s p e c t r a l c o u n t i n g f u n c t i o n for t h e w e i g h t e d p - l a p l a c i a n in one d i m e n s i o n . First, we prove t h a t all t h e eigenvalues c a n be o b t a i n e d by a m i n i m a x c h a r a c - t e r i z a t i o n a n d t h e n we s h o w t h e e x i s t e n c e of a W e y l - t y p e leading t e r m . F i n a l l y we find e s t i m a t e s for t h e r e m a i n d e r t e r m .

1. I n t r o d u c t i o n In this p a p e r we s t u d y the eigenvalue problem

(H) -(~p(~'))' = a~(~)~(~),

in a bounded open set f ~ c R , with Dirichlet or N e u m a n n b o u n d a r y conditions.

Here, the weight ~" is a real-valued, bounded, positive continuous function, ~ is a real parameter, l < p < + o c and

for s r and 0 if s = 0 .

From [7], T h e o r e m 1.1, p. 233, we know t h a t the s p e c t r u m consists of a countable sequence of nonnegative eigenvalues A1 <A2_<..._<Ak_<... (repeated according to multiplicity) tending to +oc. See also [16], where a similar result is obtained for the radial p-laplacian and for the one-dimensional p-laplacian with mixed boundary conditions. W i t h the same ideas as in [3], T h e o r e m 4.1, it is easy to prove t h a t

(1) T h e first a u t h o r is s u p p o r t e d b y U n i v e r s i d a d de B u e n o s Aires g r a n t T X 4 8 , by A N P C y T P I C T No. 03-05009. T h e s e c o n d a u t h o r is s u p p o r t e d by C O N I C E T a n d U n i v e r s i d a d de S a n A n d r e s .

(2)

268 Juligm Ferngmdez B o n d e r and J u a n Pablo Pinasco

the sequence {/~k}keN coincides with the eigenvalues obtained by the L j u s t e r n i ~ Schnirelmann theory. We recall that the variational characterization of the eigenvalues is

(1.2) where

A ~ = inf s u p f

lu'l pdz,

F E C ~ "uEF J t2

Cp = {C c M a : C compact, C = - C and 7(C) _> k},

M~'= {uE Wo'P(ft) (resp. Wl'P(ft)): ~ r(x),u,r' &c= l }

and 7(C) is the Krasnosel'skii genus (see [15] for the definition and properties of 7).

So our first result is the following theorem.

T h e o r e m 1.1.

Every eigenval~ue of problem

(1.1)

is given by

(1.2).

We define the

spectral counting function N(A, ft)

as the number of eigenvalues of problem (1.1) less t h a n a given A:

x ( a , a) = # { k : ak _< a}.

We will write

ND(A, gt)

(resp.

NN(A,

f~)) whenever we need to stress the dependence on the Dirichlet (resp. Neumann) boundary conditions.

The problem of estimating the spectral counting function has a long history, special in the linear case (p=2). See for instance [5], [9], [10], [12] and tile references therein.

However, up to our knowledge, for p•2 there is a lack of information about the behavior of N(A, f~). The only known result is due to [8]. In that paper, the authors show that the eigenvalues of the p-laplacian in R n (with r = 1) obtained by the minimax theory satisfy

(1.3) cl(a)kp/'r~ <_ xk _< c2(~)k/~.

It is easy to see t h a t this eigenvalue inequality is equivalent to

cI(~)A '~/~ <_ N(A, ~) <_ C2(~)X '~/~

for certain positive constants when A-~oo, see Lemma 3.2 below.

Our next result is concerned with the asymptotic behavior of the eigenvalues of (1.1) and begins our analysis of the function N(A, t2).

(3)

Asymptotic behavior of the eigenvalues of the one-dimensional weighted p-Laplace 269

We obtain the asymptotic expansion

( 1 . 4 )

where zc; is defined as (L5)

A1/P j ~ r 1/p dx, as A-+o c,

~0 ¹ ^ds

~P=2(p--1)I/P, (1-sP) lip"

The proof is based on variational arguments, including a suitable extension of the 'Dirichlet-Neunmnn bracketing' method, see [2]. We prove the following theorem.

Let r(x) be a real-valued, positive and bounded continuous func-

T h e o r e m 1.2.

tion in fL Then,

A1/; f

( 1 . 6 ) = - - !

rl/p dx+o(A1/P).

7rp _J~

Observe that by T h e o r e m 1.2, the asymptotic behavior of the eigenvalues (1.3) is improved. In fact, what (1.6) implies is that

A~: ~

ck p.

Once we found the first order asymptotics of N(A, 9), it is natural to t r y to improve these estimates and look for a second order term.

Following the ideas of [5], we analyze the remainder term R(A, f~)=N(A, f ~ ) -

7r; 1 f~(kr') 1/; dx.

We show that

(1.7) R(A, f~) ⁼

O(Ah/P),

where dE (0, 1] depends on the regularity of the b o u n d a r y 0f~ and on the smoothness of the weight r measured in a subtle way. To this end, let us introduce the following definitions.

Given any ~! > 0 sufficiently small, we consider a tessellation of R by a countable family of disjoint open intervals {Ir162 of length ~].

Definition

1.3. Let 12 be a bounded open set in R. Given 3 > 0 , we say that the boundary 0f~ satisfies the

3-condition,

if there exist positive constants e0 and r]0 < 1 such that for all ~!_< r]0,

(1.8)

# ( J \ I ) <_ Co~3 '

# I

(4)

270 J u l i a n Fernandez Bonder and J u a n Pab]o Pinasco

where

(].9) (1.]o)

• = { ( e z : z~ c a } , J = J(f2) = {s E z : I ~ n ~ r ~}.

It is easy to see t h a t if the set is J o r d a n contented (i.e., it is well a p p r o x i m a t e d from within and without by a finite union of intervals), then it verifies the /3- condition f o r / 3 = 1. T h e coefficient/3 allows us to measure the smoothness of 0fL

Definition 1.4. Given T > 0 , we say t h a t the function r satisfies the V-condition, if there exist positive constants Cs and r11<1 such t h a t for all ( ~ I ( f 2 ) and all 7?<~h, (1.11) ~ Ir-r~-I 1/p dx _< Cl~] ~,

where r r -~

fit

^{rl/p dx)p}is the mean value of r 1/v in Ir

Remarks 1.5. 1. T h e coefficient T enables us to measure the smoothness of r.

the larger T, the smoother r.

2. W h e n r is HSlder continuous of order 0 > 0 and is bounded away fl'om zero on 12, then it satisfies the V-condition for 0 < 7 -< 1 + 0/p.

If r is only continuous and positive on ~, then it satisfies the T-condition for 0 < 7 < 1 .

Now we are ready to state the theorem.

T h e o r e m 1.6. Let f~ be a bounded open set in R with boundary Of~ satisfying the ~3-condition for some /3>0, and let r be a bounded, positive and continuous function satisfying the T-condition for some 7 > 1 . Set u=min{/3, y - 1 } . Then, for

all 5 E [ 1 / ( , + 1 ) , 1], we have

(1.12) N(A, a ) - 1 / ~ (Ar)~/~ dz = O(Aa/~).

~rp

Finally, we end this article with some examples, where we compute the remainder t e r m explicitly.

T h e p a p e r is organized as follows. In Section 2, we introduce the genus in a version due to Krasnosel'skii, and prove the variational characterization of all the eigenvalues, together with some auxiliary lemmas. In Section 3, we prove the as- y m p t o t i c expansion (1.4). We analyze the remainder estimate in Section 4. Finally, in Section 5, we explicitly compute a nontrivial second t e r m for r - 1 and analyze the a s y m p t o t i c behavior of the eigenvalues.

(5)

2. V a r i a t i o n a l c h a r a c t e r i z a t i o n o f t h e e i g e n v a t u e s

In this section we first show t h a t every eigenvalue of (1.1) is given by a vari~- tional characterization and then we prove the D i r i c h l e t - N e u m a n n bracketing method t h a t will be the main tool in the remaining of the paper.

So let us begin with the proof of T h e o r e m 1.1.

Proof of Theorem

1.1. T h e proof follows the lines of T h e o r e m 4.1 of [3].

By [7] the s p e c t r u m is countable and we can assume t h a t it is given by the eigenvalues #i < # 2 _ < .... Given an eigenpair (u~, #k) of (1.1), we claim t h a t uk has k nodal domains. It is clear t h a t the n u m b e r of nodal domains is _<k (see, e.g., [1]).

Now the claim follows by induction, since the first eigenfhnction has exactly one nodal domain, and by [16], T h e o r e m 4.1(b), if uk has k nodal domains, then uk+l has at least k + l nodal domains.

Now, if (Uk,#k) is an eigenpair of (1.1), we can consider

w{(z)=uk(z)

if :c"

belong to the i t h nodal domain, and w i ( z ) = 0 elsewhere. Let St be the sphere in W~'P(Y~) of radius t. Then, the set C k = s p a n { w ~ , . . . ,

wk.}~St

has genus k and is an admissible set in the characterization (1.2) of the kth variational eigenvalue l k , from which it follows t h a t t k < # k and then A~=#k. []

T h e rest of this section is devoted to the proof of the so called DirichIet N e u m a n n bracketing method. We want to r e m a r k t h a t these results hold for arbi- t r a r y dimensions n_> 1 if one only considers the variational eigenvalues.

U L3 ~ i n t - - u T h e o r e m 2.1.

Let

U 1 , U 2 c I { ~

be disjoint open sets such that ~ 2 - and Iu\(glug2)l,~=o, then

ND( ),, U~ UU~) < ND( ),, U) < NN( ),, U) < NN( :~, U~ UU2).

Here [AI., stands for the n-dimensional Lebesgue measure of the set A.

Proof.

This is an easy consequence of the Ibllowing inclusions (2.1)

WI'P(U1

^u~f,2)^:

~VI'p(u1)(~WI'P(U2) C WI'P(U)

and

(2.2)

WI'P(U)

^C

WI'P(U1)@WI'P(U2)

⁼

WI'P(UI UU2),

and the variational formulation (1.2). In fact, nsing t h a t

MU(x) {uEX:./ur(X)lu]~'dx 1 } C 2 ] J U ( Y ) = { u e Y : / u r ( x ) ] u [ P d x = l }, and cU(X)cC~](Y),

where

X=IiV~'P(U1UU2) or X=W<P(U) a,nd Y:I/V~'P(U) or

Y=WI'P(UIUU2),

w e obtain the desired inequ&lity. []

T h e Dirichlet-Neumann bracketing m e t h o d is a powerf\fl too] w h e n combined with the following result.

(6)

272 Julis Ferns Bonder and Juan Pablo Pinasco

P r o p o s i t i o n 2.2. Let

ft=Uj

~2, where {Stj}j is a pairwise disjoint family of bounded open sets in R ~. Then

(2.a) x(s, a) = ~ x(x, aj).

J

Pro@ Let A be an eigenvalue of problem (1.1) in ft, and let u be the associated eigenfunction. For all v~Wd'P(ft) we have

(2A) .~ IVnlP-2vnv~ dx-~,/~ I~IP-~

d x = O.

Choosing v with compact support in ftj, we conclude that ula 5 is an eigenfunction of problem (1.1) in ftj with eigenvalue A.

For the other inclusion, it is sufficient to extend an eigenfunction u in ftj by zero outside, which gives an eigenfunction in ft. []

3. T h e function N ( A )

h this section we prove the asymptotic expansion given by Theorem 1.2.

First let us recall the iollowing lemma, which was proved in [14].

L e m m a 3.1. Let {A~}~eN be the eigenvalues of (1.1) in (O,T), with Dirichlet boundary condition and r= l. Then

~P

(3.1) Ak = ~ k p

Let {#k}keN be the eigenvalues of (1.1) in (0, T), with Neumann boundary condition and r= l. Then,

(3.2) #k= i~p (k--1)P.

~P

W i t h the aid of L e m I n a 3.1 w e c a n p r o v e the following result.

L e m m a 3.2. Let {/~k}kcN be the eigenvalues of (1.1) in (O,T) and suppose that m < r ( x ) ~ M . Then

7rP

(3.3) M TP - m TP

(7)

and

(3.4)

A s y m p t o t i c behavior of t h e eigenvalues of the one-dimensional weighted p-Laplace 273

TM1/p

r m l / ' ) a l / p _ 1 < N(A, (0, T)) <

- - A 1/p.

% %

Proof.

Equation (3.3) is an easy consequence of the sturmian comparison prin- ciple in [16], p. 182, Theorem 4.1(b) and the subsequent corollary, and the explicit formula for the eigenvalues with constant weight. Now,

(3.5) #

k:TT m_<a _ < # { k : X k _ < a } < # .

The left-hand side is greater than

T?Tzl/P ~l/p __ ], 7Cp

which gives the lower bound. In the same way, we obtain

[TM1/PA1/p ] T M 1/p

N(A, (O,T)) < < X L/;. []

k 7rp j zcp

Now we prove a proposition that is the key ingredient in the proof of Theo- rem 1.2.

P r o p o s i t i o n 3.3.

Let r(z) be a real-valued, positive continuous function in

[0, T].

Then

Xl/p .~T

_ _ rl/P dx-ko(A1/P).

(3.6) N(A, ( 0 , T ) ) - - zcp

Proof.

Let [0,

T]=UI<j< J Ij, IjNIk=O

with

]Ij I=T/J=r].

We define

ray = i n f

r(x)

and My = sup r ( x ) .

xEIj x~ij

We can choose 'q>0 such that

J T J T

j~=I?]?Tt~/P:~ 0 rl/Pd2g-s

^{a l i a}

ETjM)/P~-fo rl/pdzAr-s

j = l

with cl, c2 > 0 arbitrarily small.

(8)

274 JuliAn Fern~indez B o n d e r a n d J u a n Pablo P i n a s c o

From T h e o r e m 2.1 and Proposition 2.2, we obtain

J d

N.(:~, zj) < N(a, (0, T)) < ~ N~(a, ~).

j = l

Hence, using t h a t

7Cp

we have

j = l

- - - 1 and

7Cp

r l i p d X - C l -

J < N(A, (0, T)) <_

^{- -} r l i p d x q - c 2 .

7rp 7Cp

Letting X--~oc, we have

and the proof is complete. []

N(a, (0, T))

_{- + ] ,}

M/'~Tr P f ? rl/~ dz

Finally, we arrive at the proof of T h e o r e m 1.2.

Proof of Theorem

1.2. It is an easy consequence of Propositions 2.2 and 3.3.

g2 oo

Let =U3=1 IS, then

( 3 . 7 ) N ( A , e ) = Z ]V(/~, [ j ) ~ Z - - r lip dx - r lip dx. []

j = l J ] 7rp J 7Cp

4. R e m a i n d e r e s t i m a t e s

As we mentioned in the introduction, we now look for an improvement in the a s y m p t o t i c expansion of N(A, ~). This is the content of T h e o r e m 1.6.

Proof of Theorem

1.6. For the convenience of the reader, the proof is divided into several steps.

Moreover, we will stress tile dependence of tile spectral counting function with respect, to the weight function by writing N(A, f~, r).

(9)

Step

1. Let r/>0 be fixed. We define

(4.1)

F(k)=-~p~(Ar)l/Pdx

and

where r;=(]/r -1 fz~

rl/p dx) ~"

From Theorem 2.1 we obtain

(4.2)

~(~, r = ~ (~r

and

r 1 6 2

(4.3)

CEZ CeJ\I

We are reduced to find bounds for the left-hand side of (4.2) and for the right- hand side of (4.3).

Step

2. We can rewrite the left-hand side of (4.2) as

(4.4) Ce~ r r162

+ ~ (N. (X, I4, ,')- X~ (),, Ir ,-r

r162

Let us note that both ~r Ir rc)-g)(A, ()) and Er ~(A, ( ) - p ( A ) are negative. Now, by Lemma 3.2,

(4.5)

a s

(4.6)

IND (A, Ir r g ) - ~ ( A , ()1 <-

#([)M < If~.

r r/

We can bound

C /~ I / P # ( J \ I ) T] ]~ r ~ c /~ l / P T] ft.

Here we have used t h a t

r<M,

and that Oft satisfies the fl-condition.

(10)

276 Julign Ferngndez Bonder and Juan Pablo Pinasco

Finally, the last s u m in (4.4) can be handled using the monotonicity of the eigenvalues with respect to the weight (see [16]). Using that r_< r ( + [ r - r ( ] , a simple computation shows t h a t

No(A, I~, ~') <_ N o (A, I o ~ ) + N o ( A ,

I~, Ir-~'~ [),

w h i c h gives

E(ND(X, [(, T)--~D(X, [(, T()) ~ E ~O(~' f(' If --r(I) ~ CX1/P#(I)71~

(eI (eI

and using the same arguments as above and the fact that r satisfies the 7-condition, we obtain

(4.7)

E(No(/\, I(, 7")-No(l\ , I(, re) ) <_ cal/P71 "T-1.

(eI

Collecting (4.5), (4.6) and (4.7) we have the lower bound

(4.8)

c~1/p(719 +~_1)+ c.

Step

3. In a similar way, we can find an upper bound for the right-hand side

(4.9)

E NN(A'Ir E NN(A,I(NfLr)

(cz (E J\I

of (4.3). We only need to estimate the last sum, but

NN(;~,I;r~U,,') <_caller ~,~/P&<C(M71a) 1/~

,] Ir

and again, using the fl-condition, we have

(4.10) ~ NN(~, I~n~, ~') < C;~/% ~.

C~J\I

Hence, we obtain the upper bound

(4.1]) c.~l/p(,fljl_,'T

1 ) + s

71 for (4.3).

Step 4. From (4.S) and (4.11) we obtain

(4.12)

~V(~, ~ ) - L ~I (/~T)I/P dx ~ C/~I/P(~']}3 J[-I] "T-l) Jr-C.

7rp

We now choose 71=A -~, with 0 < a < 5 . It is clear that the last term in (4.12) is bounded by CA 5. Also, it is easy to see that, if

a>~-l(p

1 6), then A1/P71fl<A 5.

Likewise, choosing

a>_(7-1)-1(p-1-5),

we have

/\1/p71"y--1~.\5.

When fl=0, or

"/=1, we must choose

a=l/p.

This completes the proof. []

(11)

A s y m p t o t i c behavior of t h e eigenvalues of t h e one-dimensional weighted p - L a p l a c e 277

5. C o n c l u d i n g remarks

We end this p a p e r showing a family of examples with a power-like second term, and an example with an irregular second term. Finally, we discuss the a s y m p t o t i c behavior of the eigenvalues.

In the examples below, the p a r a m e t e r d provides some geometricM information a b o u t Oft. In b o t h cases, d is the interior Minkowski (or box) dimension of the boundary, we refer the reader to [41 and the references therein for the definition and properties of the Minkowski dimension.

Examples with explicit second term.

Let f t = U j ~ N

Ij,

where

IIjl=j -1/d

and 0 < d < l . We have the following a s y m p t o t i c expansion for the spectral counting function when r = 1:

(5.1) N(A, a) =

I~l~l/p+C(d)ad/p+O(),"/'(2+~)).

7Cp

T h e proof can be obtained with n u m b e r theoretic methods. We have N(A, F t ) = A 1/p = # ( m , n ) E N 2 : r n n

1/a< .

j = l t p j 71-p

In fact, for each j we can draw the vertical segment of length

j-1/aAl/V/rcp

in the plane, and the series is the n u m b e r of lattice points below the function

y(x)=

A1/'rcp lx-~/a.

See [13] for a detailed proof.

W h e n

p=2

and

]Ijl~j -1/a,

it is shown in [llJ t h a t

N ( A, a) = I~I A1/p +C(d)Ae/P +o( Ad/P),

7Fp

without the lattice point theory, the same result is valid for p r However, let us note t h a t the error in equation (5.1) is better, which enables us to obtain more precise estimates whenever we know more a b o u t the a s y m p t o t i c behavior of 113 I.

On the other hand, the result in [11] holds for more general domains t h a n the ones considered here.

Example with irregular second term.

Let ft be the complement of the t e r n a r y Cantor set, and r = l . We have

(5.2) N(~,, a ) = lal .X~/v - f ( l o g ) , ) 2 ~ ~/p log a + O ( 1 ) .

7"Cp

Here

f(x)

is a bounded, periodic function. Our proof closely follows [6], where the usual Laplace operator on a self-similar set in R '~ was studied for every n_>2.

(12)

278 J u l i g n F e r n g n d e z B o n d e r a n d J u a n P a b l o P i n a s c o

Let us define

cg(x)=x-[x],

it is evident t h a t 10(x)l~min{z, 1}. Hence, (5.a) re(A, ~ ) _ J~r A1/p = _ ~ 25

o \ ~ ] <_ cA I/p.

7Cp j : 0

It remains to prove the periodicity of f . We write the error t e r m as oo J / ~l/p "~ 1 ( A1/p

j= oc i= oo

Using t h a t I~(x)l<_ 1, the second series converges and it is bounded by a constant.

Let us introduce the new variable

log A 1/p - l o g rcp

(5.5) y = ,

log 3 which gives W=),l/p/rc> and

2s=(A1/P/rcp) d,

where

log 2

(5.6) d = log~"

Inserting this into the first t e r m in (5.4), we obtain

(5.7) ~ ~ 2J~ = - - 2J-~0(a~-J).

J - ~

\3JTrpJ 2k, rrp / j

^{- ~}

Thus, as

j - ( y - 1 ) = ( j + l ) - y ,

we deduce t h a t

f(x)

is periodic with period equal to one.

Asymptotics of eigenvalues.

From T h e o r e m 1.6 it is easy to prove the asymptotic formula for the eigenvalues

Ak ~ ck p.

This follows immediately since k ~ N ( A k ) , which gives

Ak ~ r 1/p dx

Using the D i r i c h l e t - N e u m a n n bracketing method, it is possible to improve the constants in equation (1.3). In [8] the authors only consider two cubes Q1 c f t c Q 2 , and t h e y obtain a lower and an upper bound fbr the eigenvalues in cubes which depends on the measure of the cubes Q1 and Q2 instead of the measure of ft.

(13)

Asymptotic behavior of the eigenvMues of the one-dimensional weighted p-Laplace 279

A similar a r g u m e n t to the one in [8], c h a n g i n g t h e functions {sin kcc}~=~N for {sinp ]~X}/~N, gives t h e u p p e r b o u n d

( rrp )P/~kp/,~ '

"U<-< \1~1)

where sinp k x are t h e eigenfunctions of the one-dimensionM p r o b l e m w i t h c o n s t a n t coefficients, see I3].

However, it seems difficult to improve the lower b o u n d o b t a i n e d with t h e aid of t h e B e r n s t e i n ' s lemrna.

R e f e r e n c e s

1. ANANE, A. and TSOULI, N., On the second eigenvalue of the p-Laplacian, in Nonlinear Partial Differential Equations (F&, 1994 ) (Benkirane, A. and Gossez, J.-P., eds.), Pitman Res. Notes Math. Ser. 343, pp. 1 9, Longman, Harlow, 1996.

2. COURANT, R. and HILBERT, D., Methods of Mathematical Physics, Vol. 1, Inter- science Publ., New York, 1953.

3. DRABEK, P. and MANASEVICH, R., On the closed solutions to some nonhomoge- neous eigenvalue problernes with p-laplacian, Differential Integral Equations 12 (1999), 773 788.

4. FALCONER, K., On the Minkowski measurability of fractals, Proc. Arner. Math. Soc.

123 (1995), 1115-1124.

5. FLECKINGER, J. and LAPIDUS, M., Remainder estimates for the asymptoties of ellip- tic eigenvalue problems with indefinite weights, Arch. Rational Mech. Anal.

98 (1987), 329 356.

6. FLECKINGER, J. and VASSILIEV, D., Ai1 example of a two-term asymptotics for the

"counting function" of a fractal druln, Trans. Amer'. Math. Soc. 337 (1993), 99-116.

7. Fu~fK, S., NECAS, J., SOUCEK, J. and SOU~EK, V., Spectral Analysis of Nonlinear Operators, Lecture Notes in Math. 346, Springer-Verlag, Berlin-Heidelberg, 1973.

8. GARCIA AZORERO, J. and PERAL ALONSO, I., Comportement asymptotique des valeurs propres du p-laplacien, C. R. Aead. Sci. Paris Sdr. [ Math. 307 (1988), 75-78.

9. HORMANDER, L., The Analysis of Linear Partial Differential Operators III, Springer- Verlag, Berlin Heidelberg, 1985.

10. NAC, M., Can one hear the shape of a drum?, Amer. Math. Monthly 73:4 part II (1966), 1-:23.

11. LAPIDUS, M. and POMERANCE, C., The Riemann zeta-function and the one-dimensional V~reyl Berry conjecture for fractal drums, Proc. London Math. Soc. 66 (1993), 41-69.

12. METIV1ER, G., Valeurs propres de probl~mes aux limites elliptiques irr6guli~rs, Bull.

Soc. Math. France Suppl. Mere. 5 1 - 5 2 (1977), 125-219.

(14)

280 Julis Fernandez Bonder and Juan Pablo Pinasco:

Asymptotic behavior of the eigenvalues of the one-dimensional weighted p-Laplace 13. PINASCO, J. P., Asymptotics of eigenvalues of the p-Laplace operator and lattice

points, Preprint, Universidad de Buenos Aires, 2002.

14. DEL PINO, M. and MANASEVICH, R., Multiple solutions for the p-Laplacian under global nonresonance, Proe. Amer. Math. Soc. 112 (1991), 131 138.

15. P~ABINOWITZ, P., Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conf. Ser. Math. 65, Amer. Math.

Soc., Providence, R.I., 1986.

16. WALTER, W., SturIn~Liouville theory for the radial Ap-operator, Math. Z. 227 (1998), 175-185.

Received March 25, 2002 in revised f o r m October 21, 2002

JuIigm Ferns Bonder Universidad de Buenos Aires Pabelldn I Ciudad Universitaria 1428-Buenos Aires

Argentina

emaih j f b o n d e r ~ d m . u b a . a r Juan Pablo Pinasco Universidad de San Andres Vito Dumas 284

1684-Prov. Buenos Aires Argentina

Current address:

Universidad de Buenos Aires Pabelldn I Ciudad Universita~'ia 1428-Buenos Aires

Argentina

email: jpinasco@dm.uba.ar

Asymptotic behavior of the eigenvalues of the one-dimensional weighted p-Laplace operator