Fundamental tones of clamped plates in nonpositively curved spaces

We study Lord Rayleigh’s problem for clamped plates on an arbitrary n -dimensional ( n ≥ 2) Cartan-Hadamard manifold ( M, g ) with sectional curvature K ≤ − κ 2 for some κ ≥ 0. We ﬁrst prove a McKean-type spectral gap estimate, i.e. the fundamental tone of any domain in ( M, g ) is universally bounded from below by ( n − 1) 4 16 κ 4 whenever the κ -Cartan-Hadamard conjecture holds on ( M, g ), e.g. in 2- and 3-dimensions due to Bol (1941) and Kleiner (1992), respectively. In 2- and 3-dimensions we prove sharp isoperimetric inequalities for suﬃciently small clamped plates, i.e. the fundamental tone of any domain in ( M, g ) of volume v > 0 is not less than the corresponding fundamental tone of a geodesic ball of the same volume v in the space of constant curvature − κ 2 provided that v ≤ c n /κ n with c 2 ≈ 21 . 031 and c 3 ≈ 1 . 721, respectively. In particular, Rayleigh’s problem in Euclidean spaces resolved by Nadirashvili (1992) and Ashbaugh and Benguria (1995) appears as a limiting case in our setting of of nonoptimal estimates of the fundamental tone of arbitrary clamped plates are also provided in high-dimensions. As an application, by using the sharp isoperimetric inequality for small clamped hyperbolic discs, we give necessarily and suﬃcient conditions for the existence of a nontrivial solution to an elliptic PDE involving the biharmonic Laplace-Beltrami operator.

We study Lord Rayleigh's problem for clamped plates on an arbitrary n-dimensional (n ≥ 2) Cartan-Hadamard manifold (M, g) with sectional curvature K ≤ −κ 2 for some κ ≥ 0. We first prove a McKean-type spectral gap estimate, i.e. the fundamental tone of any domain in (M, g) is universally bounded from below by (n−1) 4 16 κ 4 whenever the κ-Cartan-Hadamard conjecture holds on (M, g), e.g. in 2-and 3-dimensions due to Bol (1941) and Kleiner (1992), respectively. In 2-and 3-dimensions we prove sharp isoperimetric inequalities for sufficiently small clamped plates, i.e. the fundamental tone of any domain in (M, g) of volume v > 0 is not less than the corresponding fundamental tone of a geodesic ball of the same volume v in the space of constant curvature −κ 2 provided that v ≤ c n /κ n with c 2 ≈ 21.031 and c 3 ≈ 1.721, respectively. In particular, Rayleigh's problem in Euclidean spaces resolved by Nadirashvili (1992) and Ashbaugh and Benguria (1995) appears as a limiting case in our setting (i.e. K ≡ κ = 0). Sharp asymptotic estimates of the fundamental tone of small and large geodesic balls of lowdimensional hyperbolic spaces are also given. The sharpness of our results requires the validity of the κ-Cartan-Hadamard conjecture (i.e. sharp isoperimetric inequality on (M, g)) and peculiar properties of the Gaussian hypergeometric function, both valid only in dimensions 2 and 3; nevertheless, some

Introduction and main results
Let Ω ⊂ R n be a bounded domain (n ≥ 2), and consider the eigenvalue problem (1. 2) The minimizer of (1.2) in the plane describes the vibration of a homogeneous thin plate Ω ⊂ R 2 whose boundary is clamped, while the frequency of vibration of the plate Ω is proportional to Γ 0 (Ω) 1 2 . The famous conjecture of Lord Rayleigh [36, p.382] -formulated initially for planar domains in 1894 -states that where Ω ⊂ R n is a ball with the same measure as Ω, with equality if and only if Ω is a ball. Hereafter, ν = n 2 − 1, ω n = π n/2 /Γ(1 + n/2) is the volume of the unit Euclidean ball, while h ν is the first positive critical point of J ν I ν , where J ν and I ν stand for the Bessel and modified Bessel functions of first kind, respectively.
Assuming that the eigenfunction corresponding to Γ 0 (Ω) is sign-preserving over a simply connected domain Ω ⊂ R 2 , Szegő [38] proved (1.3) in the early fifties. As one can deduce from his paper's text, his belief on the constant-sign first eigenfunction corresponding to Γ 0 (Ω) has been based on the second-order membrane problem (called as the Faber-Krahn problem). It turned out shortly that his expectation perishes due to the construction of Duffin [19] on strip-like domains and Coffman, Duffin and Shaffer [16] on ring-shaped clamped plate, localizing nodal lines of vibrating plates. While the membrane problem involves only the Laplacian, the clamped plate problem requires the presence of the fourth order bilaplacian operator; as we know nowadays, fourth order equations are lacking general maximum/comparison principles which is unrevealed in Szegő's pioneering approach. In fact, stimulated by the papers [19] and [16], several scenarios are described for nodal domains of clamped plates, see e.g. Bauer and Reiss [3], Coffman [15], Grunau and Robert [21], from which the main edification is that eigenfunctions corresponding to (1.2) may change their sign.
In order to handle the presence of possible nodal domains, Talenti [40] developed a Schwarz-type rearrangement method on domains where the first eigenfunction corresponding to (1.2) has both positive and negative parts. In this way, a decomposition of (1.2) into a two-ball minimization problem arises which provided a nonoptimal estimate in (1.3); in fact, instead of (1.3), Talenti proved that Γ 0 (Ω) ≥ d n Γ 0 (Ω ) where the dimension-depending constant d n has the properties 1 2 ≤ d n < 1 for every n ≥ 2 and lim n→∞ d n = 1 2 . By a careful improvement of Talenti's two-ball minimization argument, Rayleigh's conjecture has been proved in its full generality for n = 2 by Nadirashvili [31,32]. Further modifications of some arguments from the papers [32] and [40] allowed to Ashbaugh and Benguria [1] to prove Rayleigh's conjecture for n = 3 (and n = 2) by exploring fine properties of Bessel functions. Roughly speaking, for n ∈ {2, 3}, the two-ball minimization problem reduces to only one ball (the other ball disappearing), while in higher dimensions the 'optimal' situation appears for two identical balls which provides a nonoptimal estimate for Γ 0 (Ω). Although asymptotically sharp estimates are provided by Ashbaugh and Laugesen [2] for Γ 0 (Ω) in high-dimensions, i.e. Γ 0 (Ω) ≥ D n Γ 0 (Ω ) where 0.89 < D n < 1 for every n ≥ 4 with lim n→∞ D n = 1, the conjecture is still open for n ≥ 4. Very recently, Chasman and Langford [6,7] provided certain Ashbaugh-Laugesentype results in Euclidean spaces endowed with a log-convex/Gaussian density, by proving that Γ w (Ω) ≥CΓ w (Ω ), where the constant C ∈ (0, 1) depends on the volume of Ω and dimension n ≥ 2, while Γ w (Ω) and Γ w (Ω ) denote the fundamental tones of the clamped plate with respect to the corresponding density function w.
Interest in the clamped plate problem on curved spaces was also increased in recent years. One of the most central problems is to establish Payne-Pólya-Weinberger-Yangtype inequalities for the eigenvalues of the problem where Ω is a bounded domain in an n-dimensional Riemannian manifold (M, g), Δ 2 g stands for the biharmonic Laplace-Beltrami operator on (M, g) and ∂ ∂n is the outward normal derivative on ∂Ω, respectively; see e.g. Chen, Zheng and Lu [9], Cheng, Ichikawa and Mametsuka [10], Cheng and Yang [11][12][13], Wang and Xia [42]. Instead of (1.2), one naturally considers the fundamental tone of Ω ⊂ M by where dv g denotes the canonical measure on (M, g), and W 2,2 0 (Ω) is the usual Sobolev space on (M, g), see Hebey [22]; in fact, it turns out that Γ g (Ω) is the first eigenvalue to (1.4). Due to the Bochner-Lichnerowicz-Weitzenböck formula, the Sobolev space H 2 0 (Ω) = W 2,2 0 (Ω) is a proper choice for (1.4), see Proposition 3.1 for details. To the best of our knowledge, no results -comparable to (1.3) -are available in the literature concerning Lord Rayleigh's problem for clamped plates on curved structures. Accordingly, the main purpose of the present paper is to identify those geometric and analytic properties which reside in Lord Rayleigh's problem for clamped plates on nonpositively curved spaces. To develop our results, the geometric context is provided by an n-dimensional (n ≥ 2) Cartan-Hadamard manifold (M, g) (i.e. simply connected, complete Riemannian manifold with nonpositive sectional curvature). Having this framework, we recall McKean's spectral gap estimate for membranes which is closely related to (1.5); namely, in an n-dimensional Cartan-Hadamard manifold (M, g) with sectional curvature K ≤ −κ 2 for some κ > 0, the principal frequency of any membrane Ω ⊂ M can be estimated as [30].
Before to state our results, we fix some notations. If κ ≥ 0, let N n κ be the n-dimensional space-form with constant sectional curvature −κ 2 , i.e. N n κ is either the hyperbolic space H n −κ 2 when k > 0, or the Euclidean space R n when κ = 0. Let B κ (L) be the geodesic ball of radius L > 0 in N n κ and if Ω ⊂ N n κ , we denote by Γ κ (Ω) the corresponding value from (1.5). By convention, we consider 1/0 = +∞ and as usual, V g (S) denotes the Riemannian volume of S ⊂ M .
Our first result provides a fourth order counterpart of McKean's spectral gap estimate, which requires the validity of the κ-Cartan-Hadamard conjecture on (M, g); the latter is nothing but the sharp isoperimetric inequality on (M, g), which is valid e.g. on hyperbolic spaces of any dimension as well as on generic 2-and 3-dimensional Cartan-Hadamard manifolds with sectional curvature K ≤ −κ 2 for some κ ≥ 0, see §2.2. Theorem 1.1. Let (M, g) be an n-dimensional Cartan-Hadamard manifold with sectional curvature K ≤ −κ 2 for some κ ≥ 0, which verifies the κ-Cartan-Hadamard conjecture.
If Ω ⊂ M is a bounded domain with smooth boundary then Moreover, for n ∈ {2, 3}, relation (1.7) is sharp in the sense that Clearly, Theorem 1.1 is relevant only for κ > 0 (as (1.7) and (1.8) trivially hold for κ = 0). Moreover, if n ∈ {2, 3} and κ > 0, and Γ l κ (Ω) denotes the lth eigenvalue of (1.4) on Ω ⊂ H n −κ 2 , then making use of (1.8) and a Payne-Pólya-Weinberger-Yang-type universal inequality on H n −κ 2 , it turns out that In particular, (1.9) confirms a claim of Cheng and Yang [12,Theorem 1.4] for n ∈ {2, 3}, where the authors assumed (1.8) itself in order to derive (1.9). In fact, one can expect the validity of (1.9) for any n ≥ 2 but some technical difficulties prevent the proof in high-dimensions; for details, see §5.3. Actually, Theorem 1.1 is just a byproduct of a general argument needed to prove the main result of our paper (for its statement, we recall that h ν is the first positive critical point of J ν I ν and ν = n 2 − 1): Some comments are in order.
The proof of Theorems 1.1 and 1.2 is based on a decomposition argument similar to the one carried out by Talenti [40] and Ashbaugh and Benguria [1] in the Euclidean framework. In fact, we transpose the original variational problem from generic nonpositively curved spaces to the space-form N n κ by assuming the validity of the κ-Cartan-Hadamard conjecture on (M, g). By a fourth order ODE it turns out that Γ κ (Ω ) is the smallest positive solution to the cross-product of suitable Gaussian hypergeometric functions (resp., Bessel functions) whenever κ > 0 (resp., κ = 0). The aforementioned decomposition argument combined with certain oscillatory and asymptotic properties of the hypergeometric function provides the proof of Theorem 1.1.
The dimensionality restriction n ∈ {2, 3} in Theorem 1.2 (and relation (1.8)) is needed not only for the validity of the κ-Cartan-Hadamard conjecture but also for some peculiar properties of the Gaussian hypergeometric function; similar phenomenon has been pointed out also by Ashbaugh and Benguria [1] in the Euclidean setting for Bessel functions. In addition, the arguments in Theorem 1.2 work only for sets with sufficiently small measure; unlike the usual Lebesgue measure in R n (where the scaling Γ 0 (B 0 (L)) = L −4 Γ 0 (B 0 (1)) holds for every L > 0), the inhomogeneity of the canonical measure on hyperbolic spaces requires the aforementioned volume-restriction. The intuitive feeling we get that eigenfunctions corresponding to Γ g (Ω) on a large domain Ω ⊂ M with strictly negative curvature may have large nodal domains whose symmetric rearrangements in H n −κ 2 produce large geodesic balls and their 'joined' fundamental tone can be definitely lower than the expected Γ κ (Ω ). In fact, our arguments show that Theorem 1.2 cannot be improved even if we restrict the setting to the model space-form H n −κ 2 . It remains an open question whether or not (1.10) remains valid for arbitrarily large domains in any dimension n ≥ 4; we notice however that some nonoptimal estimates of Γ g (Ω) are also provided for any domain in high-dimensions (see §5.4). The asymptotic property (1.11) for κ > 0 follows by an elegant asymptotic connection between hypergeometric and Bessel functions, which is crucial in the proof of (1.10) and its accuracy is shown in Table 1 (see  §5.2) for some values of L 1. Clearly, (1.11) is trivial for κ = 0 since Γ 0 (B 0 (L)) = h 4 ν /L 4 for every L > 0. A natural question arises concerning the sharp estimate of the fundamental tone on complete n-dimensional Riemannian manifolds with Ricci curvature Ric (M,g) ≥ k(n − 1) for some k ≥ 0. Some arguments based on the spherical Laplacian show that Bessel functions (when k = 0) and Gaussian hypergeometric functions (when k > 0) will play again crucial roles. Since the parameter range of the aforementioned special functions in the nonnegatively curved case is different from the present setting, further technicalities appear which require a deep analysis. Accordingly, we intend to come back to this problem in a forthcoming paper.
As an application of Theorem 1.2, we consider the elliptic problem where B κ (L) ⊂ H 2 −κ 2 is a hyperbolic disc and the range of parameters μ, γ, p, κ and L is specified below. By using variational arguments, one can prove the following result.
The paper is organized as follows. In Section 2 we recall/prove those notions/results which are indispensable in our study (space-forms, κ-Cartan-Hadamard conjecture, oscillatory properties of specific Gaussian hypergeometric functions). In Section 3 we develop an Ashbaugh-Benguria-Talenti-type decomposition from curved spaces to space-forms. In Sections 4 and 5 we provide a McKean-type spectral gap estimate (proof of Theorem 1.1) and comparison principles (proof of Theorem 1.2) for fundamental tones, respectively. In Section 6 we prove Theorem 1.3.

Space-forms
Let κ ≥ 0 and N n κ be the n-dimensional space-form with constant sectional curvature −κ 2 . When κ = 0, N n κ = R n is the usual Euclidean space, while for κ > 0, N n κ is the n-dimensional hyperbolic space represented by the Poincaré ball model N n κ = H n −κ 2 = {x ∈ R n : |x| < 1} endowed with the Riemannian metric is a Cartan-Hadamard manifold with constant sectional curvature −κ 2 . If ∇ and div denote the Euclidean gradient and divergence operator in R n , the canonical volume form, gradient and Laplacian operator on N n κ are respectively. The distance function on N n κ is denoted by d κ ; the distance between the origin and x ∈ N n κ is given by A simple change of variables gives the following useful transformation.
where r L ≥ 0 is the unique real number verifying V k (r L ) = L.

κ-Cartan-Hadamard conjecture
Let (M, g) be an n-dimensional Cartan-Hadamard manifold with sectional curvature bounded above by −κ 2 for some κ ≥ 0. The κ-Cartan-Hadamard conjecture on (M, g) (called also as the generalized Cartan-Hadamard conjecture) states that the κ-sharp isoperimetric inequality holds on (M, g), i.e. for every open bounded Ω ⊂ M one has ; moreover, equality holds in (2.2) if and only if Ω is isometric to B κ (r). Hereafter, A g and A κ stand for the area on (M, g) and N n κ , respectively. The κ-Cartan-Hadamard conjecture holds for every κ ≥ 0 on space-forms with constant sectional curvature −κ 2 (of any dimension), see Dinghas [18], and on Cartan-Hadamard manifolds with sectional curvature bounded above by −κ 2 of dimension 2, see Bol [5], and of dimension 3, see Kleiner [26]. In addition, a very recent result of Ghomi and Spruck [20] states that the 0-Cartan-Hadamard conjecture holds in any dimension; in dimension 4, the validity of the 0-Cartan-Hadamard conjecture is due to Croke [14]. In higher dimensions and for κ > 0, the conjecture is still open; for a detailed discussion, see Kloeckner and Kuperberg [28].

Gaussian hypergeometric function
For a, b, c ∈ C (c = 0, −1, −2, ...) the Gaussian hypergeometric function is defined by on the disc |z| < 1 and extended by analytic continuation elsewhere, where (a) k = Γ(a+k) Γ(a) denotes the Pochhammer symbol. The corresponding differential equation We also recall the differentiation formula Let n ≥ 2 be an integer, K > 0 be fixed, and consider the function The following result will be indispensable in our study.
Proposition 2.2. Let K > 0 be fixed. The following properties hold: Proof. For simplicity of notation, let a ± = (i) The connection formula (15.10.11) of Olver et al. [33] implies that 4 16 . First, since n 2 − a − > 0 and b − > 0, the connection formula (15.10.11) of [33] together with (2.3) implies again that By virtue of (2.4), an elementary transformation shows that w ± := w K ± verifies the ordinary differential equation It turns out that (2.6) is equivalent to where . For any τ > 0, relation (2.7) and a Sturm-type argument gives that is not oscillatory on (0, ∞) for numbers K belonging to this range.
Assume now that K > (n−1) 4 16 . Since is oscillatory. Due to the fact that , inequality (2.8) trivially holds, which concludes the proof.
Remark 2.1. Dmitrii Karp kindly pointed out that for every β ≥ 1 2 and t > 0, the function is strictly increasing on [0, ∞) from which Proposition 2.2/(ii) follows; his proof is based on fine properties of the hypergeometric functions 2 F 1 and 3 F 2 , cf. Karp [24].
3. Ashbaugh-Benguria-Talenti-type decomposition: from curved spaces to space-forms Without saying explicitly throughout this section, we put ourselves into the context of Let Ω ⊂ M be a bounded domain. Inspired by Talenti [40] and Ashbaugh and Benguria [1], we provide in this section a decomposition argument by estimating from below the fundamental tone Γ g (Ω) given in (1.5) by a value coming from a two-geodesic-ball minimization problem on the space-form N n κ . We first state: Proof. Due to Hopf-Rinow's theorem, the set Ω is relatively compact. Consequently, the Ricci curvature is bounded from below on Ω, see e.g. Bishop , is equivalent to the norm given by u → We are going to use certain symmetrization arguments à la Schwarz; namely, if U : If S ⊂ M is a measurable set, then S denotes the geodesic ball in N n κ with center in the origin such that V g (S) = V κ (S ).
Let u ∈ W 2,2 0 (Ω) be a minimizer in (1.5); since u is not necessarily of constant sign, let u + = max(u, 0) and u − = − min(u, 0) be the positive and negative parts of u, and The functions u + and u − are well-defined and radially symmetric, verifying the property that for some r t > 0 and ρ t > 0 one has For further use, we consider the sets Proposition 3.2. Let u ∈ W 2,2 0 (Ω) be a minimizer in (1.5). Then for a.e. t > 0 we have Proof. Statements (i) and (ii) are similar to those by Talenti [40,Appendix,p.278] in the Euclidean setting; for completeness, we reproduce the proof in the curved framework. By density reasons, it is enough to consider the case when u is smooth. For h > 0, Cauchy's inequality implies When h → 0, the latter relation and the co-area formula (see Chavel [8, p.86]) imply that where H n−1 is the (n − 1)-dimensional Hausdorff measure. The divergence theorem gives that which concludes the proof of (i). Similar arguments hold in the proof of (ii). Let where · # stands for the notation Proof. We first recall a Hardy-Littlewood-Pólya-type inequality, i.e. if U : Ω → [0, ∞) is an integrable function and U is defined by (3.1), one has for every measurable set moreover, if S = Ω, the equality holds in (3.6) as U being an equimeasurable rearrangement of U .
(i) Let t > 0 be fixed. In order to complete the proof, we are going to show first that and To do this, let r t > 0 be the unique real number with V k (r t ) = α(t), see (3.5). The estimate (3.7) follows by Proposition 2.1 and inequality (3.6) as The proof of (3.8) is similar; for completeness, we provide its proof. By a change of variable and Proposition 2.1 it turns out that In particular, by inequality (3.6) (together with the equality for the whole domain) and the latter relations we have which concludes the proof of (3.8).
By (3.7) and (3.8) one has which is precisely our claim. The proof of (ii) is similar.
We consider the function v : A direct computation shows that v is a solution to the problem In a similar way, the function w : Ω * = B κ (L) → R given by is a solution to In particular, by their definitions, it turns out that v ≥ 0 in B κ (a) and w ≥ 0 in B κ (b).
In fact, much precise comparisons can be said by combining the above preparatory results: Theorem 3.1. Let v and w from (3.9) and (3.11), respectively. Then 14) where a and b are from (3.2). In particular, one has In addition, Proof. We first prove (3.13). Since (M, g) verifies the κ-Cartan-Hadamard conjecture, on account of (3.

3) and (3.4), it follows that
Due to (3.3), (3.5) and (2.1), it follows that for a.e. t > 0, Combining the above relations, it yields After an integration, we obtain for every τ ∈ [0, u + L ∞ (Ω) ] that By changing the variable r t = ρ, and taking into account that r 0 = a, it follows that Let x ∈ B κ (a) be arbitrarily fixed and associate to this element the unique τ ∈ [0, u + L ∞ (Ω) ] such that d κ (x) = r τ . By the definition of u + it follows that u + (x) = τ , thus the latter inequality together with (3.9) implies that which is precisely the claimed relation (3.13). The proof of (3.14) is similar, where (3.18) is used.
The estimate in (3.15) is immediate, since where we apply (3.2) together with the estimates (3.13) and (3.14), respectively. We now prove (3.16). On one hand, by problems (3.10) and (3.12), Proposition 2.1 and a change of variables imply that On the other hand, The latter term in the above integral vanishes. Indeed, fix first 0 ≤ s < V g ({x ∈ Ω : Δ g u(x) < 0}) and let t : , a similar argument yields (Δ g u) # − (s) = 0. Therefore, by Proposition 2.1 we have which concludes the proof.
Therefore, the latter relation and Proposition 2.1 give Furthermore, by Proposition 2.1 and problems (3.9) and (3.11) we have A simple computation shows that Similar facts also hold for w; it remains to transform the above quantities into trigonometric terms.
Case 2 : αβ = 0. Since G ± are analytical functions, by continuity reason and relation (4.5) we have at once (4.6) by the previous case.
Remark 4.1. The proof of (1.8), i.e. the optimality of (1.7) in the case n ∈ {2, 3}, requires some specific properties of the hypergeometric function that are discussed in the next section; therefore, we postpone its proof to §5.3.

Comparison principles for fundamental tones: proof of Theorem 1.2 and (1.8)
In the first part of this section we establish a two-sided estimate for the first positive solution of the equation (4.5), valid on generic n-dimensional Cartan-Hadamard manifolds (verifying the κ-Cartan-Hadamard conjecture). In the second part we prove the sharp comparison principle for fundamental tones in 2-and 3-dimensions (proof of Theorem 1.2). In the third part we give the proof of (1.8) while in the last subsection we discuss the difficulties arising in high-dimensions. As before, let ν = n 2 − 1.
We now provide the approximate threshold values of L when such turnouts occur for n = 2 and n = 3, respectively. Numerical approximations show that (5.5) holds for n = 2 whenever 0 < L < 2.  which appear in the statement of the theorem.

Fundamental tones in high-dimensions: nonoptimal estimates
Our argument cannot provide sharp comparison principles for fundamental tones since inequality (5.5) fails for any choice of κ ≥ 0 and L > 0 in the n-dimensional case whenever n ≥ 4; we notice that similar phenomenon occurs also in the Euclidean setting, see Ashbaugh and Benguria [1]. However, in the case κ = 0 we can provide some weak comparison principles. To this end, if (M, g) is an n-dimensional (n ≥ 4) Cartan-Hadamard manifold and Ω ⊂ M a bounded domain with smooth boundary, a closer inspection of the proof -based on the validity of the 0-Cartan-Hadamard conjecture proved by Ghomi and Spruck [20] -gives that Although lim n→∞ D n = 1, the estimate (5.23) is not sharp since D n < 1 for every n ≥ 4.