# Estimates for Eigenvalues of the Elliptic Operator in Divergence Form on Riemannian Manifolds

Authors: Shenyang Tan; Tiren Huang; Wenbin Zhang
Journal: Advances in Mathematical Physics
 (2015)

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## Abstract

We investigate the Dirichlet weighted eigenvalue problem of the elliptic operator in divergence form on compact Riemannian manifolds(M,g,e-ϕdv). We establish a Yang-type inequality of this problem. We also get universal inequalities for eigenvalues of elliptic operators in divergence form on compact domains of complete submanifolds admitting special functions which include the Hadamard manifolds with Ricci curvature bounded below and any complete manifolds admitting eigenmaps to a sphere.


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## Body

## 1. Introduction

Let(M,〈,〉) be an n-dimensional bounded compact Riemannian manifold, ϕ∈C2(M), and dμ=e-ϕdv, where dv is the Riemannian volume measure on (M,〈,〉). Let Δ and ∇ be the Laplacian and the gradient operator on M, respectively. The witten Laplacian (or the drifting Laplacian) with respect to the weighted volume measure μ is given by
(1)Δϕ=Δ-∇ϕ,∇·.In recent years, many mathematicians have paid their attention to the eigenvalue problem of the drifting Laplacian on Riemannian manifolds (see [1–3]). They have studied the following eigenvalue problem:
(2)Δϕu=-λu,inΩ,u|∂Ω=0.
In particular in [4], Xia and Xu got a Payne-Plya-Weinberger-Yang-type inequality of the eigenvalues of this problem:
(3)∑i=1k‍λk+1-λi2≤1n∑i=1k‍λk+1-λi4λi+4ϕ0λi1/2+n2H02+ϕ02,
where H0=supΩ|H|, H is the mean curvature vector, and ϕ0=maxΩ¯|∇ϕ|.In this paper, we consider the following eigenvalue problem:(4)-div⁡A∇u+A∇ϕ,∇u+Vu=λρu,inM,u|∂M=0,
where V is a nonnegative potential function, ρ is a positive function continuous on M¯, and A is symmetric and positive definite matrices. Through integration by part, we can find
(5)∫M‍f-div⁡A∇g+A∇ϕ,∇g+Vgdμ=∫M‍g-div⁡A∇f+A∇ϕ,∇f+Vfdμ,∫M‍f-div⁡A∇g+A∇ϕ,∇gdμ=∫M‍A∇g,∇fdμ,
where f and g are smooth functions on M with f|∂M=g|∂M=0. As we know (see [5]), this problem has a real and discrete spectrum:
(6)0<λ1≤λ2≤⋯≤λr≤↗;
here each eigenvalue is repeated from its multiplicity.In Section2, we get a general inequality for the eigenvalue of the operator in divergence form -div⁡(A∇(·))+〈∇ϕ,∇(·)〉+V through the way of trial function. In Section 3, we obtain a Payne-Plya-Weinberger-Yang-type inequality through defining special trial function. In Section 4, we prove some universal inequalities for eigenvalues of the divergence operator on manifolds admitting special functions.


## 2. A General Inequality

Firstly, we give a useful inequality about the eigenvalues.Theorem 1.
Letλi be the ith eigenvalue of problem (4) and let ui be the orthonormal eigenfunction corresponding to λi; that is,
(7)-div⁡A∇ui+A∇ϕ,∇ui+Vui=λiρui,inM;∫M‍ρuiujdμ=δij,ui|∂M=0.
Then, for any h∈C3(M)∩C2(∂M) and any integer k, we have
(8)∑i=1k‍λk+1-λi2∫M‍ui2∇h,∇hdμ≤ϵ∑i=1k‍λk+1-λi2∫M‍ui2A∇h,∇hdμ+∑i=1k‍λk+1-λiϵ∫M‍1ρ∇ui,∇h+12uiΔh2dμ,
where ϵ is any positive constant.Proof.
We define a trial function(9)φi=hui-∑j=1k‍aijuj,
where aij=∫M‍ρhuiujdμ=aji, and then we have
(10)∫M‍ρφiujdμ=0,φi|∂M=0,for∀i,j=1,…,k.
If we set L=-div⁡(A∇(·))+〈A∇ϕ,∇(·)〉+V, then through direct calculation, we have
(11)Lφi=-div⁡A∇·+A∇ϕ,∇·+Vφi=hLui+uiLh-Vhui-2A∇h,∇ui-∑j=1k‍aijλjuj.
Substituting (11) into the well known Rayleigh-Ritz inequality
(12)λk+1≤∫M‍φiLφi∫M‍ρφi2dμ,
we can get
(13)λk+1-λi∫M‍ρφi2dμ≤∫M‍φiuiLh-Vhui+2A∇h,∇uidμ.
We set(14)bij=∫M‍uiLh-Vhui-2A∇h,∇uiujdμ.
Through direct calculation, we have
(15)bij=-bji,bij=λi-λjaij.
Combining with (13), we get
(16)λk+1-λi∫M‍ρφi2dμ≤-∫M‍huiuidiv⁡A∇h-A∇ϕ,∇hhhhhhhhhhhhh-2∇ui,A∇hdμ+∑j=1k‍λi-λjaij2.
Setting
(17)cij=∫M‍uj∇ui,∇h+12uiΔhdμ,
then through direct calculation, we have
(18)cij=-cji,(19)∫M‍-2φi∇ui,∇h+12uiΔhdμ=-2∫M‍hui∇ui,∇h+12uiΔhdμ+2∑j=1k‍aijcij=∫M‍ui2∇h,∇hdμ+2∑j=1k‍aijcij.
Multiplying (19) by (λk+1-λi)2, we get
(20)λk+1-λi2∫M‍ui2∇h,∇hdμ+2∑j=1k‍aijcij=λk+1-λi2·∫M‍-2ρϕi-∑j=1k‍cijρuj1ρ∇ui,∇h+12uiΔhhhhhhhhhhhhhhhhhhh-∑j=1k‍cijρujdμ≤ϵλk+1-λi3∫M‍ρϕi2dμ+λk+1-λiϵ∫M‍-∑j=1k‍cijρuj1ρ∇ui,∇h+12uiΔhhhhhhhhhhhhhhhhhhh-∑j=1k‍cijρuj2dμ=ϵλk+1-λi3∫M‍ρϕi2dμ+λk+1-λiϵ∫M‍-∑j=1k‍cij21ρ∇ui,∇h+12uiΔh2dμhhhhhhhhhhhhhhhh∫M‍-∑j=1k‍cij21ρ∇ui,∇h+12uiΔh2dμ-∑j=1k‍cij2≤ϵλk+1-λi2-∫M‍huiuidiv⁡A∇h∑j=1k‍λi-λjhhhhhhhhhhhhhhhhhhhhhhhhhhhh-A∇ϕ,∇hhhhhhhhhhhhhhhhhhhhhhhhhh+2∇ui,A∇hdμ≤ϵλk+1-λi2hh+∑j=1k‍λi-λjaij2+λk+1-λiϵ·∫M‍1ρ∇ui,∇h+12uiΔh2dμ-∑j=1k‍cij2,
where ϵ is any positive constant. Summing over i from 1 to k, we have
(21)∑j=1k‍λk+1-λi2∫M‍ui2∇h,∇hdμ-2∑i,j=1k‍λk+1-λiλi-λjaijcij≤ϵ∑i=1k‍λk+1-λi2-∫M‍huiuidiv⁡A∇hhhhhhhhhhhhhhhhhhhihhhhhhhhhh-A∇ϕ,∇hhhhhhhhhhhhhhhhhhhihhhhhhhh∫M‍huiuidiv⁡A∇h+2∇ui,A∇hdμ+∑i=1k‍λk+1-λiϵ∫M‍1ρ∇ui,∇h+12uiΔh2dμ∑i,j=1k‍λk+1-λihhhhhhhhhhhhhhhhhh-∑i,j=1k‍λk+1-λiϵλi-λj2aij2-∑i,j=1k‍λk+1-λiϵcij2.
Because of aij=aji, cij=-cji, we infer
(22)∑j=1k‍λk+1-λi2∫M‍ui2∇h,∇hdμ≤ϵ∑i=1k‍λk+1-λi2·-∫Mhuiuidiv⁡A∇hhhhhhhhhhhhhhhhh∫M‍h-A∇ϕ,∇h+2∇ui,A∇hdμ+∑i=1k‍λk+1-λiϵ∫M1ρ∇ui,∇h+12uiΔh2dμ.
Considering the property of the measure on this weighted manifold that dμ=e-ϕdv, we can refer to the fact that
(23)∫M‍hui2div⁡A∇hdμ=∫M‍hui2div⁡A∇he-ϕdv=-∫M‍A∇h,∇hui2e-ϕdv-∫M‍A∇h,∇ui2huie-ϕdv+∫M‍A∇h,∇ϕhui2e-ϕdv=-∫M‍A∇h,∇hui2dμ-∫M‍A∇h,∇ui2huidμ+∫M‍A∇h,∇ϕhui2dμ.
Substituting (23) into (22), we can finish the proof of Theorem 1.


## 3. The Main Theorem and the Proof

In this section, we give some estimates about the eigenvalues of the operator in divergence form.Lemma 2.
LetM be an n-dimensional complete Riemannian manifold and let Ω be a bounded domain with smooth boundary and let ϕ be a smooth function on Ω in M; A is a symmetry and positive definite matrix; suppose λi be the ith eigenvalue of the problem:
(24)-div⁡A∇ui+A∇ϕ,∇ui+Vui=λρui,inΩ;∫Ω‍ρuiujdμ=δij;ui|∂Ω=0.
If M is isometrically immersed in Rm with mean curvature vector H, then
(25)n∑i=1k‍λk+1-λi2∫Ω‍ui2dμ≤ϵ∑i=1k‍λk+1-λi2∫Ω‍ui2tr⁡Adμ+∑i=1k‍λk+1-λiϵ∫Ω‍1ρ∇ui2+14ui2n2H2dμ.Proof.
Letxα, α=1,2,…,m be the standard coordinate functions of Rm. Taking h=xα in (8), summing over α from 1 to m, we have
(26)∑i=1k‍λk+1-λi2∑α=1m‍∫Ω‍ui2∇xα,∇xαdμ≤ϵ∑i=1k‍λk+1-λi2∫Ω‍ui2∑α=1m‍A∇xα,∇xαdμ+∑i=1k‍λk+1-λiϵ∫Ω‍1ρ∑α=1m‍∇ui,∇xα+12uiΔxα2dμ=ϵ∑i=1k‍λk+1-λi2∫Ω‍ui2∑α=1m‍A∇xα,∇xαdμ+∑i=1k‍λk+1-λiϵ∫Ω‍1ρ∑α=1m‍∇ui,∇xα2+14ui2Δxα2hhhhhhhhhhhhhhhhhhhh+ui∇ui,Δxα∇xαdμ.
Since M is isometrically immersed in Rm, we have
(27)∑α=1m‍∇xα2=n,
and then,
(28)∫Ω‍ui2∑α=1m‍∇xα2dμ=n∫Ω‍ui2dμ.
Also, we have
(29)Δx1,x2,…,xm≡Δx1,Δx2,…,Δxm=nH.
Lete1,…,en be orthonormal tangent vector fields locally defined on M; we have
(30)∑α=1n+1‍A∇xα,∇xα=∑α=1n+1‍A∑i=1n‍∇xα,eiei,∑j=1n‍∇xα,ejej=∑i,j=1n∑α=1n+1‍eixαejxαAei,ej=∑i,j=1n‍ei,ejAei,ej=∑in‍Aei,ei=tr⁡A,
and then,
(31)∑α=1m‍∫Ω‍ui2A∇xα,∇xαdμ≤∫Ω‍ui2tr⁡Adμ,∑α=1m‍Δxα∇xα,∇ui=∑α=1m‍Δxα∇uixα=nH,∇ui=0.
Substituting (28), (29), and (31) into (26), we can finish the proof of Lemma 2.Theorem 3.
Under the same assumption of Lemma2, let τ=(sup⁡Ωρ)-1, σ=(inf⁡Ωρ)-1, V0=min⁡ΩV, |∇ϕ|≤C0, |H|≤H0, ξ1I≤A≤ξ2I, and then one has
(32)∑i=1k‍λk+1-λi2≤4ξ2σ2nτ2·∑i=1n‍λk+1-λihhhhhh·12ξ1+ξ22C02σ-4ξ1V0τ+4ξ1λi1/2ξ2C0σhhhhhhhhhhhh+ξ22C02σ-4ξ1V0τ+4ξ1λi1/2hhhhhhhhh+14σn2H0212ξ1+ξ22C02σ-4ξ1V0τ+4ξ1λi1/2ξ2C0σ.Proof.
Obviously, we have(33)σ=σ∫Ω‍ρui2dμ≥∫Ω‍ui2dμ≥τ∫Ω‍ρui2dμ=τ.
Multiplying the equation
(34)-div⁡A∇ui+A∇ϕ,∇ui+Vui=λiρui
by ui and integrating on Ω, we have
(35)λi=∫Ω‍A∇ui,∇uidμ+∫Ω‍uiA∇ϕ,∇uidμ+∫Ω‍Vui2dμ≥ξ1∫Ω‍∇ui2dμ+∫Ω‍uiA∇ϕ,∇uidμ+∫Ω‍Vui2dμ.
Considering
(36)∫Ω‍uiA∇ϕ,∇uidμ≤ξ2∫Ω‍ui∇ϕ∇uidμ≤ξ2C0∫Ω‍ui2dμ1/2∫Ω‍∇ui2dμ1/2≤ξ2C0σ∫Ω‍∇ui2dμ1/2,
then we can obtain
(37)λi≥ξ1∫Ω‍∇ui2dμ+∫Ω‍uiA∇ϕ,∇uidμ+∫Ω‍Vui2dμ≥ξ1∫Ω‍∇ui2dμ-ξ2C0σ∫Ω‍∇ui2dμ1/2+V0τ.
Solving this inequality, we have
(38)∫Ω‍∇ui2dμ≤12ξ1ξ2C0σ+ξ22C02σ2-4ξ1V0τ+4ξ1λi1/2.
Substituting (33) and (38) into (25) and taking
(39)ϵ=AB1/2,
where
(40)A=∑i=1n‍λk+1-λiσ·12ξ1+ξ22C02σ-4ξ1V0τ+4ξ1λi1/2ξ2C0σhhhhhhhhhhh+ξ22C02σ-4ξ1V0τ+4ξ1λi1/2hhhhhhh+14σn2H02,B=∑i=1n‍λk+1-λi2nξ2σ,
we can finish the proof of Theorem 3.Remark 4.
If we setϕ=constant, A=I, ρ=1 and V=0, the divergence operator becomes the usual laplace operator on Riemannian manifolds and we can find our result is sharper than the result in [4, 6].Remark 5.
For some of the recent developments about universal inequalities for eigenvalues on Riemannian manifolds, we refer to [6–12] and the references therein.


## 4. Eigenvalues on Manifolds Admitting Special Functions

In this section, we get some universal inequalities for eigenvalues of the divergence operator on manifolds admitting special functions.Theorem 6.
LetM be an n-dimensional complete Riemannian manifold and let Ω be a bounded domain with smooth boundary and let ϕ be a smooth function on Ω in M; A is a symmetry and positive definite matrix; let τ=(sup⁡Ωρ)-1, σ=(inf⁡Ωρ)-1, V0=min⁡ΩV, |∇ϕ|≤C0, |H|≤H0, ξ1I≤A≤ξ2I; suppose λi be the ith eigenvalue of the problem:
(41)-div⁡A∇ui+A∇ϕ,∇ui+Vui=λρui,inΩ;∫Ω‍ρuiujdμ=δij;ui|∂Ω=0;
if there exists a function θ:Ω→R and a constant A0 such that
(42)∇θ=1,Δθ≤A0,onΩ,
then
(43)∑i=1k‍λk+1-λi2≤8ξ2σ2nτ2∑i=1n‍λk+1-λihhhhhhhhh·12ξ1+ξ22C02σ-4ξ1V0τ+4ξ1λi1/2ξ2C0σhhhhhhhhhhhhhhh+ξ22C02σ-4ξ1V0τ+4ξ1λi1/2hhhhhhhhhh+14σA02.Proof.
Takingh=θ in (8) and considering (38), (42), and 〈∇θ,A∇θ〉≤ξ2|∇θ|2, we have
(44)τ∑i=1k‍λk+1-λi2≤∑i=1k‍λk+1-λi2∫Ω‍ui2dμ≤ϵ∑i=1k‍λk+1-λi2∫Ω‍ui2∇θ,A∇θdμ+∑i=1k‍λk+1-λiϵ∫Ω‍1ρ∇ui,∇θ+12uiΔθ2dμ≤ϵσξ2∑i=1k‍λk+1-λi2+2σ∑i=1k‍λk+1-λiϵ∫Ω‍∇ui2+A02ui24dμ≤ϵσξ2∑i=1k‍λk+1-λi2+2σ∑i=1k‍λk+1-λiϵhhhhhhhh·12ξ1ξ2C0σξ22C02σ2-4ξ1V0τ+4ξ1λi1/2A024hhhhhhhhhhhhhhh+ξ22C02σ2-4ξ1V0τ+4ξ1λi1/2hhhhhhhhhhhh+σA024.
Taking
(45)ϵ=2∑i=1kλk+1-λiξ2∑i=1k‍λk+1-λi2-1·12ξ1ξ2C0σξ22C02σ2-4ξ1V0τ+4ξ1λi1/2A024hhhhhhhhhhh∑i=1kλk+1-λi+ξ22C02σ2-4ξ1V0τ+4ξ1λi1/2+σA024·ξ2∑i=1k‍λk+1-λi2-11/2,
we can complete the proof of Theorem 6.Remark 7.
LetM be an n-dimensional connected complete Riemannian manifold; suppose its Ricci curvature satisfies Ricm≥-(n-1)c2, c≥0. If there exists a smooth function θ satisfying |∇θ|=1, then |Δθ|≤(n-1)c2. So the Bussemann functions on Cartan-Hadamard manifolds with Ricci curvature bounded below satisfy the condition in Theorem 6.Theorem 8.
LetM be an n-dimensional complete Riemannian manifold and let Ω be a bounded domain with smooth boundary; let ϕ be a smooth function on Ω in M; A is a symmetry and positive definite matrix; let τ=(sup⁡Ωρ)-1, σ=(inf⁡Ωρ)-1, V0=min⁡ΩV, |∇ϕ|≤C0, |H|≤H0, ξ1I≤A≤ξ2I; suppose λi be the ith eigenvalue of the problem:
(46)-div⁡A∇ui+A∇ϕ,∇ui+Vui=λρui,inΩ;∫Ω‍ρuiujdμ=δij;ui|∂Ω=0;
if Ω admits an eigenmap f=(f1,f2,…,fm+1):Ω→Sm corresponding to an eigenvalue η, that is
(47)Δfα=-ηfα,α=1,…,m+1,∑α=1m+1‍fα2=1,
then
(48)∑i=1k‍λk+1-λi2≤2ξ2σ2τ2·∑i=1n‍λk+1-λi·1ξ1+ξ22C02σ-4ξ1V0τ+4ξ1λi1/2ξ2C0σhhhhhhhhhhhh+ξ22C02σ-4ξ1V0τ+4ξ1λi1/2+ση21ξ1.Proof.
Because of (47), we obtain
(49)∑α=1m+1‍∇fα2=η.
Taking h=fα in (8) and summing over α from 1 to m+1, we get
(50)ητ∑i=1k‍λk+1-λi2≤∑i=1k‍λk+1-λi2∑α=1m+1‍∫Ω‍ui2∇fα2dμ≤ϵ∑i=1k‍λk+1-λi2∫Ω‍ui2∑α=1m+1‍∇fα,Afαdμ+∑i=1k‍λk+1-λiϵ∫Ω‍1ρ∑α=1m+1‍∇ui,∇fα-12ηfαui2dμ≤ϵησξ2∑i=1k‍λk+1-λi2+2σ∑i=1k‍λk+1-λiϵ∫Ω‍η∇ui2+η2ui24dμ≤ϵησξ2∑i=1k‍λk+1-λi2+σ∑i=1k‍λk+1-λiϵ·ηξ1ξ2C0σξ22C02σ2-4ξ1V0τ+4ξ1λi1/2hhhhhhhhhhhhh+ξ22C02σ2-4ξ1V0τ+4ξ1λi1/2hhhhhhhhhhh+ση22.
Taking
(51)ϵ=·ξ2∑i=1k‍(λk+1-λi)2-1∑i=1k‍λk+1-λihhhhh·1ξ1ξ22C02σ2-4ξ1V0τ+4ξ1λi1/2ξ2C0σhhhhhhhhhhh+ξ22C02σ2-4ξ1V0τ+4ξ1λi1/2+ση21ξ1∑i=1k‍λk+1-λi·ξ2∑i=1k‍(λk+1-λi)2-11/2,
then the proof of Theorem 8 is finished.Remark 9.
Any compact homogeneous Riemannian manifold admits eigenmaps to some unit sphere for the first positive eigenvalues of the Laplacian which satisfy the condition in Theorem8 [13].


## 5. Physical Interpretation

In quantum mechanics, eigenvalue is the dynamics of macro possible values. The wave function is superposition of a number of eigenstates. Different eigenstate is corresponding to the specific eigenvalue (of course there may be degenerate case; namely, the same eigenvalue corresponds to different intrinsic state). The experimental measurement of the mechanical quantity must be one of eigenvalues, and wave function in the measurement is the eigenstate of the corresponding eigenvalue. The gap between different eigenvalues means the difference between the energy levels. That is why many researchers pay much attention to this problem. In this paper, we find a relatively accurate upper bound between any two different eigenvalues.



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