Inequalities for eigenvalues of fourth order elliptic operators in divergence form on Riemannian manifolds

TL;DR

This paper establishes a general inequality for eigenvalues of fourth order elliptic operators in divergence form on Riemannian manifolds, with applications to eigenvalues on submanifolds in Euclidean space.

Contribution

It introduces a new inequality for eigenvalues of fourth order elliptic operators on Riemannian manifolds, extending previous results and applying to submanifold eigenvalue problems.

Findings

Derived a general eigenvalue inequality for fourth order elliptic operators.

Applied the inequality to eigenvalues on submanifolds in Euclidean space.

Provided insights into spectral properties of elliptic operators on manifolds.

Abstract

In this paper, we study eigenvalue of linear fourth order elliptic operators in divergence form with Dirichlet boundary condition on a bounded domain in a compact Riemannian manifolds with boundary (possibly empty) and find a general inequality for them. As an application, by using this inequality, we study eigenvalues of this operator on compact domains of complete submanifolds in a Euclidean space.

Equations

$$\{\begin{array}{ll}{\Delta }^{2}u=\lambda u,& \text{in}\Omega \\ u=\frac{\partial u}{\partial \nu }=0& \text{on}\partial \Omega ,\end{array}$$

$${\lambda }_{k+1}-{\lambda }_k\le \frac{8(n+2)}{n^2}\frac{1}{k}\sum _{i=1}^k{\lambda }_i,k=1,2,....$$

$$\sum _{i=1}^k\frac{{\lambda }_i^{\frac{1}{2}}}{{\lambda }_{k+1}-{\lambda }_i}\ge \frac{n^2{k}^{\frac{3}{2}}}{8(n+2)}{\left(\sum _{i=1}^k{\lambda }_i\right)}^{-\frac{1}{2}},k=1,2,....$$

$$\frac{n^2{k}^2}{8(n+2)}\le \left(\sum _{i=1}^k{\lambda }_i^{\frac{1}{2}}\right)\left(\sum _{i=1}^k\frac{{\lambda }_i^{\frac{1}{2}}}{{\lambda }_{k+1}-{\lambda }_i}\right),k=1,2,....$$

$${\lambda }_{k+1}-\frac{1}{k}\sum _{i=1}^k{\lambda }_i\le {\left(\frac{8(n+2)}{n^2}\right)}^{\frac{1}{2}}\frac{1}{k}\sum _{i=1}^k{\left({\lambda }_i({\lambda }_{k+1}-{\lambda }_i)\right)}^{\frac{1}{2}},k=1,2,....$$

$${\lambda }_{k+1}\le \frac{1}{k}\sum _{i=1}^k{\lambda }_i+{\left\{\frac{64}{n^2{k}^2}(\sum _{i=1}^k{\lambda }_i^{\frac{1}{2}})(\sum _{i=1}^k{\lambda }_i^{\frac{3}{2}})-\frac{1}{k}\sum _{i=1}^k({\lambda }_i-\frac{1}{k}\sum _{i=1}^k{\lambda }_i{)}^2\right\}}^{\frac{1}{2}},k=1,2,....$$

$$\{\begin{array}{ll}{\mathcal{L}}^{2}u=\lambda u,& \text{in}\Omega \\ u=\frac{\partial u}{\partial \nu }=0& \text{on}\partial \Omega ,\end{array}$$

$$\{\begin{array}{ll}{\mathcal{L}}^2{u}_i={\lambda }_i{u}_i,& \text{in}\Omega \\ u_i=\frac{\partial u_i}{\partial \nu }=0,& \text{on}\partial \Omega \\ {\int }_{\Omega }{u}_i{u}_{j}dm={\delta }_{ij}& \forall i,j=1,2,....\end{array}$$

$$\sum _{i=1}^k({\lambda }_{k+1}-{\lambda }_i{)}^2{\int }_{\Omega }{u}_i^{2}tr(T)dm\le \delta \sum _{i=1}^k({\lambda }_{k+1}-{\lambda }_i{)}^2{A}_i+\frac{1}{\delta }\sum _{i=1}^k({\lambda }_{k+1}-{\lambda }_i)B_i$$

$$A_i$$

$$B_i={\int }_{\Omega }\left\{|T(\nabla u_i){|}^2+u_i⟨T(\nabla u_i),tr(\nabla T)⟩+\frac{u_i^2}{4}(||tr(\alpha \circ T)|{|}^2+|tr(\nabla T){|}^2)\right\}dm,$$

$$\sum _{i=1}^k({\lambda }_{k+1}-{\lambda }_i{)}^2{\int }_{\Omega }{u}_i^{2}tr(T)dm\le \delta \sum _{i=1}^k({\lambda }_{k+1}-{\lambda }_i{)}^2{C}_i+\frac{1}{\delta }\sum _{i=1}^k({\lambda }_{k+1}-{\lambda }_i)D_i$$

$$C_i$$

$$D_i=||T(\nabla u_i)|{|}_{L^2(\Omega )}+T_0||T(\nabla u_i)|{|}_{L^2(\Omega )}+\frac{1}{4}((m-n)S_0^2{T}_{\ast }^2+T_0^2)$$

$$\{\begin{array}{ll}{\Delta }^2{u}_i={\lambda }_i{u}_i,& \text{in}\Omega \\ u_i=\frac{\partial u_i}{\partial \nu }=0,& \text{on}\partial \Omega \\ {\int }_{\Omega }{u}_i{u}_{j}dm={\delta }_{ij}& \forall i,j=1,2,....\end{array}$$

$$n\sum _{i=1}^k({\lambda }_{k+1}-{\lambda }_i{)}^2$$

$${\lambda }_{k+1}$$

$${\lambda }_2\le (1+\frac{17}{2n^2}){\lambda }_1.$$

$$\mathcal{L}(uv)=v\mathcal{L}u+u\mathcal{L}v+2T(\nabla u,\nabla v).$$

$${\int }_{\Omega }\mathcal{L}udm={\int }_{\partial \Omega }T(\nabla u,\nu )d\mu ,$$

$${\int }_{\Omega }v\mathcal{L}udm=-{\int }_{\Omega }T(\nabla u,\nabla v)dm+{\int }_{\partial \Omega }vT(\nabla u,\nu )d\mu .$$

$$\sum _{i=1}^k({\lambda }_{k+1}-{\lambda }_i{)}^2{w}_i$$

$$\sum _{i=1}^k({\lambda }_{k+1}-{\lambda }_i{)}^2{v}_i$$

$$w_i$$

$$p_i$$

$$v_i$$

$${\varphi }_i=h{u}_i-\sum _{j=1}^k{a}_{ij}{u}_j$$

$${\int }_{\Omega }{\varphi }_i{u}_{r}dm={\int }_{\Omega }h{u}_i{u}_{r}dm-\sum _{j=1}^k{a}_{ij}{\int }_{\Omega }{u}_r{u}_{j}d\mu =0,\forall i,r=1,2,...,k.$$

$${\lambda }_{k+1}\le \frac{{\int }_{\Omega }{\varphi }_i{\mathcal{L}}^2{\varphi }_{i}dm}{{\int }_{\Omega }{\varphi }_i^{2}dm},\forall i=1,2,...,k.$$

$$\mathcal{L}{\varphi }_i=\mathcal{L}(h{u}_i)-\sum _{j=1}^k{a}_{ij}\mathcal{L}{u}_j=h\mathcal{L}{u}_i+u_i\mathcal{L}h+2T(\nabla h,\nabla u_i)-\sum _{j=1}^k{a}_{ij}\mathcal{L}{u}_j,$$

$${\mathcal{L}}^2{\varphi }_i=\mathcal{L}h\mathcal{L}{u}_i+{\lambda }_{i}h{u}_i+2T(\nabla h,\nabla \mathcal{L}{u}_i)+\mathcal{L}(u_i\mathcal{L}h)+2\mathcal{L}T(\nabla h,\nabla u_i)-\sum _{j=1}^k{a}_{ij}{\lambda }_j{u}_j,$$

Full text

1. Introduction

In this paper, let $(M,\langle,\rangle)$ be an $n$-dimensional complete Riemannian manifold and let $\Omega\subset M$ be a bounded connected domain with smooth boundary $\partial\Omega$ in $M$. Denote by $\Delta$ the Beltrami-Laplace operator on $M$. The study of the spectrum of geometric operator is an important topic and many works have been done in this area. The clamped plate problem or the Dirichlet biharmonic operator for a connected bounded domain $\Omega\subset\mathbb{R}^{n}$ is given by

[TABLE]

where $\nu$ is the outward unit normal vector field of $\partial\Omega$. Suppose that $\{\lambda_{i}\}_{i=1}^{\infty}$ is eigenvalues of the problem (1.1). Payn et’al proved in paper [4] that

[TABLE]

In 1984, Hile and Yeh [2] generalized and showed

[TABLE]

In 1990, Hook [3]obtained the following inequality

[TABLE]

In 2006, Cheng and Yang [1] obtained the inequality

[TABLE]

In 2007, Wang and Xia [5], proved universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds, for instance they showed that, when $\Omega$ is a compact domain in $\mathbb{R}^{n}$, then

[TABLE]

The aim of the present work is to study the eigenvalues of linear fourth order elliptic operator in divergence form on Riemannian manifolds. In special case, this operator is the biharmonic operator. We prove some general inequalities for them. By using these inequalities, we obtain, when $\Omega$ is a compact domains of complete submanifolds in a Euclidean space.

Let $T$ be symmetric positive definite $(1,1)$-tensor on $M$ and $\Omega\subset M$ be a compact domain with smooth boundary $\partial\Omega$ in $M$. We will studying the eigenvalue problem

[TABLE]

where $\mathcal{L}u=div(T(\nabla u))$ and $\nabla$ is the gradient operator of $M$. If $T$ be the identity tensor the $\mathcal{L}=\Delta$. The main results of this paper are as follow

Theorem 1.1**.**

Let $\Omega$ be a domain in an $n$-dimensional complete Riemannian manifold $(M,\langle,\rangle)$ isometrically immersed in $\mathbb{R}^{m}$, $\lambda_{i}$ be the $i$th eigenvalue of (1.2) and $u_{i}$ be the corresponding orthonormal real-valued eigenfunction, that is

[TABLE]

Then for any positive constant $\delta$ and any positive integer $k$, we have

[TABLE]

where

[TABLE]

and

[TABLE]

where $I(x)=(x_{1},...,x_{m})$ for any $x=(x_{1},...,x_{m})\in\mathbb{R}^{n}$, $||f||^{2}=\int_{\Omega}f^{2}dm$, $dm$ is the volume form on $\Omega$, $\alpha$ is the fundamental form of $M$ and $\alpha\circ T=\alpha(T(.),.)$.

Theorem 1.2**.**

Let $\Omega$ be a domain in an $n$-dimensional complete Riemannian manifold $(M,\langle,\rangle)$ isometrically immersed in $\mathbb{R}^{m}$, $\lambda_{i}$ be the $i$th eigenvalue of (1.2) and $u_{i}$ be the corresponding orthonormal real-valued eigenfunction. Then for any positive constant $\delta$ and any positive integer $k$, we have

[TABLE]

where

[TABLE]

and

[TABLE]

where $S_{0}=\max\{\sup_{\bar{\Omega}}|S_{e_{k}}|:\,\,k=n+1,...,m\}$, $S_{e_{k}}$ is the Weingarten operator of the immersion with respect to $e_{k}$, $T_{*}=\sup_{\bar{\Omega}}|T|$, $T_{0}=\sup_{\bar{\Omega}}|tr(\nabla T)|$ and $I_{0}=\sup_{\bar{\Omega}}|I|$.

Theorem 1.3**.**

Let $\Omega$ be a domain in an $n$-dimensional complete Riemannian manifold $(M,\langle,\rangle)$ isometrically immersed in $\mathbb{R}^{m}$ with mean curvature $H$ and $\lambda_{i}$ be the $i$th eigenvalue of biharmonic operator, that is

[TABLE]

Then for any positive constant $\delta$ and any positive integer $k$, we have

[TABLE]

where $H_{0}=\sup_{\bar{\Omega}}|H|$ and $I_{0}=\sup_{\bar{\Omega}}|I|$.

Corollary 1.4**.**

Let $\Omega$ be a domain in an $n$-dimensional complete minimal Riemannian submanifold $(M,\langle,\rangle)$ in $\mathbb{R}^{m}$ and $\lambda_{i}$ be the $i$th eigenvalue of biharmonic operator. Then for any positive integer $k$, we have

[TABLE]

and

[TABLE]

2. Preliminaries

In this section, we describe the necessary tools about tensor $T$ and problem (1.2) which enable us to prove our results. Throughout the paper, for any vector fields $X,Y$, we denote $\langle T(X),Y\rangle$ with $T(X,Y)$. For any $u,v\in C^{\infty}(\Omega)$, straightforward computation implies that

[TABLE]

Let $d\mu$ be the volume form on the boundary induced by the outward normal vector field $\nu$ on $\partial\Omega$. The divergence theorem for operator $\mathcal{L}$ as follows

[TABLE]

then the integration by parts yields

[TABLE]

Hence, the operators $\mathcal{L}$ and $\mathcal{L}^{2}$ are self-adjoint operator in the space of all function in $L^{2}(\Omega,dm)$ that vanish on $\partial\Omega$. Therefore the eigenvalues of problem (1.2) are real and discrete.

Proposition 2.1**.**

Let $\Omega$ be a domain in a an $n$-dimensional complete Riemannian manifold $(M,\langle,\rangle)$, $\lambda_{i}$ be the $i$th eigenvalue of (1.2) and $u_{i}$ be the corresponding orthonormal real-valued eigenfunction. Then for any $h\in C^{4}(\Omega)\cup C^{3}(\partial\Omega)$ and any positive integer $k$, we have

[TABLE]

where $\delta$ is any positive constant,

[TABLE]

Proof.

For each $i$, $1\leq i\leq k$, consider the functions $\phi_{i}:\Omega\to\mathbb{R}$ given by

[TABLE]

where $a_{ij}=\int_{\Omega}hu_{i}u_{j}\,dm$. We have $\phi_{i}|_{\partial\Omega}=\frac{\partial\phi_{i}}{\partial\nu}|_{\partial\Omega}=0$ and

[TABLE]

Then by the inequality of Rayleigh-Ritz, we get

[TABLE]

Since

[TABLE]

we obtain

[TABLE]

therefore we get

[TABLE]

where $r_{ij}=\int_{\Omega}p_{i}u_{j}dm$ and

[TABLE]

Using integration by parts, we deduce that

[TABLE]

On the other hand

[TABLE]

which implies that

[TABLE]

Substituting (2.7) into (2.9), we have

[TABLE]

Moreover

[TABLE]

Combining (2.10), (2.11) and $r_{ij}=\int_{\Omega}p_{i}u_{j}dm$, we can write

[TABLE]

It follows from (2.5), (2.6) and (2.7) that

[TABLE]

where $w_{i}=\int_{\Omega}hu_{i}p_{i}\,dm$. We use that $\int_{\Omega}\phi_{i}u_{j}\,dm=0$ again to get

[TABLE]

this implies that

[TABLE]

Multiplying (2.15) by $(\lambda_{k+1}-\lambda_{i})$ and summing on $i$ from $1$ to $k$, we obtain

[TABLE]

Multiplying (2) by $(\lambda_{k+1}-\lambda_{i})^{2}$, summing on $i$ from $1$ to $k$ and $a_{ij}=a_{ji}$, we infer

[TABLE]

then

[TABLE]

which shows that (2.1) is true. In order to prove (2.1), we set

[TABLE]

Observe that

[TABLE]

and

[TABLE]

where

[TABLE]

Since for any positive constant $\delta$ and for all $x,y\in\mathbb{R}$ we have $-2xy\leq\delta x^{2}+\frac{y^{2}}{\delta}$, then multiplying (2.20) by $(\lambda_{k+1}-\lambda_{i})^{2}$, for any positive constant $\delta$, we get

[TABLE]

hence (2) implies that

[TABLE]

Now, summing over $i$ from $1$ to $k$, $a_{ij}=a_{ij}$ and $b_{ij}=-b_{ji}$ we conclude that

[TABLE]

which gives

[TABLE]

Thus (2.1) is true. Substituting (2.18) into (2) complete the proof of the proposition. ∎

Proof of Theorem 1.1.

Let $x_{1},...,x_{n}$ be the standard Euclidean coordinate of $\mathbb{R}^{m}$, $\bar{\nabla}$ be the Canonical connection of $\mathbb{R}^{m}$ and $\{e_{1},...,e_{m}\}$ be a local orthonormal geodesic frame in $p\in M$ adapted to $M$, then

[TABLE]

Therefore

[TABLE]

and

[TABLE]

Also, we have

[TABLE]

For $x=(x_{1},...,x_{n})$, we compute

[TABLE]

hence

[TABLE]

and

[TABLE]

where $\alpha\circ T=\alpha(T(.),.)\in\mathcal{X}(M)^{\perp}$. By taking $h=x_{r}$ in (2.1) we can write

[TABLE]

where

[TABLE]

Summing over $r$, we have

[TABLE]

and from (2.23) we get

[TABLE]

Also (2.25) and (2) imply that

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

Thus

[TABLE]

and

[TABLE]

Substituting (2.29), (2) and (2) into (2.27) we complete the proof of the theorem. ∎

Proof of Theorem 1.2.

Let $S_{e_{i}}$ be the Weingarten operator of the immersion with respect to $e_{i}$. Then

[TABLE]

where $S_{0}=\max\{\sup_{\bar{\Omega}}|S_{e_{k}}|:\,\,k=n+1,...,m\}$ and $T_{*}=\sup_{\bar{\Omega}}|T|$. If $T_{0}=\sup_{\bar{\Omega}}|tr(\nabla T)|$ and $I_{0}=\sup_{\bar{\Omega}}|I|$ then

[TABLE]

[TABLE]

and

[TABLE]

Also, we have

[TABLE]

and

[TABLE]

By setting

[TABLE]

and

[TABLE]

we get $A_{i}\leq C_{i}$ and $B_{i}\leq D_{i}$. Substituting these inequalities into Theorem 1.1 we complete the proof of the Theorem. ∎

Proof of Theorem 1.3.

Taking $T$ equal to identity in Theorem $1$, we obtain

[TABLE]

where

[TABLE]

and

[TABLE]

On the other hand, we have

[TABLE]

and

[TABLE]

Hence

[TABLE]

and

[TABLE]

where $H_{0}=\sup_{\bar{\Omega}}|H|$ and $I_{0}=\sup_{\bar{\Omega}}|I|$. Substituting these inequality into (2.40) we complete the proof of the theorem. ∎

Proof of Corollary 1.4.

For a minimal hypersurface we have $H=0$, therefore Theorem 3 results that

[TABLE]

Taking

[TABLE]

we get

[TABLE]

On the other hand

[TABLE]

It and (2.42) imply that

[TABLE]

solving this quadratic polynomial of $\lambda_{k+1}$, we obtain (1.7) and $(\ref{ab1})$. ∎