A note on Kuttler–Sigillito’s inequalities

We provide several inequalities between eigenvalues of some classical eigenvalue problems on compact Riemannian manifolds with C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document} boundary. A key tool in the proof is the generalized Rellich identity on a Riemannian manifold. Our results in particular extend some inequalities due to Kuttler and Sigillito from subsets of R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^2$$\end{document} to the manifold setting.


Introduction
The objective of this manuscript is to establish several inequalities between eigenvalues of the classical eigenvalue problems mentioned below. Let (M n , g) be a compact and connected Riemannian manifold of dimension n ≥ 2 with nonempty C 2  where = div∇ is the Laplace-Beltrami operator, ν is the unit outward normal vector on ∂ M, and ∂ ν denotes the outward normal derivative. The Dirichlet eigenvalues describe the fundamental modes of vibration of an idealized drum, and for n = 2, the Neumann eigenvalues appear naturally in the study of the vibrations of a free membrane; see e.g. [3,6].
We also consider the Steklov eigenvalue problem, which is an eigenvalue problem with the spectral parameter in the boundary conditions: Steklov eigenvalue problem . (1. 3) The Steklov eigenvalues encode the squares of the natural frequencies of vibration of a thin membrane with free frame, whose mass is uniformly distributed at the boundary; see the recent survey paper [11] and references therein. The last set of eigenvalue problems we consider are the so-called Biharmonic Steklov problems: The eigenvalues problems (1.4) and (1.5) for example play an important role in elastic mechanics. We refer the reader to [5,9,17,18] for some recent results on eigenvalue estimates of problem (1.4). Moreover, a physical interpretation of problem (1.4) can be found in [9,17]. Problem (1.5) was first studied in [12,13] where the main focus was on the first nonzero eigenvalue, which appears as an optimal constant in a priori inequality; see [12] for more details.
It is well-known that the spectra of the eigenvalue problems (1.2)-(1.5) are discrete and non-negative, see e.g. [2,6,9,10,12,17]. We thus arrange their eigenvalues in increasing order, where we repeat an eigenvalue as often as its multiplicity requires. The k-th eigenvalue of one of the above eigenvalue problems will be denoted by the corresponding letter for the eigenvalue with a subscript k, e.g. the k-th Neumann eigenvalue will be denoted by μ k . Note that μ 1 = σ 1 = ξ 1 = 0.
There is a variety of literature on the study of bounds on the eigenvalues of each problem mentioned above in terms of the geometry of the underlying space [11,15,17,22]. However, instead of studying each eigenvalue problem individually, it is also interesting to explore relationships and inequalities between eigenvalues of different eigenvalue problems. Among this type of results, one can mention the relationships between the Laplace and Steklov eigenvalues studied in [14,21,24], and various inequalities between the first nonzero eigenvalue of problems (1.2)-(1.5) on bounded domains of R 2 obtained by Kuttler and Sigillito in [13]; see Table 1 (Note that there was a misprint in Inequality VI in [13]. The correct version of the inequality is stated in Table 1.).
We extend Kuttler-Sigillito's results in two ways. Firstly, we consider compact manifold M with C 2 boundary of any dimension n ≥ 2. Secondly, we also prove inequalities between higher-order eigenvalues.
Our first theorem provides lower bounds for ξ k in terms of Neumann and Steklov eigenvalues. Theorem 1.1 Let (M n , g) be a compact manifold of dimension n ≥ 2 with C 2 boundary. For every k ∈ N we have (a) μ k σ 2 ≤ ξ k , and (b) μ 2 σ k ≤ ξ k . Compared to inequality (b), inequality (a) gives a better lower bound for ξ k for large k. For k = 2 and M ⊂ R 2 , Theorem 1.1 was previously proved in [13]. Kuttler in [12] also obtained an inequality between some higher order eigenvalues ξ k and μ k for a rectangular domain in R 2 using symmetries of the eigenfunctions. In order to state our next results, we need to introduce some notation first. For any given p ∈ M, consider the distance function and one half of the square of the distance function, Furthermore, we set where we borrowed the notation from [13].

Remark 1.2
Note that ρ p is not necessarily differentiable on the cut locus of p. However, the direction derivative denoted by ∇ρ p (x), ζ , ζ ∈ T x M always exists and is given by is the unit tangent vector of a geodesic joining x to p}.
We shall see that under the assumption of a lower Ricci curvature bound, there exists a lower bound on the first nonzero Steklov eigenvalue σ 2 in terms of μ 2 on star shaped manifolds. A manifold M with C 2 boundary is called a star shaped manifold if there exist p ∈ M and a star shaped domain in R n ∼ = T p M such that exp p is defined on and exp p ( ) = M. This implies that ∇ρ p (x), ν(x) ≥ 0 for every x ∈ ∂ M. Theorem 1.3 Let (M n , g) be a compact, star shaped Riemannian manifold whose Ricci curvature Ric g satisfies Ric g ≥ (n − 1)κ. Then we have where C 0 := C 0 (n, κ, r max ) is a positive constant depending only on n, κ and r max .
When M is a subdomain of R n , inequality (1.6) was stated in [13] with C 0 = 2.
In the following theorem we provide several inequalities for eigenvalues of (1.2)-(1.5) on star shaped manifolds under the assumption of bounded sectional curvature. Here and hereafter, we make use of the notation for all A, B ∈ R, and the convention c/0 = +∞, c ∈ R {0}. Theorem 1.4 Let (M n , g) be a compact, star shaped Riemannian manifold of dimension n whose sectional curvature K g satisfies κ 1 ≤ K g ≤ κ 2 . Moreover, assume that there exists p ∈ M such that M is star shaped with respect to p and the cut locus of p in M is the empty set. Then there exist constants C i := C i (n, κ 1 , κ 2 , r max ), i = 1, 2, depending only on n, κ 1 , κ 2 and r max and C 3 = C 3 (n, κ 1 , r max ) such that Here, m is the multiplicity of λ k .
Note that the constants C i , i = 1, 2, 3 are not positive in general. However, there exists r 0 := r 0 (n, κ 1 , κ 2 ) > 0 such that for r max ≤ r 0 these constants are positive; see Sect. 4 for details. In inequality (ii), we have a non trivial upper bound only if When M is a domain in R n , the quantity M d 2 p dv g is called the second moment of inertia; see Example 4.2. The proof of Theorem 1.4 also leads to a non-sharp lower bound on η 1 This in particular shows that the right-hand side of the inequality in part i) is always positive.
The proof of Theorem 1.1 is based on using the variational characterization of the eigenvalues and alternative formulations thereof. Apart from the Laplace and Hessian comparison theorems, and the variational characterization of the eigenvalues, the key tool in the proof of Theorems 1.3 and 1.4 is a generalization of the classical Rellich identity to the manifold setting. This is the content of the next theorem. Let us denote M \ ∂ M by M • .
The classical Rellich identity was first stated by Rellich in [23]. A special case of Theorem 3.1, called the generalized Pohozaev identity, was proved in [21,25] in order to get some spectral inequalities between the Steklov and Laplace eigenvalues. The paper is structured as follows. In Sect. 2, we recall tools needed in later sections, namely the Hessian and Laplace comparison theorems. Moreover, we give variational characterizations and alternative representations for the eigenvalues of problems (1.2)-(1.5). Sect. 3 contains the deduction of the Rellich identity on manifolds, as well as several applications thereof. Finally, we prove the main theorems in Sect. 4.

Preliminaries
In this section we provide the basic tools needed in later sections. Namely, we give the variational characterizations and alternative representations of the eigenvalues of problems (1.2)-(1.5) in the first subsection. In the second subsection, we recall the Hessian and Laplace comparison theorems.

Variational characterization and alternative representations
Below, we list the variational characterization of eigenvalues of (1.1)-(1.5) and their alternative representations. We refer to [2,6] for the variational characterization of (1.1)-(1.3), and to Appendix for (1.4) and (1.5). For the special case of the first nonzero eigenvalues of (1.1)-(1.5), their alternative representations are contained in [13]. The general proofs of their alternative representations follow along the same lines of the proofs in [13] and are therefore omitted. Dirichlet eigenvalues: (2.1) Neumann eigenvalues: Steklov eigenvalues: where H(M) is the space of harmonic functions on M.

Hessian and Laplace comparison theorems
The idea of comparison theorems is to compare a given geometric quantity on a Riemannian manifold with the corresponding quantity on a model space. Below we recall the Hessian and Laplace comparison theorems. For more details we refer the reader to [4,7,20]. For any κ ∈ R, denote by H κ : [0, ∞) → R the function satisfying the Riccati equation With this preparation at hand we can now state the Hessian comparison theorem.

Then (a) d p satisfies the inequalities
Furthermore, we have Here A ∧ B := min{A, B} and A ∨ B := max{A, B} for A, B ∈ R.
(b) ρ p satisfies the inequalities Next, we state the Laplace comparison theorem.
hold at smooth points of d p . Moreover the above inequalities hold on the whole manifold in the sense of distribution. (b) Under the same assumption and notations of Theorem 2.1, the following inequalities hold.
Notice that part (b) in the above theorems is an immediate consequence of part (a), since the distance function d p and one half of the square of the distance function ρ p satisfy

Generalized Rellich identity
An important identity which is used in the study of eigenvalue problems is the Rellich identity. To our knowledge it was first stated and used by Rellich [23] in the study of the Dirichlet eigenvalue problem. Some versions of the Rellich identity are also referred to as the Pohozaev identity; see [8,21,25] for more details and its applications. In this section, we provide the generalized Rellich identity on Riemannian manifolds, i.e. Theorem 1.5, and its higher order version. Some applications of this result can be found in the last subsection and in Sect. 4.

Rellich identity on manifolds
The next theorem states the Rellich identity on Riemannian manifolds.
In [21,25], the authors proved the above identity when w is harmonic and λ = 0. The proof of the general version follows the same line of argument. For the sake of completeness we give the whole argument.
Proof of Theorem 3. 1 We calculate M w F, ∇w dv g and M λw F, ∇w dv g separately. In order to calculate the latter, we apply the divergence theorem to obtain Thus, we get For the other term, using integration by parts, we obtain For further simplification, we observe that Plugging this identity into (3.1) we get This completes the proof.

Higher order Rellich identities
In this section, we provide a higher order Rellich identity. Throughout the section, M is a compact Riemannian manifold with nonempty C 2 boundary. The following preparatory lemma is a simple consequence from Theorem 3.1. For the special case M ⊂ R n , the identity stated in the lemma was first proven by Mitidieri in [19].
Proof Replacing w by w + v in Theorem 3.1 and set λ = 0 we get the identity.
The following theorem states the higher order Rellich identity.

Theorem 3.3 Let the boundary of M be C 2 smooth. Then for
By the divergence theorem we have which together with Theorem 3.1 establishes the claim.
For the special case M ⊂ R n and λ = 0, the statement of Theorem 3.3 is contained in [19].

Applications of the Rellich identities
In 1940, Rellich [23] dealt with the Dirichlet eigenvalue problem on sets M ⊂ R n . For this special case he used the identity derived in Theorem 3.1 to express the Dirichlet eigenvalues in terms of an integral over the boundary. One decade ago, Liu [16] extended Rellich's result to the Neumann eigenvalue problem, the clamped plate eigenvalue problem and the buckling eigenvalue problem, each on sets M ⊂ R n . In the latter two cases Liu (implicitly) applied the higher order Rellich identity.
Recall that for any compact Riemannian manifold M with C 2 boundary ∂ M, the clamped plate eigenvalue problem and the buckling eigenvalue problem are given by respectively. Below we reprove the result of Liu for the case of the buckling eigenvalue problem. Note there is no new idea for the proof, however, our proof is shorter and clearer since we do not carry out the calculations in coordinates. One can proceed similarly for the clamped plate eigenvalue problem.

Lemma 3.4 ([16]) Let M ⊂ R n be a bounded domain with C 2 smooth boundary.
(i) Let w be an eigenfunction corresponding to the eigenvalue of the buckling eigenvalue problem. Then we have where r 2 = x 2 1 + · · · + x 2 n and x i are Euclidean coordinates. (ii) Let w be an eigenfunction corresponding to the eigenvalue of the clamped plate eigenvalue problem. Then we have Proof In order to prove (i) we apply Theorem 3.3 for the special case M ⊂ R n and where F is given by the gradient of the distance function. In this case we have D F( · , · ) = g( · , · ) and divF = n. Note furthermore that w |∂ M = 0 implies ∇w = ∂ ν wν on ∂ M. Since we have ∂ ν w |∂ M = 0 by assumption, ∇w vanishes along the boundary of M.
Plugging the above information into Theorem 3.3 we get Applying the divergence theorem once more, we thus obtain The variational characterization of asserts that for an eigenfunction w corresponding to we have Furthermore, the identities and w = ∂ 2 νν w hold on the boundary of M. Thus the claim is established. The proof of (ii) is omitted since it is similar to the one of (i).

Remark 3.5 In Lemma 3.4 (i)
, when normalizing the eigenfunction w such that M |∇w| 2 dv g = 1, we obtain i.e. is expressed in terms of an integral over the boundary. A similar remark holds for Lemma 3.4 (ii).
Finally we use the Rellich identities to get some estimates on eigenvalues. Note that from now on we do not assume anymore that M is a subset of the Euclidean space. However, we assume that M is a manifold with C 2 smooth boundary and that there exists a Lipschitz vector field F on M satisfying the following properties: (A) There exist some positive constants c 1 , c 2 ∈ R + such that wherever F is differentiable. The following lemma is an easy consequence of Theorems 3.1 and 3.3, respectively. It establishes upper estimates for eigenvalues in terms of integrals over the boundary ∂ M and α.

Lemma 3.7 Let M be a manifold with C 2 smooth boundary. Assume that there exists a Lipschitz vector field F on M satisfying properties A and B above. Then
(i) the eigenvalue λ corresponding to eigenfunction w of the Dirichlet eigenvalue problem satisfies (ii) the eigenvalue corresponding to eigenfunction w of the buckling eigenvalue problem satisfies Proof We start by proving (i). Theorem 3.1 and Condition A imply Since w ≡ 0 on ∂ M we have ∇w = ∂ ν wν on ∂ M. Combining this with Condition B we obtain The latter inequality implies the claim. Below, we prove (ii). Theorem 3.3 implies Here, we made use of which is a consequence of the divergence theorem. Applying (3.4) yields and thus the claim is established.

Proof of the Main Theorems
In this section, we prove the main theorems. The key ingredients of the proof are the comparison theorems and the Rellich identity. (a) and (b) are an immediate consequence of the variational characterizations of μ k , σ k and ξ k given in (2.2), (2.3) and (2.5). Indeed, let V be the space generated by eigenfunctions associated with ξ 2 , . . . , ξ k . Then by the variational characterization (2.2) we get

Proof of Theorem 1.1 Inequalities
The proof of part (b) is similar. Let V be given as in part (a). By the variational characterization, we obtain This completes the proof.
Proof of Theorem 1.3 Let p ∈ M be a point such that M is star shaped centered at p. We use the following identity 1 2 ∂ M w 2 ν, ∇ρ p ds g = M w ∇w, ∇ρ p dv g + 1 2 M w 2 ρ p dv g which follows easily from integration by parts. Using the Laplace comparison theorem, we thus get Thus, from inequality (4.1), we get Let u be an eigenfunction associated to the eigenvalue σ 2 and choose w to be Then we have Combining this inequality with (4.2), we finally get establishes the claim.

Proof of Theorem 1.4
Throughout the proof we repeatedly use the Hessian and Laplace comparison theorems as well as the generalized Rellich identity, i.e. Theorem 3.1.
(i) We start by proving the first inequality in (i), namely C 1 η m /h max ≤ λ k . Let E k be the eigenspace associated with λ k and let u 1 , . . . , u m be an orthonormal basis for E k . We first show that ∂ ν u 1 , . . . , ∂ ν u m are linearly independent functions on ∂ M. We prove it by contradiction. Let assume that there exists u ∈ Span(∂ ν u 1 , · · · , ∂ ν u m ) such that ∂ ν u = 0. LetM be a Riemannian manifold such that M admits an isometric embedding. Let N be a Riemannian manifold obtained by doublingM along its boundary (if ∂M = ∅), endowed with the induced metric fromM. More precisely, N ∼ =M M / ∼, where ∼ identifies the two boundaries by the identity map. We smooth out the metric along the image of ∂M without changing the metric on the two copies of M in N . Then we define Therefore, it is also a strong solution.
Since v ≡ 0 on N \ M, we get v ≡ 0 on N by the unique continuation theorem. This in particular shows that dim(E k /H 2 0 (M)) = k. Thus, we can consider E k as a test functional space in (2.4).
Next we bound the denominator from below. Applying Theorem 3.1 with λ = 0 and F = ∇ρ p yields ii) Let φ > 0 be a continuous function on ∂ M. For every l ∈ N set We also have Therefore, Thanks to the Laplace and Hessian comparison theorem, the vector field F = ∇ρ p satisfies properties A and B (see page 10) on M with α = 1, and Taking we get which completes the proof. Finally, we provide examples for Theorem 1.4 (ii) in which vector fields satisfying conditions A and B arise naturally. The first example is just a special case of Theorem 1.4 (ii).

Example 4.2
Let M be a star-shaped domain in R n with respect to the origin. Thus F(x) = x satisfies properties A and B on M for α = 1 and c 1 = c 2 = n. Then by Theorem 1.4 (ii) we have where m is the multiplicity of μ k and I 2 (M) = M |x| 2 dv g is the second moment of inertia. If in addition the origin is also the centroid of M, i.e. M xdv g = 0, then we have where m 0 denotes the multiplicity of μ 2 . Combining this inequality with Theorem 1.1 (b) we get These two last inequalities has been previously obtained in [13] for the special case n = 2.

Example 4.3
Let B n+1 be the unit ball in R n+1 centered at the origin, and M be a free boundary minimal hypersurface in B n+1 . Consider F(x) = x, or equivalently ρ 0 (x) = ρ(x) = |x| 2 2 . It is well-known that the coordinate functions of R n+1 are harmonic on M. Hence divF = ρ = n.
Thus, condition A on page 10 is satisfied. Also, by the definition of a free boundary minimal hypersurface, we have ∇ρ, ν = 1 on ∂ M. To verify condition B, one can show that the eigenvalues of ∇ 2 ρ at point x ∈ M are given by 1 − κ i x, N (x) , i = 1, . . . , n, where N (x) is the unit normal to the M such that N | ∂ M = ν, and κ i are principal curvatures. Indeed, let X , Y ∈ T x M. Then we have where ·, · is the Euclidian inner product, ∇ is the induced connection on M, D is the Euclidean connection (or simply the differentiation) on R n+1 , and S(x) is the shape operator In dimension two, α > 0 is equivalent to |κ i | x, N (x) < 1. By results in [1], if |κ i | x, N (x) < 1 then x, N (x) ≡ 0 on M, and M is the equilateral disk. Hence, there is no nontrivial 2-dimensional minimal surface satisfying Properties A and B. It is an intriguing question whether there are non-trivial minimal hypersurfaces with α > 0 in higher dimensions.
We proceed with a discussion on the proof of the variational characterization for biharmonic Steklov problem II, see (1.5). To prove this theorem, we first need to state a counterpart of Theorem 4.5 for the eigenvalue problem (1.5). Although the argument is classic and standard, for the sake of completeness, we state the theorem and we include a brief discussion on its proof. Consider (4.14)

Theorem 4.7 For every k ∈ N we have
We denote by V 1 /R the completion of Z 1 /R with respect to inner product (4.9) and by W 1 the completion of Z 1 with respect to inner product (4.14). The Hilbert space Since, for every f ∈ W 1 , we have where V 1 is the dual space, and let The linear map L is an isomorphism. Thus, the linear operator K = π • L −1 • i 1 • i : V 1 /R → V 1 /R is a positive compact self-adjoint operator with strictly positive eigenvalues. Here π : V 1 → V 1 /R is the orthogonal projection onto V 1 /R. The inverse of eigenvalues of K give the positive eigenvalues of (1.5), and the eigenfunctions are the same. We summarize this discussion in the following theorem. Theorem 4.8 Let M be a manifold with C 2 boundary. Then the spectrum of eigenvalue problem (1.5) consists of a countable set of non-negative eigenvalues 0 = ξ 1 < ξ 2 ≤ · · · ≤ ξ k ≤ · · · with finite multiplicities, and the corresponding eigenfunctions {ψ k } form a complete orthogonal system for V 1 . Now, the proof of Theorem 4.7 is similar to that of Theorem 4.6. We leave the details of the proof to the interested reader.