Eigenvalue inequalities for the buckling problem of the drifting Laplacian of arbitrary order

In this paper, we investigate the buckling problem of the drifting Laplacian of arbitrary order on a bounded connected domain in complete smooth metric measure spaces (SMMSs) supporting a special function, and successfully get a general inequality for its eigenvalues. By applying this general inequality, if the complete SMMSs considered satisfy some curvature constraints, we can obtain a universal inequalities for eigenvalues of this buckling problem.


Introduction
Let Ω be a bounded domain in an n-dimensional complete Riemannian manifold M, and let ∆ be the Laplace operator acting on functions on M. Consider the following eigenvalue problems (−∆) m u = −Λ∆u in Ω, u = ∂ u ∂ ν = · · · = ∂ m−1 u ∂ ν m−1 = 0 on ∂ Ω, (1.1) where ν is the outward unit normal vector field of the boundary ∂ Ω, l is an arbitrary positive integer and m is an arbitrary positive integer no less than 2. They are called the buckling problem of arbitrary order and the eigenvalue problem of polyharmonic operator, respectively. The buckling problem (1.1) is used to describe the critical buckling load of a clamped plate subjected to a uniform compressive force around its boundary. 0 Denote by 0 < Λ 1 ≤ Λ 2 ≤ Λ 3 ≤ · · · , 0 < λ 1 ≤ λ 2 ≤ λ 3 ≤ · · · the successive eigenvalues for (1.1) and (1.2) respectively, where each eigenvalue is repeated according to its multiplicity. An important theme in Geometric Analysis is to estimate these (and other) eigenvalues.
If m = 2 in the bucking problem (1.1) and Ω is a bounded domain in an n-dimensional Eucliden space R n , Cheng and Yang [6] proved the following universal inequality which gives an answer to a long standing question proposed by Payne, Pólya and Weinberger [26,27]. If m = 2 in (1.1) and Ω is a bounded domain in an n-dimensional unit sphere S n (1), Wang and Xia [29] successfully obtained the following universal inequality where δ is an arbitrary positive constant. Later, Cheng and Yang [7] gave an improvement for the universal inequalities (1.3) and (1.4) as follows is an arbitrary positive non-increasing monotone sequence. For arbitrary m, when Ω is a bounded domain in a Euclidean space or a unit sphere, Jost, Li-Jost, Wang and Xia [20] obtained some universal inequalities for eigenvalues of the buckling problem (1.1), which have been improved by Cheng, Qi, Wang and Xia [2] already. For bounded domains of some special Ricci flat manifolds considered in [12] and of product manifolds M × R considered in [30] (with M a complete Riemannian manifold), universal inequalities for eigenvalues of the buckling problem (1.1) have been obtained therein. For some recent developments about universal inequalities for eigenvalues of the eigenvalue problem (1.2) on Riemannian manifolds, we refer to [4,5,12,21] and the references therein.
A smooth metric measure space (also known as the weighted measure space, and here written as SMMS for short) is actually a Riemannian manifold equipped with some measure which is conformal to the usual Riemannian measure. More precisely, for a given complete n-dimensional Riemannian manifold (M, , ) with the metric , , the triple (M, , , e −φ dv) is called a SMMS, where φ is a smooth real-valued function on M and dv is the Riemannian volume element related to , (sometimes, we also call dv the volume density). On a SMMS (M, , , e −φ dv), we can define the the so-called drifting Laplacian (also called weighted Laplacian) L φ as follows where ∇ is the gradient operator on M, and, as before, ∆ is the Laplace operator. Some interesting results concerning eigenvalues of the drifting Laplacian can be found, for instance, in [8,9,14,16,22,23,24]. On the SMMS (M, , , e −φ dv), we can also define the so-called ∞-Bakry-Émery Ricci tensor Ric φ given by which is also called the weighted Ricci curvature. Here Ric, Hess are the Ricci tensor and the Hessian operator on M, respectively. The equation Ric φ = κ , for some constant κ is just the gradient Ricci soliton equation, which plays an important role in the study of Ricci flow. For κ = 0, κ > 0, or κ < 0, the gradient Ricci soliton (M, , , e −φ dv, κ) is called steady, shrinking, or expanding respectively. We refer readers to [1] for some recent interesting results about Ricci solitons.
Let Ω be a bounded domain in a complete SMMS (M, , , e −φ dv). Consider the following eigenvalue problem of the drifting Laplacian where, as before, ν is the outward unit normal vector field of the boundary ∂ Ω and l is an arbitrary positive integer. We know that L φ l is self-adjoint on the space of functions with respect to the inner product and so the eigenvalue problem (1.7) has a discrete spectrum whose elements are called eigenvalues and can be listed increasingly as follows where each eigenvalue is repeated with its multiplicity. For the eigenvalue problem (1.7), when l = 1, Xia-Xu [31] investigated the eigenvalues of the Dirichlet problem of the drifting Laplacian on compact manifolds and got some universal inequalities; when l = 2, Du, Wu, Li and Xia [13] obtained some universal inequalities of Yang type for eigenvalues of the bi-drifting Laplacian problem either on a compact Riemannian manifold with boundary (possibly empty) immersed in a Euclidean space, a unit sphere or a projective space, or on bounded domains of complete manifolds supporting some special function; when l is an arbitrary integer no less than 2, Pereira, Adriano and Pina [25] gave some universal inequalities on bounded domains in a Euclidean space or a unit sphere, while Du, Mao, Wang and Wu [10] successfully obtained some universal inequalities on bounded domains in the Guassian and cylinder solitons.
In this paper, we will consider the following buckling problem (of the drifting Laplacian) of arbitrary order where, as before, ν is the outward unit normal vector field of the boundary ∂ Ω and m is an arbitrary positive integer no less than 2. The eigenvalue problem (1.8) has discrete spectrum (see Section 2 for the details), which can be listed increasingly as follows where each eigenvalue is repeated with its multiplicity. For the eigenvalue problem (1.8), we can prove: ) be a complete connected Riemannian manifold having weighted Ricci curvature Ric φ ≥ 0 for some φ ∈ C 2 (M), which is bounded above uniformly on M, and containing a line. Then we have (1.9) (3) When m = 2, the buckling problem (1.8) degenerates into the one considered in [11] where Du, Mao, Wang and Wu firstly obtained universal inequalities on bounded connected domains on the Gaussian shrinking soliton R n , , can , e − 1 4 |x| dv, 1 2 , with x ∈ R n , and on the gradient Ricci soliton Σ × R, , , e − κt 2 2 dv, κ , with Σ an Einstein manifold of constant Ricci curvature κ, x ∈ Σ and t ∈ R. In this sense, our universal inequality (1.9) here can be seen as a continuation of those in [11].

Preliminaries
In this section, firstly, inspired by Cheng and Yang [6], let us construct trial functions for the buckling problem (1.8).
Let Ω be a bounded domain with smooth boundary in the complete SMMS (M, g, e −φ dv). Since any complete Riemannian manifold can be isometrically embedded in some Euclidaen space, we can treat our M as a submanifold of some R q . Let us denote by , the canonical metric on R q as well as that induced on M. As before, dµ = e −φ dv, denote by ∆ and ∇ the Laplacian and the gradient operator of M, respectively. Let u i be the i-th orthonormal eigenfunctions of the buckling (2.1) For functions f and g on Ω, the Dirichlet inner product ( f , g) D of f and g is given by The Dirichlet norm of a function f is defined by Let ∇ k be the denote the k-th covariant derivative operator on M, defined in the usual weak sense.
For a function f on Ω, the squared norm of ∇ k f is defined as (cf. [17]) where e 1 , · · · , e n are orthonormal vector fields locally defined on Ω. Define the Sobolev space Then H 2 m (Ω) is a Hilbert space with respect to the inner product , : where .
The poly-drifting Laplacian operator L φ m defines a self-adjoint operator acting on H 2 m,D (Ω) with discrete eigenvalues 0 ≤ Λ 1 ≤ · · · ≤ Λ k ≤ · · · for the buckling problem (1.8) and the eigenfunctions The norm of F is given by Let H 2 m (Ω) be the Hilbert space of vector-valued functions given by with norm · 1 : Observe that a vector field on Ω can be regarded as a vector-valued function from Ω to R q . Let where h i ∈ H 2 m,D (Ω), ∇h i is the projection of g∇u i in H 2 m,D (Ω) and W i ⊥H 2 m,D (Ω). Thus, we have, for any function h ∈ C 1 (Ω) ∩ L 2 (Ω), At the end of this section, we would like to mention two facts. First, a simple calculation gives the following Bochner formula for the drifting Laplacian (see [28]): for any f ∈ C 3 (Ω), Hence, on the SMMS (M, g, e −φ dv), for any functions f , g ∈ C 3 (Ω), we have (2.5) Second, using a similar calculation as that in the proof of [20, Lemma 2.1], we can get the following fact: Lemma 2.1. Let Λ i be the i-th, i = 1, 2, · · ·, eigenvalue of the eigenvalue problem (1.8), and u i be the orthonormal eigenfunction corresponding to Λ i . Then

A universal inequality of the buckling problem (of the drifting Laplacian) of arbitrary order
In this section, first, we will give a general inequality on a bounded domain in SMMSs supporting a special function.
Proof. Consider the function ψ i : Ω → R given by where b i j = Ω g ∇u i , ∇u j dµ = b ji , and h i is determined by (2.3). It is easy to check that ψ i satisfies ∂ Ω = 0, and Ω ∇ψ i , ∇u j dµ = 0 for any j = 1, · · · , k. It therefore follows from the Rayleigh-Ritz inequality that Observing that Ω ψ i L φ u j dµ = − Ω ∇ψ i , ∇u j dµ = 0, and

5)
and It follows from L φ g = 0 and (2.5) that Combining (3.4)-(3.7), we have On one hand, On the other hand, Then we infer from (3.9) and (3.10) that Hence By a direct computation, we have (3.12) and 14) It is easy to see that Substituting the above equalities into (3.14) yields Since Ω ∇ ∇g, ∇u i , W i dµ = 0, we can get Putting (3.16) and (3.17) into (3.15), we have Setting Z i = ∇ ∇g, ∇u i , we can obtain On the other hand, Then by the Schwarz inequality and (3.18)- (3.20), for any positive constant δ , we can get holds. This completes the proof of Theorem 3.1. (3.24) When m = 2p, p ∈ Z + with Z + the set of all positive integers, we have Together with (3.26), we have It follows from (3.26) and (3.27) that The proof is finished.
Using ( . Now, we would like to give the proof of Theorem 1.1. However, before that, we need the following fact: • FACT ([15, Theorem 1.1]) Let (M, , ) be a complete connected Riemannian manifold with weighted Ricci curvature Ric φ ≥ 0 for some φ ∈ C 2 (M) which is bounded above uniformly on M. Then it splits isometrically as N ×R l , where N is some complete Riemannian manifold without lines and R l is the l-Euclidean space. Furthermore, the function φ is constant on each R l in this splitting.
Hence, one can find a special function g on M satisfying (3.28) also (if Ric φ ≥ 0 for some bounded φ and M contains a line), which implies that a general inequality in Theorem 3.1 can be used in the setting. In fact, if Ric φ ≥ 0 for some bounded φ and M contains a line, we know that (M, , , dµ) splits isometrically as N × R. Let x = (t, x 1 ) be the standard coordinate functions of N × R, where t ∈ N and x 1 ∈ R. As (3.28), for any vector field Y ∈ X (M), we have L φ x 1 = ∆x 1 − ∇φ , ∇x 1 = 0, |∇x 1 | = 1, Ric φ (∇x 1 ,Y ) = 0.
Proof of Theorem 1.1. Clearly, (3.28) shows the existence of special function mentioned in Theorem 3.1 if the constraints on the weighted Ricci curvature Ric φ and the weighted function φ in Theorem 1.1 were satisfied. Hence, by using Lemma 3.3 (since M contains a line, l cannot be zero), the universal inequality (1.9) for the buckling problem (1.8) follows directly.