Abstract
Let \(\Omega \) be a bounded domain in a n-dimensional Euclidean space \(\mathbb {R}^{n}\). We study eigenvalues of an eigenvalue problem of a system of elliptic equations of the drifting Laplacian
Estimates for eigenvalues of the above eigenvalue problem are obtained. Furthermore, a universal inequality for lower order eigenvalues of the problem is also derived. Finally, we prove an universal inequality type Ashbaugh and Benguria for the drifting Laplacian on Riemannian manifold immersed in an unit sphere or a projective space.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Let \((M,<,>)\) be an n-dimensional compact Riemannian manifold with boundary (possibly empty), \(\phi \in C^2(M)\) and \(d\mu = e^{-\phi }d\nu \), where \(d\nu \) is the Riemannian volume measure on M. The drifting Laplacian with respect to the weighted volume measure \(\mu \) is given by
Interesting results for the eigenvalues of the drifting Laplacian have been obtained in recent years, for example in the works of Ma-Liu [13, 14] and Ma-Du [12]. In 2014, Xia-Xu [19] have investigated the eigenvalues of the Dirichlet problem of the drifting Laplacian on compact manifolds and got some universal inequalities for them. Besides, at the same year, Xia et al. [6] have drawn universal inequalities of Yang type for eigenvalues of the bi-drifting Laplacian problem on a compact Riemannian manifold with boundary (possibly empty) immersed in: an Euclidean space, a unit sphere or a projective space. Recently, Pereira et al. [15] have given some universal inequalities for the poly-drifting laplacian on bounded domains in a Euclidean space or a unit sphere.
Let \(\Omega \) a bounded domain with smooth boundary in an n-dimensional Euclidean space \(\mathbb {R}^{n}\). Consider an eigenvalue problem of a system of n elliptic equations
where \(\mathbb {L_{\phi }}\) is the drifting laplacian in \(\mathbb {R}^{n}\), \(\mathbf{u} = (u_1,u_2,\ldots ,u_n)\) is a vector-valued function from \(\Omega \) to \(\mathbb {R}^{n}\), \(\alpha \) is a non-negative constant, \(\mathrm {div}{} \mathbf{u}\) denotes the divergence of \(\mathbf{u}\) and \(\nabla f\) is the gradient of a function f.
Let
be the eigenvalues of the problem (1). Here each eigenvalue is repeated according to its multiplicity. When \(\phi \equiv 0\) and \(n=3\), the problem (1) describes the behavior of the elastic vibration [17]. The literature about this eigenvalue problem is extensive, more informations can be found in [4, 5, 7, 10].
This paper is organized as follows. In Sect. 2, we shall establish some general estimates for eigenvalues of the problem (1), in particular, we shall give an universal inequality wich for \(\phi \equiv 0\) is the same got in [4]. In the last section, we shall use an algebraic argument for to prove a universal inequality, without Rayleigh-Ritz, for the problem (1) and for the drifting-Dirichlet problem. Besides, we shall prove this inequality for Riemannian manifold immersed in either an euclidean space, a unit sphere or a projective space.
2 The first results
In this section we prove some general estimates for the problem (1). We shall prove the inequality wich for \(\phi \equiv 0\) coincides with the inequality obtained in [4].
Lemma 2.1
Let \(\Omega \) be a bounded domain in an n-dimensional Euclidean space \(\mathbb {R}^{n}\). Denote by \(\mathbb {L_{\phi }}\) the drifting operator of \(\mathbb {R}^{n}\). Let \(\bar{\sigma }_{i}\) denote the i-th eigenvalue of the eigenvalue problem (1) and \(\mathbf{u_{i}}\) be the orthonormal vector-valued eigenfunction corresponding to \(\bar{\sigma }_{i}\). For any function \(f\in C^2(\Omega )\cap C^{1}(\partial \Omega )\), we have
and, for any positive constant \(\delta \),
where \(P_{i}(f) = 2 \ \nabla f\cdot \nabla (\mathbf{u}_{i}) +\mathbf{u}_{i}\mathbb {L_{\phi }}f + \alpha \{\nabla (\nabla f\cdot \mathbf{u}_{i}) + \nabla f \mathrm {div}(\mathbf{u}_{i}) - \nabla \phi (\nabla f\cdot \mathbf{u}_{i})\}\) and \(\nabla f \cdot \nabla (\mathbf{u}_{i})\) is defined by
Proof
We define the vector-valued functions \(\mathbf{v}_{i}\) by \(\mathbf{v}_{i}= f\mathbf{u}_{i} - \sum _{j=1}^{k}a_{ij}\mathbf{u}_{j}\), where \(a_{ij} = \int _{\Omega }f\mathbf{u}_{i}\mathbf{u}_{j}d\mu \).
Since \(\mathbf{v}_{i}|_{\partial \Omega } = 0 \) and \(\int _{\Omega }\mathbf{v}_{i}\mathbf{u}_{j}d\mu = 0\), then it follows from the Rayleigh-Ritz inequality [9] that
From the definition of \(\mathbf{v}_{i}\), we derive
We have that
Therefore, from (4), we have
Define \(b_{ij} = \displaystyle \int _{\Omega }\left( \nabla f \cdot \nabla \mathbf{u}_{i} + \frac{1}{2}\mathbf{u}_{i}\mathbb {L_{\phi }}f\right) \mathbf{u}_{j}d\mu \). From Stokes’ theorem, we infer
and
With the inequalities obtained in (7) and (8), we have
Moreover, we derive
Thus,
Hence,
Summing over \(i=1,\ldots ,k\), we infer
Since \(a_{ij}\) is anti-symmetric, it follows
This finishes the proof of inequality (2).
In order to obtain the inequality (3), we derive
Thus,
Since \(a_{ij}\) is symmetric and \(b_{ij}\) is anti-symmetric, we have
With the results obtained in (11) and (12), moreover that
we infer that
By a simple computation, we get (3). \(\square \)
The last lemma gives us the tools for the proof of next theorem.
Theorem 2.2
Let \(\Omega \) be a bounded domain in an n- dimensional Euclidean space \(\mathbb {R}^{n}\). We consider \(\phi \) a smooth function in \(\Omega \) with \(C_{0}= \max _{\bar{\Omega }}|\nabla \phi |\). Let \(\mathbb {L_{\phi }}\) be the drifting laplacian in \(\mathbb {R}^{n}\). Denote by \(\bar{\sigma }_{i}\) the ith eigenvalue of the problem (1) and \(\mathbf{u}_{i}\) the orthonormal vector-valued eigenfunction corresponding to \(\bar{\sigma }_{i}\). Then
where
Proof
Let \(x^1,\ldots , x^n\) be standard coordinates functions in \(\mathbb {R}^{n}\). Setting \(f=x^p\), in (3) we have
Summing over \(p=1,\ldots ,n\), we infer
By a simple calculation, we infer
Replacing (14) in (13), we have
Putting \(\delta = \left\{ \frac{\sum _{i=1}^{k}(\bar{\sigma }_{k+1} - \bar{\sigma }_{i})\left( \bar{\sigma }_{i} - \alpha ||\mathrm {div}\mathbf{u}_{i}||^2 + C_{0}\sqrt{\bar{\sigma }_{i} - \alpha ||\mathrm {div}\mathbf{u}_{i}||^2} +\frac{C_{0}^2}{4}\right) }{\sum _{i=1}^{k}(\bar{\sigma }_{k+1} - \bar{\sigma }_{i})^2(n+\alpha )}\right\} ^{\frac{1}{2}}\), we get
Therefore,
On the other hand, taking \(f=x^{p}\) in (2) and summing over \(p=1,\ldots ,n\), we have
A straightforward computation yields
Hence
where
If \(\alpha ^2 - (n+2)\alpha -4\ge 0\), namely, \(\alpha \ge \frac{n + 2 + \sqrt{(n+2)^2 + 16}}{2},\) we have
If \(\alpha ^2 - (n+2)\alpha -4< 0\), namely, \(0\le \alpha <\frac{n + 2 + \sqrt{(n+2)^2 + 16}}{2}\) , we have
From (15) and (17), it follows
where \( L = \frac{(4 + (n +2)\alpha - \alpha ^2)n^2}{4(n+\alpha )^2}.\)
Thus, for any \(\alpha \ge 0\)
Making a comparison between (15) and (18), we finish the proof. \(\square \)
3 Lower order estimates
In this section, we shall give many informations about lower order estimates.
Theorem 3.1
Let \(\bar{\sigma }_{i}\) denote the i-th eigenvalue of the problem (1), \(i=1,\ldots ,n\). Then, we have
Proof
Let \(\{x^{i}\}_{i=1}^{n}\) be the standard coordinate functions of \(\mathbb {R}^{n}\). Let us define a \(n\times n\) matrix \(C:=(c_{ij})\) where \(c_{ij} = \int _{\Omega }x^{i}\mathbf{u}_{1}\mathbf{u}_{j+1}d\mu .\) Using the orthogonalization of Gram and Schmidt, we know that there exists an upper triangle matrix B and an orthogonal matrix T such that \(B=TC\), namely
for \(1\le j <i\)
Putting \(y_{i} = \sum _{k=1}^{n}t_{ik}x^{k}\), we have \(\int _{\Omega } y_{i}\mathbf{u}_{1}\mathbf{u}_{j+1}d\mu = 0\) for \(1\le j <i\).
We define a vector-valued function \(\mathbf{w}_{i} = (y_{i} - a_{i})\mathbf{u}_{1}\) where \(a_{i} = \int _{\Omega }y_{i}{\mathbf{u}_{1}^2}d\mu \). We infer that \(\mathbf{w}_{i}|_{\partial \Omega } = 0\) and \(\int _{\Omega }\mathbf{w}_{i}\mathbf{u}_{j+1}d\mu = 0\) for any \(j=1,\ldots ,i-1\). From the Rayleight-Ritz inequality we have
We derive that
Replacing the above identity in (19), we have
where \(P_{1}(y_{i}):=\mathbf{u}_{1}\mathbb {L_{\phi }}y_{i} + 2 \nabla y_{i}\cdot \nabla \mathbf{u}_{1} + \alpha \bigl [ \nabla (\nabla y_{i}\cdot \mathbf{u}_{1}) + \nabla y_{i}\mathrm {div}\mathbf{u}_{1} - \nabla \phi (\nabla y_{i}\cdot \mathbf{u}_{1})\bigr ].\)
By a simple computation, we derive that
and
So, from (20), we infer
On the other hand, we have
Summing over \(i=1,\ldots ,n\), we conclude
Since
putting \(\delta = \left\{ \frac{\bar{\sigma }_{1} - \alpha ||\mathrm {div}\mathbf{u}_{1}||^2 + C_{0}\sqrt{\bar{\sigma }_{1} - \alpha ||\mathrm {div}\mathbf{u}_{1}||^2} + \frac{{C_{0}^2}}{4}}{\sum _{i=1}^{n}(\bar{\sigma }_{i+1} - \bar{\sigma }_{1})(1+\alpha )}\right\} ^{\frac{1}{2}}\), we infer
\(\square \)
It is not difficult to see that, when \(\phi \equiv 0\), the inequality (24) is the same obtained in [5].
The theorem below and the corollary can be found in [11].
Theorem 3.2
(Algebraic) Let \(\mathcal {H}\) and \(\mathcal {G}\) be self-adjoint operators with domains \(D_{\mathcal {H}}\) and \(D_{\mathcal {G}}\) respectively, such that \(\mathcal {G}(D_{\mathcal {H}})\subseteq D_{\mathcal {H}} \subseteq D_{\mathcal {G}}\). Let \(\bar{\sigma }_{j}\) and \(\mathbf{v}_{j}\) be eigenvalues and eigenvectors of \(\mathcal {H}\); then, for each j
Corollary 3.3
Under conditions of Theorem 3.2
With the help of the above corollary, we obtain an estimate about the gap from any consecutive eigenvalues.
Theorem 3.4
Let \(\Omega \) be a bounded domain in \(\mathbb {R}^{n}\). We consider \(\phi \) a smooth function in \(\Omega \) with \(C_{0}= \max _{\bar{\Omega }}|\nabla \phi |\). Denote by \(\bar{\sigma }_{i}\) the ith eigenvalue of the problem (1) and \(\mathbf{u}_{i}\) the orthonormal vector-valued eigenfunction corresponding to \(\bar{\sigma }_{i}\). Then
Proof
We denote \(N = (-\mathbb {L_{\phi }})\) and \(M = \nabla \phi \;\mathrm {div}- \nabla \mathrm {div}\), so \(\mathcal {H}= N + \alpha M\). This operator is associated to the eigenvalue problem (1). We consider \(G_{p} = x^{p} \), then from (25), we have
We derive
Notice that
Hence, it follows from Theorem 2.2
Inserting the above inequality in (26), we have
When \(\phi \equiv 0\), we have that (27) is the same inequality obtained in Example 4.4 [11]. \(\square \)
In the next theorem our aim is to get a result similar to Ashbaugh and Benguria in [1]. This result is very interesting because it doesn’t make use the Rayleigh-Ritz inequality.
Theorem 3.5
Let \(M^{n}\)be a complete Riemannian manifold and let \(\Omega \) be a bounded domain with smooth boundary in M. Let \(\phi \) be a smooth function in \(\Omega \) with \(C_{0} = \max _{\Omega }{\mid }{\nabla \phi }{\mid }\). Denote by \(\lambda _{j}\) the j-th eigenvalue of the drifting problem, namely
If M is isometrically immersed in \(\mathbb {R}^{m}\) with mean curvature vector \(\vec {H}\), then
where \(H_{0} = \displaystyle \sup _{\Omega }\mid \vec { H} \mid ,\ l\in \mathbb {N}\).
Proof
Let \(\{x_{\alpha }\}_{\alpha =1}^{m}\) be the standard coordinate functions in \(\mathbb {R}^{m}\). For each j fixed, we considerer a \(m\times m\) matrix, C where \(c_{\alpha \beta } = \left<\left[ \mathcal {H}, x_{\alpha }\right] u_{j},u_{j+\beta }\right>\)
Using the orthogonalization process we have that there exist \(B:=(b_{\alpha \beta })\) an upper triangle matrix and \(T:=(t_{\alpha \beta })\) an orthogonal matrix such that \(B = TC\), that is
for \(1\le \beta < \alpha \le m \).
Defining \(g_{\alpha } = \sum _{\gamma =1}^{m}t_{\alpha \gamma }x_{\gamma }\), we have that \(\left<\left[ \mathcal {H}, g_{\alpha }\right] u_{j},u_{j+\beta }\right> = 0\) for \(1\le \beta < \alpha \).
At this moment, we analyze the term
Observe that, for \(k = j + 1, \ldots , j + (\alpha - 1)\), we obtain
For \(k\ge j + \alpha \), we have that \(\lambda _{k} \ge \lambda _{j + \alpha }\). Hence,
When \(k = j\), in the algebraic lemma we assume \(\frac{0}{0} = 0\). For the case in that \(k < j\), we have \(\lambda _{k} - \lambda _{j} < 0\). So,
Therefore, for any positive integer k, we have
Summing over k, we get
In the other hand, the Parseval’s Identity gives us
In Algebraic Lemma 3.2, we assume \(\mathcal {G} = g_{\alpha }\) and we use the inequalities (29), (30). Thus, we can conclude that
Here, we assume that \(\mathcal {H} = (-\mathbb {L_{\phi }})\) and we obtain
Summing over \(\alpha \) in (31)
We remember that M is isometrically immersed in \(\mathbb {R}^{n}\), namely,
Consequently, we have
Furthermore, from (32), we obtain
Since \(\mid \nabla g_{\alpha } \mid ^2 \le 1\) then
Therefore,
\(\square \)
For \(j=1\) and \(\phi \) constant, the inequality derived covers Ashbaugh and Benguria in [1].
Corollary 3.6
Under the same assumptions as in Theorem 3.5. We have
-
i)
If M is isometrically immersed in the unit sphere \(\mathbb {S}^{m-1} \subset \mathbb {R}^{m}\) with mean curvature vector \(\overrightarrow{H}\) then
$$\begin{aligned} \sum _{k=1}^{n}\lambda _{l+k} \ \le \ (n+4)\lambda _{l} + n^2\left( H_1^2 +1\right) + C_{0}^2 + 4C_{0}\lambda _{l}^{\frac{1}{2}}, \end{aligned}$$where \(H_{1} = \sup _{\Omega }\mid \overrightarrow{ H }\mid \).
-
ii)
If M is isometrically immersed in a projective space \(\mathbb {F}P^{m}\) with mean curvature \(\overrightarrow{H}\), then
$$\begin{aligned} \sum _{k=1}^{n}\lambda _{l+k} \ \le \ (n+4)\lambda _{l} + n^2\left( H_2^2 +\frac{2(n + d)}{n}\right) + C_{0}^2 + 4C_{0}\lambda _{l}^{\frac{1}{2}}, \end{aligned}$$where \(H_{2} = \sup _{\Omega }\mid \overrightarrow{H} \mid \) and \(d= \dim _{\mathbb {F}} = \left\{ \begin{array}{ll} 1, &{} \mathbb {F} = \mathbb {R} \\ 2, &{} \mathbb {F} = \mathbb {C} \\ 4, &{} \mathbb {F} = \mathcal {Q} \end{array} \right. \)
Proof
-
i)
Let f be standard embedding from M into an unit sphere \(\mathbb {S}^{m-1}\) and let j be the inclusion map. Denote by \(\overrightarrow{H}\) the mean curvature vector of f and by \(\overrightarrow{H'}\) the mean curvature vector of \(j\circ f\). Since \(j\ : \ \mathbb {S}^{m-1}\hookrightarrow \mathbb {R}^{m}\) has \(- I\) like the Gauss normal map then the shape operator \(S_{N} = I\). Hence,
$$\begin{aligned} \mid H' \mid ^2 = \mid H \mid ^2 + 1. \end{aligned}$$Using the Theorem 3.5, we obtain
$$\begin{aligned} \sum _{k=1}^{n}\lambda _{l+k} \ \le \ (n+4)\lambda _{l} + n^2\left( H_1^2 +1\right) + C_{0}^2 + 4C_{0}\lambda _{l}^{\frac{1}{2}} \end{aligned}$$ -
ii)
Let \(\overrightarrow{H}\) be the mean curvature vector of isometric immersion f and let \(\overrightarrow{H'}\)be the mean curvature vector of \(\varphi \circ f\). Let \(\varphi \) be the standard immersion of \(\mathbb {F}P^{m}\) into \(H(m+1; \mathbb {F})\), where \(H(m+1; \mathbb {F})\) is the vector space of \((m+1)\times (m+1)\) Hermitian matrices with coefficients in the field \(\mathbb {F}\). We’re going to use the Lemma 2.2 that is shown us in [6], namely
$$\begin{aligned} \mid H' \mid ^2 \ \le \ \mid H \mid ^2 + \frac{2(n+d)}{n}. \end{aligned}$$Replacing in Theorem 3.5 we obtain the result required.
\(\square \)
See [3] for more informations about the standard imbeddings of projective spaces.
References
Ashbaugh, M.S., Benguria, R.D.: More bounds on Eigenvalue ratios for Dirichlet Laplacian in \(n\)- dimension. SIAM J. Math. Anal. 24, 1622–1651 (1993)
Chavel, I.: Eigenvalues in Riemannian geometry. Academic Press (1984)
Chen, B.Y.: Total mean curvature and submanifolds of finite type. World Scientific (1984)
Chen, D., Cheng, Q.-M., Wang, Q., Xia, C.: On eigenvalues of a system of elliptic equations and of the biharmonic operator. J. Math. Anal. Appl. 387, 1146–1159 (2012)
Cheng, Q.-M., Yang, H.C.: Universal inequalities for eigenvalues of a system of elliptic equations Proc. R. Soc. Edinburgh Sect. A 139, 273–285 (2009)
Du, F., Wu, C., Li, G., Xia, C.: Estimates for eigenvalues of the bi-drifting Laplacian operator Z. Angew. Math. Phys. 66, 703–726 (2015)
Hook, S.M.: Domain independent upper bounds for eigenvalues of elliptic operator. Trans. Am. Math. Soc. 318, 615–642 (1990)
Jost, J., Li-Jost, X., Wang, Q., Xia, C.: Universal bounds for eigenvalues of the polyharmonic operators Trans. Am. Math. Soc. 363, 1821–1854 (2011)
Kawohl, B., Sweers, G.: Remarks on eigenvalues and eigenfunctions of elliptic system. Z. Angew. Math. Phys. 38, 730–740 (1987)
Levine, H.A., Protter, M.H.: Unrestricted lower bounds for eigenvalues of elliptic equations and systems of equations with applications to problems in elasticity. Math. Methods Appl. Sci. 7, 210–222 (1985)
Levitin, M., Parnovski, L.: Commutators, Spectral trace identities and Universal Estimates for Eigenvalues. J. Funct. Anal. 192, 425–445 (2002)
Ma, L., Du, S.H.: Extension of Reilly formula with applications to eigenvalue estimates for drifting Laplacians. C.R. Math. Acad. Sci. Paris. 348, 1203–1206 (2010)
Ma, L., Liu, B.Y.: Convex eigenfunction of a drifting Laplacian operator and the fundamental gap. Pacific J. Math. 240, 343–361 (2009)
Ma, L., Liu, B.Y.: Convexity of the first eigenfunction of the drifting Laplacian operator and its applications. New York J. Math. 14, 393–401 (2008)
Pereira, R.G., Adriano, L., Pina, R.: Universal bounds for eigenvalues of the poly-drifting Laplacian operator in compact domains in the \(\mathbb{R}^{n}\) and \(\mathbb{S}^{n}\). Ann. Glob. Anal. Geom. 47, 373–397 (2015)
Perelman, G.: Ricci flow with surgery on three manifolds, arXiv:math/0303109
Pleijel, A.: Propriets asymptotique des fonctions fondamentales du problems des vibrations dans un corps lastique. Ark. Mat. Astron. Fys 26, 1–9 (1939)
Sun, H.J., Zeng, L.Z.: Universal Inequalities for Lower Order Eigenvalues of Self-Adjoint Operators and the Poly-Laplacian. Acta Math. Sin. English Series 29(11), 2209–2218 (2013)
Xia, C., Xu, H.: Inequalities for eigenvalues of the drifting Laplacian on Riemannian manifolds. Ann. Glob. Anal. Geom. 45, 155–166 (2014)
Acknowledgments
The authors are very grateful to the referee for the valuable suggestions which lead to improvements in the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Jüngel.
Rosane Gomes Pereira supported in part by CAPES/REUNI. Levi Adriano supported by CAPES/PNPD and FAPEG.
Rights and permissions
About this article
Cite this article
Pereira, R.G., Adriano, L. & Cavalheiro, A. Universal inequalities for eigenvalues of a system of elliptic equations of the drifting Laplacian. Monatsh Math 181, 797–820 (2016). https://doi.org/10.1007/s00605-015-0875-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-015-0875-8