Neumann eigenvalues of the biharmonic operator on domains: geometric bounds and related results
aa r X i v : . [ m a t h . SP ] J u l NEUMANN EIGENVALUES OF THE BIHARMONIC OPERATOR ONDOMAINS: GEOMETRIC BOUNDS AND RELATED RESULTS
BRUNO COLBOIS AND LUIGI PROVENZANO
Abstract.
We study an eigenvalue problem for the biharmonic operator with Neu-mann boundary conditions on domains of Riemannian manifolds. We discuss the weakformulation and the classical boundary conditions, and we describe a few propertiesof the eigenvalues. Moreover, we establish upper bounds compatible with the Weyl’slaw under a given lower bound on the Ricci curvature. Introduction
Let (
M, g ) be a complete n -dimensional smooth Riemannian manifold and let Ω ⊂ M be a bounded domain, i.e., a bounded connected open set, with boundary ∂ Ω. We considerthe following Neumann eigenvalue problem for the biharmonic operator:(1.1) ∆ u = µu, in Ω , ∂ u∂ν = 0 , on ∂ Ω , div ∂ Ω ( ∇ ν ∇ u ) ∂ Ω + ∂ ∆ u∂ν = 0 , on ∂ Ω . in the unknowns u (the eigenfunction) and µ (the eigenvalue). Here ∆ u = div( ∇ u ) is theLaplacian (or Laplace-Beltrami operator) of u on ( M, g ), ∆ u = ∆(∆ u ), ν denotes theouter unit normal to ∂ Ω, ∂ u∂ν = h∇ ν ∇ u, ν i is the second normal derivative, div ∂ Ω is thedivergence on ∂ Ω with respect to the induced metric, and F ∂ Ω denotes the projection of F ∈ T M on T ∂
Ω.We recall that in the case M = R n with the Euclidean metric, problem (1.1) is well-known and has increasingly gained attention in recent years. We refer to [3, 8, 9, 11, 12,13, 29, 35, 36, 37] for the eigenvalue problem and to [39] for the corresponding boundaryvalue problem. Problem (1.1) in dimension two represents Kirchhoff’s solution to theproblem of describing the transverse vibrations of a thin elastic plate with free edges. Werefer to [7, 22, 31, 32] for more details and for historical information.We also note that the corresponding Dirichlet problem for the biharmonic operator,which for planar domains is related to the study of the transverse vibrations of a thin elas-tic plate with clamped edges [20], has been extensively studied not only in the Euclideansetting, but also for domains in Riemannian manifolds, see e.g., [16, 40, 41]. In particular,the Dirichlet problem on Euclidean domains and the analogous problem on Riemannianmanifolds share many properties which can be derived by using similar arguments.On the other hand, we have not been able to find the analogue of the biharmonic Neu-mann problem on Riemannian manifolds in the literature. The first aim of the present Mathematics Subject Classification.
Key words and phrases.
Biharmonic operator, Neumann boundary conditions, Riemannian manifolds,eigenvalues, eigenvalue bounds, spectral geometry. paper is to introduce problem (1.1) on domains of Riemannian manifolds in a suitable way,derive the boundary conditions as well as the variational formulation. The problem whichwe obtain turns out to be the genuine generalization of the biharmonic Neumann prob-lem for Euclidean domains. We remark that the standard technique used to derive theboundary conditions and the variational formulation of problem (1.1) in the Euclideancase it to multiply the eigenvalue equation ∆ u = µu by a test function φ ∈ C ∞ (Ω),integrate the resulting equality over Ω and perform suitable integrations by parts. Com-putations become easy since we can exchange the order of partial derivatives. This is nomore possible in the case of Riemannian manifolds, hence we have to take a longer path,described in Subsection 3.1. An essential tool is Reilly’s identity. It turns out that thestrategy described in Subsection 3.1 allows to define alternatively problem (1.1) also inthe Euclidean case.Then, we prove that problem (1.1) admits an increasing sequence of eigenvalues offinite multiplicity diverging to + ∞ of the form −∞ < µ ≤ µ ≤ · · · ≤ µ j ≤ · · · ր + ∞ . If Ω is a bounded domain of R n it is known that the eigenvalues are non-negative andsatisfy the Weyl’s asymptotic law lim j → + ∞ µ j j n = 16 π ( ω n | Ω | ) n , that is(1.2) µ j ∼ π ω n n (cid:18) j | Ω | (cid:19) n , as j → + ∞ , where ω n denotes the volume of the unit ball in R n and | Ω | denotes the Lebesgue measureof Ω, see e.g., [29].An important question regarding the eigenvalues of Neumann-type problems is that offinding upper bounds which are compatible with the Weyl’s law. One of the main purposesof the present paper is that of finding upper bounds for µ j which can be compared with(1.2) and which contain the correct geometric information.In the case of Euclidean domains, Weyl-type upper bounds for µ j are well-known andare of the form(1.3) µ j ≤ A n (cid:18) j | Ω | (cid:19) n with A n = (cid:0) n (cid:1) /n π ω /nn , see e.g., [29]. The proof in [29] is in the spirit of the analogousresult of Kr¨oger for the Neumann eigenvalues m j of the Laplacian on Euclidean domains,namely(1.4) m j ≤ B n (cid:18) j | Ω | (cid:19) n , where B n = (cid:0) n (cid:1) /n π ω /nn , see [28]. Note that (1.4) is compatible with the Weyl’s law(1.5) m j ∼ π ω n n (cid:18) j | Ω | (cid:19) n , as j → + ∞ , EUMANN EIGENVALUES OF THE BIHARMONIC OPERATOR ON DOMAINS 3
The proofs of (1.3) and (1.4) rely on harmonic analysis techniques and are hardly adapt-able to the case of manifolds.In the case of the eigenvalues of the Laplacian on manifolds, one of the first result inthis direction is presented in [10], where it is proved that(1.6) m j ≤ ( n − κ + C n (cid:18) j | M | (cid:19) n . Here m j denote, with abuse of notation, the eigenvalues of the Laplacian on a compactmanifold (without boundary) M with Ric ≥ − ( n − κ , κ ≥
0. Results on domains havebeen obtained more recently. In [18] the authors prove the following upper bound(1.7) m j ≤ A n (cid:18) j | Ω | (cid:19) n + B n κ , for the Neumann eigenvalues m j of the Laplacian on a domain Ω of a complete Riemannianmanifold with Ric ≥ − ( n − κ , κ ≥
0. The authors adopt a metric approach for theproof of (1.7).We will use this approach also in the present paper in order to obtain upper boundsfor µ j . In view of (1.2), (1.5) and (1.7), it is natural to conjecture that the inequality(1.8) µ j ≤ A n (cid:18) j | Ω | (cid:19) n + B n κ holds for any bounded domain of a complete Riemannian manifold with Ric ≥ − ( n − κ , κ ≥
0. We are able to prove (1.8) for certain classes of domains and manifolds. Inparticular we prove (1.8) for domains of manifolds with non-negative Ricci curvature and n = 2 , , µ j ≤ A n (cid:18) j | Ω | (cid:19) n + C ( g )(see Theorem 5.7), where C ( g ) has an explicit form and depends on κ, r inj, Ω , | ∂ Ω | , where r inj, Ω is the injectivity radius relative to Ω (see (5.3) for the definition) and | ∂ Ω | is the n − M . In particular we provide more refined estimatesin the case of domains with sufficiently small diameter in manifolds with non-negativeRicci curvature (see Theorem 5.14) and in the case of Cartan-Hadamard manifolds (seeTheorem 5.18).It is important to remark that a bound of the form (1.9) is good in the sense that thegeometry of the domain and of the manifold appears as an additive constant in front ofthe term encoding the asymptotic behavior, which has the correct form compared withthe asymptotic law (1.2).As already mentioned, in order to prove the upper bounds, we adopt a metric approach.In particular, we exploit a result of decomposition of a metric measure space by disjoint capacitors , see [23, 25] for more details, see also [17, 18]. Namely, given a domain Ω, wefind, for each j ∈ N , a family of j disjointly supported sets A i , i = 1 , ..., j , in Ω withsufficient volume. Associated to each set A i we build a function u i to test in the min-max BRUNO COLBOIS AND LUIGI PROVENZANO formula for the eigenvalues (see formula (3.33)). Since the u i are disjointly supported,from (3.33) we deduce that it is sufficient to bound the Rayleigh quotient of each u i inorder to upper bound µ j . Hence the functions u i have to be constructed in a proper way.We remark that test functions for the biharmonic operator need to belong to the stan-dard Sobolev space H (Ω). Usually, test functions are built in terms of distance-typefunctions, which are Lipschitz, but are not in general H (Ω) functions. In particular,test functions for the Laplacian eigenvalues are cut-off functions which are just Lipschitzregular. The application of the technique used in [18] for the Laplacian is no straightfor-ward in our situation, in fact it is notoriously a difficult task to build cut-off functionsenjoying precise estimates for first and second derivatives, see e.g., [5, 15, 24]. We paythe price of the fact that we need cut-off functions in H (Ω) with well-behaved gradientand Laplacian by introducing into the estimates the quantities r inj, Ω and | ∂ Ω | . Gettingrid of these quantities in the general case seems a very difficult issue.Looking at (1.2) and (1.5), one may wonder if there is some sort of relationship between µ j and m j and if it is possible, in general, to recover upper estimates for µ j from upperestimates on m j . The answer is negative in general, in fact we provide examples showingthat the ratio µ j m j may be made arbitrarily large or arbitrarily close to zero.Another interesting feature of problem (1.1) is that it is possible to produce negative eigenvalues. This does not happen with the eigenvalues of the biharmonic Dirichlet prob-lem on domains of manifolds. In particular, in subsection 4.3 we prove that any domainof the standard hyperbolic space H n admits at least n negative eigenvalues. Moreover, weprove that there exist domains with an arbitrarily large number of negative eigenvalues,the absolute value of which can be made arbitrarily large. On the other hand, for domainsin manifolds with positive Ricci curvature we prove a lower bound for the eigenvalues µ j in terms of m j , η j and a lower bound on the Ricci curvature (see Theorem 4.3), where η j denote the eigenvalues of the rough Laplacian on Ω.The present paper is organized as follows. In Section 2 we recall some preliminariesand introduce the notation. In Section 3 we describe the classical Neumann boundaryconditions in (1.1) and derive the weak formulation of the problem, proving that it is well-posed and characterizing its spectrum. In Section 4 we discuss a few properties of theeigenvalues. In particular we provide examples where the ratio µ j m j can be made arbitrarilylarge or close to zero. We prove that any domain of the hyperbolic space admits at least n negative eigenvalues, and that there exists domains with an arbitrarily large number ofnegative eigenvalues with arbitrarily large absolute value. We also prove a lower boundfor µ j for domains on manifolds with positive Ricci curvature. In Section 5 we recall themain technical results of decomposition of a metric measure space by capacitors, whichallow to prove the upper estimates for the eigenvalues µ j presented in the same section.2. Preliminaries and notation
Let (
M, g ) be a complete n -dimensional smooth Riemannian manifold. For a boundeddomain Ω in M , by L (Ω) we denote the space of measurable functions f on Ω such that R Ω u dv < ∞ .The Sobolev space H (Ω) is the completion of C ∞ (Ω) with respect to the norm(2.1) k f k H (Ω) := (cid:18)Z Ω (cid:0) | D f | + |∇ f | + f (cid:1) dv (cid:19) . EUMANN EIGENVALUES OF THE BIHARMONIC OPERATOR ON DOMAINS 5
The space L (Ω) is a Hilbert space when endowed with the standard scalar product(2.2) h f, g i L (Ω) := Z Ω f gdv. The space H (Ω) is a Hilbert space when endowed with the standard scalar product(2.3) h f, g i H (Ω) := Z Ω (cid:0) h D f, D g i + h∇ f, ∇ g i + f g (cid:1) dv, which induces the norm (2.1).The space H (Ω) is the completion of C ∞ c (Ω) with respect to (2.1), there C ∞ c (Ω) isthe space of functions in C ∞ (Ω) compactly supported in Ω. We refer to [26] for anintroduction to Sobolev spaces on Riemannian manifolds.We denote here by dv the Riemannian volume element of M and by dσ the induced n − ∂ Ω.Through all the paper, we denote by h· , ·i the inner product on the tangent spaces of M associated with the metric g , and, with abuse of notation, we shall denote by h· , ·i also the induced metric on ∂ Ω. Let ∇ , D and ∆ denote the gradient, the Hessian andthe Laplacian on M , respectively, and let ∇ ∂ Ω , div ∂ Ω and ∆ ∂ Ω denote the gradient, thedivergence and the Laplacian on ∂ Ω with respect to the induced metric, respectively. Wedenote by ν the outer unit normal to ∂ Ω. The shape operator of ∂ Ω, denoted by S , isdefined for any X ∈ T ∂
Ω as S ( X ) := ∇ X ν , where ∇ X ν is the covariant derivative of ν along a vector field X . The second fundamental form of ∂ Ω, denoted by II ( X, Y ), isdefined as II ( X, Y ) := h S ( X ) , Y i for all X, Y ∈ T ∂
Ω. We recall that the eigenvalues of S are the principal curvatures of ∂ Ω. We will denote by H := n − tr S = n − div ν | ∂ Ω themean curvature of ∂ Ω. Let Ric( · , · ) denote the Ricci tensor of M . Finally, for an open set E ∈ M we denote by | E | the standard Lebesgue measure of E . For a closed set G ∈ M offinite n − | G | the n − G .We recall Bochner’s formula:(2.4) | D f | + Ric( ∇ f, ∇ f ) = 12 ∆ (cid:0) |∇ f | (cid:1) − h∇ ∆ f, ∇ f i , holding pointwise for smooth functions f on Ω.It is also useful to recall Reilly’s formula, see [38]:(2.5) Z Ω (∆ f ) dv − Z Ω | D f | + Ric( ∇ f, ∇ f ) dv = Z ∂ Ω ( n − H (cid:18) ∂f∂ν (cid:19) + 2∆ ∂ Ω f ∂f∂ν + II ( ∇ ∂ Ω f, ∇ ∂ Ω f ) dσ, holding for smooth functions f on Ω.We also recall Green’s identity for smooth functions f, g :(2.6) Z Ω (∆ f g − f ∆ g ) dv = Z ∂ Ω (cid:18) ∂f∂ν g − f ∂g∂ν (cid:19) dσ. We recall that for any smooth vector field F on T ∂
Ω and any function f defined on ∂ Ω,(2.7) Z ∂ Ω h F, ∇ ∂ Ω f i dσ = Z ∂ Ω div ∂ Ω F f dσ,
BRUNO COLBOIS AND LUIGI PROVENZANO that is, the divergence operator is the adjoint of the gradient. In particular, (2.7) holdswith ∂ Ω replaced by any complete smooth (boundaryless) Riemannian manifold (
M, g ),and div ∂ Ω , ∇ ∂ Ω replaced by the divergence and gradient on M , respectively.Finally, by N we denote the set of positive natural numbers.3. The eigenvalue problem for the biharmonic operator with Neumannboundary conditions
In this section we describe the classical Neumann boundary conditions in (1.1), as wellas the weak formulation of the problem. This is done in Subsection 3.1.Then we prove that problem (3.1) is well-posed under suitable assumptions on thedomain, and admits an increasing sequence of eigenvalues of finite multiplicity boundedfrom below and diverging to + ∞ . This is done in Subsection 3.2.3.1. Classical Neumann boundary conditions and weak formulation.
We con-sider the following variational problem:(3.1) Z M h D u, D φ i + Ric( ∇ u, ∇ φ ) dv = µ Z M uφdv , ∀ φ ∈ H (Ω) , in the unknowns u ∈ H (Ω) and µ ∈ R . Problem (3.1) is the variational (weak) for-mulation of problem (1.1), as stated in the following theorem. We remark that it is notstraightforward to recognize the left-hand side of (3.1) as the right quadratic form foran eigenvalue problem for the biharmonic operator with Neumann boundary conditions.One would like to take the simpler quadratic form R Ω ∆ u ∆ φdv , which however providesan ill-defined problem, see Remark 3.3. Theorem 3.1.
Let ( M, g ) be a complete n -dimensional smooth Riemannian manifoldand let Ω be a smooth ( C ∞ ) bounded domain in M . Given a solution ( u, µ ) of problem (3.1) such that u ∈ C (Ω) ∩ C (Ω) , then ( u, µ ) solves problem (1.1) . Vice versa, anysolution ( u, µ ) of problem (1.1) is a solution of problem (3.1) . Actually, we will prove that (3.1) is the weak formulation of the following eigenvalueproblem:(3.2) ∆ u = µu, in Ω , ( n − H ∂u∂ν + ∆ ∂ Ω u − ∆ u = 0 , on ∂ Ω , ∆ ∂ Ω (cid:0) ∂u∂ν (cid:1) − div ∂ Ω S ( ∇ ∂ Ω u ) + ∂ ∆ u∂ν = 0 , on ∂ Ω , in the unknowns u (the eigenfunction) and µ (the eigenvalue). Then, we will show thatthe two boundary conditions in (3.2) coincide with those of (1.1), namely we will provethe following lemma. Lemma 3.2.
Let ( M, g ) be a complete n -dimensional smooth Riemannian manifold andlet Ω be a smooth bounded domain in M . Then, for any u ∈ C (Ω)(3.3) ( n − H ∂u∂ν + ∆ ∂ Ω u − ∆ u = − ∂ u∂ν and (3.4) ∆ ∂ Ω (cid:18) ∂u∂ν (cid:19) − div ∂ Ω S ( ∇ ∂ Ω u ) + ∂ ∆ u∂ν = div ∂ Ω ( ∇ ν ∇ u ) ∂ Ω + ∂ ∆ u∂ν . EUMANN EIGENVALUES OF THE BIHARMONIC OPERATOR ON DOMAINS 7
Proof of Theorem 3.1.
Assume that a function u ∈ C (Ω) ∩ C (Ω) and a real number µ are solution of the eigenvalue equation(3.5) ∆ u = µu , in Ω . We multiply both sides of (3.5) by a function φ ∈ C ∞ and integrate over Ω, obtainingthanks to (2.6)(3.6) Z Ω ∆ uφdv = Z Ω ∆ u ∆ φdv + Z ∂ Ω (cid:18) ∂ ∆ u∂ν φ − ∆ u ∂φ∂ν (cid:19) dσ = µ Z Ω uφdv. We set, for a function f ∈ C (Ω)(3.7) Q Ω ( f ) := Z Ω (∆ f ) dv − Z Ω (cid:0) | D f | + Ric( ∇ f, ∇ f ) (cid:1) dv and(3.8) Q ∂ Ω ( f ) := Z ∂ Ω ( n − H (cid:18) ∂f∂ν (cid:19) + 2∆ ∂ Ω f ∂f∂ν + II ( ∇ ∂ Ω f, ∇ ∂ Ω f ) dσ. We note that(3.9) 14 ( Q Ω ( u + φ ) − Q Ω ( u − φ )) = Z Ω ∆ u ∆ φdv − Z Ω (cid:0) h D u, D φ i + Ric( ∇ u, ∇ φ ) (cid:1) dv and that(3.10) 14 ( Q ∂ Ω ( u + φ ) − Q ∂ Ω ( u − φ ))= Z ∂ Ω ( n − H ∂u∂ν ∂φ∂ν + ∆ ∂ Ω u ∂φ∂ν + ∂u∂ν ∆ ∂ Ω φ + II ( ∇ ∂ Ω u, ∇ ∂M φ ) dσ. Reilly’s formula (2.5) implies that Q Ω ( u + φ ) − Q Ω ( u − φ ) = Q ∂ Ω ( u + φ ) − Q ∂ Ω ( u − φ ),thus from (3.9) and (3.10) we deduce that(3.11) Z Ω ∆ u ∆ φdv − Z Ω (cid:0) h D u, D φ i − Ric( ∇ u, ∇ φ ) (cid:1) dv = Z ∂ Ω ( n − H ∂u∂ν ∂φ∂ν + ∆ ∂ Ω u ∂φ∂ν + ∂u∂ν ∆ ∂ Ω φ + II ( ∇ ∂ Ω u, ∇ ∂M φ ) dσ, hence(3.12) Z Ω ∆ u ∆ φdv = Z Ω h D u, D φ i + Ric( ∇ u, ∇ φ ) dv = Z ∂ Ω ( n − H ∂u∂ν ∂φ∂ν + ∆ ∂ Ω u ∂φ∂ν + ∂u∂ν ∆ ∂ Ω φ + II ( ∇ ∂ Ω u, ∇ ∂M φ ) dσ. Using (3.12) in (3.6), we can re-write (3.6) as(3.13) Z Ω ∆ uφdv = Z Ω h D u, D φ i + Ric( ∇ u, ∇ φ ) dv + Z ∂ Ω (cid:18) ( n − H ∂u∂ν + ∆ ∂ Ω u − ∆ u (cid:19) ∂φ∂ν dσ + Z ∂ Ω (cid:18) II ( ∇ ∂ Ω u, ∇ ∂ Ω φ ) + ∂ ∆ u∂ν φ + ∂u∂ν ∆ ∂ Ω φ (cid:19) dσ = µ Z Ω uφdv. BRUNO COLBOIS AND LUIGI PROVENZANO
We note now that(3.14) Z ∂ Ω II ( ∇ ∂ Ω u, ∇ ∂ Ω φ ) dσ = Z ∂ Ω h S ( ∇ ∂ Ω u ) , ∇ ∂ Ω φ i dσ = − Z ∂ Ω div ∂ Ω S ( ∇ ∂ Ω u ) φdσ, where the second equality follows from (2.7), and that(3.15) Z ∂ Ω ∂u∂ν ∆ ∂ Ω φdσ = Z ∂ Ω ∆ ∂ Ω (cid:18) ∂u∂ν (cid:19) φdσ. Thanks to (3.14) and (3.15), (3.13) can be rewritten as follows(3.16) Z Ω ∆ uφdv = Z Ω h D u, D φ i + Ric( ∇ u, ∇ φ ) dv + Z ∂ Ω (cid:18) ( n − H ∂u∂ν + ∆ ∂ Ω u − ∆ u (cid:19) ∂φ∂ν dσ + Z ∂ Ω (cid:18) ∆ ∂ Ω (cid:18) ∂u∂ν (cid:19) − div ∂ Ω S ( ∇ ∂ Ω u ) + ∂ ∆ u∂ν (cid:19) φdσ = µ Z Ω uφdv. Assume now that the function u satisfies the boundary conditions in (3.2). Then(3.17) Z Ω h D u, D φ i + Ric( ∇ u, ∇ φ ) dv = µ Z Ω uφdv , ∀ φ ∈ C ∞ (Ω) . From the definition of H (Ω) we deduce the validity of (3.1).On the other hand, assume that there exist a solution ( u, µ ) ∈ ( C (Ω) ∩ C (Ω)) × R to (3.1). From (3.16), by taking test functions φ ∈ C ∞ (Ω) we immediately deduce that u solves the differential equation (3.5) as well as the boundary conditions in (3.2), thusthe pair ( u, µ ) is a solution of (3.2). This concludes the proof. (cid:3) We prove now Lemma 3.2
Proof of Lemma 3.2.
We start by proving (3.3). Let { E i } ni =1 an orthonormal frame in aneighborhood of a point p ∈ ∂ Ω such that { E i } n − i =1 is a orthonormal frame of ∂ Ω and E n = ν is the outward unit normal to ∂ Ω. For a Lipschitz vector field F in a neighborhoodof ∂ Ω, we denote by F ∂ Ω := P n − i =1 h F, E i i E i , hence in pF = F ∂ Ω + h F, ν i ν. Note that h∇ ν F, ν i = h∇ ν ( F ∂ Ω + h F, ν i ν ) , ν i = h∇ ν F ∂ Ω , ν i + h∇ ν h F, ν i ν, ν i = h∇ ν h F, ν i ν, ν i , EUMANN EIGENVALUES OF THE BIHARMONIC OPERATOR ON DOMAINS 9 where we have used the fact that h∇ ν F ∂ Ω , ν i = 0. Moreover, P n − i =1 h∇ E i ν, E i i = div ν =( n − H . Thus we have(3.18) div F | ∂ Ω = n X i =1 h∇ E i F, E i i = n X i =1 h∇ E i ( F ∂ Ω + h F, ν i ν ) , E i i = n − X i =1 h∇ E i F ∂ Ω , E i i + n X i =1 h∇ E i h F, ν i ν, E i i = div ∂ Ω F ∂ Ω + n − X i =1 h∇ E i h F, ν i ν, E i i + h∇ ν h F, ν i ν, ν i = div ∂ Ω F ∂ Ω + h F, ν i n − X i =1 h∇ E i ν, E i i + h∇ ν F, ν i = div ∂ Ω F ∂ Ω + ( n − Hh F, ν i + h∇ ν F, ν i , Now, noting that ∇ u | ∂ Ω = ∇ ∂ Ω u + ∂u∂ν ν, and that, by definition ∂ u∂ν = h∇ ν ∇ u, ν i . we immediately obtain from (3.18) the following identity∆ u | ∂ Ω = div ∂ Ω ∇ ∂ Ω u + ( n − H ∂u∂ν + h∇ ν ∇ u, ν i = ∆ ∂ Ω u + ( n − H ∂u∂ν + ∂ u∂ν . This proves (3.3).We prove now (3.4). Let us consider the second boundary condition in (3.2). We needto show that ∆ ∂ Ω (cid:18) ∂u∂ν (cid:19) − div ∂M S ( ∇ ∂ Ω u ) = div ∂ Ω ( ∇ ν ∇ u ) ∂ Ω , which can be re-written asdiv ∂ Ω (cid:18) ∇ ∂ Ω (cid:18) ∂u∂ν (cid:19) − S ( ∇ ∂ Ω u ) − ( ∇ ν ∇ u ) ∂ Ω (cid:19) = 0 . Actually we will prove that ∇ ∂ Ω (cid:18) ∂u∂ν (cid:19) − S ( ∇ ∂ Ω u ) − ( ∇ ν ∇ u ) ∂ Ω = 0We note that for any vector fiel X ∈ T M h∇ ( h∇ u, ν i ) , X i = h∇ X ∇ u, ν i + h∇ X ν, ∇ u i = h∇ ν ∇ u, X i + h∇ ∇ u ν, X i , since D u and ∇ ν are symmetric. Thus ∇ ( h∇ u, ν i ) = ∇ ν ∇ u + ∇ ∇ u ν . We have then ∇ ∂ Ω (cid:18) ∂u∂ν (cid:19) − S ( ∇ ∂ Ω u ) − ( ∇ ν ∇ u ) ∂ Ω = ∇ ∂ Ω ( h∇ u, ν i ) − ∇ ∇ ∂ Ω u ν − ∇ ν ∇ u + h∇ ν ∇ u, ν i ν = ∇ ( h∇ u, ν i ) − h∇ ( h∇ u, ν i ) , ν i ν − ∇ ∇ ∂ Ω u ν − ∇ ν ∇ u + h∇ ν ∇ u, ν i ν = ∇ ν ∇ u + ∇ ∇ u ν − h∇ ν ∇ u, ν i ν − h∇ ∇ u ν, ν i ν − ∇ ∇ ∂ Ω u ν − ∇ ν ∇ u + h∇ ν ∇ u, ν i ν = ∇ ∇ u ν − ∇ ∇ ∂ Ω u ν − h∇ ∇ u ν, ν i ν = 0 , since ∇ ∇ u ν = ∇ ∇ ∂ Ω u ν + ∇ h∇ u,ν i ν ν = ∇ ∇ ∂ Ω u ν and h∇ ∇ u ν, ν i = h∇ ∇ ∂ Ω u ν, ν i + h∇ h∇ u,ν i ν ν, ν i = 0 . In fact ∇ ν ν = 0 and h∇ ∇ ∂ Ω u ν, ν i = 0. This proves (3.4). The proof is now concluded. (cid:3) Since we will be interested in the variational problem (3.1), we can relax the hypothesison the smoothness of Ω. A sufficient condition for the solvability of (3.1) is, e.g., that Ωis of class C , see Subsection 3.2. Remark . By looking at (3.6) it is natural to ask whathappens if we consider in the left-hand side of (3.1) the more familiar quadratic form(3.19) Z Ω ∆ u ∆ φdv. The corresponding variational problem would read(3.20) Z Ω ∆ u ∆ φdv = µ Z Ω uφdv , ∀ φ ∈ H (Ω) , in the unknowns u ∈ H (Ω), µ ∈ R . We note that this problem is not well-posed: it isimmediate to see that all harmonic functions in H (Ω) are eigenfunctions correspondingto the eigenvalue µ = 0 . This is due to the fact that the quadratic form (3.19) is notcoercive in H (Ω), indeed we can add to the quadratic form (3.19) a term γ R Ω uφdv with γ > H (Ω), see also Lemma 3.4. In [37] it is proved that (3.20) hasan infinite kernel consisting of all harmonic functions in H (Ω). Moreover, if we rule outthe kernel, problem (3.20) admits an increasing sequence of positive eigenvalues of finitemultiplicity which coincide with the Dirichlet eigenvalues of the biharmonic operator. It isnot difficult to adapt the results of [37] to the case of domains in a Riemannian manifold.The classical formulation of problem (3.20) reads(3.21) ∆ u = µu , in Ω , ∆ u = 0 , on ∂ Ω , ∂ ∆ u∂ν = 0 , on ∂ Ω . We remark that Neumann boundary conditions are usually called “natural boundaryconditions” and in a certain sense arises from “solving a variational problem on thelargest possible energy space”, which in this case is H (Ω). In this space, problem (3.20)is evidently not well posed. EUMANN EIGENVALUES OF THE BIHARMONIC OPERATOR ON DOMAINS 11
We also remark that the situation is completely different if we impose Dirichlet bound-ary conditions, namely if we consider problem(3.22) ∆ u = Λ u , in Ω ,u = 0 , on ∂ Ω , ∂u∂ν = 0 , on ∂ Ω , in the unknowns u (the eigenfunction) and Λ (the eigenvalue). In this case, the corre-sponding weak formulation is(3.23) Z Ω ∆ u ∆ φdv = Λ Z Ω uφdv , ∀ φ ∈ H (Ω) , in the unknowns u ∈ H (Ω), Λ ∈ R . In this case boundary conditions are no more “nat-ural” but are “imposed” with the choice of the energy space H (Ω). Actually, problem(3.23) can be written in the form (3.1) with the space H (Ω) replaced by H (Ω). In fact,it is easy to see that(3.24) Z Ω ∆ u ∆ φdv = Z Ω h D u, D φ i + Ric( ∇ u, ∇ φ ) dv for all u, φ ∈ H (Ω), see (3.6) and (3.16). It turns out that (3.1) and (3.20) are equivalentin H (Ω).The situation is similar if Ω = M is a compact complete (boundaryless) smooth Rie-mannian manifold. In this case H ( M ) = H ( M ) (see [26]), hence (3.24) holds for all u, φ ∈ H ( M ). Thus, the weak formulation of the biharmonic closed problem on M isfairly simple, and actually it turns out that the eigenvalues of the biharmonic operator on M coincide with the squares of the eigenvalues of the Laplacian on M , the eigenfunctionsbeing the same. We refer to Subsection 5.8 for more details.Finally, we remark that one can also consider the variational problem(3.25) Z Ω h D u, D φ i dv = µ Z Ω uφdv , ∀ φ ∈ H (Ω) , in the unknowns u ∈ H (Ω), µ ∈ R . As it is done in Subsection 3.2 it is possibleto prove that problem (3.25) is well-posed and admits an increasing sequence of non-negative eigenvalues of finite multiplicity. However, it is not always possible to recoveran eigenvalue problem of the form (1.1) starting from a smooth solution of (3.25) as inthe proof of Theorem 3.1, except for few particular cases. In fact, by following the proofof Theorem 3.1, we are left to deal with the term R Ω Ric( ∇ u, ∇ φ ) dv , and we would liketo have an identity of the form Z Ω Ric( ∇ u, ∇ φ ) dv = Z Ω L ( u ) φdv + Z ∂ Ω B ( u ) φdσ + Z ∂ Ω B ( u ) ∂φ∂ν dσ , ∀ φ ∈ H (Ω) , for suitable differential operators L, B , B . It is not in general possible to have explicitform for L, B , B (they exist by Riesz Theorem). If ( M, g ) is an Einstein manifold, thatis, Ric = Kg , then L ( u ) = − K ∆ u , B ( u ) = K ∂u∂ν and B ( u ) = 0. Thus, any smoothsolution of (3.25) solves(3.26) ∆ u + K ∆ u = µu, in Ω , ∂ u∂ν = 0 , on ∂ Ω , div ∂ Ω ( ∇ ν ∇ u ) ∂ Ω + ∂ ∆ u∂ν + K ∂u∂ν = 0 , on ∂ Ω . Problem (3.26) contains lower order terms in the eigenvalue equation and in the secondboundary condition.3.2.
Neumann eigenvalues of the biharmonic operator.
We prove here that, undersuitable hypothesis on Ω, problem (3.17) admits an increasing sequence of eigenvalues offinite multiplicity bounded from below and diverging to + ∞ . To do so we recast problem(3.1) into an eigenvalue problem for a compact self-adjoint operator acting on a Hilbertspace. First we note that (3.1) can be re-written as(3.27) Z Ω h D u, D φ i + Ric( ∇ u, ∇ φ ) + γuφdv = Γ Z Ω uφdv , ∀ φ ∈ H (Ω) , where γ ∈ R is fixed, in the unknowns u ∈ H (Ω) and Γ ∈ R . Clearly a pair ( u, µ ) ∈ H (Ω) × R is a solution of (3.1) if and only if the pair ( u, µ + γ ) ∈ H (Ω) × R is a solutionof (3.27). We will study the eigenvalue problem in the equivalent formulation (3.27) forsuitable choices of γ .We consider on H (Ω) the bilinear form(3.28) h f, g i H (Ω) := Z Ω (cid:0) h D f, D g i + Ric( ∇ f, ∇ g ) + γf g (cid:1) dv, with γ >
0. We denote by H (Ω) the space H (Ω) endowed with the form (3.28). Wealso set(3.29) k f k H (Ω) := Z Ω (cid:0) | D f | + Ric( ∇ f, ∇ f ) + γf (cid:1) dv. We state the following lemma, whose proof we postpone at the end of this section.
Lemma 3.4.
Let ( M, g ) be a complete n -dimensional smooth Riemannian manifold andlet Ω be a bounded domain in M with C boundary. There exist γ > such that forall γ > γ , the bilinear form (3.28) defines a scalar product in H (Ω) which induces on H (Ω) a norm which is equivalent to the standard one. Through all this subsection, we fix once and for all a positive number γ > γ , where γ is as in Lemma 3.4.Then we define the operator P as an operator from H (Ω) to its dual H (Ω) ′ by setting(3.30) P ( u )[ φ ] := Z Ω (cid:0) h D u, D φ i + Ric( ∇ u, ∇ φ ) + γuφ (cid:1) dv , ∀ u, φ ∈ H (Ω) . By the Riesz Theorem it follows that P is surjective isometry. Then we consider theoperator J from H (Ω) ⊂ L (Ω) to H (Ω) ′ defined by(3.31) J ( u )[ φ ] := Z Ω uφdv , ∀ u, φ ∈ H (Ω) . If the embedding H (Ω) ⊂ L (Ω) is compact, then the operator J is compact. Finally,we set(3.32) T = P ( − ◦ J. If J is compact, since P is bounded, then also T is compact. Moreover h T ( u ) , φ i H (Ω) = h u, φ i L (Ω) , for all u, φ ∈ H (Ω). Hence T is self-adjoint. Note that Ker T = Ker J = { } andthe non-zero eigenvalues of T coincide with the reciprocals of the eigenvalues of (3.27), EUMANN EIGENVALUES OF THE BIHARMONIC OPERATOR ON DOMAINS 13 the eigenfunctions being the same. If Ω is of class C , then the embeddings H (Ω) ⊂ H (Ω) ⊂ L (Ω) are compact (see e.g., [4, § Theorem 3.5.
Let ( M, g ) be a smooth n -dimensional Riemannian manifold and let Ω be a bounded domain in M with C boundary. Then the eigenvalues of (3.1) have finitemultiplicity and are given by a non-decreasing sequence of real numbers { µ j } ∞ j =1 boundedfrom below defined by (3.33) µ j = min U ⊂ H ( M )dim U = j max = u ∈ U R Ω | D u | + Ric( ∇ u, ∇ u ) dv R Ω u dv , where each eigenvalue is repeated according to its multiplicity.Moreover, there exists a Hilbert basis of { u j } ∞ j =1 of H (Ω) of eigenfunctions u j asso-ciated with the eigenvalues µ j . By normalizing the eigenfunctions with respect to (3.29) ,then (cid:8) u j √ µ j + γ (cid:9) ∞ j =1 define a Hilbert basis of L (Ω) with respect to its standard scalar prod-uct.Proof. By the Hilbert-Schmidt Theorem applied to the compact self-adjoint operator T it follows that T admits an increasing sequence of positive eigenvalues { q j } ∞ j =1 , boundedfrom above, converging to zero and a corresponding Hilbert basis { u j } ∞ j =1 of eigenfunc-tions of H (Ω). Since q = 0 is an eigenvalue of T if and only if µ = q − γ is an eigenvalueof (3.1) with the same eigenfunction, we deduce the validity of the first part of the state-ment. In particular, formula (3.33) follows from the standard min-max formula for theeigenvalues of compact self-adjoint operators.To prove the final part of the theorem, we recast problem (3.27) into an eigenvalueproblem for the compact self-adjoint operator T ′ = i ◦ P ( − ◦ J ′ , where J ′ denotes themap from L (Ω) to the dual of H (Ω) defined by(3.34) J ′ ( u )[ φ ] := Z Ω uφdv , ∀ u ∈ L (Ω) , φ ∈ H (Ω) , and i denotes the embedding of H (Ω) into L (Ω). We apply again the Hilbert-SchmidtTheorem and observe that T and T ′ admit the same non-zero eigenvalues, and that theeigenfunctions of T ′ can be chosen in H (Ω) and coincide with the eigenfunctions of T .From (3.27) we deduce that the normalized eigenfunction u j of T with respect to (3.29),divided by √ µ j + γ , form a orthonormal basis of L (Ω). This concludes the proof. (cid:3) We prove now Lemma 3.4.
Proof of Lemma 3.4.
It is easy to see that there exists
C > M and γ ) such that for any u ∈ H (Ω) k u k H (Ω) ≤ C k u k H (Ω) , in fact we can trivially take C = max (cid:8) , k Ric k L ∞ (Ω) , γ (cid:9) .We prove now the opposite inequality(3.35) k u k H (Ω) ≥ C k u k H (Ω) , possibly re-defining the constant C . We note that for ε ∈ (0 , k u k H (Ω) ≥ ε k u k H (Ω) + (1 − ε ) Z Ω (cid:18) | D u | − k Ric k L ∞ (Ω) + ε − ε |∇ u | + γ − ε − ε u (cid:19) dv Hence, in order to prove (3.35) is is sufficient to prove that for any fixed
B >
A > Z Ω (cid:0) | D u | − B |∇ u | + Au (cid:1) dv ≥ . We argue by contradiction and assume that such constant does not exists. We find asequence { u k } ∞ k =1 ⊂ H (Ω) such that Z Ω (cid:0) | D u k | + ku k (cid:1) dv ≤ B Z Ω |∇ u k | dv. We normalize the functions u k by setting R Ω |∇ u k | dv = 1. Hence R Ω | D u k | dv ≤ B and R Ω u k dv ≤ Bk , thus the sequence { u k } ∞ k =1 is bounded in H (Ω). Passing to a sub-sequence, we have that u k ⇀ ¯ u in H (Ω) as k → + ∞ (we have re-labeled the elementsof the subsequence as u k ) and u k → ¯ u in H (Ω) by the compactness of the embed-ding H (Ω) ⊂ H (Ω). Hence R Ω |∇ ¯ u | dv = lim k → + ∞ R Ω |∇ u k | dv = 1 and R Ω ¯ u dv =lim k → + ∞ R Ω u k dv = 0. Then we have found a function ¯ u ∈ H (Ω) such that R Ω |∇ ¯ u | dv =1 and R Ω ¯ u dv = 0, a contradiction. This concludes the proof of (3.35) and of thelemma. (cid:3) A few properties of Neumann eigenvalues
In this section we investigate a few properties of the eigenvalues µ j of problem (3.1). Inparticular we study the behavior of the ratio µ j m j , where m j are the Neumann eigenvaluesof the Laplacian on Ω. In fact, in view of the asymptotic laws (1.2) and (1.5), it is naturalto compare µ j with m j . In particular we show that this ratio can be arbitrarily large orarbitrarily close to zero. We denote by0 = m < m ≤ · · · ≤ m j ≤ · · · ր + ∞ the Neumann eigenvalues of the Laplacian on Ω, which are given by(4.1) m j = min U ⊂ H (Ω)dim U = j max = u ∈ U R Ω |∇ u | dv R Ω u dv . Here H (Ω) denotes the closure of C ∞ (Ω) with respect to the norm R Ω |∇ u | + u dv .We also consider the sign of the eigenvalues, proving that in some situations negativeeigenvalues may appear. As a consequence we also provide examples where the ratio µ j m j can be made negative and with arbitrarily large absolute value. In order to producesuitable examples, we restrict our analysis to the Euclidean space, to manifolds withRic ≥ ( n − K > H n . EUMANN EIGENVALUES OF THE BIHARMONIC OPERATOR ON DOMAINS 15
Domains of the Euclidean space.
Through this subsection (
M, g ) is the standardEuclidean space R n . It is well-known that if Ω is a bounded Lipschitz domain, then0 = µ = µ = · · · = µ n +1 < µ n +2 ≤ · · · ≤ µ j ≤ · · · ր + ∞ , and the eigenspace corresponding to the eigenvalue µ = 0 is generated by { , x , ..., x n } ,see e.g., [37].We have the following theorem. Theorem 4.1.
For all N ∈ N there exists a sequence { Ω ε,N } ε ∈ (0 ,ε ) such that lim ε → + µ j m j → , for all N + 2 ≤ j ≤ ( N + 1)( n + 1) ,Proof. The domains providing the result are obtained by connecting with thin junctionsa fixed domain Ω to N domains Ω , ..., Ω N , N ∈ N , of fixed volume and disjoint from Ω,and by letting the size of the junctions go to zero. We will prove the theorem for n = 2and N = 1.For ε ∈ (0 , ε, = Ω ε := Ω L ∪ Ω R ∪ R ε , where Ω L = ( − , × (0 , R =(1 , × (0 , R ε := (cid:8) x ∈ R : 0 ≤ x ≤ , < x < ε (cid:9) .Let { m j } ∞ j =1 denote the Neumann eigenvalues of Ω L ∪ Ω R , let { ξ j } ∞ j =1 denote the eigen-values of − f ′′ ( t ) = ξf ( t ) in (0 ,
1) with Dirichlet boundary conditions, and let (cid:8) m εj (cid:9) ∞ j =1 denote the Neumann eigenvalues of Ω ε . It is known that the sequence (cid:8) m εj (cid:9) ∞ j =1 convergesto the sequence { m j } ∞ j =1 ∪ { ξ j } ∞ j =1 , where in the union the eigenvalues have been orderedincreasingly, see e.g., [2]. In particular, m ε = 0, m ε → ε → + , and m εj are uniformlybounded from below by some positive constant independent on ε for j ≥ µ εj denote the eigenvalues of (3.1) in Ω ε . We prove that µ εj ≤ Cε for j ≤ C > ε .To do so, let φ L ( x , x ) ∈ C ( R ) be such that φ L ( x , x ) = 1 if x <
0, 0 ≤ φ L ( x , x ) ≤ ≤ x ≤ , and φ L ( x , x ) = 0 if x > . By construction | D φ L ( x , x ) | ≤ c for some c >
0. We define analogously φ R ( x , x ) ∈ C ( R ) which issupported in (cid:8) x > (cid:9) by setting φ R ( x , x ) = φ L (1 − x , x ).We set u L = φ L | Ω ε , u L = x · φ L | Ω ε , u L = x · φ L | Ω ε , u R = φ R | Ω ε , u R = x · φ R | Ω ε , u R = x · φ R | Ω ε . These functions are linearly independent and belong to H (Ω ε ). More-over, any u in the space generated by u L , u L , u L , u R , u R , u R with R Ω ε u dv = 1 is easilyseen to satisfy Z Ω | D u | dv ≤ Cε, with C independent of ε . This implies from (3.33) that µ εj ≤ Cε for j ≤
6. The proof isnow complete in the case n = 2 and N = 1.The proof for n > N > (cid:3)
Remark . Let us consider a domain Ω ε,N as in the proof of Theorem 4.1. Such adomain is usually called a N + 1 -dumbbell . We observe that if N > n , we have that m j , µ j → ε → + for n + 2 ≤ j < N + 2. It is well-known that m = 0 and m j = O ( ε n +1 ) as ε → + for 2 ≤ j ≤ N + 1, see e.g., [1, 27]. Moreover, it is possibleto show that in the case of a sufficiently regular N + 1-dumbbell domain Ω ε,N , µ j → ε → + for n + 2 ≤ j ≤ ( N + 1)( n + 1), and µ ( N +1)( n +1)+1 is bounded away from zero, uniformly in ε . We refer to [3] for the proof in the case N = 1. Thanks to this fact,with the same arguments of [1] (see also [27]) it is possible to prove that µ j = O ( ε n − ) as ε → + for n + 2 ≤ j ≤ ( N + 1)( n + 1). We omit the details of the computations whichare standard but quite technical and go beyond the scopes of the present article. Anyway,we have that µ j m j → + ∞ as ε → n + 2 ≤ j < N + 2. Thus in the Euclidean casedumbbell domains provide examples where either µ j < m j (for certain j ∈ N ) or µ j > m j (for other j ∈ N ).4.2. Domains in manifolds with
Ric ≥ ( n − K > . Through all this subsection(
M, g ) will be a complete n -dimensional smooth Riemannian manifold with Ric ≥ ( n − K >
0. We note that for any domain Ω of class C of M we have µ = 0 and µ > µ j ≥ j ∈ N and that µ = 0is an eigenvalue with corresponding eigenfunctions the constant functions. Constantfunctions are the only eigenfunctions associated with µ = 0. In fact, any eigenfunction u corresponding to the eigenvalue µ = 0 satisfies Z Ω | D u | + Ric( ∇ u ∇ u ) dv = 0 , hence |∇ u | = 0, thus u is a constant. This implies that µ > ≤ η ≤ η ≤ · · · ≤ η j ≤ · · · ր + ∞ the eigenvalues of the rough Laplacian on Ω with Neumann boundary conditions. Theyare characterized by(4.2) η j = min W⊂H (Ω)dim W = j max = ω ∈W R Ω |∇ ω | dv R Ω ω dv , where H (Ω) is the space of 1-forms of class H (Ω), see e.g., [19] for more information onthe eigenvalues of the rough Laplacian. We have the following. Theorem 4.3.
Let ( M, g ) be a complete n -dimensional smooth Riemannian manifoldwith Ric ≥ ( n − K > and let Ω be a bounded domain in M with C boundary. Then µ j ≥ ( η + ( n − K ) m j for all j ∈ N .Proof. The inequality is trivially true for j = 1. Hence we assume j ≥
2. We observethat for any non-constant u ∈ H (Ω)(4.3) R Ω | D u | + Ric( ∇ u, ∇ u ) dv R Ω u dv = R Ω | D u | dv R Ω |∇ u | dv · R Ω |∇ u | dv R Ω u dv + R Ω Ric( ∇ u, ∇ u ) dv R Ω u dv ≥ η R Ω |∇ u | dv R Ω u dv + ( n − K R Ω |∇ u | dv R Ω u dv = ( η + ( n − K ) R Ω |∇ u | dv R Ω u dv . Hence, for any subspace U ⊂ H ( M ) of dimension j ≥ = u ∈ U R Ω | D u | + Ric( ∇ u, ∇ u ) dv R Ω u dv ≥ ( η + ( n − K ) max = u ∈ U R Ω |∇ u | dv R Ω u dv EUMANN EIGENVALUES OF THE BIHARMONIC OPERATOR ON DOMAINS 17
This implies(4.4) µ j = min U ⊂ H (Ω)dim U = j max = u ∈ U R Ω | D u | + Ric( ∇ u, ∇ u ) dv R Ω u dv ≥ ( η + ( n − K ) min U ⊂ H (Ω)dim U = j max = u ∈ U R Ω |∇ u | dv R Ω u dv ≥ ( η + ( n − K ) min U ⊂ H (Ω)dim U = j max = u ∈ U R Ω |∇ u | dv R Ω u dv = ( η + ( n − K ) m j , where in the last inequality we have used the fact that H (Ω) ⊂ H (Ω), hence theminimum decreases. The proof is now complete. (cid:3) Remark . Note that if Ω is a bounded domain with II ≥
0, we have that m ≥ nK, with equality if and only if Ω is isometric to an n -dimensional Euclidean hemisphereof curvature K , see e.g., [21]. This result is in the spirit of the well-known Obata-Lichnerowicz inequality, see [14, 30, 33]. Hence, for any bounded domain Ω with II ≥ µ ≥ ( η + ( n − K ) nK. It is natural to conjecture that(4.5) µ ≥ n K . Open problem.
Prove (4.5).Thanks to Theorem 4.3 we have the following inequality for all j ∈ N , j ≥ µ j m j ≥ ( η + ( n − K ) m j . We recall now that for any N ∈ N there exists a sequence { Ω ε,N } ε ∈ (0 ,ε ) such that m j ≤ Cε for all j ≤ N . These domains are obtained by connecting to a fixed domain Ω, N domains of fixed volume and disjoint from Ω with thin junctions, and by letting thewidth of the channels, represented by the parameter ε >
0, go to zero. This is a standardconstruction (see [1, 2], see also Subsection 4.1).This implies the validity of the following theorem.
Theorem 4.5.
Let ( M, g ) be a complete n -dimensional smooth Riemannian manifoldwith Ric ≥ ( n − K > . For all N ∈ N there exist a sequence { Ω ε,N } ε ∈ (0 ,ε ) of domainssuch that (4.7) lim ε → + µ j m j = + ∞ , for all ≤ j ≤ N . On the other hand, if Ω is such that the second fundamental form of its boundary isnon-negative, that is, II ≥
0, we have that(4.8) µ j m j ≤ . We refer to Subsection 5.9 for the proof of (4.8).4.3.
Domains of the hyperbolic space.
Given an eigenvalue µ of (3.1) and a corre-sponding eigenfunction u µ ∈ H (Ω), we have µ = R Ω | D u µ | + Ric( ∇ u µ , ∇ u µ ) dv R Ω u µ dv . It is well-known, from Bochner’s formula (2.4) and integration by parts, that, for any u ∈ H (Ω) Z Ω | D u | + Ric( ∇ u, ∇ u ) dv = Z Ω (∆ u ) dv ≥ . Note that for a complete, compact (boundaryless) smooth Riemannian manifold M Z M | D u | + Ric( ∇ u, ∇ u ) dv = Z M (∆ u ) dv ≥ , for all u ∈ H ( M ).In view of this, the natural question arises whether the biharmonic Neumann eigen-values µ j can be negative or not. Clearly, a necessary condition for the appearance ofnegative eigenvalues is that Ric
0. In this section we consider domains of the standardhyperbolic space H n . We have the following theorem. Theorem 4.6.
Let Ω be a bounded domain of the hyperbolic space H n with C boundary.Then Ω admits at least n strictly negative eigenvalues.Proof. We start by proving the result for n = 2. To do so, we will use Fermi coordinatesfor H . In this case the metric is given by g ( x, y ) = dx + cosh ( x ) dy . The Christoffel symbols areΓ , = Γ , = Γ , = Γ , = 0 , Γ , = tanh( x ) , Γ , = − x ) sinh( x ) . For any smooth function u ( x, y ) we have ∇ u = u x dx + u y dy, hence |∇ u | = u x + u y cosh( x ) . Moreover, D u = u xx dx ⊗ dx + ( u xy − tanh( x ) u y ) ( dx ⊗ dy + dy ⊗ dx )+ ( u yy + cosh( x ) sinh( x ) u x ) dy ⊗ dy, therefore | D u | = u xx + 2 ( u xy − tanh( x ) u y ) cosh( x ) + ( u yy + cosh( x ) sinh( x ) u x ) cosh( x ) A natural test function for the Rayleigh quotient in (3.33) is ˜ u ( x, y ) = x , which is thesigned distance from the geodesic x = 0. We have then | D u | + Ric( ∇ u, ∇ u ) = | D u | − |∇ u | = tanh( x ) − < . This implies that µ = min u ∈ H (Ω) R Ω | D u | + Ric( ∇ u, ∇ u ) dv R Ω u ≤ R Ω | D ˜ u | + Ric( ∇ ˜ u, ∇ ˜ u ) dv R Ω ˜ u < . Thus, µ <
0. Now, fixed a domain Ω, let µ < u be an associated eigenfunction. Let p ∈ Ω, v = 0 be a vector in T p H , and let { γ θ } θ ∈ [0 , π ) be the family of geodesics with γ θ (0) = p and with the angle between γ ′ θ (0) and v equalsto θ . Let h θ be the signed distance to γ θ . For all θ ∈ [0 , π ), | D h θ | +Ric( ∇ h θ , ∇ h θ ) < x = h θ . Moreover, we have that h π = − h .Let us consider the function θ ρ ( θ ) defined by ρ ( θ ) := R Ω h θ u dv . The function ρ iscontinuous and satisfies ρ (0) = − ρ ( π ). Therefore, there exists 0 ≤ θ ≤ π with ρ ( θ ) = 0.Hence, there exists at least one function h θ with strictly negative Rayleigh quotient andorthogonal to u . From (3.33) we deduce that µ = min = u ∈ H (Ω) R Ω uu dv =0 | D u | + Ric( ∇ u, ∇ u ) dvu dv ≤ | D h θ | + Ric( ∇ h θ , ∇ h θ ) dvh θ dv < . This proves the existence of a second strictly negative eigenvalue. This concludes theproof in the case n = 2.The proof for n ≥ n ≥ g : S n → R n is an odd function (that is, g ( p ) = − g ( − p ) where − p isthe antipodal point to p in S n ), then there exists p ∈ S n such that g ( p ) = 0.Let q ∈ Ω and let H be an hyperplane containing q . Let f be the signed distancefrom H . Then | D f | + Ric( ∇ f , ∇ f ) dv < . The proof is analogous to that of the case n = 2. It follows by explicit computationsin Fermi coordinates ( x , ..., x n ) where x represents the signed distance from H and( x , ..., x n ) are normal coordinates on H = H n − (in this system q = (0 , ..., µ < u a corresponding eigenfunction.Let π be a fixed plane in T q H n and let v ∈ π be a non-zero vector. For θ ∈ [0 , π ),let H θ be the hyperplane whose tangent space at q is normal to the vector v θ ∈ π ,where v θ is a unit vector which forms with v an angle of width θ in π . Let f θ be thesigned distance from H θ . The Rayleigh quotient of this function is again strictly negative.Moreover, we have that f = − f π and if we define ρ ( θ ) := R Ω f θ u dv , we find out thatthere exists θ ∈ [0 , π ) such that ρ ( θ ) = 0. Thus we deduce the existence of a functionwith strictly negative Rayleigh quotient orthogonal to u . As in the case n = 2, we deducethat µ <
0. Assume now that we have µ , ..., µ k <
0, with k < n , and with associatedeigenfunctions u , ..., u k . We prove that µ k +1 < π k +1 be a fixed k + 1-dimensional subspace of T q H n and let v k be a non-zerovector in π k +1 . Let, for θ = ( θ , ..., θ k ) ∈ S k , v θ be a vector in π k +1 forming a directionalangle θ = ( θ , ..., θ k ) with v k . Let H θ be the hyperplane whose tangent space at q is normal to the vector v θ ∈ π k +1 . Let f θ be the signed distance from H θ . The Rayleighquotient of this function is strictly negative. Moreover, we have that f θ = − f − θ forall θ ∈ S k . We define now ρ k : S k → S k by ρ k ( θ ) := ( R Ω f θ u dv, ..., R Ω f θ u k dv ). Byconstruction ρ k is continuous and odd, hence there exists θ ∈ S k such that ρ k ( θ ) = 0,hence R Ω f θ u dv = · · · = R Ω f θ u k dv = 0. Therefore there exists a function in H (Ω) withstrictly negative Rayleigh quotient and orthogonal to u , ..., u k . From (3.33) we deducethat µ k +1 < (cid:3) Remark . We can give an upper bound on the number of negative eigenvalues of Ωin terms of the number of eigenvalues of the rough Laplacian smaller than one. Indeed,if we have µ , ..., µ N negative eigenvalues with corresponding eigenfunctions u , ..., u N ,then any u = P Ni =1 α i u i is such that Z Ω | D u | + Ric( ∇ u, ∇ u ) dv = Z Ω | D u | − |∇ u | dv < . To end this section we show that there exists domains with an arbitrary number ofarbitrarily large (in absolute value) negative eigenvalues.
Theorem 4.8.
For any N ∈ N and M > there exist a bounded domain Ω M,N of thehyperbolic space H n with | Ω M,N | = 1 and such that µ j ≤ − M, for all j ≤ N .Proof. We start by proving the theorem with N = 1. Let γ be a simple geodesic in H n and let γ δ a δ -neighborhood of γ , that is γ δ := { p ∈ H n : dist( p, γ ) < δ } . In γ δ weconsider Fermi coordinates ( x , ..., x n ), where ( x , , ...,
0) correspond to the points on γ and (0 , x , ..., x n ) correspond to a normal neighborhood of 0. Moreover, g ij ( p ) = δ ij and ∂g ij ∂x k ( p ) = 0 = Γ kij ( p ) for all p ∈ γ .Given ε >
0, we find δ > | g ij ( p ) − δ ij | < ε , (cid:12)(cid:12)(cid:12) ∂g ij ∂x k ( p ) (cid:12)(cid:12)(cid:12) < ε and | Γ kij ( p ) | < ε for all p ∈ γ δ .On the domain D δ,L := { p ∈ γ δ : 0 < x < L } we consider n test functions x i , i =2 , ..., n . We have, for all i = 2 , ...n | D x i | = |∇ dx i | ≤ C ′ ε for some C ′ > ε , andRic( ∇ x i , ∇ x i ) = − . Therefore Z D δ,L | D x i | + Ric( ∇ x i , ∇ x i ) dv ≤ ( Cε − Lδ n − , while Z D δ,L x i dv ≥ C ′′ Lδ n − δ , for some constant C ′′ > ε, δ , since in γ δ \ γ δ/ , | x i | ≥ δ . By choosing ε > µ ≤ − Cδ . EUMANN EIGENVALUES OF THE BIHARMONIC OPERATOR ON DOMAINS 21
Moreover, by choosing L = δ n − we have that | D δ,L | = O (1) as δ → + . This proves thestatement for N = 1.Let N ∈ N be fixed. Consider the domain D δ,NL . Let the points p , ..., p N ∈ γ be givenby p i = (cid:0) L (cid:0) i − (cid:1) , , ..., (cid:1) for i = 1 , ..., N and let B i := B (cid:0) p i , L (cid:1) and 2 B i := B (cid:0) p i , L (cid:1) .The balls 2 B i are disjoint. Associated with each 2 B i we define cut-off functions φ i by φ i ( p ) = , if p ∈ B i , (cid:16) p,p i ) L − (cid:17) (cid:16) p,p i ) L − (cid:17) , if p ∈ B i \ B i , , if p ∈ H n \ B i . The functions u i := x k φ i , i = 1 , ...N (for some k = 2 , ..., n ) are N disjointly supportedfunctions in H ( D δ,L ), hence from (3.33) we deduce that µ N ≤ max i =1 ,...,N R D δ,NL | D u i | + Ric( ∇ u i , ∇ u i ) dv R D δ,NL u i dv . The same computations above show that the Rayleigh quotient of each u i is boundedabove by − Cδ , hence we have N negative eigenvalues with arbitrary large absolute value.By choosing N L = δ n − we have also that | D δ,NL | = O (1) as δ → + . This concludesthe proof. (cid:3) Remark . A consequence of Theorem 4.8 is that we can always locally perturb afixed domain of the hyperbolic space H n in order to obtain an arbitrary large number ofnegative eigenvalues with arbitrary large absolute value. This is done exactly as for theNeumann Laplacian, in which case we can deform locally a domain in order to have anarbitrary large number of eigenvalues arbitrarily close to zero. Indeed, it is sufficient tojoin to the domain a sufficient number of small balls by sufficiently thin junctions. Remark . A natural problem is to find classes of domains of the hyperbolic spacewhich have exactly n negative eigenvalues. A first immediate conjecture is that hyperbolicballs admit exactly n negative eigenvalues. However, a simple proof of this fact is currentlyunavailable and we leave this as an open question. Nevertheless, it is possible to provethe result in the case of sufficiently small balls. In fact, from Remark 4.7 we deduce thatif η n +1 >
1, where η n +1 is the n + 1-th eigenvalue of the rough Laplacian on Ω, thenΩ admits exactly n negative eigenvalues. Using normal coordinates with origin in thecenter of a ball B ε of radius ε it is possible to prove by means of explicit computationsand max-min formula for the eigenvalues that η n +1 → + ∞ as ε → + . Thus we deducethe existence of ε > ε admits exactly n negative eigenvalues. Open problem.
Prove that hyperbolic balls admit exactly n negative eigenvalues.5. Upper estimates for eigenvalues
In this section we provide upper bounds for the eigenvalues µ j of (3.1) which arecompatible with Weyl’s law (1.2). As we have highlighted in Section 4, there is in generalno monotonicity between the eigenvalues µ j and the squares of the Neumann eigenvaluesof the Laplacian m j , hence in general there is no hope to recover upper bounds for µ j from the known upper bounds on m j . We remark that the situation is very different if we think of Dirichlet eigenvalues ofthe Laplacian and the biharmonic operator. Indeed, if we denote by λ j and by Λ j theeigenvalues of the Laplace and biharmonic operator respectively, with Dirichlet boundaryconditions on a domain Ω of a complete n -dimensional smooth Riemannian manifold,then λ j ≤ Λ j , for all j ∈ N . This is an immediate consequence of the min-max principle for λ j and Λ j .Hence, lower bounds for Λ j can be obtained from lower bounds on λ j .5.1. Decomposition of a metric measure space by capacitors.
In this subsectionwe present the main technical tools which will be used to prove upper bounds for eigen-values. We start with some definitions.We denote by ( X, dist , ς ) a metric measure space with a metric dist and a Borel measure ς . We will call capacitor every couple ( A, D ) of Borel sets of X such that A ⊂ D . By anannulus in X we mean any set A ⊂ X of the form A = A ( a, r, R ) = { x ∈ X : r < dist( x, a ) < R } , where a ∈ X and 0 ≤ r < R < + ∞ . By 2 A we denote2 A = 2 A ( a, r, R ) = n x ∈ X : r < dist( x, a ) < R o . Moreover, for any F ⊂ X and r > r -neighborhood of F by F r , namely F r := { x ∈ X : dist( x, F ) < r } . We recall the following metric construction of disjoint capacitors from [23].
Theorem 5.1 ([23, Theorem 1.1]) . Let ( X, dist , ς ) be a metric-measure space with ς anon-atomic finite Borel measure. Assume that the following properties are satisfied:i) there exists a constant Γ such that any metric ball of radius r can be covered by atmost Γ balls of radius r ;ii) all metric balls in X are precompact sets.Then for any integer j there exists a sequence { A i } ji =1 of j annuli in X such that, forany i = 1 , ..., j ς ( A i ) ≥ c ς ( X ) j , and the annuli A i are pairwise disjoint. The constant c depends only on the constant Γ in i). As we shall see, Theorem 5.1 is not easy to use in many concrete cases, e.g., when X is a domain in the standard hyperbolic space H n and ς is the restriction of the Lebesguemeasure on X . In fact, hypothesis i ) fails to hold with Γ depending only on the di-mension because of the exponential growth of the volume of balls. We state now thefollowing lemma, which improves [17, Lemma 4.1]. We postpone its proof at the end ofthis subsection. Lemma 5.2.
Let ( X, dist , ς ) be a compact metric measure space with a finite measure ς .Assume that for all s > there exists an integer N ( s ) such that each ball of radius s can EUMANN EIGENVALUES OF THE BIHARMONIC OPERATOR ON DOMAINS 23 be covered by N ( s ) balls of radius s . Let β > satisfying β ≤ ς ( X )2 and let r > be suchthat for all x ∈ X ς ( B ( x, r )) ≤ β N ( r ) . Then there exist two open sets A and D of X with A ⊂ D such that:i) A = B ( x , r ) ∪ · · · ∪ B ( x l , r ) with dist( x i , x j ) ≥ r if i = j ;ii) D = A r = B ( x , r ) ∪ · · · ∪ B ( x l , r ) ;iii) ς ( A ) ≥ β N ( r ) , ς ( D ) ≤ β and dist( A, D c ) ≥ r . A consequence of Lemma 5.2 is the following result providing a decomposition of ametric measure space by capacitors which is alternative of that of Theorem 5.1.
Lemma 5.3.
Let ( X, dist , ς ) be a compact metric measure space with a finite measure ς .Assume that for all s > there exists an integer N ( s ) such that each ball of radius s canbe covered by N ( s ) balls of radius s . If there exists an integer k > and a real number r > such that, for each x ∈ X ς ( B ( x, r )) ≤ ς ( X )4 N ( r ) k , then there exist k ς -measurable subsets A , ..., A k of X such that ς ( A i ) ≥ ς ( X )2 N ( r ) k , for all i ≤ k , dist( A i , A j ) ≥ r for i = j , and A i = B ( x i , r ) ∪ · · · ∪ B ( x il i , r ) . The proof of Lemma 5.3 is a consequence of Lemma 5.2 and follows exactly the samelines of the proof of [17, Lemma 2.1]. We remark that [17, Lemma 2.1] provides adecomposition of a metric measure space by capacitors given by union of balls. In Lemma5.3 the decomposition is given by unions of disjoint balls.A clever merging of Theorem 5.1 and Lemma 5.3 allows to obtain the following The-orem, which provides a further construction of disjoint families of capacitors. This is aconstruction which we will widely use in the next subsections. Its proof follows exactlythe same lines as those of [25, Theorem 2.1]. In fact, the substantial difference is the useof Lemma 5.3 instead of [25, Lemma 2.3] (see also [17, Lemma 2.1] and [18, Corollary2.3]).
Theorem 5.4.
Let ( X, dist , ς ) be a compact metric-measure space with ς a non-atomicfinite Borel measure and let a > . Assume that there exists a constant Γ such that anymetric ball of radius < r ≤ a can be covered by at most Γ balls of radius r . Then, forevery j ∈ N there exists two families { A i } ji =1 and { D i } ji =1 of Borel subsets of X suchthat A i ⊂ D i , with the following properties:i) ς ( A i ) ≥ c ς ( X ) j , where c depends only on Γ ;ii) D i are pairwise disjoint;iii) the two families have one of the following form:a) all the A i are annuli and D i = 2 A i , with outer radii smaller than a , orb) all the A i are of the form A i = B ( x i , r ) ∪ · · · ∪ B ( x il i , r ) , G i = A r i and dist( x ik , x il ) ≥ r , where r = a . We remark that for a sufficiently large integer j it is always possible to apply theconstruction of Theorem 5.1 and obtain a decomposition of the metric measure space byannuli (Theorem 5.4 i ), ii ) and iii )- a )). In particular we have the following. Lemma 5.5.
Assume that the hypothesis of Theorem 5.4 hold. Then there exists aninteger j X such that for every j ≥ j X there exists two families { A i } ji =1 and { D i } ji =1 ofBorel subsets of X such that A i ⊂ D i satisfying i ) , ii ) and iii ) - a ) of Theorem 5.4. We refer to [25, Proposition 2.1] for the proof of Lemma 5.5.We state now a useful corollary of Theorem 5.1 which gives a lower bound of the innerradius of the annuli of the decomposition, see [23, Remark 3.13].
Corollary 5.6.
Let the assumptions of Theorem 5.1 hold. Then each annulus A i haseither internal radius r i such that (5.1) r i ≥
12 inf { r ∈ R : V ( r ) ≥ v j } , where V ( r ) := sup x ∈ X ς ( B ( x, r )) and v j = c ς ( X ) j , or is a ball of radius r i satisfying (5.1) . It turns out that Corollary 5.6 applies to the case iii )- a ) of Theorem 5.4, see also [25].We conclude this subsection with the proof of Lemma 5.2 Proof of Lemma 5.2.
Step 1.
We construct the points x i by induction. Let Ω = X .The point x is such that ς ( B ( x , r ) ∩ Ω ) = max { ς ( B ( x, r ) ∩ Ω ) : x ∈ X } .Let Ω = X \ B ( x , r ). The point x is such that ς ( B ( x , r ) ∩ Ω ) = max { ς ( B ( x, r ) ∩ Ω ) : x ∈ X } . Note that this definition implies that dist( x , x ) ≥ r . If dist( x , x ) < r , B ( x, r ) ∩ Ω = ∅ . Suppose that we have constructed x , ..., x j . Let Ω j +1 = X \ ( B ( x , r ) ∪ ... ∪ B ( x j , r ).Suppose Ω j +1 = ∅ . The point x j +1 is such that ς ( B ( x j +1 , r ) ∩ Ω j +1 ) = max { ς ( B ( x, r ) ∩ Ω j +1 ) : x ∈ X } . Note that this definition implies that dist( x j +1 , x i ) ≥ r . If dist( x, x i ) < r , B ( x, r ) ∩ Ω j +1 = ∅ .By compactness, the process has to stop: there exist only finitely many points on X such that dist( x i , x j ) ≥ r . Let ( x , ..., x k ) the set of points we have constructed. Wehave ς ( X \ ( B ( x , r ) ∪ ... ∪ B ( x k , r ))) = 0 otherwise we could do another iteration. Then ς ( X ) = ς ( B ( x , r ) ∪ ... ∪ B ( x k , r )) . Step 2.
We write B ( x , r ) ∪ ... ∪ B ( x k , r ) = ( B ( x , r ) ∩ Ω ) ∪ ( B ( x , r ) ∩ Ω ) ∪ ... ∪ ( B ( x k , r ) ∩ Ω k ) . Indeed, if x ∈ B ( x i , r ) and x B ( x i , r ) ∩ Ω i , we have by construction that x ∈ B ( x , r ) ∪ ... ∪ B ( x i − , r )). Let j be the smallest integer such that x ∈ B ( x j , r ). Then x ∈ Ω j . In fact, if it not the case, x ∈ B ( x , r ) ∪ ... ∪ B ( x j − , r )) and this wouldcontradict the fact that j was minimal.As this is a disjoint union, we get ς ( X ) = ς ( B ( x , r ) ∪ ... ∪ B ( x k , r )) = ς ( B ( x , r ) ∩ Ω ) + ... + ς ( B ( x k , r ) ∩ Ω k ) . Step 3.
For each i , let us show that ς ( B ( x i , r ) ∩ Ω i ) ≤ N ( r ) ς ( B ( x i , r ) ∩ Ω i ) . EUMANN EIGENVALUES OF THE BIHARMONIC OPERATOR ON DOMAINS 25
In fact by definition, ς ( B ( x i , r ) ∩ Ω i ) = max { ς ( B ( x, r ) ∩ Ω i ) : x ∈ X } and B ( x i , r ) is covered by N ( r ) balls of radius r . Let B ( y, r ) one of these balls. We have ς ( B ( y, r ) ∩ Ω i ) ≤ ς ( B ( x i , r ) ∩ Ω i ) , so that ς ( B ( x i , r ) ∩ Ω i ) ≤ N ( r ) ς ( B ( x i , r ) ∩ Ω i ). Step 4.
We deduce(5.2) ς ( X ) = ς ( B ( x , r ) ∩ Ω ) + ... + ς ( B ( x k , r ) ∩ Ω k ) ≤ N ( r )( ς ( B ( x , r ) ∩ Ω ) + ... + ς ( B ( x k , r ) ∩ Ω k ) ≤ N ( r )( ς ( B ( x , r ) + ... + ς ( B ( x k , r )) , and ς ( B ( x , r )) + ... + ς ( B ( x k , r )) ≥ ς ( X ) N ( r ) . By hypothesis β ≤ ς ( X )2 . As ς ( B ( x , r ) + ... + ς ( B ( x k , r )) ≥ ς ( X ) N ( r ) ≥ βN ( r ) , we choose l such that ς ( B ( x , r )) + ... + ς ( B ( x l , r )) ≥ β N ( r )and ς ( B ( x , r )) + ... + ς ( B ( x l − , r )) ≤ β N ( r ) . Let A = B ( x , r ) ∪ ... ∪ B ( x l , r ) and D = B ( x , r ) ∪ ... ∪ B ( x l , r ). We have by constructiondist( x i , x j ) ≥ r and we immediately deduce that ς ( A ) ≥ β N ( r ) , dist( A, D r ) ≥ r . Itremains to show ς ( D ) ≤ β. We can argue as before and deduce that ς ( D ) = ς ( B ( x , r )) + ... + ς ( B ( x l , r ) = ς ( B ( x , r ) ∩ Ω ) + ... + ς ( B ( x l , r ) ∩ Ω l ) ≤≤ N ( r )( ς ( B ( x , r ) + ... + ς ( B ( x l , r ))) ≤ β . This concludes the proof. (cid:3)
A first general estimate.
In this subsection we prove an upper bound whichholds for any domain with C boundary of a complete n -dimensional smooth Riemannianmanifold ( M, g ) with a given lower bound on the Ricci curvature of the form Ric ≥− ( n − κ , κ ≥ r inj ( p ) the injectivity radius ofthe manifold ( M, g ) at p . We denote by r inj the injectivity radius of the manifold ( M, g ),which is defined as the infimum of r inj ( p ) for p ∈ M . We will use also the injectivityradius relative to Ω ⊂ M , defined by:(5.3) r inj, Ω := inf p ∈ Ω r inj ( p ) . If Ω is bounded, then the infimum in (5.3) is actually a minimum and it is strictly positive.We are ready to state the main result of this subsection.
Theorem 5.7.
Let ( M, g ) be a complete n -dimensional smooth Riemannian manifoldwith Ric ≥ − ( n − κ , κ ≥ and let Ω be a bounded domain of M of class C . Let a := min (cid:8) κ , r inj, Ω (cid:9) . Then (5.4) µ j ≤ A n (cid:18) j | Ω | (cid:19) n + B n | ∂ Ω | | Ω | + C n a . for all j ∈ N , where A n , B n , C n are positive constants which depend only on the dimension. The strategy of the proof of Theorem 5.7 is to build, for each j ∈ N , j disjointlysupported functions u , ..., u j ∈ H (Ω). Hence, the linear space U j spanned by u , ..., u j is j -dimensional and we can use U j in the min-max formula (3.33). The fact that thefunctions u , ..., u j have disjoint support makes easy to estimate the Rayleigh quotient ofany function in U j : it is in fact sufficient to estimate the Rayleigh quotient of each of the u i .Suitable test functions for the Rayleigh quotient in (3.33) are built, in this subsection,in terms of the Riemannian distance function from a point p ∈ M . For any x, p ∈ M wedenote by δ p ( x ) the function δ p ( x ) := dist( x, p ) . We also denote by cut( p ) the cut-locus of a point p ∈ M . We will make use of theLaplacian Comparison Theorem, see e.g., [34, §
9] for details.
Theorem 5.8.
Let ( M, g ) be a complete n -dimensional smooth Riemannian manifoldsatisfying Ric ≥ − ( n − κ , κ ≥ and let p ∈ M . Then, for any x ∈ M \ ( { p } ∪ cut( p )) i) ∆ δ p ( x ) ≤ ( n − κ coth( κδ p ( x )) if κ > ;ii) ∆ δ p ( x ) ≤ n − δ p ( x ) if κ = 0 . We prove now the following lemma.
Lemma 5.9.
Let ( M, g ) be a complete n -dimensional smooth Riemannian manifold with Ric ≥ − ( n − κ , κ ≥ . Then, for any p ∈ M and any x ∈ B (cid:16) p, r inj ( p )2 (cid:17) we havei) | ∆ δ p ( x ) | ≤ ( n − κ coth( κδ p ( x )) if κ > ;ii) | ∆ δ p ( x ) | ≤ n − δ p ( x ) if κ = 0 .In particular, for any κ ≥ | ∆ δ p ( x ) | ≤ n − δ p ( x ) + ( n − κ. Proof.
We prove point ii ). Let p ∈ M and let x ∈ B (cid:16) p, r inj ( p )2 (cid:17) . Let p ′ be the uniquepoint such that δ p ( p ′ ) = r inj ( p )2 and x belongs to the geodesic joining p and p ′ . FromTheorem 5.8 it follows that ∆ δ p ( x ) ≤ n − δ p ( x ) . Moreover, since x belongs to the geodesic connecting p with p ′ we see that∆ δ p ( x ) = ∆ (cid:18) r inj ( p )2 − δ p ′ ( x ) (cid:19) = − ∆ δ p ′ ( x ) ≥ − n − δ p ′ ( x )= − n − r inj ( p )2 − δ p ( x ) ≥ − n − δ p ( x ) . EUMANN EIGENVALUES OF THE BIHARMONIC OPERATOR ON DOMAINS 27
Point ii ) is now proved. Point i ) is proved exactly in the same way. The last statementfollows by observing that cosh( t ) ≤ t − t >
0, which implies (5.5). (cid:3)
We note that the restriction x ∈ B (cid:16) p, r inj ( p )2 (cid:17) is somehow natural. We may think ofthe unit 2-dimensional standard sphere and coordinates ( θ, φ ) ∈ [0 , π ] × [0 , π ) and p beingthe north pole ( θ = 0). In this case r inj = π . Hence δ p ( θ, φ ) = θ and ∆ δ p ( θ, φ ) = θ ) .Then, for any ( θ, φ ) ∈ (cid:2) , π (cid:3) × [0 , π ), 0 ≤ ∆ δ p ( θ, φ ) ≤ δ p ( θ,φ ) , which in particular implies | ∆ δ p ( θ, φ ) | ≤ δ p ( θ,φ ) . This last inequality is no longer true for any ( θ, φ ) ∈ (cid:0) π , π (cid:3) × [0 , π ).Actually, it stills remains true for θ ∈ (cid:0) π , θ ∗ (cid:3) , for some θ ∗ ∈ (cid:0) π , π (cid:1) ( θ ∗ ≈ . θ ∈ ( θ ∗ , π ].We are now ready to prove Theorem 5.7. Proof of Theorem 5.7.
We first apply Theorem 5.4 with a = min (cid:8) κ , r inj, Ω (cid:9) (if κ = 0,then we take a = r inj, Ω ). We take X = Ω endowed with the induced Riemannian distance,and with the measure ς defined as the restriction to Ω of the Lebesgue measure of M ,namely ς ( E ) = | E ∩ Ω | for all measurable set E . Step 1 (large j ). From Lemma 5.5 we deduce that there exists j Ω ∈ N such that for all j ≥ j Ω there exists a sequence { A i } ji =1 of 4 j annuli such that 2 A i are pairwise disjointand(5.6) | Ω ∩ A i | ≥ c | Ω | j . The constant c depends only on Γ of Theorem 5.4, hence it depends only on the dimensionand can be determined explicitly (see [25]). Since we have 4 j annuli, we can pick at least2 j of them such that(5.7) | Ω ∩ A i | ≤ | Ω | j . Among these last 2 j annuli, we can pick at least j of them such that(5.8) | ∂ Ω ∩ A i | ≤ | ∂ Ω | j . We take this family of j annuli, and denote it by { A i } ji =1 .Subordinated to this decomposition we construct a family of j disjointly supportedfunctions u , ..., u j . If u , ..., u j belong to H (Ω), then from (3.33)(5.9) µ j ≤ max i =1 ,...,j R Ω | D u i | + Ric( ∇ u i , ∇ u i ) dv R Ω u i dv , Thus, in order to estimate µ j it is sufficient to estimate the Rayleigh quotient of each ofthe test functions.Let f : [0 , ∞ ) → [0 ,
1] be defined as follows:(5.10) f ( t ) = t − (4 t − , t ∈ (cid:2) , (cid:3) , , t ∈ (cid:2) , (cid:3) , , t ∈ [1 , + ∞ [ . By construction f ∈ C , [0 , + ∞ ). Moreover f ∈ C ( (cid:2) , (cid:3) ). We consider test functionsof the form f ( ηδ p ( x )) for some η ∈ R and p ∈ M . We note that(5.11) ∇ f ( ηδ p ( x )) = ηf ′ ( ηδ p ( x )) ∇ δ p ( x )and(5.12) ∆ f ( ηδ p ( x )) = η f ′′ ( ηδ p ( x )) + ηf ′ ( ηδ p ( x ))∆ δ p ( x ) . In (5.12) we have used the fact that |∇ δ p ( x ) | = 1 for almost all x ∈ M , the equalityholding pointwise in M \ ( { p } ∪ cut( p ))). Standard computations show that(5.13) | f ′ ( t ) | ≤ | f ′′ ( t ) | ≤ . Let now A i be an annulus of the family { A i } ji =1 . We have two possibilities. Either A i isa proper annulus with 0 < r i < R i ≤ r inj, Ω , or is a ball of radius 0 < r i ≤ r inj, Ω . Case a (ball).
Assume that A i is a ball of radius 0 < r i ≤ r inj, Ω and center p i . Associ-ated to A i we define a function u i as follows(5.15) u i ( x ) = , ≤ δ p i ( x ) ≤ r i f ( δ pi ( x )2 r i ) , r i ≤ δ p i ( x ) ≤ r i , otherwise . By construction, u i | Ω ∈ H (Ω). Standard computations (see (5.11)-(5.14)) andLemma 5.9 show that(5.16) |∇ u i | ≤ r i and(5.17) | ∆ u i | ≤ r i + 3 r i | ∆ δ p i ( x ) | ≤ n + 1) r i + 3( n − κr i . When estimating the Rayleigh quotient of u i we will also need to estimate |∇|∇ u i | | . We have that |∇ u i ( x ) | = f ′ ( δ p i ( x ) / r i ) r i , hence(5.18) |∇|∇ u i ( x ) | | = | f ′ ( δ p i ( x ) / r i ) f ′′ ( δ p i ( x ) / r i ) | r i ≤ r i . Case b (annulus).
Assume that A i is a proper annulus of radii 0 < r i < R i ≤ r inj, Ω and center p i . Associated to A i we define a function u i as follows(5.19) u i ( x ) = − f ( δ pi ( x ) r i ) , r i ≤ δ p i ( x ) ≤ r i , r i ≤ δ p i ( x ) ≤ R i f ( δ pi ( x )2 R i ) , R i ≤ δ p i ( x ) ≤ R i , otherwise . EUMANN EIGENVALUES OF THE BIHARMONIC OPERATOR ON DOMAINS 29
By construction, u i | Ω ∈ H (Ω). Standard computations (see (5.11)-(5.14)) showthat(5.20) |∇ u i ( x ) | ≤ R i , R i ≤ δ p i ( x ) ≤ R i , r i , r i ≤ δ p i ( x ) ≤ r i , , otherwise . In any case then(5.21) |∇ u i ( x ) | ≤ r i . Moreover, from Lemma 5.9 we have(5.22) | ∆ u i ( x ) | ≤ n +1) R i + n − κR i , R i ≤ δ p i ( x ) ≤ R i , n +1) r i + n − κr i , r i ≤ δ p i ( x ) ≤ r i , , otherwise . In any case then(5.23) | ∆ u i ( x ) | ≤ n + 1) r i + 6( n − κr i . We will also need an estimate on |∇|∇ u i | | . As for (5.18) we find that(5.24) |∇|∇ u i ( x ) | | ≤ ≤ R i , R i ≤ δ p i ( x ) ≤ R i , ≤ r i , r i ≤ δ p i ( x ) ≤ r i , , otherwise . In any case then(5.25) |∇|∇ u i ( x ) | | ≤ r i . We also need an upper bound for the volume of 2 A i . Since the outer radius of 2 A i is by construction smaller than κ , we have, from the volume comparison and standardcalculus that(5.26) | A i | ≤ n sinh(1) n − ω n R ni . From Bochner’s formula (2.4) we deduce that(5.27) Z Ω ∩ A i | D u i | + Ric( ∇ u i , ∇ u i ) dv = Z Ω ∩ A i
12 ∆( |∇ u i | ) − h∇ ∆ u i , ∇ u i i dv = Z ∂ Ω ∩ A i ∂ |∇ u i | ∂ν − ∆ u i ∂u i ∂ν dσ + Z Ω ∩ A i (∆ u i ) dv ≤ Z ∂ Ω ∩ A i |∇|∇ u i | | + | ∆ u i ||∇ u i | dσ + Z Ω ∩ A i (∆ u i ) dv We note that the boundary integrals are taken on ∂ Ω ∩ A i since by construction u i ∈ H (2 A i ). From (5.8), (5.16), (5.17), (5.18), (5.21), (5.23) and (5.25) we deduce that(5.28) Z Ω ∩ A i | D u i | + Ric( ∇ u, ∇ u ) dv ≤ Z Ω ∩ A i (∆ u i ) dv + | ∂ Ω | j (cid:18) n + 5) r i + 36( n − κr i (cid:19) ≤ Z Ω ∩ A i (∆ u i ) dv + | ∂ Ω | j (cid:18) n + 4 r i (cid:19) . where the last inequality follows from the fact that r i ≤ R i ≤ κ .Corollary 5.6 gives us information on the size of the radius r i , in fact(5.29) r i ≥
12 ˜ r := 12 inf B where(5.30) B := (cid:26) r ∈ R : V ( r ) ≥ c | Ω | j (cid:27) . We observe that each r ∈ B is such that c | Ω | j ≤ V ( r ) = sup x ∈ Ω | B ( x, r ) ∩ Ω | ≤ | B ( x, r ) | ≤ | B ( p ′ , r ) | κ by volume comparison, where | B ( p ′ , r ) | κ denotes the volume of the ball of radius r in thespace form of constant curvature − κ . If κ = 0 then each r ∈ B is such that c | Ω | j ≤ ω n r n . Hence any r ∈ B is such that r ≥ (cid:18) c | Ω | ω n j (cid:19) n , therefore(5.31) r i ≥ (cid:18) c | Ω | ω n j (cid:19) n . If κ >
0, then ˜ r ≤ r i ≤ κ by construction, and since ˜ r = inf B , from volume comparisonand standard calculus (see also (5.26)) c | Ω | j ≤ sinh(2) n − ω n ˜ r n . Therefore(5.32) r i ≥ ˜ r ≥ (cid:18) c | Ω | sinh(2) n − ω n j (cid:19) n . We note that (5.31) implies (5.32) which holds true for any κ ≥
0. We conclude that(5.33) Z Ω ∩ A i | D u i | + Ric( ∇ u, ∇ u ) dv ≤ Z Ω ∩ A i (∆ u i ) dv + α n | ∂ Ω | j (cid:18) j | Ω | (cid:19) n EUMANN EIGENVALUES OF THE BIHARMONIC OPERATOR ON DOMAINS 31 where(5.34) α n = 144(5 n + 4) (cid:18) ω n sinh(2) n − c (cid:19) n In order to complete the estimates, it remains to bound the term R Ω ∩ A i (∆ u i ) dv . Weneed to distinguish the case n = 2 , , n > Case a’ (lower dimensions).
Let n ≤
4. We note that in this case it irrelevant toknow that | A i ∩ Ω | ≤ | Ω | j . This fact is crucial only for higher dimensions.If A i is a ball of radius r i ≤ κ , and hence | A i | ≤ n sinh(1) n − ω n r ni , wehave(5.35) Z Ω ∩ A i (∆ u i ) dv ≤ Z A i (∆ u i ) dv ≤ (cid:18) n + 1) r i + 3( n − κr i (cid:19) | A i |≤ (cid:18) n + 1) r i + 3( n − κr i (cid:19) n ω n sinh(1) n − r ni ≤ (cid:18) ( n + 1) r − ni + ( n − κ r − ni (cid:19) n sinh(1) n − ω n ≤ (cid:18) ( n + 1) r − ni + ( n − r − ni (cid:19) n sinh(1) n − ω n = 18(4( n + 1) + ( n − )2 n − ω n sinh(1) n − r − ni , where in the last line we have used the fact 2 r i ≤ κ if κ >
0. From (5.32) weobtain that(5.36) Z Ω ∩ A i (∆ u i ) dv ≤ β ′ n (cid:18) j | Ω | (cid:19) n − , where we set(5.37) β ′ n = 72 ω n sinh(1) n − (4( n + 1) + ( n − ) (cid:18) sinh(2) ω n c (cid:19) n − . In the very same way it is possible to prove that (5.36) holds if A i is aproper annulus, possibly with a different β ′ n , but still dependent only on n . It issufficient to split the integral R A i (∆ u i ) dv as the sum of the integrals of (∆ u i ) on the annulus r i ≤ δ p i ( x ) ≤ r i and on the annulus R i ≤ δ p i ( x ) ≤ R i and use(5.22) and (5.26) in each case. Case b’ (higher dimensions).
Let n >
4. Let A i be a ball of radius r i . Then, byH¨older’s inequality,(5.38) Z Ω ∩ A i (∆ u i ) dv ≤ | Ω ∩ A i | − n (cid:18)Z Ω ∩ A i (∆ u i ) n dv (cid:19) n ≤ | Ω ∩ A i | − n (cid:18)Z A i (∆ u i ) n dv (cid:19) n ≤ (cid:18) | Ω | j (cid:19) − n (cid:18) n + 1) r i + 3( n − κr i (cid:19) | A i | n ≤ (cid:18) | Ω | j (cid:19) − n n + 1) + ( n − ) r i ( ω n sinh(1) n − n r ni ) n = β ′′ n (cid:18) | Ω | j (cid:19) − n , where(5.39) β ′′ n = 72(4( n + 1) + ( n − )( ω n sinh(1) n − ) n . In the same way it is possible to prove that (5.38) holds if A i is a proper annulus,possibly with a different β ′′ n , but still dependent only on n . It is sufficient tosplit the integral R A i (∆ u i ) dv as the sum of the integrals of (∆ u i ) on theannulus r i ≤ δ p i ( x ) ≤ r i and on the annulus R i ≤ δ p i ( x ) ≤ R i and use (5.22)and (5.26) on each annulus.We have then proved that, for all dimensions n ≥ Z Ω ∩ A i (∆ u i ) dv ≤ β n (cid:18) j | Ω | (cid:19) n − , where β n is a constant which depends only on n and is explicitly computable. Thisconcludes the estimate of the numerator of the Rayleigh quotient for u i . As for thedenominator we have Z Ω ∩ A i u i dv ≥ Z Ω ∩ A i u i dv = | Ω ∩ A i | ≥ c | Ω | j . Then, we have for all i = 1 , ..., j (5.41) R Ω | D u i | + Ric( ∇ u i , ∇ u i ) dv R Ω u i dv ≤ β n c (cid:18) j | Ω | (cid:19) n + 4 α n c | ∂ Ω || Ω | (cid:18) j | Ω | (cid:19) n ≤ (cid:18) β n + 3 α n c (cid:19) (cid:18) j | Ω | (cid:19) n + α n c | ∂ Ω | | Ω | , where we have used Young’s inequality in the last passage. We have proved then that(5.42) µ j ≤ A n (cid:18) j | Ω | (cid:19) n + B n | ∂ Ω | | Ω | , for all j ≥ j Ω , where(5.43) A n := (cid:18) β n + 3 α n c (cid:19) EUMANN EIGENVALUES OF THE BIHARMONIC OPERATOR ON DOMAINS 33 and(5.44) B n := 2 α n c . Step 2 (small j ). Let now j < j Ω be fixed. By using Theorem 5.4 as in Step 1, we findthat there exists a sequence of 4 j sets { A i } ji =1 such that | Ω ∩ A i | ≥ c | Ω | j . If the sets A i are annuli, we can proceed as in Step 1 and deduce the validity of (5.42). Assume nowthat j is such that the sets A i of the decomposition are of the form A i = B ( x i , r ) ∪ · · · ∪ B ( x il i , r ) , where r = a , D i = A r i are pairwise disjoint, and δ x il ( x ik ) ≥ r if l = k . Bydefinition D i = B ( x i , r ) ∪ · · · ∪ B ( x il i , r ) and 5 r ≤ r inj, Ω . Since we have 4 j disjointsets D i , we can pick j of them such that | Ω ∩ D i | ≤ | Ω | j and | ∂ Ω ∩ D i | ≤ | ∂ Ω | j . Wetake from now on this family of j capacitors. Note that D i is a disjoint union of l i balls B ( x i , r ) , · · · , B ( x il i , r ) of radius 5 r . Associated to each B ( x ik , r ), k = 1 , ..., l weconstruct test functions u ik as in (5.15). Then we define the function u i associated withthe capacitor ( A i , D i ) by setting u i = u ik on B ( x ik , r ). We have j disjointly supportedtest functions in H (Ω). We estimate the Rayleigh quotient of each of the u i as in Step1. As in (5.16), (5.17) and (5.18) we estimate |∇ u i | , | ∆ u i | and |∇|∇ u i | | . In particular,we find a universal constant c such that(5.45) |∇ u i | ≤ c r , | ∆ u i | ≤ c r + c κr , |∇|∇ u i | | ≤ c r . By using (5.27), as we have done for (5.33), (5.36) and (5.38), we find that R Ω | D u i | + Ric( ∇ u i , ∇ u i ) dv R Ω u i dv ≤ A n a + B n a | ∂ Ω || Ω | , for all i = 1 , ..., j , which implies, by using Young’s inequality as in (5.41), that(5.46) µ j ≤ A n a + B n a | ∂ Ω | | Ω | . The proof of (5.4) follows by combining (5.42) and (5.46), possibly re-defining theconstants A n , B n . (cid:3) Remark . We point out that in the proof of Theorem 5.7 the inequality (5.46) appears.Apparently this may look like a nonsense, in fact the right-hand side of the inequality doesnot depend on j . However, note that this situation may occur only for a finite numberof eigenvalues µ j , since, starting from a certain j Ω (of which it is possible in principle togive a lower bound), the capacitors of the decomposition given by (5.4) are of the form iii ) − a ), hence the estimate (5.42) holds starting from a certain j Ω .In the next subsections we will present more refined estimates under additional as-sumptions.5.3. Manifolds with
Ric ≥ and n = 2 , , . In this subsection we establish upperbounds for bounded domains in Riemannian manifolds satisfying Ric ≥
0. In this casewe will use Theorem 5.1, noting that the constant Γ depends only on n . Moreover, testfunctions in this case are not given in terms of the distance from a point, thus avoidingthe obstruction of the lack of regularity in correspondence of the cut-locus.We state the main theorem of this subsection. Theorem 5.11.
Let ( M, g ) be a complete n -dimensional smooth Riemannian manifoldswith Ric ≥ and n = 2 , , . Let Ω be a bounded domain of M with C boundary. Then (5.47) µ j ≤ A n (cid:18) j | Ω | (cid:19) n , for all j ∈ N , the constant A n depending only on the dimension. The proof is similar to that of Theorem 5.7. However we will use different test functions.To do so, we adapt a construction originally contained in [15, Theorem 6.33]. We havethe following lemma (see [24, Theorem 2.2], see also [5]).
Lemma 5.12.
Let ( M, g ) be a complete n -dimensional smooth Riemannian manifoldwith Ric ≥ . Then, for any p ∈ M and r > there exists a function φ r : M → [0 , , φ ∈ C ∞ ( M ) , such that(1) φ r ≡ on B (cid:0) p, r (cid:1) ;(2) supp( φ r ) ⊂ B ( p, r ) ;(3) |∇ φ r | ≤ C ( n ) r ;(4) | ∆ φ r | ≤ C ( n ) r ,the constant C ( n ) depending only on the dimension.Proof. Let p ∈ M and r > M , namely g r := r g . In this new metric B ( p, r ) = B g r ( p, B (cid:16) p, r (cid:17) = B g r (cid:18) p, (cid:19) . Here we denote by B ( p, r ) and B (cid:0) p, r (cid:1) the balls of center p and radius r and r in theoriginal metric g . Moreover, Ric g r ≥
0, hence from Theorem 6.33 of [15] we deduce thatthere exists a constant c ( n ) and φ : M → [0 , φ ∈ C ∞ ( M ), such that(1) φ ≡ B g r (cid:0) p, (cid:1) ;(2) supp( φ ) ⊂ B g r ( p, |∇ φ | ≤ c ( n );(4) | ∆ φ | ≤ c ( n ).Since ∆ g r = r ∆ and | ω | g r = r | ω | for all 1-forms ω on M , we conclude that |∇ φ | ≤ c ( n ) r and | ∆ φ | ≤ c ( n ) r . This concludes the proof by taking φ r := φ . (cid:3) Proof of Theorem 5.11.
We use Theorem 5.1, which provides, for all indexes j ∈ N , afamily { A i } ji =1 of annuli such that | A i ∩ Ω | ≥ c | Ω | j and the annuli 2 A i are pairwise disjoint. Moreover, from Corollary 5.6 we deduce thateach annulus A i has either internal radius r i satisfying (5.1) or is a ball of radius r i EUMANN EIGENVALUES OF THE BIHARMONIC OPERATOR ON DOMAINS 35 satisfying (5.1). In this case B = n r ∈ R : V ( r ) ≥ c | Ω | j o and V ( r ) := sup x ∈ Ω | B ( x, r ) ∩ Ω | .Since Ric ≥
0, from volume comparison we know that every ball B ( p, r ) ⊂ M satisfies | B ( x, r ) | ≤ ω n r n . Hence any r ∈ B is such that(5.48) c | Ω | j ≤ V ( r ) ≤ ω n r n ⇐⇒ r ≥ (cid:18) c | Ω | jω n (cid:19) n . Subordinated to the family { A i } ji =1 we build a family of test functions { u i } ji =1 in H (Ω)in the following way. If A i is a ball of radius r i , we take u i = φ r i , where φ r i is defined inLemma 5.12. Hence u i is supported in 2 A i and u i ≡ A i . If A i is a proper annulusof radii 0 < r i < R i , we take u i = φ R i − φ r i . Again, u i is supported in 2 A i and u i ≡ A i . The functions (cid:8) u i | Ω (cid:9) ji =1 are disjointly supported and belong to H (Ω). Hence, from(3.33) we deduce that µ j ≤ max i =1 ,...,j R Ω | D u i | + Ric( ∇ u i , ∇ u i ) dv R Ω u i dv . We estimate now the Rayleigh quotient of each of the u i . Assume that A i is a ball ofradius r i . We have, for the numerator(5.49) Z Ω | D u i | + Ric( ∇ u i , ∇ u i ) dv = Z Ω ∩ A i | D u i | + Ric( ∇ u i , ∇ u i ) dv ≤ Z A i | D u i | + Ric( ∇ u i , ∇ u i ) dv = Z A i (∆ u i ) dv ≤ | A i | c ( n ) r i ≤ c ( n ) ω n r n − i ≤ c ( n ) ω n n c − n (cid:18) | Ω | j (cid:19) − n . In the first inequality we have estimated R Ω ∩ A i | D u i | + Ric( ∇ u i , ∇ u i ) dv with the inte-gral on the whole ball 2 A i , being the integrand a non-negative function. Moreover, fromBochner’s formula, since u i ∈ H (2 A i ), we have that R A i | D u i | + Ric( ∇ u i , ∇ u i ) dv = R A i (∆ u i ) dv ≤ | A i | c ( n ) r i . Finally we have used (5.48) since n ≤ Z Ω u i dv = Z Ω ∩ A i u i dv ≥ Z Ω ∩ A i u i dv = | Ω ∩ A i | ≥ c | Ω | j . From (5.49) and (5.50) we deduce(5.51) R Ω | D u i | + Ric( ∇ u i , ∇ u i ) dv R Ω u i dv ≤ c ( n ) ω n n c n (cid:18) j | Ω | (cid:19) n . In the very same way it is possible to prove that (5.49) holds if A i is a proper annulus,possibly with a different dimensional constant in front of the term (cid:16) j | Ω | (cid:17) n . It is sufficientto split the integral R A i (∆ u i ) dv as the sum of the integrals of (∆ u i ) on the annulus r i ≤ δ p i ( x ) ≤ r i and on the annulus R i ≤ δ p i ( x ) ≤ R i .The proof is now complete. (cid:3) Remark . We remark that, differently from the proof of Theorem 5.7, we did notchoose the annuli of the decomposition in such a way that | Ω ∩ A i | ≤ | Ω | j . In fact,in the case n = 2 , ,
4, an upper bound on the size of the supports of test functions seems to be irrelevant in the estimates. Moreover, being Ric ≥
0, the quadratic form | D u | + Ric( ∇ u, ∇ u ) is always non-negative, hence we can estimate its integral overΩ ∩ A i with the whole integral over 2 A i and use Bochner’s formula. Of course we cando this passage also for n >
4. However, in this case we would obtain(5.52) Z Ω ∩ A i | D u i | + Ric( ∇ u i , ∇ u i ) dv ≤ Z A i | D u i | + Ric( ∇ u i , ∇ u i ) dv = Z A i (∆ u i ) dv ≤ | A i | c ( n ) r i ≤ c ( n ) ω n r n − i , but n − >
0, and inequality (5.52) is useless to obtain uniform estimates.In the case n > { A i } ji =1 annuli such that | Ω ∩ A i | ≥ c | Ω | j and then choose j annuli among the 4 j of the family in such a way that | Ω ∩ A i | ≤ | Ω | j and | ∂ Ω ∩ A i | ≤ | ∂ Ω | j , as in the proofof Theorem 5.7. We build then test functions u i as in the proof of Theorem 5.11. FromBochner’s formula, as in (5.27), we have Z Ω ∩ A i | D u i | + Ric( ∇ u i , ∇ u i ) dv ≤ Z ∂ Ω ∩ A i |∇|∇ u i | | + | ∆ u i ||∇ u i | dσ + Z Ω ∩ A i (∆ u i ) dv. Assume that A i is a ball of radius r i . The term R ∂ Ω ∩ A i | ∆ u i ||∇ u i | dσ is estimated by Z ∂ Ω ∩ A i | ∆ u i ||∇ u i | dσ ≤ | ∂ Ω | j c ( n ) r i ≤ | ∂ Ω | j c ( n ) ω n n c n (cid:18) j | Ω | (cid:19) n For the term R Ω ∩ A i (∆ u i ) dv we have(5.53) Z Ω ∩ A i (∆ u i ) dv ≤ (cid:18)Z Ω ∩ A i (∆ u i ) n dv (cid:19) n | Ω ∩ A i | − n ≤ (cid:18)Z A i (∆ u i ) n dv (cid:19) n (cid:18) | Ω | j (cid:19) − n ≤ c ( n ) | A i | n r i (cid:18) | Ω | j (cid:19) − n ≤ c ( n ) ω n n (cid:18) | Ω | j (cid:19) − n . Hence, as for (5.41), we find that(5.54) R Ω | D u i | + Ric( ∇ u i , ∇ u i ) dv R Ω u i dv ≤ A n (cid:18) j | Ω | (cid:19) n + B n | ∂ Ω | | Ω | + C n k∇|∇ u i | k ∞ , for some constants A n , B n , C n which depend only on the dimension. In the case that A i is a proper annulus, inequality (5.54) still holds, with possibly different dimensionalconstants. Unfortunately an estimate of the form |∇|∇ φ r | | ≤ c ( n ) r is not availablefor a function φ r as in Lemma 5.12. If such an inequality would hold, then we wouldimmediately have(5.55) R Ω | D u i | + Ric( ∇ u i , ∇ u i ) dv R Ω u i dv ≤ A n (cid:18) j | Ω | (cid:19) n + B n | ∂ Ω | | Ω | , EUMANN EIGENVALUES OF THE BIHARMONIC OPERATOR ON DOMAINS 37 and therefore(5.56) µ j ≤ A n (cid:18) j | Ω | (cid:19) n + B n | ∂ Ω | | Ω | . Open problem.
Prove (5.56) for domains in complete smooth manifolds with Ric ≥ n >
4. Prove inequality (5.47) for domains in complete smooth manifolds withRic ≥ n ≥ Manifolds with
Ric ≥ and small diameter. If Ric ≥ r inj, Ω , it is possible to build test functions for the Rayleighquotient in terms of the distance function. In particular, we have the following. Theorem 5.14.
Let ( M, g ) be a complete n -dimensional smooth Riemannian manifoldwith Ric ≥ and let Ω be a bounded domain in M with C boundary and diameter D . If D < r inj, Ω , then (5.57) µ j ≤ A n (cid:18) j | Ω | (cid:19) n + B n | ∂ Ω | | Ω | , for all j ∈ N , where A n , B n depend only on the dimension. If n ≤ we can choose B n = 0 .Proof. In order to prove Theorem 5.14 we exploit Theorem 5.1 and Corollary 5.6. Wefind, as in the proof of Theorem 5.7, for all j ∈ N a family of j annuli { A i } ji =1 suchthat | Ω ∩ A i | ≥ c | Ω | j , | ∂ Ω ∩ A i | ≤ | ∂ Ω | j , | Ω ∩ A i | ≤ | Ω | j and the annuli 2 A i are pairwisedisjoint. Moreover, since we have taken D < r inj, Ω , the annuli 2 A i can be chosen suchthat the outer radius is strictly smaller that r inj, Ω . Associated with A i we then build testfunctions u i of the form (5.15) if A i is a ball or (5.19) if A i is a proper annulus. It followsthen, as for the proof of (5.41) R Ω | D u i | + Ric( ∇ u i , ∇ u i ) dv R Ω u i dv ≤ A n (cid:18) j | Ω | (cid:19) n + B n | ∂ Ω | | Ω | , where the constants A n , B n depend only on the dimension. This implies (5.57) for all j ∈ N . The last statement follows immediately from Theorem 5.11. This concludes theproof. (cid:3) Domains on the sphere.
In this section we obtain bounds for domains of thestandard sphere S nR , namely we have the following theorem. Theorem 5.15.
Let ( M, g ) = S nR be the sphere of radius R with standard round metricand let Ω be a domain in S n with C boundary. Then µ j ≤ A n (cid:18) j | Ω | (cid:19) n , for all j ∈ N .Proof. Through all the proof we assume n >
4. The validity of the theorem for n ≤ p , p ∈ S nR , the maximal distance amongthem is attained when they are antipodal points. In this case δ p ( p ) = πR . Moreover, µ = 0 and the corresponding eigenfunctions are the constant functions on Ω, and µ > We apply Theorem 5.1 with X = Ω with the Riemannian distance, ς ( A ) = | A ∩ Ω | .Since we have positive Ricci curvature, point i ) of Theorem 5.1 is satisfied for some Γwhich depends only on n . Points ii) and iii) are easily seen to hold. Hence we deducethat there exists c which depends only on the dimension such that for any j ≥
2, thereexists A , ..., A j annuli withi) | A i ∩ Ω | ≥ c | Ω | j ;ii) 2 A i are pairwise disjoint;iii) | A i ∩ Ω | ≤ | Ω | j ;iv) each annulus 2 A i has outer radius less than πR .Points iii) and iv) follow from the fact that we shall actually apply Theorem 5.1 with2 j + 1 and by observing that we can choose first 2 j among the 2 j + 1 annuli given by theconstruction so that iv) holds. Indeed, if an annulus A i with center p i has outer radiusstrictly greater than πR , the set S nR \ A i is B ( p , r ) ∪ B ( p ′ i , r ′ ), where p ′ i is the antipodalpoint of p i , r, r ′ < πR ( B ( p ′ , r ′ ) = ∅ if A i is a ball), and all other annuli 2 A k of thedecomposition need belong to B ( p , r ) ∪ B ( p ′ i , r ′ ). Hence, no more than one annulus cansatisfy iv). Moreover, among the remaining 2 j annuli, we can chose j annuli such thatiii) holds (see also the proof of Theorem 5.7). We shall denote by r i and R i the inner andouter radius of A i , if A i is an actual annulus, while we shall denote by r i the radius, if A i is a ball. By p i we denote the center of the annuli A i .Associated to each of the j annuli A , ..., A j satisfying i)-iv) we construct test functions u i as in (5.15) (if A i is a ball) or in (5.19) if A i is a proper annulus. The functions u i areof the form u i ( x ) = f ( δ p ( x )).We apply now Bochner’s formula to a function of the form f ( δ p ( x )), and we use thefact that |∇ δ p ( x ) | = 1 almost everywhere. We have(5.58) 12 ∆ (cid:0) |∇ f ( δ p ( x )) | (cid:1) = 12 ∆ (cid:0) | f ′ ( δ p ( x )) ∇ δ p ( x ) | (cid:1) = 12 ∆ (cid:0) f ′ ( δ p ( x )) (cid:1) = ∇ ( f ′ ( δ p ( x )) f ′′ ( δ p ( x ))) · ∇ δ p ( x ) + f ′ ( δ p ( x )) f ′′ ( δ p ( x ))∆ δ p ( x )= f ′′ ( δ p ( x )) + f ′ ( δ p ( x )) f ′′′ ( δ p ( x )) + f ′ ( δ p ( x )) f ′′ ( δ p ( x ))∆ δ p ( x ) . On the other hand(5.59) ∇ ∆ f ( δ p ( x )) · ∇ f ( δ p ( x ))= f ′ ( δ p ( x )) ∇ δ p ( x ) · ∇ ( f ′′ ( δ p ( x )) + f ′ ( δ p ( x ))∆ δ p ( x ))= f ′ ( δ p ( x )) f ′′′ ( δ p ( x )) + f ′ ( δ p ( x )) f ′′ ( δ p ( x ))∆ δ p ( x )+ ( f ′ ( δ p ( x ))) ∇ δ p ( x ) · ∇ ∆ δ p ( x ) . We deduce then(5.60) | D f ( δ p ( x )) | + Ric( ∇ f ( δ p ( x )) , ∇ f ( δ p ( x )))= f ′′ ( δ p ( x )) − ( f ′ ( δ p ( x ))) ∇ δ p ( x ) · ∇ ∆ δ p ( x ) . EUMANN EIGENVALUES OF THE BIHARMONIC OPERATOR ON DOMAINS 39
Moreover,(5.61) ∇ δ p ( x ) · ∇ ∆ δ p ( x ) = ∇ (cid:18) ( n − R cot (cid:18) δ p ( x ) R (cid:19)(cid:19) · ∇ δ p ( x )= − n − R sin ( δ p ( x ) /R ) |∇ δ p ( x ) | = − n − R sin ( δ p ( x ) /R )for all x = p, p ′ , where p ′ is the antipodal point to p , andsin ( δ p ( x ) /R ) ≤ δ p ( x ) π R , for all x such that 0 ≤ δ p ( x ) ≤ πR .Now, since | D f ( δ p ( x )) | + Ric( ∇ f ( δ p ( x )) , ∇ f ( δ p ( x ))) ≥ n > Z Ω ∩ A i | D f ( δ p ( x )) | + Ric( ∇ f ( δ p ( x )) dv ≤ | Ω ∩ A i | − n (cid:18)Z Ω ∩ A i (cid:0) | D f ( δ p ( x )) | + Ric( ∇ f ( δ p ( x )) , ∇ f ( δ p ( x ))) (cid:1) n dv (cid:19) n ≤ | Ω ∩ A i | − n (cid:18)Z A i (cid:0) | D f ( δ p ( x )) | + Ric( ∇ f ( δ p ( x )) , ∇ f ( δ p ( x ))) (cid:1) n dv (cid:19) n = | Ω ∩ A i | − n (cid:18)Z A i (cid:0) f ′′ ( δ p ( x )) − ( f ′ ( δ p ( x ))) ∇ δ p ( x ) · ∇ ∆ δ p ( x ) (cid:1) n dv (cid:19) n = | Ω ∩ A i | − n Z A i (cid:18) f ′′ ( δ p ( x )) + ( f ′ ( δ p ( x ))) ( n − R sin ( δ p ( x ) /R ) (cid:19) n dv ! n ≤ | Ω ∩ A i | − n Z A i (cid:18) f ′′ ( δ p ( x )) + ( n − π ( f ′ ( δ p ( x ))) δ p ( x ) (cid:19) n dv ! n ≤ A ′ n | Ω ∩ A i | − n ≤ A ′ n (cid:18) | Ω | j (cid:19) − n , where the constant A ′ n depends only on the dimension. For the denominator, we have Z Ω ∩ A i u i dv ≥ Z Ω ∩ A i u i dv = | Ω ∩ A i | ≥ c | Ω | j . The proof follows now the same lines as that of Theorem 5.7. (cid:3)
Remark . We remark that explicit constructions like the one just presented for do-mains on the sphere are difficult already in other cases of manifolds for which we knowthe exact structure of the cut-locus of a point. This is the case of domains on an infinitelylong cylinder. In this case, obtaining good estimates with the technique used in the proofof Theorem 5.15 seems quite involved.5.6.
Domains of the hyperbolic space.
In this subsection we provide estimates fordomains of the standard hyperbolic space H nκ . We have the following theorem. Theorem 5.17.
Let ( M, g ) = H nκ be the standard n -dimensional hyperbolic space ofcurvature − κ , κ > and let Ω be a bounded domain in H nκ with C boundary. Then (5.63) µ j ≤ A n (cid:18) j | Ω | (cid:19) n + B n κ , for all j ∈ N .Proof. We apply Theorem 5.4 with a = κ . Step 1.
There exists j ∈ N such that for all j ≥ j Ω we find a sequence of j annuli { A i } ji =1 such that | Ω ∩ A i | ≥ c | Ω | j , | Ω ∩ i | ≤ | Ω | j and 2 A i pairwise disjoint.Associated to A i we construct test functions u i given by (5.15) if A i is a ball, or by(5.19) if A i is an a proper annulus. The distance function δ p is smooth on all H nκ \ { p } , forall p ∈ H nκ , hence by construction u i | Ω ∈ H (Ω). We estimate now the Rayleigh quotientof the u i . From (5.60) and from the fact that ∇ δ p · ∇ ∆ δ p = − ( n − κ sinh ( κδ p ) ≤ − ( n − δ p we deduce, as in (5.62), that(5.64) Z Ω ∩ A i | D f ( δ p ( x )) | + Ric( ∇ f ( δ p ( x )) dv ≤ | Ω ∩ A i | − n Z A i (cid:18) f ′′ ( δ p ( x )) + ( f ′ ( δ p ( x ))) ( n − δ p ( x ) (cid:19) n dv ! n ≤ A ′ n | Ω ∩ A i | − n ≤ A ′ n (cid:18) | Ω | j (cid:19) − n , if n > n = 2 , , A i is a ball of radius r i (5.65) Z Ω ∩ A i | D f ( δ p ( x )) | + Ric( ∇ f ( δ p ( x )) dv = Z Ω ∩ A i f ′′ ( δ p ( x )) + ( f ′ ( δ p ( x ))) ( n − δ p ( x ) dv ≤ A ′ n | Ω ∩ A i | r i ≤ A ′ n | A i | r i ≤ A ′′ n r n − i ≤ A ′′′ n (cid:18) | Ω | j (cid:19) − n , where we have used (5.31). Analogous computations show that inequality (5.65) holdsalso in the case that A i is a proper annulus, possibly with a different value of the constant A ′′′ n .For the denominator of the Rayleigh quotient we have Z Ω ∩ A i u i dv ≥ Z Ω ∩ A i u i dv = | Ω ∩ A i | ≥ c | Ω | j . EUMANN EIGENVALUES OF THE BIHARMONIC OPERATOR ON DOMAINS 41
Therefore(5.66) µ j ≤ A n (cid:18) j | Ω | (cid:19) n , for all j ≥ j Ω . Step 2. If j < j Ω , we proceed as in Step 2 of the proof of Theorem 5.7. By applyingTheorem 5.4 we find that there exists a family { A i } ji =1 of sets with the property | Ω ∩ A i | ≥ c | Ω | j . If the A i are annuli, we proceed as in Step 1 and deduce the validity of (5.66).Assume now that Theorem 5.4 provides 2 j sets such that A i = B ( x i , r ) ∪ · · · ∪ B ( x il i , r ) , with r = a , D i = A r i are pairwise disjoint, and δ x il ( x ik ) ≥ r if l = k . Since wehave 2 j disjoint sets, we can pick j of them such that | Ω ∩ D i | ≤ | Ω | j . Note that each D i is a disjoint union of balls B ( x i , r ) , ..., B ( x il i , r ). Associated with each B ( x ik , r ) wedefine a test function u ik as in (5.15). Then, for any i = 1 , ..., j we define the function u i by setting u i = u ik on B ( x ik , r ). Now, analogous computations as those in Step 1 allowto conclude that(5.67) µ j ≤ B n a . Since a = κ , from (5.66) and (5.67) we deduce the validity of (5.63). This concludes theproof. (cid:3) Domains of Cartan-Hadamard manifolds.
A Cartan-Hadamard manifold is acomplete, simply-connected Riemannian manifold (
M, g ) with non-positive sectional cur-vature. As a corollary of Theorem 5.7 we have the following.
Theorem 5.18.
Let ( M, g ) be a n -dimensional Cartan-Hadamard manifold with Ric ≥− ( n − κ , κ > and let Ω be a bounded domain in M of class C . Then µ j ≤ A n (cid:18) j | Ω | (cid:19) n + B n | ∂ Ω | | Ω | + C n κ , for all j ∈ N Proof.
We note that we can choose a = κ in Theorem 5.7, since for any p ∈ M , δ p issmooth on the whole M \ { p } . Moreover, ∆ δ p ≥ p ∈ M , see e.g., [6]. (cid:3) Remark . A way of getting rid of the term | ∂ Ω | | Ω | is to have explicit expressions forthe right-hand side of (5.60). This is the case of domains of standard spheres or for thehyperbolic space. However, we note that ∇ δ p · ∇ ∆ δ p is exactly ∂ H ( r ) ∂r , where H ( r ) is themean curvature of the sphere centered at p in M and r is the radial direction. FromBochner’s formula we can just recover ∇ δ p · ∇ ∆ δ p = −| D δ p | − Ric( ∇ δ p , ∇ δ p ) ≤ − (∆ δ p ) n + ( n − κ . However, a lower bound for such quantity is needed. Otherwise, we necessarily have topass through an integration by parts as in Theorem 5.7, and this involves boundary terms.
Manifolds without boundary.
In this subsection Ω = M , with ( M, g ) a compactcomplete n -dimensional smooth Riemannian manifold (without boundary) and Ric ≥− ( n − κ , κ ≥ Z M | D u | + Ric( ∇ u, ∇ u ) dv = Z M (∆ u ) dv for all u ∈ H ( M ). In particular we have that0 = µ < µ ≤ · · · ≤ µ j ≤ · · · ր + ∞ . In fact, one easily checks that all the eigenvalues are non-negative, and that there is onlyone zero eigenvalue with associated eigenfunctions the constant functions on M .We prove now that the eigenvalues µ j of (3.1) are exactly the squares of the eigenvaluesof the Laplacian on M . Recall that the weak formulation of the closed eigenvalue problemfor the Laplacian is(5.68) Z M h∇ u, ∇ φ i dv = m Z M uφdv , ∀ φ ∈ H ( M ) , in the unknowns ( u, m ) ∈ H ( M ) × R . Problem (5.68) admits an increasing sequence ofnon-negative eigenvalues of finite multiplicity0 = m < m ≤ · · · ≤ m j ≤ · · · ր + ∞ and the corresponding eigenfunctions can be chosen to form a orthonormal basis of L ( M ).We have the following theorem. Theorem 5.20.
Let ( M, g ) be a compact complete n -dimensional smooth Riemannianmanifold. Let { m j } ∞ j =1 denote the eigenvalues of the Laplacian on M . Then for all j ∈ N µ j = m j . and the corresponding eigenfunctions can be chosen to be the same.Proof. Let v i denote the eigenfunctions associated with m i normalized by R Ω v i v k dv = δ ik . Since the metric is smooth, we have that v i ∈ H ( M ) and − ∆ v i = m i v i in L ( M ). Hence, by setting V := < v , ..., v j > , we have that V is a j -dimensionalsubspace of H ( M ) and a function v ∈ V is of the form v = P ji =1 α i v i for some α , ....α j ∈ R . Hence, from (3.33) we have µ j ≤ max v ∈ V R M (∆ v ) dv R M v dv = max ( α ,...,α j ) ∈ R j P ji =1 α i m i P ji =1 α i = m j . On the other hand, the well-known min-max principle for the eigenvalues of the Laplacianon M states that m j = min U ⊂ H ( M )dim U = j max u ∈ Uu =0 R M |∇ u | dv R M u dv . We choose U := < u , ..., u j > where u , ..., u j are the eigenfunctions associated withthe eigenvalues µ , ..., µ j of the biharmonic operator on M normalized by R M u i u k dv = EUMANN EIGENVALUES OF THE BIHARMONIC OPERATOR ON DOMAINS 43 δ ik . Then R M ∆ u i ∆ u k dv = µ i δ ik . Any u ∈ U is of the form u = P ji =1 α i u i for some α , ..., α j ∈ R . We note that(5.69) Z M |∇ u | dv = − Z M u ∆ udv ≤ (cid:18)Z M u dv (cid:19) (cid:18)Z M (∆ u ) dv (cid:19) = j X i =1 α i ! j X i =1 α i µ i ! / , hence m j ≤ max ( α ,...,α j ) ∈ R j P ji =1 α i µ i P ji =1 α i ! = µ j . The rest of the proof is straightforward. (cid:3)
From Theorem 5.20 and from (1.6) we deduce the following corollary.
Corollary 5.21.
Let ( M, g ) be a compact complete n -dimensional smooth Riemannianmanifold (without boundary) with Ric ≥ − ( n − κ , κ ≥ . Then for all j ∈ N µ j ≤ ( n − κ + C n (cid:18) j | Ω | (cid:19) n ! . Domains with convex boundary.
In this last subsection we shall present a casein which upper bounds for biharmonic Neumann eigenvalues µ j can be deduced directlyby comparison with Neumann eigenvalues of the Laplacian and by (1.7).Let ( M, g ) be a complete n -dimensional smooth Riemannian manifold with Ric ≥− ( n − κ , κ ≥
0, and let Ω be a bounded domain in M with C boundary. If II ≥ M replaced by Ω. Neumann eigen-values of the Laplacian have finite multiplicity, are non-negative and form an increasingsequence 0 = m < m ≤ · · · ≤ m j ≤ · · · ր + ∞ . The associated eigenfunctions are denoted by { v i } ∞ i =1 can be chosen to form a orthonormalbasis of L (Ω). We have the following Theorem 5.22.
Let ( M, g ) be a complete n -dimensional smooth Riemannian manifoldand let Ω will be a bounded domain of M of class C with II ≥ . Then µ j ≤ m j , for all j ∈ N Proof.
We have seen that for all u, φ ∈ H (Ω) (see (3.6) and (3.13))(5.70) Z Ω h D u, D φ i + Ric( ∇ u, ∇ φ ) dv = Z Ω ∆ u ∆ φdv − Z ∂ Ω (cid:18) ( n − H ∂u∂ν + ∆ ∂ Ω u (cid:19) ∂φ∂ν dσ − Z ∂ Ω (cid:18) II ( ∇ ∂ Ω u, ∇ ∂ Ω φ ) + ∂u∂ν ∆ ∂ Ω φ (cid:19) dσ. Since the domain is of class C , by standard elliptic regularity we have that the eigenfunc-tions v i of the Neumann Laplacian belong to H (Ω). Therefore − ∆ v i = m i v i in L (Ω)and ∂v i ∂ν = 0 in L ( ∂ Ω). We deduce that for any linear combination v = P ji =1 α i v i with α i ∈ R (5.71) Z Ω | D v | + Ric( ∇ v, ∇ v ) dv = Z Ω (∆ v ) dv − Z ∂ Ω II ( ∇ ∂ Ω v, ∇ ∂ Ω v ) dσ ≤ Z Ω (∆ v ) dv = Z Ω j X i =1 α i m i v i ! dv = j X i =1 α i m i . Consider then V := < v , ..., v j > the j -dimensional space spanned by the first j eigen-functions of the Neumann Laplacian. This is a subspace of H (Ω) of dimension j . Each v ∈ V is of the form v = P ji =1 α i v i for some α , ..., α j ∈ R . Moreover Z Ω v dv = j X i =1 α i . From (3.33) we have that µ j ≤ max v ∈ V R Ω | D v | + Ric( ∇ v, ∇ v ) dv R Ω v dv ≤ max ( α ,...,α j ) ∈ R j P ji =1 α i m i P ji =1 α i = m j . This concludes the proof. (cid:3)
Theorem 5.22 and inequality (1.7) imply the following corollary.
Corollary 5.23.
Let ( M, g ) be a complete n -dimensional smooth Riemannian manifoldwith Ric ≥ − ( n − κ , κ ≥ , and let Ω be a bounded domain of M of class C with II ≥ . Then (5.72) µ j ≤ A n κ + B n (cid:18) j | Ω | (cid:19) n ! , for all j ∈ N . We remark that this bound holds independently of the size of Ω, its diameter and theinjectivity radius of M . Hence it is natural to pose the following question, whose answerseems quite complicated at this stage. Open problem.
Prove inequality (5.72) for any bounded domain Ω with C bound-ary in a complete n -dimensional smooth Riemannian manifold with Ric ≥ − ( n − κ , κ ≥ EUMANN EIGENVALUES OF THE BIHARMONIC OPERATOR ON DOMAINS 45
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Bruno Colbois, Universit`e de Neuchˆatel, Institute de Math´ematiques, Rue Emile Argand11, 2000 Neuchˆatel, Switzerland
E-mail address : [email protected] Luigi Provenzano, Universit`a degli Studi di Padova, Dipartimento di Matematica, Via Tri-este 63, 35121 Padova, Italy
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