Eigenvalues of elliptic operators with density

We consider the eigenvalue problem

on a connected bounded open subset $\Omega$ of $\mathbb{R}^{N}$, where $\rho$ is a positive function bounded away from zero and infinity and where we impose Neumann boundary conditions on $u$. Under suitable regularity assumptions on the boundary of $\Omega$ (e.g., if $\Omega$ has a Lipschitz boundary) it is standard to prove that problem (1.1) admits an increasing sequence of non-negative eigenvalues of finite multiplicity

where $d_{N,m}$ denotes the dimension of the space of polynomials of degree at most $m-1$ in $\mathbb{R}^{N}$.

In this paper we will prove a few results on the dependence of the eigenvalues $\mu_{j}[\rho]$ upon variation of $\rho$. In particular we will consider the problem of finding upper bounds for $\mu_{j}[\rho]$ among all positive and bounded densities $\rho$ satisfying suitable constraints. We shall also consider the issue of lower bounds, which also presents some interesting features.

Keeping in mind important problems for the Laplace and the biharmonic operators in linear elasticity (see e.g., [16]) we shall think of the weight $\rho$ as a mass density of the body $\Omega$ and we shall refer to the quantity $M=\int_{\Omega}\rho dx$ as to the total mass of $\Omega$. In fact, when $N=2$ the eigenvalues $\mu_{j}[\rho]$ describe the vibrations of a non-homogeneous membrane with free edge when $m=1$ (see e.g., [28, § 9]) and of a non-homogeneous plate with free edge when $m=2$ (see [9, 12]).

Relevant questions on the dependence of the eigenvalues $\mu_{j}[\rho]$ upon $\rho$ are whether it is possible to minimize or maximize the eigenvalues under the assumption that the total mass is fixed, or whether it is possible to have uniform upper or lower bounds for the eigenvalues (i.e., bounds which depend only on the total mass, the dimension and the eigenvalue index) under the same constraint, and which have the correct behavior in $j\in\mathbb{N}$ as described by the Weyl’s asymptotic law.

Most of the existing literature treats the case of the Laplace operator with Dirichlet boundary conditions. In particular, we recall the famous result of Krein [31] on the eigenvalues of the Dirichlet Laplacian in one dimension (fixed string) which completely answers the questions raised above. In fact he finds sharp upper and lower bounds which depend only on $j,M,H,l$ for all the eigenvalues of the Laplacian on the string $]0,l[$ upon densities $0\leq\rho\leq H$ for which $M=\int_{0}^{l}\rho dx$ is fixed (see Remark 3.24). We refer also to the extensive work of Banks and collaborators for generalizations and extensions of Krein’s results (see [2, 3, 5, 6, 26] and the references therein). We mention also [21, § 5] which contains a detailed analysis of the eigenvalues of Sturm-Liouville problems with Dirichlet conditions with density (and also other types of weight). In particular, in [21, § 5], the authors provide estimates (upper and lower bounds) under various type of linear and non-linear constraints on the weights. Existence of minimizers and maximizers under mass constraint in higher dimensions for the Dirichlet Laplacian has been investigated in [11, 17, 18, 19, 24], where the authors impose the additional constraint that admissible densities are uniformly bounded from below and above by some fixed constants. We refer to [28, § 9] and to the references therein for further discussions on eigenvalue problems for inhomogeneous strings and membranes with fixed edges.

As for Neumann boundary conditions, much less is known. Very recently the problem of finding uniform upper bounds for the Neumann eigenvalues of the Laplacian with weights has been solved (for $N\geq 2$) by Colbois and El Soufi [14] in the more general context of Riemannian manifolds, by exploiting a general result of decomposition of a metric measure space by annuli (see [27], see also [29]). The authors have not considered the case $N=1$ and, in fact, as we shall see in the present paper, upper bounds with mass constraint do not exists in dimension one.

There are very few results for what concerns higher order operators. We recall [4, 35], where the authors consider the case of the biharmonic operator in one dimension with intermediate boundary conditions (hinged rod) and [7], where the author considers the case of the biharmonic operator with Dirichlet conditions in dimension one (clamped rod) and two (clamped plate). We also refer to [21, § 7.9] where it is possible to find some estimates for the eigenvalues of elliptic operators of order $2m$ with density subject to Dirichlet boundary conditions. We refer again to [28, § 11] for a more detailed discussion on eigenvalue problems for inhomogeneous rods and plates with hinged and clamped edges. Up to our knowledge, there are no results in the literature on the existence and characterization of upper and lower bounds with respect to mass densities for higher order operators subject to Neumann boundary conditions (already for the biharmonic operator or the Laplacian in dimension one).

Finally, we refer to [32] where the authors prove continuity and differentiability results for the dependence of the eigenvalues of a quite wide class of higher order elliptic operators and homogeneous boundary conditions upon variation of the mass density and in most of the cases (except, again, that of Neumann boundary conditions), they establish a maximum principle for extremum problems related to mass density perturbations which preserve the total mass. We remark that in [32] partial results are obtained in the case of Neumann boundary conditions only for the Laplace operator.

In this paper we shall primarily address the issue of finding upper bounds for the eigenvalues $\mu_{j}[\rho]$ of the polyharmonic operators with Neumann boundary conditions which are consistent with the power of $j$ in the Weyl’s asymptotic formula (see (2.9)), among all densities which satisfy a suitable constraint. In particular, we consider two very natural constraints: $\int_{\Omega}\rho dx={\rm const.}$ and $\int_{\Omega}\rho^{\frac{N}{2m}}dx={\rm const.}$. This second constraint arises naturally since it is well-known (see e.g., [23]) that if we set $N(\mu):=\left\{\#\mu_{j}:\mu_{j}<\mu\right\}$, then $N(\mu)\sim\omega_{N}(2\pi)^{\frac{1}{N}}\mu^{\frac{N}{2m}}\int_{\Omega}\rho^{\frac{N}{2m}}dx$. This means that $\|\rho\|_{L^{\frac{N}{2m}}(\Omega)}$ describes the asymptotic distribution of the eigenvalues of problem (1.1) (and in particular implies the Weyl’s law (2.9)). Most of the literature mentioned above considers only the fixed mass constraint.

In view of the physical interpretation of problem (1.1) when $m=1$ and $N=1$ or $N=2$, it is very natural to ask whether it is possible to redistribute a fixed amount of mass on a string (of fixed length) or on a membrane (of fixed shape) such that all the eigenvalues become arbitrarily large when the body is left free to move, or, on the contrary, if there exists uniform upper bounds for all the eigenvalues. As highlithed in [14], uniform upper bounds with mass constraint exist if $N\geq 2$. In this paper, by using the techniques of [14] we prove that if $N\geq 2m$, uniform upper bounds exist (see Theorem 3.4), namely we prove that if $N\geq 2m$

where $C_{N,m}$ depends only on $m$ and $N$. Surprisingly, in lower dimensions, uniform upper bounds do not hold. In fact we find explicit examples of densities with fixed mass and arbitrarily large eigenvalues (see Theorem 3.29). In this case, however, we are able to find upper bounds which depend also on $\|\rho\|_{L^{\infty}(\Omega)}$ (see Theorem 3.22), namely we prove that if $N<2m$

where again $C_{N,m}$ depends only on $m$ and $N$ and the exponent of $\|\rho\|_{L^{\infty}(\Omega)}$ is sharp. We remark that this inequality holds when $m=1$ for $N=1$, and it is the analogue of the upper bounds (3.25) proved by Krein [31] for the Dirichlet Laplacian on an interval (up to a universal constant). We note that in order to prove that certain eigenvalue bounds under some natural constraints do not hold, one has to provide counterexamples. It is then natural to ask whether it is possible to find ‘weaker’ bounds which include the correct quantities that explain the counterexamples. This is the case of the bounds (1.3).

We note that the interplay between the dimension of the space and the order of the operator plays a crucial role in the existence of uniform upper bounds for the eigenvalues of problem (1.1) under mass constraint. We can summarize our first result in this way:

“If $N\geq 2m$ there exist uniform upper bounds with mass constraint for all the eigenvalues of (1.1), while if $N<2m$ we can always redistribute a fixed amount of mass such that all the eigenvalues of (1.1) become arbitrarily large”.

As for the the non-linear constraint $\int_{\Omega}\rho^{\frac{N}{2m}}dx={\rm const.}$, in view of the fact that $N(\mu)\sim\omega_{N}(2\pi)^{\frac{1}{N}}\mu^{\frac{N}{2m}}\int_{\Omega}\rho^{\frac{N}{2m}}dx$, it is natural to ask whether upper bounds of the form

hold. We will call bounds of the form (1.4) “Weyl-type bounds”. Clearly, for $N=2m$ inequality (1.4) is equivalent to (1.2). For $N>2m$ we are able to find densities with fixed $L^{\frac{N}{2m}}$-norm and which produce arbitrarily large eigenvalues (see Theorem 4.16). However, we are able to prove upper bounds for all the eigenvalues which involve both $\|\rho\|_{L^{\frac{N}{2m}}(\Omega)}$ and $\|\rho\|_{L^{\infty}(\Omega)}$ (see Theorem 4.11), namely we prove that if $N>2m$ then

where $C_{N,m}$ depends only on $m$ and $N$ and the exponent of $\|\rho\|_{L^{\infty}(\Omega)}$ is sharp. Since (1.4) holds for $N=2m$ we are led to conjecture that it must hold for any $N<2m$. We are still not able to prove (1.4) for $N<2m$, and actually it seems to be a quite difficult issue. However we can prove the weaker inequality

We leave the validity of (1.4) for $N<2m$ as an open question. We refer to Remark 4.10 where we discuss relevant examples in support of the validity of our conjecture. In particular, we note that if (1.4) holds true for $N<2m$, when $m=1$ and $N=1$ we would find uniform upper bounds for the eigenvalues of the Neumann Laplacian under the constraint that $\int_{\Omega}\sqrt{\rho}dx={\rm const.}$. We can summarize our second result as follows:

“If $N>2m$, we can always find a density $\rho$ with fixed $\|\rho\|_{L^{\frac{N}{2m}}(\Omega)}$ such that all the eigenvalues of (1.1) are arbitrarily large, while we have uniform Weyl-type upper bounds when $N=2m$. We conjecture the existence of uniform Weyl-type upper bounds when $N<2m$”.

We also mention [28, § 9.2.3] where it is considered a spectral optimization problem for the Dirichlet Laplacian with the non-linear constraint $\int_{\Omega}\rho^{p}dx={\rm const.}$, where $p>N/2$ and $N\geq 2$ (see also [21, § 5]).

We have also considered the issue of lower bounds and we have found that ‘surprisingly’ the interplay between the space dimension $N$ and the order $m$ of the operator plays a fundamental role also in the existence of lower bounds. In fact we are able to prove the following facts (see Theorems 5.1,5.4,5.13 and 5.15):

“If $N<2m$ there exists a positive constant $C$ which depends only on $m,N$ and $\Omega$ such that the first positive eigenvalue of problem (1.1) is bounded from below by $C\left(\int_{\Omega}\rho dx\right)^{-1}$, while if $N\geq 2m$, for all $j\in\mathbb{N}$ we can always redistribute a fixed amount of mass such that the first $j$ eigenvalues of (1.1) are arbitrarily close to zero”

“If $N>2m$ there exists a positive constant $C$ which depends only on $m,N$ and $\Omega$ such that the first positive eigenvalue of problem (1.1) is bounded from below by $C\|\rho\|_{L^{\frac{N}{2m}}(\Omega)}^{-1}$, while if $N\leq 2m$ for all $j\in\mathbb{N}$ we can always find densities with fixed $L^{\frac{N}{2m}}$-norm such that the first $j$ eigenvalues of (1.1) are arbitrarily close to zero”.

We note that lower bounds for the first eigenvalue under one of the two constraints exist in the case that upper bounds with the same constraint do not exist. We remark that the situation is very different if we consider for example the issue of the minimization of the eigenvalues of (1.1) with $\rho\equiv 1$ among all bounded domains with fixed measure: it is standard to prove that there exist domains with fixed volume and such that the first $j$ eigenvalues can be made arbitrarily close to zero, in any dimension $N\geq 2$.

Finally we remark that all the results of this paper can be adapted to the more general eigenvalue problem

with Neumann boundary conditions, where $\mathcal{L}$ is defined by

and is an elliptic operator of order $2m$, under suitable assumptions on the domain $\Omega$ and the coefficients of $A_{\alpha\beta}$. We refer to [32] for a detailed description of eigenvalue problems for higher order elliptic operators with density (see also [21, § 7]).

The present paper is organized as follows: Section 2 is dedicated to some preliminaries. In Section 3 we consider the problem of finding uniform upper bounds with mass constraint. In particular in Subsection 3.1 we prove uniform upper bounds (1.2) for $N\geq 2m$, in Subsection 3.2 we prove upper bounds (1.3) for $N<2m$ and in Subsection 3.3 we provide counterexamples to uniform upper bounds in dimension $N<2m$. In Section 4 we investigate the existence of upper bounds with the non-linear constraint $\int_{\Omega}\rho^{\frac{N}{2m}}dx={\rm const.}$. In particular in Subsections 4.1 and 4.2 we prove upper bounds (1.6) and (1.5), respectively, while in Subsection 4.3 we provide counterexamples to uniform upper bounds (1.4) for $N>2m$. In Subsection 4.1 we state the open question whether bounds of the form (1.4) hold if $N<2m$. In Section 5 we consider lower bounds and in particular we discuss how the constraint, the space dimension and the order of the operator influence their existence. At the end of the paper we have two appendices, Appendix A and Appendix B. In Appendix A we discuss Neumann boundary conditions for higher order operators and develop some basic spectral theory for such operators. In Appendix B we prove some useful functional inequalities which are crucial in the proof of Theorem 3.29 in Subsection 3.3.

Let $\Omega$ be a bounded domain (i.e., an open connected bounded set) of $\mathbb{R}^{N}$. By $H^{m}(\Omega)$ we shall denote the standard Sobolev space of functions in $L^{2}(\Omega)$ with weak derivatives up to order $m$ in $L^{2}(\Omega)$, endowed with its standard norm defined by

for all $u\in H^{m}(\Omega)$, where

In what follow we will use the standard multi-index notation. Hence, for $\alpha\in\mathbb{N}^{N}$, $\alpha=(\alpha_{1},...,\alpha_{N})$, we shall denote by $|\alpha|$ the quantity $|\alpha|=\alpha_{1}+\cdots+\alpha_{N}$. Moreover, for $\alpha,\beta\in\mathbb{N}^{N}$, $\alpha+\beta=(\alpha_{1}+\beta_{1},...,\alpha_{N}+\beta_{N})$ and $\alpha!=\alpha_{1}!\cdots\alpha_{N}!$. For $x\in\mathbb{R}^{N}$, $x=(x_{1},...,x_{N})$, we will write $x^{\alpha}=x_{1}^{\alpha_{1}}\cdots x_{N}^{\alpha_{N}}$. For a function $u$ of class $C^{|\alpha|}$, we write $\partial^{\alpha}u=\frac{\partial^{|\alpha|}u}{\partial_{x_{1}}^{\alpha_{1}}\cdots\partial_{x_{N}}^{\alpha_{N}}}$. Finally, for a function $U:\mathbb{R}\rightarrow\mathbb{R}$ and $l\in\mathbb{N}$, we shall write $U^{(l)}(x)$ to denote the $l$-th derivative of $U$ with respect to $x$.

In the sequel we shall assume that the domain $\Omega$ is such that the embedding of $H^{m}(\Omega)$ into $L^{2}(\Omega)$ is compact (which is ensured, for example, if $\Omega$ is a bounded domain with Lipschitz boundary). By $\mathcal{R}$ we shall denote the subset of $L^{\infty}(\Omega)$ of those functions $\rho\in L^{\infty}(\Omega)$ such that ${\rm ess}\inf_{\Omega}\rho>0$.

We shall consider the following eigenvalue problem:

in the unknowns $u\in H^{m}(\Omega)$ (the eigenfunction), $\mu\in\mathbb{R}$ (the eigenvalue), where

We note that problem (2.1) is the weak formulation of the following eigenvalue problem:

in the unknowns $u\in C^{2m}(\Omega)\cap C^{2m-1}(\overline{\Omega})$ and $\mu\in\mathbb{R}$. Here $\mathcal{N}_{j}$ are uniquely defined ‘complementing’ boundary operators of degree at most $2m-1$ (see [25] for details), which we will call Neumann boundary conditions (see Appendix A.1).

In Appendix A.1 we discuss in more detail boundary conditions for problems (2.2) and (2.5) and, more in general, Neumann boundary conditions for the polyharmonic operators.

It is standard to prove (see Theorem A.5) that the eigenvalues of (2.1) are non-negative, have finite multiplicity and consist of a sequence diverging to $+\infty$ of the form

where

The eigenfunctions associated with the eigenvalue $\mu=0$ are the polynomials of degree at most $m-1$ in $\mathbb{R}^{N}$ (the dimension of the space spanned by the polynomials of degree at most $m-1$ in $\mathbb{R}^{N}$ is exactly $d_{N,m}$). We note that we have highlithed the dependence of the eigenvalues upon the density $\rho$, which is the main object of study of the present paper.

By standard spectral theory, we deduce the validity of the following variational representation of the eigenvalues (see [16, § IV] for more details):

We conclude this section by recalling the asymptotic behavior of the eigenvalues as $j\rightarrow+\infty$, which is described by the Weyl’s law.

We refer to [23] for a proof of Theorem 2.8.

In this section we consider the problem of finding uniform upper bounds for the $j$-th eigenvalue $\mu_{j}[\rho]$ among all mass densities $\rho\in\mathcal{R}$ which preserve the mass (that is, among all $\rho\in\mathcal{R}$ such that $\int_{\Omega}\rho dx={\rm const.}$), and which show the correct growth in the power of $j$ with respect to the Weyl’s law (2.9). In particular, in Subsection 3.1 we prove that such bounds exist if $N\geq 2m$ (see Theorem 3.4), while in Subsection 3.3 we will give counter-examples in dimension $N<2m$ (see Theorem 3.29). Moreover, in Subsection 3.2 we establish upper bounds in the case $N<2m$ which involve also a suitable power of $\|\rho\|_{L^{\infty}(\Omega)}$ (see Theorem 3.22) which turns out to be sharp.