Eigenvalue inequalities of elliptic operators in ...

Zhu et al. Journal of Inequalities and Applications (2016) 2016:191 DOI 10.1186/s13660-016-1130-0

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Eigenvalue inequalities of elliptic operators in weighted divergence form on smooth metric measure spaces Yuming Zhu, Gusheng Liu and Feng Du* *

Correspondence: [email protected] School of Mathematics and Physics Science, Jingchu University of Technology, Jingmen, 448000, China

Abstract In this paper, we study the eigenvalue problem of elliptic operators in weighted divergence form on smooth metric measure spaces. First of all, we give a general inequality for eigenvalues of the eigenvalue problem of elliptic operators in weighted divergence form on compact smooth metric measure space with boundary (possibly empty). Then applying this general inequality, we get some universal inequalities of Payne-Pólya-Weinberger-Yang type for the eigenvalues of elliptic operators in weighted divergence form on a connected bounded domain in the smooth metric measure spaces, the Gaussian shrinking solitons, and the general product solitons, respectively. MSC: 35P15; 53C20; 53C42 Keywords: eigenvalue; universal inequalities; drifting Laplacian; elliptic operators in weighted divergence form; Gaussian shrinking soliton; general product soliton

1 Introduction A smooth metric measure space is actually a Riemannian manifold equipped with some measure which is absolutely continuous with respect to the usual Riemannian measure. More precisely, for a given complete n-dimensional Riemannian manifold (M, , ) with the metric , , the triple (M, , , e–f dν) is called a smooth metric measure space, where f is a smooth real-valued function on M and dν is the Riemannian volume element related to , (sometimes, we also call dν the volume density). Let be a bounded domain in a smooth metric measure space (M, , , e–f dν), and let A : → End(T) be a smooth symmetric and positive definite section of the bundle of all endomorphisms of T, we can define the elliptic operator in weighted divergence form as Lf = – divf A∇,

(.)

where divf X = ef div(e–f X) is the weighted divergence of vector fields X, and ∇ is the gradient operator. When A is an identity map, –Lf becomes the drifting Laplacian f , for the drifting Laplacian, some universal inequalities have been given in [–]. When f is a constant, Lf becomes the elliptic operator in divergence form, for some recent developments about universal inequalities of the eigenvalue of elliptic operator in divergence form © 2016 Zhu et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Zhu et al. Journal of Inequalities and Applications (2016) 2016:191

Page 2 of 15

on Riemannian manifolds, we refer to [–] and the references therein. As briefly mentioned above, it is a natural problem how to get the universal inequalities of the eigenvalues of elliptic operator in weighted divergence form. Actually, in this paper, we first consider the eigenvalue problem as follows: (Lf + V )u = λρu, in , (.) u = , on ∂, where is a bounded domain in a complete smooth metric measure space (M, , , e–f dν), V is a non-negative continuous function on M, and ρ is a weight function which is positive and continuous on M. For the eigenvalues of (.), we can give the following universal inequalities. Theorem . Let be a connected bounded domain in an n-dimensional complete smooth metric measure space (M, , , e–f dν) . Assume that ξ I ≤ A, tr(A) ≤ nξ throughout , and ρ ≤ ρ(x) ≤ ρ , |∇f |(x) ≤ C , ∀x ∈ , here I is the identity map, ξ , ξ , ρ , ρ , C are positive constants and tr(A) denotes the trace of A. Let λi be the ith eigenvalue of the eigenvalue problem (.), then we have k

(λk+ – λi )

i=

k ξ ρ λi – ρ– V λi – ρ– V  n H + C (λ – λ ) + C + , k+ i  ξ ξ ρ nρ i= 

≤

(.)

where H = supx∈ |H|(x), V = minx∈ V (x), and H is the mean curvature vector field of M in a Euclidean space Rm . Remark . From inequality (.), we can get some results which are given in [, ], for example, if f is a constant, then C = , (.) becomes (.) in []. For the fourth-order elliptic operator in weighted divergence, we can consider the following eigenvalue problem: Lf u = u, in , u = ∂u = , on ∂, ∂ν

(.)

we also give some universal inequalities for the eigenvalues of (.) as follows. Theorem . Let be a connected bounded domain in an n-dimensional complete smooth metric measure space (M, , , e–f dν). Assume that ξ I ≤ A ≤ ξ I throughout , and |∇f |(x) ≤ C , ∀x ∈ , here I is the identity map, ξ , ξ , C are positive constants. Let i be the ith eigenvalue of the eigenvalue problem (.), then we have k    ξ ( k+ – i ) ≤ ( k+ – i ) (n + ) i + C ξ i + ξ n H + C nξ i= i=

k



k    ( k+ – i )  i + C ξ i + ξ n H + C × i=

 ,

(.)


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where H = supx∈ |H|(x), and H is the mean curvature vector field of M in a Euclidean space Rm . Remark . From inequality (.), we can get some results which are given in [, ], for example, if A is an identity map, then ξ = ξ = , (.) becomes (.) in []. On smooth metric measure spaces, we can also define the so-called weighted Ricci curvature Ricf given by Ricf = Ric + Hess f , which is also called the ∞-Bakry-Émery Ricci tensor. The equation Ricf = κ, for some constant κ is just the gradient Ricci soliton equation, which plays an important role in the study of Ricci flow. We refer the reader to [] for some recent progress about Ricci solitons. For κ > , κ = , or κ < , the gradient Ricci soliton (M, , , e–f dν, κ) is called shrinking, steady, or expanding, respectively. In the following, we would like to give two examples of Ricci solitons. 



Example  The Gaussian shrinking soliton (Rn , , can , e–  |x| dν,  ), where , can is the standard Euclidean metric on Rn , f =  |x| , x ∈ Rn , and Ricf =  , can . Example  More generally, consider the Riemannian product × Rn , where (; , ) is an Einstein manifold satisfying Ric = κ, . Define the smooth function f : × Rn → R by setting f (p; x) = κ |x| + x · a + b, for any a ∈ Rn \ {} and b ∈ R. Then ( × Rn , , + , can , e–f dν ⊗ dνRn , κ) is a non-trivial gradient Ricci soliton. Similarly, one could even construct gradient Ricci solitons with a warped product structure. More details of the product solitons can be found in the Remark . in []. In the following, we will give some universal inequalities for the Dirichlet eigenvalues in a connected bounded domain on the Gaussian shrinking solitons and general product solitons. Theorem . Let be a connected bounded domain in the Gaussian shrinking soliton   (Rn , , can , e–  |x| dν,  ), and assume that ξ I ≤ A, tr(A) ≤ nξ throughout , here I is the identity map, ξ , ξ are positive constants and tr(A) denotes the trace of A. Let λi be the ith eigenvalue of the Dirichlet problem Lf u = λu, in , u = , on ∂,

(.)

where f =  |x| , then we have k  ξ ξ . n – min |x| (λk+ – λi ) ≤ (λk+ – λi ) λi + nξ i=  x∈ i=

k



(.) 

Remark . (i) For a self-shrinker, the drifting Laplacian f with f = |x| is actually the operator L := –  x, ∇(·), which was introduced by Colding-Minicozzi [] to study self-shrinker hypersurfaces. For the Dirichlet problem of the operator L, some univer-


Page 4 of 15

sal inequalities have been obtained by Cheng and Peng []. In this case, our results can be regarded as conclusions for the Dirichlet problem of the elliptic operator in weighted divergence form. (ii) Let b = ξξ , using the recursive formula in Cheng and Yang [], we can infer from (.) that λk+ +

ξ ξ b , n – min |x| ≤ C (n, k)k n λ + n – min |x|   x∈ x∈

where C (n, k) ≤  +

b n

(.)

is a constant (see []).

Theorem . Let be a connected bounded domain in the gradient product Ricci soliton κt 

( × R, , , e–  dν, κ), where is an Einstein manifold with constant Ricci curvature κ. Set x = (x, t) ∈ , where x ∈ , t ∈ R, and assume that ξ I ≤ A ≤ ξ I throughout , here I is the identity map, ξ , ξ are positive constants. Let λi be the ith eigenvalue of the eigenvalue  problem (.), where f = κt , then we have k

(λk+ – λi ) ≤

i=

k ξ (λk+ – λi ) λi + ξ – κ  ξ min |t| . ξ i= (x,t)∈

(.)

2 A general inequality In this section, we will prove a general inequality, which will play a key role in the proof of our main results which are listed in Section . Lemma . Let (M, , , e–f dν) be an n-dimensional compact smooth metric measure space with boundary ∂M (possibly empty), and let a, b be the random non-negative constants and a + b = . Let λi be the ith eigenvalue of the eigenvalue problem of the fourth-order elliptic operator in weighted divergence form with weight ρ such that (aLf + bLf + V )u = λρu, in M, u = ∂u = , on ∂M, ∂ν and ui be the orthonormal eigenfunction corresponding to λi , that is, ⎧  ⎪ ⎨(aLf + bLf + V )ui = λi ρui , in M, i ui = ∂u = , on ∂M, ∂ν ⎪ ⎩ ∀i, j = , , . . . , M ρui uj dμ = δij , where dμ = e–f dν. Then, for any h ∈ C  (M), we have k

(λk+ – λi ) M

i=

≤

k

ui |∇h| dμ

δ(λk+ – λi )

+

hui pi dμ M

i=

k λk+ – λi i=

δ

M

ui f h   ∇h, ∇ui + dμ, ρ 

(.)


Page 5 of 15

where δ is any positive constant and pi = –a ∇h, A∇(Lf ui ) + aLf hLf ui – aLf ∇h, A∇ui + aLf (ui Lf h) – b∇h, A∇ui + bui Lf h. Proof Let ϕi = hui – Then we have ϕi |∂M =

k

j= aij uj ,

∂ϕi = , ∂ν ∂M

here k ≥  is any integer and aij =

k

j= M ρhui uj dμ

= aji .

and

ρϕi uj dμ = ,

∀i, j = , . . . , k.

M

By the Rayleigh-Ritz inequality, we get

λk+ M

ρϕi dμ ≤

M

ϕi aLf + bLf + V ϕi dμ.

(.)

From the definition of Lf , we have Lf (hui ) = – divf A∇(hui ) = –ef div e–f A(h∇ui + ui ∇h) = – div A(h∇ui + ui ∇h) – ∇f , A(h∇ui + ui ∇h) = –h divf (A∇ui ) – ∇h, A∇ui – ui divf (A∇h) – ∇ui , A∇h = hLf ui – ∇h, A∇ui + ui Lf h

(.)

and Lf (hui ) = Lf hLf ui – ∇h, A∇ui + ui Lf h = hLf ui –  ∇h, A∇(Lf ui ) + Lf hLf ui – Lf ∇h, A∇ui + Lf (ui Lf h).

(.)

It follows from (.) and (.) that

aLf + bLf + V (hui ) = λi ρhui + pi ,

where pi is defined by pi = –a ∇h, A∇ Lf (ui ) + aLf hLf ui – aLf ∇h, A∇ui + aLf ui L(h) – b∇h, A∇ui + bui Lf h. Let us compute M

ϕi aLf + bLf + V ϕi dμ

= M

ϕi aLf + bLf + V (hui ) dμ

(.)


Page 6 of 15

= λi

ϕi ρhui dμ +

ϕi pi dμ

M

M

= λi M

ρϕi dμ +

hui pi dμ –

k

M

aij bij ,

(.)

j=

where bij is defined by bij = M pi uj dμ. On the other hand, by (.) and (.), we have

(λk+ – λi ) M

ρϕi dμ ≤

hui pi dμ –

k

M

aij bij .

(.)

j=

By a similar computation to (.)-(.) in [], we have

Lf uj A∇h, ∇ui dμ – 

 M

=– M

hui Lf uj dμ +

M

M

uj Lf ui Lf h dμ,

(.)

M

Lf ui ∇h, A∇uj dμ +

M

uj Lf (ui Lf h) dμ = M

M

Lf uj ∇h, A∇ui dμ –

=

ui Lf uj Lf h dμ –

uj ∇h, A∇(Lf ui ) dμ M

huj Lf ui dμ +

uj Lf ∇h, A∇ui dμ + M

Lf ui A∇h, ∇uj dμ M

uj Lf ui Lf h dμ,

(.)

M

(.)

ui Lf uj Lf h dμ, M

and

uj –∇h, A∇ui + ui Lf h dμ =

M

hui Lf uj dμ – M

huj Lf ui dμ.

(.)

M

Combining (.)-(.) and a similar calculation to (.) in [], we get bij =

pi uj dμ = (λj – λi )aij .

(.)

M

We infer from (.) and (.) that

(λk+ – λi ) M

Setting tij =

ρϕi dμ ≤

hui pi dμ – M

M uj (∇h, ∇ui +

k (λj – λi )aij .

(.)

j=

ui f h ) dμ, 

thus cij = –cji and

ui f h dμ –ϕi ∇h, ∇ui +  M k –hui ∇h, ∇ui – hui f h dμ +  aij cij = M

j=

= M

ui |∇h| dμ + 

k j=

aij cij .

(.)


Page 7 of 15

Using (.), (.), and the Schwarz inequality, we can get (λk+ – λi )

 M

ui |∇h| dμ +  √



– ρϕi M

≤ δ(λk+ – λi ) M

aij cij

M

k ui f h  √ – cij ρuj dμ √ ∇h, ∇ui + ρ  j=

ρϕi dμ

λk+ – λi + δ

 k ui f h  √ – cij ρuj dμ √ ∇h, ∇ui + ρ  j=

≤ δ(λk+ – λi )



hui pi dμ –

k

M

λk+ – λi + δ

j=

= (λk+ – λi )

k

M

(λj – λi )aij

j=

k ui f h    ∇h, ∇ui + – cij , ρ  j=

(.)

where δ is any positive constant. Summing over i from  to k in (.) and noticing aij = aji , cij = –cji , we have k

(λk+ – λi )

 M

i=

≤

k

ui |∇h| dμ – 

δ(λk+ – λi )



hui pi dμ + M

k

(λk+ – λi )(λi – λj )aij cij

i,j=

i=

–

k


δ(λk+ – λi )(λj – λi ) aij –

i,j=

δ

k λk+ – λi i,j=

δ

M

cij ,

ui f h   ∇h, ∇ui + dμ ρ  (.)

which implies that k

(λk+ – λi )

M

i=

≤



k i=

ui |∇h| dμ

δ(λk+ – λi )

hui pi dμ + M

This completes the proof of Lemma ..


δ

M

ui f h   ∇h, ∇ui + dμ. ρ 

3 Proof of Theorem 1.1 and Theorem 1.3 In this section, we will give the proof of Theorem . and Theorem . by using Lemma .. Proof of Theorem . From the Nash embedding theorem, we know that there exists an isometric immersion from a complete Riemannian manifold M into a Euclidean space Rm . Thus, M can be considered as an n-dimensional complete isometrically immersed submanifold in Rm . Let y , y , . . . , ym be the standard coordinate functions of Rm . Then we


Page 8 of 15

have m

|∇yα | = n,

(.)

α=

(y , y , . . . , ym ) = (y , y , . . . , ym ) = nH, m

∇yα , ∇f  =

m

α= m

∇yα , ∇ui  =

= |∇f | ,

(.)

m  ∇ui (yα ) = |∇ui | ,

(.)

α=

∇yα , ∇ui ∇yα , ∇f =

α= m



α=

α= m

∇f (yα )

(.)

m

∇ui (yα )∇f (yα ) = ∇ui , ∇f ,

(.)

α=

yα ∇yα , ∇ui =

α=

m

yα ∇ui (yα ) = nH, ∇ui = ,

(.)

α=

and m

yα ∇yα , ∇f =

α=

m

yα ∇f (yα ) = nH, ∇f = .

(.)

α=

Then we infer from (.)-(.) that m

f yα ∇yα , ∇ui =

α=

m yα – ∇yα , ∇f ∇yα , ∇ui = ∇ui , ∇f

(.)

α=

and m

(f yα ) =

α=

m

 yα – ∇yα , ∇f

α=

=

m

(yα ) – yα ∇yα , ∇f + ∇yα , ∇f 

α=

= n |H| + |∇f | .

(.)

Let a = , b =  in (.), then taking h = yα and summing over α, and noticing ρ– ≤ ui  ≤ ρ– , we get nρ–

k i=

(λk+ – λi ) ≤ 

k

δ(λk+ – λi )



yα ui pαi dμ

α=

i=

+

m

k m ui f yα  λk+ – λi  ∇yα , ∇ui + dμ, δ  ρ α= i=

where pαi = –∇yα , A∇ui + ui Lf yα . Since

yα ui ∇yα , A∇ui dμ =

–

ui ∇yα , A∇yα dμ –

yα ui Lf yα dμ,

(.)


we infer from above equality and m

yα ui pαi dμ =

m

α=

Page 9 of 15

m

α= ∇yα , A∇yα = tr(A) ≤ nξ

that

yα ui –∇yα , A∇ui + ui Lf yα dμ

α=

=

m α=

ui ∇yα , A∇yα dμ ≤ nξ ui  ≤ nξ ρ– .

(.)

From ρ– ≤ ui  ≤ ρ– and A ≥ ξ I, we have

λi =

ui (Lf + V )ui dμ =

–ui divf (A∇ui ) dμ +

∇ui , A∇ui dμ +

=

Vui dμ

Vui dμ ≥ ξ ∇ui  + ρ– V ,

which implies ∇ui  ≤

λi – ρ– V . ξ

(.)

Using the Schwarz inequality and the above inequality, we have

∇f , ∇ui dμ ≤



 λi – ρ– V  |∇f ||∇ui | dμ ≤ C ∇ui   ≤ C . ξ

(.)

Combining (.), (.), (.), (.), and (.), we have

m ui f yα   dμ ∇yα , ∇ui + ρ α= 

m ui (f yα )   ∇yα , ∇ui + ui f yα ∇yα , ∇ui + = dμ  ρ α=  ui     dμ |∇ui | + ∇f , ∇ui + n |H| + |∇f | =  ρ   λi – ρ– V    λi – ρ– V  n H + C . ≤ + C + ρ ξ ξ ρ

(.)

Substituting (.) and (.) into (.), we have k i=

(λk+ – λi ) ≤

k i=

δ(λk+ – λi )

ρ ξ ρ

k λi – ρ– V  λk+ – λi ρ λi – ρ– V + + C δ nρ ξ ξ i= n H + C . + ρ 


Page 10 of 15

Taking k

i= (λk+

δ=

λ –ρ – V

– λi ){ i ξ k

+ C (

i= (λk+

λi –ρ– V  ) ξ

+

n H +C  }  ρ

– λi ) nξ

,

we can get (.). This completes the proof of Theorem ..

Proof of Theorem . Let a = , b = , V ≡ , ρ ≡  in (.), then taking h = yα and summing over α, where {yα }m α= are defined as above, we get

n

k k m ( k+ – i ) ≤ δ( k+ – i ) yα ui pαi dμ i=

α=

i=

+

k m ui f yα  k+ – i ∇yα , ∇ui + dμ, δ  α= i=

(.)

where pαi = –∇yα , A∇(Lf ui ) + Lf yα Lf ui – Lf (∇yα , A∇ui ) + Lf (ui Lf yα ). By a direct computation, we have yα ui pαi dμ

yα ui – ∇yα , A∇(Lf ui ) + Lf yα Lf ui – Lf ∇yα , A∇ui

=

+ Lf (ui Lf yα ) dμ

=  ui Lf ui ∇yα , A∇yα + yα Lf ui ∇ui , A∇yα – yα ui Lf yα Lf ui dμ

+

yα ui Lf yα Lf ui dμ

Lf yα ui + yα Lf ui – ∇yα , A∇ui –∇yα , A∇ui + ui Lf yα dμ

+

ui Lf ui ∇yα , A∇yα dμ +

=

∇yα , A∇ui  dμ

ui Lf yα ∇yα , A∇ui dμ +

–

(ui Lf yα ) dμ.

(.)

Since ξ I ≤ A ≤ ξ I, we can infer from (.)-(.) that m α=

ui Lf ui ∇yα , A∇yα dμ ≤ nξ

ui Lf ui dμ



 ≤ nξ ui  Lf ui   = nξ i , m ∇yα , A∇ui  dμ = A∇ui  ≤ ξ ∇ui , A∇ui

α=

(.)



= ξ

ui Lf ui dμ ≤ ξ i ,

(.)


Page 11 of 15

m – ui Lf yα ∇yα , A∇ui dμ

α=

≤ ξ ui f yα ∇yα , A∇ui dμ

= ξ ui ∇f , A∇ui dμ

ui |∇f ||∇ui | dμ

≤ ξ

 ≤ C ξ ui  A∇ui    

= C ξ

   ∇ui , A∇ui dμ ≤ C ξ i ,

(.)

and m α=

(ui Lf yα ) dμ ≤ 

ξ

m α=

= ξ ≤

ξ

ui (f yα ) dμ

ui n |H| + |∇f | dμ

  n H + C .

(.)

Combining (.)-(.), we have m α=

   yα ui pαi dμ ≤ ξ (n + ) i + C ξ i + ξ n H + C .

Since ∇ui ≤ 

 ξ

∇ui , A∇ui dμ

=

 ξ

(.)



ui Lf ui dμ

≤

i ξ

, then from (.), we have

m ui f yα  dμ ∇yα , ∇ui +  α= u |∇ui | + ∇f , ∇ui + i n |H| + |∇f | dμ =  



 C i n H + C ≤ i + . +  ξ  ξ

(.)

Taking (.) and (.) into (.), we have

n

k k    ( k+ – i ) ≤ δ( k+ – i ) ξ (n + ) i + C ξ i + ξ n H + C i=

i=

k k+ – i  

+

i

i=

δ

ξ



+

C i 

ξ

+

n H + C . 

(.)


Page 12 of 15

Let 

k

i= ( k+ – i ){

δ= k

i ξ



+

C i  ξ



+ 

n H +C } 





,

       i= ( k+ – i ) ξ ((n + ) i + C ξ i + ξ (n H + C ))

we can infer from (.) that k    ξ ( k+ – i ) ≤ ( k+ – i ) (n + ) i + C ξ i + ξ n H + C nξ i= i=

k



 k    ( k+ – i )  i + C ξ i + ξ n H + C , ×

(.)

i=

this completes the proof of Theorem ..

4 Proof of Theorem 1.5 and Theorem 1.7 In this section, applying Lemma ., we will give the proof of Theorem . and Theorem .. Proof of Theorem . Let a = , b = , V ≡ , ρ ≡  in (.), then taking h = xα and summing over α, where {xα }nα= are the coordinate functions of Rn , we have n

k k n (λk+ – λi ) ≤ δ(λk+ – λi ) xα ui pαi dμ i=

i=

+

α=

k n ui f xα  λk+ – λi ∇xα , ∇ui + dμ, δ  α= i=

(.)

where pαi = –∇xα , A∇ui + ui Lf xα . By a similar computation to (.), we have n

xα ui pαi dμ ≤ nξ .

(.)

α=

Since f =

|x| , 

we have

  |x| f xα = xα – ∇ , ∇xα = – xα ,   hence, we infer from the above equality that n ui f xα  ∇xα , ∇ui + dμ  α= n u (f xα ) ∇xα , ∇ui  + ui f xα ∇xα , ∇ui + i dμ  α= n n ∂ui xα ui  |∇ui | + xα ui + = dμ. ∂xα α=  α=

=

(.)


Page 13 of 15

From integration by parts, we have n

ui xα

α=

∂ui – |x| e  dν ∂xα

n    ∂ui – |x|  – |x|   – |x|  – ui e dν – ui xα e dν + u x e dν = ∂xα  i α α= n |x|  ∂ui – |x|  a ui xα e dν + ui |x| e–  dν, = –n – ∂xα  α= which implies that

–

n α=

∂ui  ui xα dμ = n – ∂x  α

ui |x| dμ.

(.)

By a similar computation to (.), we have ∇ui  ≤

λi . ξ

(.)

Using (.), we have n n ui xα   ∂ui ui xα  dμ ∇xα , ∇ui – ∇xα , ∇ui – dμ = ui xα +   ∂xα  α= α= n  = + |∇ui | – ui |x| dμ   ≤

 n λi + – min |x| .  ξ  x∈

(.)

Taking (.) and (.) into (.), we have

n

k k (λk+ – λi ) ≤ δ(λk+ – λi ) nξ i=

i=

+

k (λk+ – λi ) n i=

δ



+

λi  min |x| . – ξ  x∈

(.)

Taking k δ=

i= (λk+

– λi ){ n + λξi –  minx∈ {|x| }} k  i= (λk+ – λi ) nξ

 ,

we obtain (.). This finishes the proof of Theorem ..

Proof of Theorem . Let a = , b = , V ≡ , ρ ≡  in (.), Set x = (x, t) ∈ , where x ∈ , t ∈ R. By a direct computation, we know that |∇t| = , f t = –κt, then taking h = t, we


Page 14 of 15

have k

(λk+ – λi ) ≤ 

i=

k

δ(λk+ – λi )



+

tui pi dμ

i=

k κtui  λk+ – λi dμ, ∇t, ∇ui – δ  i=

(.)

where pi = –∇t, A∇ui + ui Lf t. In the following, let us estimate the right side of (.), first of all, by a similar computation to (.), we infer from A ≤ ξ I that

tui –∇t, A∇ui + ui Lf t dμ =

tui pi dμ =

ui ∇t, A∇t dμ ≤ ξ .

(.)

By a direct computation,

tui ∇t, ∇ui dμ = –

ui |∇t| dμ –

tui ∇t, ∇ui dμ –

tui f t dμ,

which implies that

tui ∇t, ∇ui dμ = – + κ



t  ui dμ.

(.)

From the above equality, we have 

κtui  κ ∇t, ∇ui – ∇t, ∇ui  – κtui ∇t, ∇ui + t  ui dμ dμ =    κ ∇t, ∇ui  +  – t  ui dμ =  ≤  + ∇ui  – ≤+

κ min |t|  (x,t)∈

λi κ  min |t| . – ξ  (x,t)∈

(.)

Taking (.) and (.) into (.), we have k

(λk+ – λi ) ≤

i=

k

δ(λk+ – λi ) ξ

i=

+

k λk+ – λi λi κ  + – min |t| . δ ξ  (x,t)∈ i=

(.)

Taking k δ=

i= (λk+



– λi ){ + λξi – κ min(x,t)∈ {|t| }} k  i= (λk+ – λi ) ξ



we obtain (.). This finishes the proof of Theorem ..

,


Page 15 of 15

Competing interests All authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to this work. All authors read and approved the final manuscript. Acknowledgements Y Zhu and G Liu were partially supported by NSF of Hubei Provincial Department of Education (Grant No. Q20154301). F Du was partially supported by the NSF of China (Grant No. 11401131), and NSF of Hubei Provincial Department of Education (Grant No. Q20154301). Received: 9 March 2016 Accepted: 20 July 2016 References 1. Du, F, Mao, J, Wang, Q, Wu, C: Universal inequalities of the poly-drifting Laplacian on the Gaussian and cylinder shrinking solitons. Ann. Glob. Anal. Geom. 48, 255-268 (2015) 2. Du, F, Mao, J, Wang, Q, Wu, C: Eigenvalue inequalities for the buckling problem of the drifting Laplacian on Ricci solitons. J. Differ. Equ. 260, 5533-5564 (2016) 3. Du, F, Wu, C, Li, G, Xia, C: Estimates for eigenvalues of the bi-drifting Laplacian operator. Z. Angew. Math. Phys. 66(3), 703-726 (2015) 4. Pereira, RG, Adriano, L, Pina, R: Universal bounds for eigenvalues of the polydrifting Laplacian operator in compact domains in the Rn and Sn . Ann. Glob. Anal. Geom. 47, 373-397 (2015) 5. Xia, C, Xu, H: Inequalities for eigenvalues of the drifting Laplacian on Riemannian manifolds. Ann. Glob. Anal. Geom. 45, 155-166 (2014) 6. Du, F, Wu, C, Li, G: Estimates for eigenvalues of fourth-order elliptic operators in divergence form. Bull. Braz. Math. Soc. 46(3), 437-459 (2015) 7. Du, F, Wu, C, Li, G, Xia, C: Estimates for eigenvalues of fourth-order elliptic operators in divergence form on complete Riemannian manifolds. Acta Math. Sin. 58(4), 635-648 (2015) (in Chinese) 8. do Carmo, MP, Wang, Q, Xia, C: Inequalities for eigenvalues of elliptic operators in divergence form on Riemannian manifolds. Ann. Math. 189, 643-660 (2010) 9. Sun, HJ, Chen, D: Inequalities for lower order eigenvalues of second order elliptic operators in divergence form on Riemannian manifolds. Arch. Math. (Basel) 101(4), 381-393 (2013) 10. Wang, Q, Xia, C: Inequalities for the Navier and Dirichlet eigenvalues of elliptic operators. Pac. J. Math. 251(1), 219-237 (2011) 11. Wang, Q, Xia, C: Inequalities for eigenvalues of a clamped plate problem. Calc. Var. Partial Differ. Equ. 40(1-2), 273-289 (2011) 12. Cao, HD: Recent progress on Ricci solitons. In: Recent Advances in Geometric Analysis. Adv. Lect. Math., vol. 11, pp. 1-38. International Press, Somerville (2010) 13. Colding, TH, Minicozzi, WP II: Generic mean curvature flow I; generic singularities. Ann. Math. 175(2), 755-833 (2012) 14. Cheng, QM, Peng, Y: Estimates for eigenvalues of L operator on self-shrinkers. Commun. Contemp. Math. 15(6), 1350011 (2013) 15. Cheng, QM, Yang, HC: Bounds on eigenvalues of Dirichlet Laplacian. Math. Ann. 58(2), 159-175 (2007)