Geometric Estimates of the First Eigenvalue of(p,q)-elliptic Quasilinear System Under Integral Curvature Condition

HABIBI VOSTA KOLAEI Mohammad Javad,AZAMI Shahroud

Department of pure Mathematics,Faculty of Science, Imam Khomeini International University,Qazvin,Iran.

Abstract. Consider (M,g) as a complete, simply connected Riemannian manifold.The aim of this paper is to provide various geometric estimates in different cases for the first eigenvalue of(p,q)-elliptic quasilinear system in both Dirichlet and Neumann conditions on Riemannian manifold. In some cases we add integral curvature condition and maybe we prove some theorems under other conditions.

Key Words: Eigenvalue;(p,q)-elliptic quasilinear system;geometric estimate;integral curvature.

1 Introduction

Finding the bound of the eigenvalue for the Laplacian on a given manifold is a key aspects in Riemannian geometry. A major objective of this purpose is to study eigenvalue that appears as solutions of the Dirichlet or Neumann boundary value problems for curvature functions.In the recent years,because of the theory of self-adjoint operators,the spectral properties of linear Laplacian studied extensively.As an important example,mathematicians generally are interested in the spectrum of the Laplacian on compact manifolds with or without boundary or noncompact complete manifolds due to in these two cases the linear Laplacian can be uniquely extended to self-adjoint operators(see[1,2]).

Since the study of the properties of spectrum of Laplacian(specially in Dirichlet condition)in infinitely stretched regions has applications in elasticity,electromagnetism and quantum physics, it attracts attention of many mathematicians and physicists. Recently Mao has proved the existence of discrete spectrum of linear Laplacian on a class of 4-dimensional rotationally symmetric quantum layers, which are noncompact noncomplete manifolds in[3].

ConsiderMas a compact domain in a complete,simply connected Riemannian manifold M. Letu:M?→be a smooth function onMoru∈W1,p(M) whereW1,p(M) is the Sobolev space. Thep-Laplacian ofufor 1

Although the regularity theory of thep-Laplacian is very different from the usual Laplacian, many of the estimates for the first eigenvalue of the Laplacian (for example whenp=2)can be generalized to generalp. As an important example in[4], you can find remarkable results in a case of closed manifolds with bounded Ricci curvature by(n?1)KwhereK>0. The special caseK=0 and general caseK∈are studied in [5] and [6],respectively. Also you can find some similar topics in[7].

In a case ofp-Laplacian also you can find various results in[8]in which Bakery-Emery curvature has a positive lower bound for weightedp-Laplacian and you can find useful results in[9]and[10]to general metric measure space.

Generally studying the eigenvalues of geometric operators are important tools for understanding the Riemannian manifolds. There are many mathematicians who work and give important results in a term of geometric quantities for the spectrum of Laplacian(you can see[11–13]).

1.1 The first eigenvalue of(p,q)-elliptic quasilinear system

In this paper we are going to obtain some geometric estimates for the first eigenvalue of(p,q)-elliptic quasilinear system which is defined as

In this situationλis called an eigenvalue of system (1.2) and (u,v) are eigenfunctions corresponding toλ.

In the term of the first nontrivial eigenvalue of the (p,q)-elliptic quasilinear system(1.2),there are two cases.The first case is Neumann eigenvalue which is defined as below

and A is as same as above.

Zographopoulos in [14] has discussed the existence and uniqueness of the solution of the (p,q)-elliptic quasilinear system (1.2). This type of systems have been found in different cases in physics. For example to the study of transport of electron temperature in a confined plasma and also to the study of electromagnetic phenomena in nonhomogeneous super conductors,you can see[15,16]. Also for more details in electrochemistry and nuclear reaction,you can find useful results in[17]or[18],respectively.

In this section,we also introduce Cheeger’s isoperimetric constant and give the theorem below

Theorem 1.3.LetMn be a Riemannian manifold and consider M?Mas a compact domain. IfC(M)denotes the Cheeger’s isoperimetric constant and p≥q,then for the first eigenvalue of the system(1.2)we have

is obtained from the Faber-Krahn type estimate.

2 Cheng-type estimate

In this section we generalize Cheng-type estimate for the first Dirichlet eigenvalue of the system(1.2). First of all, we have to give some preliminaries, then we are going to give a proof for our first theorem. Considerxas a point in Mn, letρ(x) denote the smallest eigenvalue forRic:TxM?→TxM whereRicis Ricci tensor on Riemannian manifold M.In this case we define

3 Faber-Krahn estimate and Cheeger’s isoperimetric constant

It was shown before by Faber [21] and Krahn [22], the first eigenvalue of the Laplacian on a bounded open setM?2of given area attains it’s minimum value if and only ifMis a disk.

Also you can find as similar as this paper’s results in[23]in which second author has worked in a case of (p,q)-Laplacian system. Here we present our theorem in a term of Faber-Krahn estimate and also we will give an independent corollary of this theorem.

From[24]we have this proposition which help us in giving proof for Theorem 1.2.

Proof of Theorem 1.2. Consideruandvas eigenfunctions corresponding toλ1,p,q(M).Without loss of generality, we can suppose thatuandvare Morse functions to ensure that the level sets ofuandvare closed regular hypersurfaces for almost all values. In this case we start by justuthen we can show similar terms forv,also like before we use ˉuandˉvas eigenfunctions in model space. LetMt:={x∈M|u>t} whereMis a domain in M and consider the re-arrangement ofudefined by

There are another way where for a constant sectional curvature Riemannian manifold Mnwe would be able to give a proof for the inequality (3.2) independently from Theorem 1.2.

Theorem 3.1.Let M be a domain in a complete, simply connected Riemannian manifoldMof constant sectional curvature. Let BK(x)be a geodesic ball of radius K,centred at x∈M,such thatvol(M)=vol(BK(x)),then we have

Proof.We use some identities from co-area formula[13]. Suppose the sets

Consideruandvas a positive eigenfunctions associated withλ1,p,q(M). We define a radially decreasing function ˉu:BK→+,also we introduce the setsMtand?Mtas same as we have seen before. Let?ˉMt={x∈BK:ˉu(x)=t}such that vol(Mt)=vol(BK)for eacht,also d(?Mt)and d(?ˉMt)tare the(n?1)-dimensional volume elements of sets?Mtand?ˉMtrespectively.

By H¨older’s inequality we get

which is what we were trying to prove.

3.1 Cheeger’s isoperimetric constant

Consider Mnas a Riemannian manifold andMas a compact domain in M,from[25]we define Cheeger’s isoperimetric constant C(M)as

where the infimum is taken all over the manifoldMcwith compact closure inMand?Mcis assumed to be smooth.

Proof of Theorem 1.3. Let(u,v)be the pair of eigenfunctions corresponding toλ1,p,q(M)whereu>0,v>0. By applying co-area formula forupwe have

4 Lichnerowicz-type estimate

In this section we are going to focus on giving the proof for our theorem in term of Lichnerowicz estimate. For this proof,we need the following Sobolev inequality in a term of proposition. You can find useful proof of this proposition in[26], for integral curvature and in[27]for Aubry’s diameter estimate.

So we conclude that

which finally implies

In comparison geometry, as a research area, one of the most interesting questions is finding relationship between first eigenvalue in the terms of Dirichlet and Neumann condition.

From [4], if you consideruas a eigenfunction corresponding toλ1,pwhereλ1,pis the first Dirichlet eigenvalue of ?pu, the nodal domain ofudefined as the connected components of Mu?1(0)in which M is the Riemannian manifold.

We have below proposition from[4]

Proposition 4.2.Let u be a first eigenfunction for?p onM, p>1and consider A1=u?1(+)and A2=u?1(?)as nodal domains of u,then

In the case of (p,q)-elliptic quasilinear system (the system (1.2)), if we denoteλ1,p,qas a Dirichlet first eigenvalue then similarly the nodal domain of eigenfunctions (u,v)corresponding to Dirichlet first eigenvalueλ1,p,qis defined as

Proof of Corollary 1.1. Applying the Faber-Krahn type estimate(Theorem 1.2)to the nodal domain,we get the proof.