Download (0)

Full text



JUAN PABLO PINASCO Received 14 April 2003

We present sharp lower bounds for eigenvalues of the one-dimensionalp-Laplace oper- ator. The method of proof is rather elementary, based on a suitable generalization of the Lyapunov inequality.

1. Introduction

In [9], Krein obtained sharp lower bounds for eigenvalues of weighted second-order Sturm-Liouville differential operators with zero Dirichlet boundary conditions. In this paper, we give a new proof of this result and we extend it to the one-dimensional p- Laplacian

u(x)p2u(x)=λr(x)u(x)p2u(x), x(a,b),

u(a)=0, u(b)=0, (1.1)

whereλis a real parameter,p >1, andris a bounded positive function. The method of proof is based on a suitable generalization of the Lyapunov inequality to the nonlinear case, and on some elementary inequalities. Our main result is the following theorem.

Theorem1.1. Letλnbe thenth eigenvalue of problem (1.1). Then, 2pnp

(ba)p1abr(x)dxλn. (1.2) We also prove that the lower bound is sharp.

Eigenvalue problems for quasilinear operators of p-Laplace type like (1.1) have re- ceived considerable attention in the last years (see, e.g., [1,2,3,5,8,13]). The asymptotic behavior of eigenvalues was obtained in [6,7].

Lyapunov inequalities have proved to be useful tools in the study of qualitative nature of solutions of ordinary linear differential equations. We recall the classical Lyapunov’s inequality.

Copyright©2004 Hindawi Publishing Corporation Abstract and Applied Analysis 2004:2 (2004) 147–153 2000 Mathematics Subject Classification: 34L15, 34L30 URL:http://dx.doi.org/10.1155/S108533750431002X


Theorem1.2 (Lyapunov). Letr: [a,b]Rbe a positive continuous function. Letube a solution of

u(x)=r(x)u(x), x(a,b),

u(a)=0, u(b)=0. (1.3)

Then, the following inequality holds:


ar(x)dx 4

ba. (1.4)

For the proof, we refer the interested reader to [10,11,12]. We wish to stress the fact that those proofs are based on the linearity of (1.3), by direct integration of the differential equation. Also, in [12], the special role played by the Green functiong(s,t) of a linear differential operatorL(u) was noted, by reformulating the Lyapunov inequality for

L(u)(x)r(x)u(x)=0 (1.5)



ar(x)dx 1

Maxg(s,s) :s(ba). (1.6) The paper is organized as follows.Section 2 is devoted to the Lyapunov inequality for the one-dimensional p-Laplace equation. InSection 3, we focus on the eigenvalue problem and we proveTheorem 1.1.

2. The Lyapunov inequality

We consider the following quasilinear two-point boundary value problem:

|u|p2u =r|u|p2u, u(a)=0=u(b), (2.1) wherer is a bounded positive function and p >1. By a solution of problem (2.1), we understand a real-valued functionuW01,p(a,b), such that


a|u|p2uv= b

ar|u|p2uv for eachvW01,p(a,b). (2.2) The regularity results of [4] imply that the solutionsuare at least of classCloc1,αand satisfy the differential equation almost everywhere in (a,b).

Our first result provides an estimation of the location of the maxima of a solution in (a,b). We need the following lemma.


Lemma2.1. Letr: [a,b]Rbe a bounded positive function, letube a solution of problem (2.1), and letcbe a point in(a,b)where|u(x)|is maximized. Then, the following inequali- ties hold:


ar(x)dx 1

ca p/q

, b

c r(x)dx 1

bc p/q

, (2.3)

whereqis the conjugate exponent ofp, that is,1/ p+ 1/q=1.

Proof. Clearly, by using H¨older’s inequality, u(c)=


au(x)dx(ca)1/q c


u(x)pdx 1/ p

. (2.4)

We note thatu(c)=0. So, integrating by parts in (2.1) after multiplying byugives c


u(x)pdx= c

ar(x)u(x)pdx. (2.5)


u(c)(ca)1/q c

ar(x)u(x)pdx 1/ p

(ca)1/qu(c) c

ar(x)dx 1/ p



Then, the first inequality follows after cancellingu(c) in both sides while the second is

proved in a similar fashion.

Remark 2.2. The sum of both inequalities shows thatccannot be too close toaorb. We haveabr(x)dx <, but


1 ca


+ 1

bc p/q

= lim


1 ca


+ 1

bc p/q

= ∞. (2.7) Our next result restates the Lyapunov inequality.

Theorem2.3. Letr: [a,b]Rbe a bounded positive function, letube a solution of prob- lem (2.1), and letqbe the conjugate exponent ofp(1, +). The following inequality holds:

2p (ba)p/q


ar(x)dx. (2.8)

Proof. For everyc(a,b), we have 2u(c)=


au(x)dx+ b

c u(x)dx b


u(x)dx. (2.9)


By using H¨older’s inequality,

2u(c)(ba)1/q b


u(x)pdx 1/ p

=(ba)1/q b

ar(x)u(x)pdx 1/ p



We now choosecin (a,b) such that|u(x)|is maximized. Then, 2u(c)(ba)1/qu(c)

b a r(x)dx

1/ p

. (2.11)

After cancelling, we obtain

2p (ba)p/q


ar(x)dx, (2.12)

and the theorem is proved.

Remark 2.4. We note that, forp=2=q, inequality (2.8) coincides with inequality (1.4).

3. Eigenvalues bounds

In this section, we focus on the following eigenvalue problem:

|u|p2u =λr|u|p2u, u(a)=0=u(b), (3.1) whererL(a,b) is a positive function,λis a real parameter, andp >1.

Remark 3.1. The eigenvalues could be characterized variationally:

λk(Ω)= inf





r|u|p, (3.2)


Ck =

CM:Ccompact,C= −C,γ(C)k, M=

uW01,p(Ω) :


, (3.3)

andγN∪ {∞}is the Krasnoselskii genus,

γ(A)=minkN, there exist f CA,Rk\ {0} , f(x)= −f(x). (3.4) The spectrum of problem (1.1) consists of a countable sequence of nonnegative eigen- values λ1 < λ2 <··· < λk <···, and coincides with the eigenvalues obtained by Ljusternik-Schnirelmann theory.


Now, we prove the lower bound for the eigenvalues of problem (3.1) for every p (1, +). We now prove our main result,Theorem 1.1.

Proof of Theorem 1.1. Letλnbe thenth eigenvalue of problem (3.1) and letunbe an as- sociate eigenfunction. As in the linear case,unhasnnodal domains in [a,b] (see [2,13]).

Applying inequality (2.8) in each nodal domain, we obtain n



xkxk1 p/qλn

n k=1






a r(x)dx, (3.5) wherea=x0< x1<···< xn=bare the zeros ofunin [a,b].

Now, the sum on the left-hand side is minimized when all the summands are the same, which gives the lower bound

2pn n

ba p/q



ar(x)dx. (3.6)

The theorem is proved.

Finally, we prove that the lower bound is sharp.

Theorem3.2. LetεRbe a positive number. There exist a family of weight functionsrn,ε such that


λn,ε 2pnp (ba)p1abrn,ε

=0, (3.7)

whereλn,εis thenth eigenvalue of

|u|p2u =λrn,ε|u|p2u, u(a)=0=u(b). (3.8) Proof. We begin with the first eigenvalueλ1. We fixabr(x)dx=M, and letcbe the mid- point of the interval (a,b).

Letr1be the delta functionc(x). We obtain λ1= min



a|u|p b

aδcup = min


2ac|u|p Mup(c) =


M , (3.9)

whereµ1is the first Steklov eigenvalue in [a,c],


u(c)p2u(c)=µu(c)p2u(c), u(a)=0.


A direct computation gives

µ1= 2p1

(ba)p1. (3.11)


Now, we define the functionsr1,ε:


0 forx

a,a+b 2 ε

, M

2ε forx a+b

2 ε,a+b 2 +ε

, 0 forx

a+b 2 +ε,b



and the result follows by testing, in the variational formulation (3.2), the first Steklov eigenfunction


xa ifx

a,a+b 2

, bx ifx

a+b 2 ,b



Thus, the inequality is sharp forn=1.

We now consider the casen2. We divide the interval [a,b] innsubintervalsIi of equal length, and letcibe the midpoint of theith subinterval.

By using a symmetry argument, thenth eigenvalue corresponding to the weight rn(x)=M

n n i=1

δci(x), (3.14)

restricted toIi, is the first eigenvalue in this interval, that is, λn=2nµ1

M =


M(ba)p1. (3.15)

The proof is now completed.


This work has been supported by Fundacion Antorchas and ANPCyT PICT Grant 03- 05009. We would like to thank Prof. R. Duran and Prof. N. Wolanski for interesting con- versations.


[1] A. Anane,Simplicit´e et isolation de la premi`ere valeur propre dup-Laplacien avec poids[Simplic- ity and isolation of the first eigenvalue of thep-Laplacian with weight], C. R. Acad. Sci. Paris S´er. I Math.305(1987), no. 16, 725–728 (French).

[2] A. Anane, M. Moussa, and O. Chakrone,Spectrum of one dimensionalp-Laplacian operator with indefinite weight, Electron. J. Qual. Theory Differ. Equ. (2002), no. 17, 1–11.

[3] M. Del Pino, P. Dr´abek, and R. Man´asevich,The Fredholm alternative at the first eigenvalue for the one-dimensionalp-Laplacian, C. R. Acad. Sci. Paris S´er. I Math.327(1998), no. 5, 461–465.

[4] E. DiBenedetto,C1+αlocal regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal.7(1983), no. 8, 827–850.


[5] P. Dr´abek and R. Man´asevich,On the closed solution to some nonhomogeneous eigenvalue prob- lems withp-Laplacian, Differential Integral Equations12(1999), no. 6, 773–788.

[6] J. Fernandez Bonder and J. P. Pinasco,Asymptotic behavior of the eigenvalues of the one dimen- sional weightedp-Laplace operator, Ark. Mat.41(2003), 267–280.

[7] J. Garc´ıa Azorero and I. Peral Alonso,Comportement asymptotique des valeurs propres dup- laplacien[Asymptotic behavior of the eigenvalues of thep-Laplacian], C. R. Acad. Sci. Paris S´er. I Math.307(1988), no. 2, 75–78 (French).

[8] M. Guedda and L. V´eron,Bifurcation phenomena associated to the p-Laplace operator, Trans.

Amer. Math. Soc.310(1988), no. 1, 419–431.

[9] M. G. Krein,On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability, Amer. Math. Soc. Transl. Ser. 21(1955), 163–187.

[10] A. Liapounoff,Probl`eme G´en´eral de la Stabilit´e du Mouvement, Annals of Mathematics Studies, no. 17, Princeton University Press, New Jersey, 1947 (French).

[11] W. T. Patula,On the distance between zeroes, Proc. Amer. Math. Soc.52(1975), 247–251.

[12] W. T. Reid,A generalized Liapunov inequality, J. Differential Equations13(1973), 182–196.

[13] W. Walter,Sturm-Liouville theory for the radialp-operator, Math. Z.227(1998), no. 1, 175–


Juan Pablo Pinasco: Departamento de Matem´atica, Universidad de Buenos Aires, Pabellon 1, Ciu- dad Universitaria, 1428 Buenos Aires, Argentina

Current address: Instituto de Ciencias, Universidad Nacional de General Sarmiento, J.M. Gutierrez 1150, Los Polvorines, 1613 Buenos Aires, Argentina

E-mail address:jpinasco@dm.uba.ar


Special Issue on Space Dynamics

Call for Papers

Space dynamics is a very general title that can accommodate a long list of activities. This kind of research started with the study of the motion of the stars and the planets back to the origin of astronomy, and nowadays it has a large list of topics. It is possible to make a division in two main categories: astronomy and astrodynamics. By astronomy, we can relate topics that deal with the motion of the planets, natural satellites, comets, and so forth. Many important topics of research nowadays are related to those subjects.

By astrodynamics, we mean topics related to spaceflight dynamics.

It means topics where a satellite, a rocket, or any kind of man-made object is travelling in space governed by the grav- itational forces of celestial bodies and/or forces generated by propulsion systems that are available in those objects. Many topics are related to orbit determination, propagation, and orbital maneuvers related to those spacecrafts. Several other topics that are related to this subject are numerical methods, nonlinear dynamics, chaos, and control.

The main objective of this Special Issue is to publish topics that are under study in one of those lines. The idea is to get the most recent researches and published them in a very short time, so we can give a step in order to help scientists and engineers that work in this field to be aware of actual research. All the published papers have to be peer reviewed, but in a fast and accurate way so that the topics are not outdated by the large speed that the information flows nowadays.

Before submission authors should carefully read over the journal’s Author Guidelines, which are located athttp://www .hindawi.com/journals/mpe/guidelines.html. Prospective au- thors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking Sy- stem athttp://mts.hindawi.com/according to the following timetable:

Manuscript Due July 1, 2009 First Round of Reviews October 1, 2009 Publication Date January 1, 2010

Lead Guest Editor

Antonio F. Bertachini A. Prado,Instituto Nacional de Pesquisas Espaciais (INPE), São José dos Campos, 12227-010 São Paulo, Brazil;prado@dem.inpe.br

Guest Editors

Maria Cecilia Zanardi,São Paulo State University (UNESP), Guaratinguetá, 12516-410 São Paulo, Brazil;


Tadashi Yokoyama,Universidade Estadual Paulista (UNESP), Rio Claro, 13506-900 São Paulo, Brazil;


Silvia Maria Giuliatti Winter,São Paulo State University (UNESP), Guaratinguetá, 12516-410 São Paulo, Brazil;


Hindawi Publishing Corporation http://www.hindawi.com




Related subjects :