Estimates of eigenvalues of an elliptic differential system in divergence form

Let $\mathbb{R}^{n}$ be the $n$-dimensional Euclidean space with its canonical metric $\langle,\rangle$, and $\Omega\subset\mathbb{R}^{n}$ be a bounded domain with smooth boundary $\partial\Omega$. Let us consider a symmetric positive definite $(1,1)$-tensor $T$ on $\mathbb{R}^{n}$ and a function $\eta\in C^{2}(\mathbb{R}^{n})$, so that we can define a second-order elliptic differential operator $\mathscr{L}$ in the $(\eta,T)$-divergence form as follows:

where $\mathrm{div}$ stands for the divergence operator and $\nabla$ for the gradient operator. Since $\Omega$ is bounded, there exist two positive real constants $\varepsilon$ and $\delta$, such that $\varepsilon I\leq T\leq\delta I$, where $I$ is the $(1,1)$-tensor identity on $\mathbb{R}^{n}$.

The analysis of the sequence of the eigenvalues of elliptic differential operators in divergence forms in bounded domains in $\mathbb{R}^{n}$ is an interesting topic in both mathematics and physics. In particular, problems linking the shape of a domain to the spectrum of an operator are among the most fascinating of mathematical analysis. One of the reasons which make them so attractive is that they involve different fields of mathematics such as spectral theory, Riemannian geometry, and partial differential equations. Not only the literature about this subject is already very rich, but also it is not unlikely that operators in divergence forms may play a fundamental role in the understanding of countless physical facts.

In this paper, we address the eigenvalue problem for an operator which is a second-order perturbation of $\mathscr{L}$. More precisely, we compute universal estimates of the eigenvalues of the coupled system of second-order elliptic differential equations, namely:

where ${\bf u}=(u^{1},u^{2},\ldots,u^{n})$ is a vector-valued function from $\Omega$ to $\mathbb{R}^{n}$, the constant $\alpha$ is non-negative and $\mathscr{L}{\bf u}=(\mathscr{L}u^{1},\mathscr{L}u^{2},\ldots,\mathscr{L}u^{n})$.

We will see that $\mathscr{L}+\alpha\nabla\mathrm{div}_{\eta}$ is a formally self-adjoint operator in the Hilbert space $\mathbb{L}^{2}(\Omega,dm)$ of all vector-valued functions that vanish on $\partial\Omega$ in the sense of the trace. It follows from inner product induced by Eqs. (3.3) and (3.4) in Section 3. Thus the eigenvalue problem (1.2) has a real and discrete spectrum

where each $\sigma_{i}$ is repeated according to its multiplicity.

A special case that we can obtain from Problem 1.2 occurs when $T$ is divergence-free, see Problem 6.2. For the sake of convenience, we address this case in Section 6. Some results from Problems 1.4 and 1.5 below are particular cases of this section. However, these two latter problems still remain prototype for us. In the next two paragraphs, we make brief comments about them.

When $\eta$ is a constant and $T$ is the identity operator $I$ on $\mathbb{R}^{n}$, Problem (1.2) becomes

where $\Delta{\bf u}=(\Delta u^{1},\ldots,\Delta u^{n})$ and $\Delta$ is the Laplacian operator on $C^{\infty}(\Omega)$. The operator $\Delta+\alpha\nabla\mathrm{div}$ is known as Lamé’s operator. In the $3$-dimensional case it shows up in the elasticity theory and $\alpha$ is determined by the positive constants of Lamé, so the assumption $\alpha\geq 0$ is justified. For further details on this issue, the interested reader can consult Pleijel [16] or Kawohl and Sweers [11]. It is worth mentioning here the works of Levine and Protter [12], Livitin and Parnovski [13], Hook [10], Cheng and Yang [5] and Chen et al. [3] in which we can find some interesting estimates of the eigenvalues of Problem (1.4). We will be more precise later when we will discuss the three latter papers.

When $\eta$ is not necessarily constant and $T=I$, Problem (1.2) is rewritten as

where $\Delta_{\eta}{\bf u}=(\Delta_{\eta}u^{1},\ldots,\Delta_{\eta}u^{n})$ and $\Delta_{\eta}=\mathrm{div}_{\eta}\nabla$ is the drifted Laplacian operator on $C^{\infty}(\Omega)$. The drifted Laplacian as well as the Bakry-Emery Ricci tensor $Ric+\nabla^{2}\eta$ are the most appropriate geometric objects to study the smooth metric measure spaces $(M^{n},g,e^{-\eta}dvol_{g})$. In particular, the Bakry-Emery Ricci tensor has been especially studied in the theory of Ricci solitons, since a gradient Ricci soliton $(M^{n},g,\eta)$ is characterized by $Ric+\nabla^{2}\eta=\lambda g$, for some constant $\lambda$.

In Corollary 2.5, we show an interesting case of rigidity inequalities of eigenvalues of the Laplacian in a countable family of bounded domains in Gaussian shrinking soliton $(\mathbb{R}^{n},\delta_{ij},\frac{\lambda}{2}|x|^{2})$ by taking a specific isoparametric function as being the drifting function $\eta$, see Remarks 2.2 and 2.3. We address the Gaussian expanding soliton case in Corollaries 2.6 and 2.7.

Throughout this paper $dm=e^{-\eta}d\Omega$ stands for the weight volume form on $\Omega$ and $|\cdot|$ for the Euclidean norm. Moreover, let us define

so that

Henceforth, since there is no danger of confusion, we are using the same notation $\|\cdot\|$ for the norm in (1.6) as well as for the canonical norm of a real-valued function in $L^{2}(\Omega,dm)$.

Our proofs will be facilitated by analyzing the more general setting in which the function $\eta$ is not necessarily constant and $T$ is not necessarily the identity. In this case, we prove a universal quadratic estimate for the eigenvalues of Problem (1.2), which is an essential tool to obtain some of our estimates.

We identify the quadratic estimate in Theorem 1.1 as the most appropriate inequality for the applications of our results. In particular, the constant $C_{0}$ in (1.7) has a crucial importance for us.

Theorem 1.1 is an extension for $\mathscr{L}+\alpha\nabla\mbox{div}_{\eta}$ on vector-valued functions of the well-known Yang’s estimate of the eigenvalues of the Laplacian on real-valued functions. Its proof is motivated by the corresponding results for the Laplacian on real-valued functions case by Yang [20], for $\Delta+\alpha\nabla\mathrm{div}$ on vector-valued functions case by Chen et al. [3, Theorem 1.1], and for $\mathscr{L}$ on real-valued functions case by Gomes and Miranda [9].

We also prove an estimate for the sum of lower order eigenvalues in terms of the first eigenvalue and its correspondent eigenfunction.

Theorem 1.2 is an extension for $\mathscr{L}+\alpha\nabla\mbox{div}_{\eta}$ on vector-valued functions of a stronger result obtained by Cheng and Yang [5, Theorem 1.2] for lower order eigenvalues of Problem (1.4). Its proof is motivated by the corresponding results for $\Delta+\alpha\nabla\mathrm{div}$ case in [5] and for $\Delta_{\eta}+\alpha\nabla\mathrm{div}_{\eta}$ case in [8, Theorem 1.3].

We begin this section by defining a known class of the functions which is closely related to our applications. A nonconstant smooth function $f:\mathbb{R}^{n}\to\mathbb{R}$ is called transnormal function if

for some smooth function $b$ on the range of $f$ in $\mathbb{R}$. The function $f$ is called an isoparametric function if it moreover satisfies

for some continuous function $a$ on the range of $f$ in $\mathbb{R}$.

Eq. (2.1) implies that the level set hypersurfaces of $f$ are parallel hypersurfaces and it follows from Eq. (2.2) that these hypersurfaces have constant mean curvature. Isoparametric functions appear in the isoparametric hypersurfaces theory (i.e., has constant principal curvatures) systematically developed by Cartan [2] on space forms. Wang [18] considered the problem of extending this theory to a general Riemannian manifold and studied some properties of (2.1) and (2.2) more closely. Notice that isoparametric functions exist on a large class of spaces (e.g. symmetric spaces) other than space forms. Currently, new examples of isoparametric functions on Riemannian manifolds have been discovered, for instance, the potential function of any noncompact gradient Ricci soliton $(M^{n},g,\eta)$ with constant scalar curvature $R$ is an isoparametric function, since we can assume that $\eta$ (after a possible rescaling) satisfies $|\nabla\eta|^{2}=2\lambda\eta-R$ and $\Delta\eta=\lambda n-R$, see, e.g., Chow et al. [7]. In particular, the potential function of the Gaussian shrinking soliton $(\mathbb{R}^{n},\delta_{ij},\frac{\lambda}{2}|x|^{2})$ is an isoparametric function, see Example 2.1. This latter fact and a brief analysis of the constant $C_{0}$ in (1.7) were the main motivations to consider the isoparametric function $\eta(x)=\frac{\lambda}{2}|x|^{2}$ to give some applications of our results. The quadratic estimate below is a basic result for it.

The following corollary is an immediate consequence of Theorem 1.2.

Notice that the appearance of the constant $C_{0}$ is natural, since we did not impose any restriction on the function $\eta$. We highlight that this constant has an unexpected geometric interpretation. Indeed, let us consider the warped metric $g=g_{0}+e^{-\eta}d\theta^{2}$ on the product $\Omega\times\mathbb{S}^{1}$, where $g_{0}$ stands for the canonical metric in the domain $\Omega\subset\mathbb{R}^{n}$, whereas $d\theta^{2}$ is the canonical metric of the unit sphere $\mathbb{S}^{1}$, so that the scalar curvature of $g$ is given by $\frac{1}{2}\Delta\eta-\frac{1}{4}|\nabla\eta|^{2}$. Hence, we can obtain $C_{0}$ as supremum of the scalar curvature on the warped product $\Omega\times_{e^{-\eta}}\mathbb{S}^{1}$. Moreover, we ask the following natural question:

We give an answer to this question by using a specific family of domains in Gaussian shrinking soliton. More precisely, we consider a countable family of bounded domains in $\mathbb{R}^{n}$ that makes the behavior of known estimates of eigenvalues of the Laplacian invariant by a first-order perturbation of the Laplacian, see Corollary 2.5.

Coming back to Corollary 2.1, we define

so that, from (2.3), we get

Notice that $\sigma_{i}+D_{0}>0.$

Now, as mentioned in the introduction, we immediately recover the following inequality:

which has been obtained by Chen et al. [3, Corollary 1.2] for Problem (1.4). Indeed, it follows from (2.5) and (2.6), since $\alpha\geq 0$ and we can take $\eta$ to be a constant. Moreover, Inequality (2.7) implies Theorem 1.1 in Cheng and Yang [5], whereas [5, Theorem 1.1] implies Theorem 10 in Hook [10]. However, we highlight that Inequality (2.6) provides an estimate for the eigenvalues of Problem (1.4) which is better than Inequality (2.7).

In the case of Problem (1.5), we can see that Inequality (2.4) is better than Inequality (1.7) in Du and Bezerra [8]; whereas Inequality (2.6) is better than Inequality (1.3) again in [8].

Besides, from Inequality (2.6) and following the steps of the proof of [9, Theorem 3], we obtain the inequalities:

Again from (2.6) and by applying the recursion formula of Cheng and Yang [4], we obtain the following corollary.

From the classical Weyl’s asymptotic formula for the eigenvalues [19], we know that estimate (2.11) is optimal in the sense of the order on $k$.

Example 2.1 below is a special case of $C_{0}=0$. To see this, let us consider an isoparametric function $\eta(x)=\frac{\lambda}{2}(x_{1}^{2}+\cdots+x_{k}^{2})$ on $\mathbb{R}^{n}$, where $\lambda$ is any nonzero real number, $k$ an integer with $0<k\leq n$ and $x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}$. It is easy to verify that

In particular, if $k=n$, the function $\eta(x)=\frac{\lambda}{2}|x|^{2}$ is the potential function of the Gaussian shrinking ($\lambda>0$) or expanding ($\lambda<0$) soliton on $\mathbb{R}^{n}$. We now take $\eta(x)=\frac{\lambda}{2}|x|^{2}$ into the equation of $C_{0}$ in Corollary 2.1, so that,

With these considerations in mind, we write the next two examples.

We are now in the right position to give the interesting applications that we had promised.

Now, we will address the expanding case.

This section is brief and serves to set the stage, introducing some basic notation and describing what is meant by the properties of a $(1,1)$-tensor in a bounded domain $\Omega\subset\mathbb{R}^{n}$ with smooth boundary $\partial\Omega$.

Throughout the paper, we will be constantly using the identification of a $(0,2)$-tensor $T:\mathfrak{X}(\Omega)\times\mathfrak{X}(\Omega)\to C^{\infty}(\Omega)$ with its associated $(1,1)$-tensor $T:\mathfrak{X}(\Omega)\to\mathfrak{X}(\Omega)$ by the equation

In particular, the tensor $\langle,\rangle$ will be identified with the identity $I$ in $\mathfrak{X}(\Omega)$. From the definition of $\eta$-divergence of $X$ (see Eq. (1.1)) and the usual properties of divergence of vector fields, one has

for all $f\in C^{\infty}(\Omega)$.

Notice that the $(\eta,T)$-divergence form of $\mathscr{L}$ on $\Omega$ allows us to check that the divergence theorem remains true in the form

In particular, for $X=T(\nabla f)$,

where $dm=e^{-\eta}d\Omega$ and $d\mu=e^{-\eta}d\partial\Omega$ are the weight volume form on $\Omega$ and the volume form on the boundary $\partial\Omega$ induced by the outward unit normal vector $\nu$ on $\partial\Omega$, respectively. Thus, the integration by parts formula is given by

for all $\ell,f\in C^{\infty}(\Omega)$. Hence, $\mathscr{L}$ is a formally self-adjoint operator in the space of all real-valued functions in $L^{2}(\Omega,dm)$ that vanish on $\partial\Omega$ in the sense of the trace. Furthermore, from (3.1) and (3.2) we obtain

for all vector-valued function ${\bf u}=(u^{1},u^{2},\ldots,u^{n})$ and ${\bf v}=(v^{1},v^{2},\ldots,v^{n})$ both from $\Omega$ to $\mathbb{R}^{n}$, with ${\bf v}$ vanishing on $\partial\Omega$.

We conclude that $\mathscr{L}+\alpha\nabla\mathrm{div}_{\eta}$ is a formally self-adjoint operator in the Hilbert space $\mathbb{L}^{2}(\Omega,dm)$ of all vector-valued functions that vanish on $\partial\Omega$ in the sense of the trace, with inner product induced by Eqs. (3.3) and (3.4). Thus the eigenvalue problem (1.2) has a real and discrete spectrum $0<\sigma_{1}\leq\sigma_{2}\leq\cdots\leq\sigma_{k}\leq\cdots\to\infty$, where each $\sigma_{i}$ is repeated according to its multiplicity.

It is worth mentioning here the paper by Gomes and Miranda [9, Section 2] from which we know some geometric motivations to work with the operator $\mathscr{L}$ in the $(\eta,T)$-divergence form in bounded domains in Riemannian manifolds. They showed that it appears as the trace of a $(1,1)$-tensor on a Riemannian manifold $M$, and computed a Bochner-type formula for it. An interesting fact is that Eq.  $(2.3)$ of [9] relates the operator $\mathscr{L}$ to Cheng and Yau’s operator $\Box$ defined in [6]. In particular, if we take $T$ to be the Einstein tensor, then this latter relation is likely to have applications in physics.

For a vector-valued function ${\bf u}=(u^{1},u^{2},\ldots,u^{n})$ from $\Omega$ to $\mathbb{R}^{n}$, we define

Now, we are considering two definitions for $T$, and since there is no danger of confusion, we are using the same notation $T$ for both definitions. Let $X$ be a vector field on $\Omega$, we define

and

Moreover, for all real-valued functions $f,\ell\in C^{\infty}(\Omega)$ it is immediate from the properties of $\mathrm{div}_{\eta}$ and the symmetry of $T$ that

So, the following equation is well understood for a vector-valued function ${\bf u}$ and a real-valued function $f\in C^{\infty}(\Omega)$

Besides, we are using the classical norms: $|{\bf u}|^{2}=\sum_{i=1}^{n}(u^{i})^{2}$ and $\|{\bf u}\|^{2}=\int_{\Omega}|{\bf u}|^{2}dm$. We observe that $\varepsilon I\leq T\leq\delta I$ implies

In particular, we obtain

In order to prove our first theorem we will need three technical lemmas. The first one is motivated by the corresponding results for Problem (1.4) proven by Chen et al. [3, Lemma 2.1] and for Problem (1.5) proven by Du and Bezerra [8, Lemma 2.1]. Here, we follow the steps of the proof of Lemma 2.1 in [3] with appropriate adaptations for $\mathscr{L}+\alpha\nabla\mathrm{div}_{\eta}$.