Isoperimetric bounds for Wentzel-Laplace eigenvalues on Riemannian manifolds
aa r X i v : . [ m a t h . M G ] M a y Isoperimetric bounds for Wentzel-Laplaceeigenvalues on Riemannian manifolds
A¨ıssatou M. NDIAYE
Abstract.
In this paper, we investigate eigenvalues of the Wentzel-Laplace operator on a bounded domain in some Riemannian manifold. Weprove asymptotically optimal estimates, according to the Weyl’s lawthrough bounds that are given in terms of the isoperimetric ratio of thedomain. Our results show that the isoperimetric ratio allows to controlthe entire spectrum of the Wentzel-Laplace operator in various ambientspaces.
Let n > M, g ) be an n -dimensional Riemannian manifold. LetΩ ⊂ M be a bounded domain with smooth boundary Γ. We denote by ∆and ∆ Γ the Laplace-Beltrami operators acting on functions on M and Γ,respectively. Notice that, in conformance with conventions in computationalgeometry, we define the Laplacian with negative sign, that is the negativedivergence of the gradient operator. The gradient operators on M and Γ willbe denoted by ∇ and ∇ Γ respectively, the outer normal derivative on Γ by ∂ n . Throughout the paper we denote by d M and d Γ the Riemannian volumeelements of M and Γ.Given an arbitrary constant β ∈ IR, consider the following eigenvalueproblem on Ω: ( ∆ u = 0 in Ω ,β ∆ Γ u + ∂ n u = λu on Γ . (Wentzel Problem) (1.1)In what follows, we will assume that β , which we refer to as the bound-ary parameter, is non-negative. In this case, the Wentzel eigenvalues form a . . . A. Ndiayediscrete sequence that can be arranged as0 = λ βW, < λ βW, λ βW, · · · λ βW,k · · · ր ∞ . (1.2)We adopt the convention that each eigenvalue is repeated according to itsmultiplicity.The boundary condition in (1.2), which we call Wentzel boundary con-dition, was initially introduced in [15], in order to find the most generalboundary conditions for which the associated operator generates a Markov-ian semigroup. It is often considered in a more general form cf.[6, (1.2)],[7,(2.32)]. Or sometimes, it subordinates the heat equation as in [11, (1.3)] seealso [6]. A good discussion on motivations and the physical interpretation ofWentzel boundary conditions can be found in [10].The present paper we use valuable tools to find bounds in terms ofgeometric quantities in order to estimate Wentzel eigenvalues. These boundsare optimal according to the asymptotic behaviour of the eigenvalues givenby the Weyl law (2.8).The eigenvalue problem of the Laplacian with Wentzel boundary con-dition has only recently been significantly investigated. When β = 0, theeigenvalue problem (1.1) reduced to the so called Steklov eigenvalue prob-lem. An advanced reference providing an overwiew on the Steklov problem,is [9]. As in [7], the problem (1.1) can be viewed as a perturbed ( as oppositeto unperturbed when β = 0) Steklov problem.The most relevant works on bounds for eigenvalues of the Wentzel-Laplace operator have been done in [4, 17, 5]. Dambrine, Kateb and Lamboley[4] obtained a first upper bound for the first non-trivial eigenvalue λ βW, interms of purely geometric quantities if Ω is a bounded domain in IR n :Let ∧ (Ω) denote the spectral radius of the matrix P (Ω) def = (cid:18)Z Γ δ ij − n i n j d Γ (cid:19) i,j =1 ,...,n . The following inequality holds: λ βW, Vol(Ω) + β ∧ (Ω) ω − n n Vol(Ω) n +1 n (cid:20) c n (cid:16) Vol(Ω)∆ B Vol( B ) (cid:17) (cid:21) , c n := ( n √ − n + 1)4 n . (1.3)Here, B is the ball having the same volume as M and with the same centerof mass than Γ and ω n denotes the volume of the n -dimensional Euclideanunit ball. Equality holds in (1.3) if M is a ball. In [17], Wang and Xia provedthe following bound for the same eigenvalue: λ βW, n Vol(Ω) + β ( n − n Vol(Ω)(Vol(Ω) ω − n ) n . (1.4)They also present a bound for λ βW, in non-Euclidean case, when the Riccicurvature of M and the principle curvatures of Γ are bounded. Going further,Du-Wang-Xia provided the following isoperimetric bound for the first n ( n igenvalues of the Wentzel-Laplace operator 3being the dimension) eigenvalues, when M is immersed in an Euclidean spaceIR N equipped with the canonical Euclidean metric. If H is the mean curvaturevector field of Γ in IR N then one has1 n − n − j =1 λ βW,j q [ n Vol( M ) + ( n − β Vol(Γ)] R Γ | H | d Γ Vol(Γ) . (1.5)When N = n , that is, M is a bounded domain of IR N , then equality holds in(1.5) if and only if M is a ball.The aim of this work is to go even further and provide uniformal isoperi-metric bounds for all the eigenvalues of (1.1). If Ω is a domain of an n -dimensional complete Riemannian manifold ( M, g ), with boundary Γ, theisoperimetric ratio of Ω is defined by I (Ω) := Vol(Γ)Vol(Ω) n − n . In the numeratorVol stands for the ( n − n -Riemannianvolume from g in the denominator.Our first result provides an upper bound in the case of Euclidean do-mains. We respectively denote ω n and ρ n − = nω n the volumes of the unitball and the unit sphere in the n -dimensional Euclidean space. Theorem 1.1.
Let n > and Ω ⊂ IR n be a bounded euclidean domain withsmooth boundary Γ . Then, for every k > , one has λ βW,k (Ω) ζ ( n ) (cid:18) Vol(Ω)Vol(Γ) (cid:19) − n (cid:18) k Vol(Γ) (cid:19) n + ζ ( n ) I (Ω) n − (cid:20) Vol(Ω)Vol(Γ) + β (cid:21) (cid:18) k Vol(Γ) (cid:19) n − , (1.6) where ζ ( n ) := 2 n +1) ω n n and ζ ( n ) := n +3) n ω n n . Corollary 1.2.
Let Ω ⊂ IR n be a bounded euclidean domain of dimension n > with smooth boundary Γ . Then, for every k > , we have λ βW,k (Ω) C (Ω , β ) + C (Ω , β ) (cid:18) k Vol(Γ) (cid:19) n − . (1.7) Here C (Ω , β ) and C (Ω , β ) are geometric constants given by : C (Ω , β ) = ζ ( n ) I (Ω) n − (cid:20) Vol(Ω)Vol(Γ) + β (cid:21) + 1 C (Ω , β ) = ζ n ( n ) (cid:18) Vol(Ω)Vol(Γ) (cid:19) n − C (Ω , β ) . The constants ζ ( n ) and ζ ( n ) are the same as in Theorem 1.1. For bounded domains in Riemannian manifold with Ricci curvaturebounded from below, we have an isoperimetric upper bound, which also de-pends on the infimum isoperimetric ratio that we define as follows: A. Ndiaye
Definition 1.3.
Let (
M, g ) be a complete Riemannian manifold of dimension n M . The infimum isoperimetric ratio of Ωis the quantity I (Ω) := inf { I ( U ) : U open set in Ω } . In particular, if Ω is anEuclidean domain, one has I (Ω) = I (IR n ) = nω n n . Theorem 1.4.
Let ( M, g ) be a complete Riemannian manifold of dimension n > with non-negative Ricci curvature. Let Ω ⊂ M be a bounded domainwith smooth boundary Γ . Then for every k > , we have λ βW,k (Ω) c ( n ) (cid:18) Vol(Ω)Vol(Γ) (cid:19) − n (cid:18) k Vol(Γ) (cid:19) n + c ( n ) (cid:18) I (Ω) I (Ω) (cid:19) n − (cid:20) Vol(Ω)Vol(Γ) + β (cid:21) (cid:18) k Vol(Γ) (cid:19) n − , (1.8) where c ( n ) := 2 n +1) ω n n and c ( n ) := 2 n +5) ρ n − n − .Remark . It is not usually simple to gauge this quantity I (Ω). It is noteasy to determinate the best constant in the isoperimetric inequality for do-mains in many complete Riemmannian manifolds. For example, as we see inCorollary 1.7, the longstanding conjecture known as the Cartan-Hadamardconjecture, is about sharp isoperimetric inequalities in complete Riemannianmanifolds with negative sectional curvature. Theorem 1.6.
Let ( M, g ) be a complete Riemannian manifold of dimension n > with Ricci curvature bounded from below by − ( n − κ , κ ∈ IR > . Let Ω ⊂ M be a bounded domain with smooth boundary Γ . Then for every k > ,we have λ βW,k (Ω) A (Ω , β ) + B (Ω , β ) k Vol( ˜Γ) ! n − , (1.9) where A (Ω , β ) = κ ζ ( n ) ( (cid:18) κ Vol(Ω)Vol(Γ) (cid:19) − n + (cid:18) I (Ω) I (Ω) (cid:19) n − (cid:20) κ Vol(Ω)Vol(Γ) + β (cid:21)) ,B (Ω , β ) = ζ ( n ) ( (cid:18) I (Ω) I (Ω) (cid:19) n − (cid:20) κ Vol(Ω)Vol(Γ) + β (cid:21) ) ,ζ ( n ) being a constant depending only on the dimension n . Theorems 1.4 and 1.6 emanate from a generic result (Theorem 3.6) thatwe prove in Section 3.Besides the Euclidean case, we have at least one other situation where weknow something about the quantity I (Ω). The so called Cartan-Hadamardconjecture, proved in dimensions n = 2 by Weil [16], n = 3 by Kleiner [12]and n = 4 by Croke [3], states that any bounded domain in a smooth Cartan-Hadamard manifold of dimension n satisfies I (Ω) > C ( n )igenvalues of the Wentzel-Laplace operator 5where C ( n ) is a dimensional constant. Very recently, Ghomi and Spruck(2019) in [8] proposed a solution in all dimensions. This leads to the fol-lowing corollary. Corollary 1.7.
Let ( M, g ) be a smooth Cartan-Hadamard manifold of dimen-sion n > with Ricci curvature bounded from below by − ( n − κ , κ ∈ IR > and Ω ⊂ M a bounded domain with smooth boundary Γ . Then for every k > , we have λ βW,k (Ω) A (Ω , β ) + B (Ω , β ) k Vol( ˜Γ) ! n − , (1.10) where A (Ω , β ) = κ ζ ( n ) ( (cid:18) κ Vol(Ω)Vol(Γ) (cid:19) − n + I (Ω) n − (cid:20) κ Vol(Ω)Vol(Γ) + β (cid:21)) ,B (Ω , β ) = ζ ( n ) ( I (Ω) n − (cid:20) κ Vol(Ω)Vol(Γ) + β (cid:21) ) ,ζ ( n ) being a constant depending on the dimension n . Plan of the paper.
The proofs of Theorems 1.1 1.4 and 1.6 are presented inSection 4. We present the proof of Theorem 3.6 in Section 3, following sometechnical results. We devote Section 2 to briefly summarize properties of theWentzel-Laplace eigenvalues.
Consider the map ∧ : L (Γ) −→ L (Ω) related to the Dirichlet problem ( ∆ u = 0 in Ω ,u | Γ = f on Γ , (2.1)which associate to any f ∈ L (Γ) its harmonic extension i.e. the uniquefunction u in L (Ω) satisfying (2.1). This map is well defined from L (Γ)(respectively, H (Γ)) to L (Ω) (respectively, H (Ω)). See [14, p. 320, Prop1.7], for more details. By H s (Ω) and H s (Γ), we denote the Sobolev spacesof order s on Ω and Γ, respectively, and u | Γ ∈ H (Γ) stands for the trace of u ∈ H (Ω) at the boundary Γ. This will also be denoted simply by u , if noambiguity can result.Then the Dirichlet-to-Neumann operator is defined byN D : H (Γ) H − (Γ) (2.2) f ∂ n ( ∧ f ) . Again ∂ n stands for the normal derivative at the boundary Γ of Ω with unitnormal vector n pointing outwards. A. NdiayeFor all u ∈ H (Γ), we define the operators B = N D (in the operatorsense). For β >
0, we define for all u ∈ H (Γ) C β u def = β ∆ Γ u and B β def = B +C β . The eigenvalue sequence { λ βW,k } ∞ k =0 given in (1.2) can be interpreted asthe spectrum associated with the operator B β and is subject to the followingmin-max characterization (see e.g., [13, Thm1.2] and [7, (2.33)]): Min-max principle.
Let V ( k ) denotes the set of all k -dimensional subspacesof V β which is defined by V def = { ( u, u Γ ) ∈ H (Ω) × H (Γ) : u Γ = u | Γ } , V β def = { ( u, u Γ ) ∈ H (Ω) × H (Γ) : u Γ = u | Γ } , β > . Of course, for all β >
0, we have V β ⊂ V . For every k ∈ N , the k Theigenvalue of the Wentzel-Laplace operator B β satisfies λ βW,k = min V ∈ V ( k ) max = u ∈ V R β ( u ) , k > , (2.3)where R β ( u ), the Rayleigh quotient for B β , is given by R β ( u ) def = R Ω |∇ u | d M + β R Γ |∇ Γ u | d Γ R Γ u d Γ , for all u ∈ V β \{ } . (2.4) Asymptotic behaviour.
The eigenvalues for the Dirichlet-to-Neumann map B = N D are those of the well known Steklov problem. ( ∆ u = 0 , dans Ω ,∂ n u = λ S u, sur Γ . (2.5)A good discussion of this problem can be found in [9]. The Steklov eigenvaluesare then { λ W,k } ∞ k =0 which we shall denote equivalently as { λ S,k } ∞ k =0 . Theybehave according to the following asymptotic formula: λ S,k = C n k n − + O ( k n − ) , k → ∞ . (2.6)where C n = π ( ω n − Vol(Γ) n − . The reader can refer to [13, section 4]. For β > λ βW,k can be deduced directly from properties ofperturbed forms using the asymptotic behaviour of the spectrum of C β byH¨ormander: λ C β,k = βC n k n − + O ( k n − ) , k → ∞ . (2.7)The Weyl law for eigenvalues on problem (1.1) reads λ βW,k = βC n k n − + O ( k n − ) , k → ∞ . (2.8)A complete and detailed discussion about the spectral properties of theWentzel Laplacian can be found in [7, Section 2] and for the asymptoticin (2.8), the reader can refer to [7, Prop 2.7 and (2.37)].igenvalues of the Wentzel-Laplace operator 7 In this section, we establish some needed technical results and the ma-jor result in this paper used to prove our main theorems. Let n > M, g ) be an n -dimensional Riemannian manifold. Let Ω ⊂ M a boundeddomain with smooth boundary Γ. Let r ∈ IR > , we denote by B ( x, r ) = { p ∈ M, d ( x, p ) < r } the metric ball of radius r centered at x ∈ M , where d is theRiemannian distance associated to the metric g . We assume Γ satisfies thefollowing hypothesis:( H ): There exists a radius r − (Γ) > C ∈ N > such that forall x ∈ Γ and r < r − (Γ), one hasVol( B ( x, r )) < Cω n r n and Vol( ∂B ( x, r )) < Cρ n − r n − . (3.1)Here ∂B ( x, r ) denotes the geodesic sphere of radius r centered at x . Lemma 3.1.
Let ( M, g ) , Ω and Γ be as above. For every K ∈ N , let r K be anassociated “maximal” radius defined by r K := (cid:18) Vol(Ω)2 (cid:19) n (cid:18) I (Ω) KCρ n − (cid:19) n − . (3.2) Let { x j } Kj =1 be an arbitrary set of points in Γ . Then for every r > satisfyingboth r < r − (Ω) and r r K , one has Vol Γ \ K [ j =1 B ( x j , r ) > . (3.3) Proof.
We denote by Ω (respectively Γ ) the subset Ω \ S Kj =1 B ( x j , r ) (re-spectively Γ \ S Kj =1 B ( x j , r )). One can think of Ω as a holed cheese. Sincethe boundary of Ω , that we denote by ∂ Ω , is contained in the unionΓ S (cid:16)S Kj =1 Vol( ∂B ( x j , r )) (cid:17) , one hasVol(Γ ) > Vol( ∂ Ω ) − K X j =1 Vol( ∂B ( x j , r ))= I (Ω )Vol(Ω ) n − n − K X j =1 Vol( ∂B ( x j , r )) , where ∂B ( x j , r ) = { p ∈ M | d ( x j , p ) = 2 r } .Then, since 2 r < r − (Γ), one hasVol(Γ ) + KCρ n − (2 r ) n − > I (Ω )Vol(Ω ) n − n . (3.4)Now, we assume that I (Ω) nn − Vol(Ω) − KCρ nn − n − (2 r ) n > . (3.5) A. NdiayeNoticing that I (Ω) I (IR n ), we have then I (Ω )Vol(Ω ) n − n > I (Ω )[Vol(Ω) − KCω n (2 r ) n ] n − n > [ I (Ω) nn − Vol(Ω) − KCρ nn − n − (2 r ) n ] n − n . Replacing in (3.4), this leads to the following inequality:Vol(Γ ) > [ I (Ω) nn − Vol(Ω) − KCρ nn − n (2 r ) n ] n − n − KCρ n − (2 r ) n − . (3.6)The right hand side is non-negative if (cid:18) I (Ω) KCρ n − (cid:19) nn − Vol(Ω)( KC ) − n − + 1 > (2 r ) n . (3.7)We Notice that KC ) − n − +1 > . Inequality (3.7) is then satisfied whenever r (cid:18) I (Ω) KCρ n − (cid:19) n − (cid:18) VolΩ)2 (cid:19) n = 12 r K . This implies that Vol(Γ ) >
0, under the assumption in (3.5). However, (3.5)can be written as r (cid:18) Vol(Ω) KC (cid:19) n (cid:18) I (Ω) ρ n − (cid:19) n − = 2 n KC ) n ( n − r K . (3.8)Since 1 n ( KC ) n ( n − , (3.8) is satisfied by assumption. This ends theproof. (cid:3) Lemma 3.2.
Let the assumptions of Lemma 3.1 be fulfilled. We define K := $ I (Ω) Cρ n − r − (Γ) n − (cid:18) Vol(Ω)2 (cid:19) n − n % + 1 , (3.9) where ⌊ ⌋ denotes the floor function, so that r K < r − (Γ) if K > K . Let { x j } Kj =1 be an arbitrary set of points in Γ . Then, for every K > K and < r r k , we have Vol Γ \ K [ j =1 B ( x j , r ) > (cid:18) rr K (cid:19) n − n I (Ω)Vol(Ω) n − n . (3.10) Proof.
From (3.6) in the proof of Lemma 3.1, one hasVol(Γ ) > [ I (Ω) nn − Vol(Ω) − KCρ nn − n (2 r ) n ] n − n − KCρ n − (2 r ) n − . Setting α := r K r (we notice that α > since r r K ), we have(2 r ) n = (cid:18) α r K (cid:19) n = (cid:18) α (cid:19) n Vol(Ω)2 (cid:18) I (Ω) KCρ n − (cid:19) nn − KCρ nn − n − n − n α ! n I (Ω) nn − Vol(Ω) , (3.11)igenvalues of the Wentzel-Laplace operator 9where we have used that KC >
1. On the other hand,(2 r ) n − = (cid:18) α r K (cid:19) n − = (cid:18) α (cid:19) n − (cid:18) Vol(Ω)2 (cid:19) n − n I (Ω) KCρ n − KCρ n − n − n α ! n − I (Ω) nn − Vol(Ω) . (3.12)From inequalities (3.11) and (3.12), we getVol Γ \ K [ j =1 B ( x j , r ) > − n − n α ! n ! n − n − n − n α ! n − I (Ω)Vol(Ω) n − n . (3.13)We notice that, since α > − n − n α ! n ! n − n − n − n α ! n − > (cid:0) − α − (cid:1) n − n − α − n − n = α − n − n h ( α − n − n − i > α − n − n h n − n − i . It follows thatVol Γ \ K [ j =1 B ( x j , r ) > α − n − n I (Ω)Vol(Ω) n − n , since 15 n − n > n > (cid:3) Let (
M, g ), Ω and Γ be as described above and r ∈ IR > . The externalcovering number N extr (Γ) of Γ in M with respect to r is defined as the fewestnumber of points x , . . . , x N ∈ M such that the balls B ( x , r ) , . . . , B ( x N , r )cover Γ. Lemmas 3.1 and 3.2 imply the following principal lemma. Lemma 3.3.
Let n > and ( M, g ) be an n -dimensional Riemannian manifold.Let Ω ⊂ M a bounded domain with smooth boundary Γ . Then for every K > K and < r r K ,i. K < N extr (Γ) . ii. If in addition r r K then for every arbitrary set of points { x j } Kj =1 in M , one has Vol(Γ \ K [ j =1 B ( x j , r )) > (cid:18) rr K (cid:19) n − n I (Ω)Vol(Ω) n − n . (3.14)0 A. Ndiaye Proof.
Suppose N extr (Γ) K and let { B ( x j , r ) } N extr (Γ) j =1 be a minimal coveringof Γ. By the minimality assumption, every B ( x j , r ) intersects Γ. For j ∈{ , . . . , N extr (Γ) } , let x ′ j ∈ B ( x j , r ) ∩ Γ, one has B ( x j , r ) ⊂ B ( x ′ j , r ) , for every i ∈ { , . . . , N extr (Γ) } . This impliesVol Γ \ N extr (Γ) [ j =1 B ( x ′ j , r ) Vol Γ \ N extr (Γ) [ j =1 B ( x j , r ) . We complete the family { B ( x ′ j , r ) } N extr (Γ) j =1 to { B ( x ′ j , r ) } Kj =1 by setting x ′ j := x ′ for N extr (Γ) < j K . Then, applying Lemma 3.1, we haveVol(Γ \ N extr (Γ) [ j =1 B ( x j , r )) > Vol(Γ \ N extr (Γ) [ j =1 B ( x ′ j , r ))= Vol(Γ \ K [ j =1 B ( x ′ j , r )) > . Hence, it is contradictory to Γ ⊂ S N extr (Γ) j =1 B ( x j , r )). To prove (ii.), we noticethat if B ( x j , r ) ∩ Γ = ∅ then B ( x j , r ) ⊂ B ( x ′ j , r ) with x ′ ∈ Γ. The inequalityfollows the applying Lemma 3.2. (cid:3)
The next lemma of Colbois and Maerten [2] provides the final ingredientto prove the most technical results in this paper presented in Theorems 3.5and 3.6.Let (
X, d ) be a complete, locally compact metric space. Let ε ∈ N and N : (0 , ρ ] −→ N > an increasing function. We say that ( X, d ) satisfies the(
N, ε )-covering property if each ball of radius r can be covered by N ( r ) ballsof radius rε . In order to simplify notation, we will write N r instead of N ( r ).We denominate capacitor any couple ( A, B ) of subsets such that ∅ 6 = A ⊂ B ⊂ X . Two capacitors ( A , B ) and ( A , B ) are disjoint if B ∩ B = ∅ .A family of capacitors is a finite set of capacitors in X that are pairwisedisjoint. Lemma 3.4 (Colbois-Maerten, ). Let ( X, d, µ ) be a complete, locallycompact metric measure space satisfying the ( N, -covering property with N : (0 , diam( X )] −→ N > a discrete positive function. Let r > and K ∈ N such that for every x ∈ X , µ ( B ( x, r )) µ ( X )4 KN r . Then there exists a family of K capacitors { ( A i , B i ) } i K with the following properties for i, j K µ ( A i ) > µ ( X )2 N r K ,2. B i = A ri := { x ∈ X, d ( x, A i ) < r } is the r -neighbourhood of A i and d ( B i , B j ) > r whenever i = j . igenvalues of the Wentzel-Laplace operator 11 Theorem 3.5.
Let ( M, g ) be a complete Riemannian manifold of dimension n > . Let Ω ⊂ M be a bounded domain whose boundary Γ is a smooth hy-persurface satisfying ( H ) . We assume that M , with respect to the distanceassociated to the metric g satisfies the ( N, -covering property for some dis-crete positive function N .Then, for every integer k > K ( K is the same as in (3.9) ), one has λ βW,k (Ω) C (cid:18) Vol(Ω)Vol(Γ) (cid:19) − n (cid:18) k Vol(Γ) (cid:19) n + C (cid:18) I (Ω) I (Ω) (cid:19) n − (cid:20) Vol(Ω)Vol(Γ) + β (cid:21) (cid:18) k Vol(Γ) (cid:19) n − , (3.15) where C = 2 ( Cω n ) n N r , C = 2 ( Cρ n − ) n − N r and r := r (4 k ) .Proof. The methods we use in this proof are inspired by [1]. We consider themetric measure space (
M, d, µ ), where d is the distance from the metric g and µ is the Borel measure with support Γ defined for each Borelian A of M by µ ( A ) := Z A ∩ Γ d Γ . Fix K = 4 k and choose in M a family of points { x j } Kj =1 satisfying ( B ( x j , r ) ∩ B ( x i , r ) = ∅ for all 1 i = j K,µ ( B ( x , r )) > µ ( B ( x , r )) > . . . > µ ( B ( x K , r )) > µ ( B ( x, r )) , (3.16)for all x ∈ M := M \ S Kj =1 B ( x j , r ). This can be done inductively, selectingthe point x such that µ ( B ( x , r )) = sup { µ ( B ( x, r )) , x ∈ M } , and the points x j , for j = 2 , . . . , K , such that µ ( B ( x j , r )) = sup { µ ( B ( x, r )) , x ∈ M \ j − [ i =1 B ( x i , r ) } . There are two possible cases:
Case µ ( B ( x K , r )) µ ( M )4 KN r . We consider the metric measure space( M , d, µ ) where µ is defined by µ ( A ) := Z A ∩ Γ d Γ , Γ := Γ \ K [ i =1 B ( x i , r ) . for every Borelian A in M . Since 4 r = r K , it follows from Lemma 3.3, that µ ( M ) = Vol(Γ ) > n − n I (Ω)Vol(Ω) n − n . µ ( B ( x, r )) µ ( B ( x, r )) µ ( M )4 KN r for every x ∈ M .Applying Lemma 3.4, we have a family of K capacitors { ( A i , B i ) } i K with the following properties for 1 i, j K :1. µ ( A i ) > µ ( M )2 N r K ,2. B i = A ri = { x ∈ X, d ( x, A i ) < r } is the r -neighborhood of A i and d ( B i , B j ) > r whenever i = j .We notice that µ ( M ) = Vol(Γ ).For each 1 j K , we consider the function ϕ j supported in A rj defined by ϕ j ( x ) := ( − d ( A j ,x ) r ∀ x ∈ A rj , ∀ x ∈ M \ A rj . It follows that R β ( ϕ j ) R Ω ∩ Arj |∇ ϕ j | d M + β R Γ ∩ Arj |∇ ϕ j | d Γ R Γ ∩ Aj ϕ j d Γ . i) One has Z Ω ∩ A rj |∇ ϕ j | d M r Vol(Ω ∩ A rj ) . The A rj ’s are pairwise disjoint then P kj =1 Vol(Ω ∩ A rj ) Vol(Ω). Wededuce that at least 2 k of A rj ’s satisfyVol(Ω ∩ A rj ) Vol(Ω) k . (3.17)Up to re-ordering, we assume that for the first 2 k of the A rj ’s we have(3.17). Hence, Z Ω ∩ A rj |∇ ϕ j | r Vol(Ω) k , ∀ j k. ii) By the same arguments, at least k of the A rj ’s satisfyVol(Γ ∩ A rj ) Vol(Γ) k . (3.18)Up to re-ordering, we assume that for the first k of the A rj ’s (3.18) holds.Hence, Z Γ ∩ A rj |∇ f i | r Vol(Γ) k , ∀ j k. Since R Γ ∩ A j ϕ j d Γ > R Γ ∩ A j d Γ = µ ( A j ) > Vol(Γ )8 N r k , we have R β ( ϕ j ) N r k Vol(Γ ) (cid:20) r Vol(Ω) k + β r Vol(Γ) k (cid:21) = 8 N r r Vol(Γ ) [Vol(Ω) + β Vol(Γ)] . However, 1 r = (cid:18) r (4 k ) (cid:19) = 2 (cid:18) (cid:19) n (cid:18) kCρ n − I (Ω) (cid:19) n − igenvalues of the Wentzel-Laplace operator 13and Vol(Γ ) > n − n I (Ω)Vol(Ω) n − n . Thus, 1 r Vol(Γ ) ( kCρ n − ) n − Vol(Ω) n I (Ω) n − . We get R β ( ϕ j ) N r ( kCρ n − ) n − Vol(Ω) n I (Ω) n − [Vol(Ω) + β Vol(Γ)] N r ( Cρ n − ) n − (cid:18) k Vol(Γ) (cid:19) n − Vol(Γ) n − Vol(Ω) n I (Ω) n − (cid:20) Vol(Ω)Vol(Γ) + β (cid:21) = 2 N r ( Cρ n − ) n − (cid:18) k Vol(Γ) (cid:19) n − (cid:18) I (Ω) I (Ω) (cid:19) n − " Vol(Γ) n − I (Ω) n − + β . Hence, λ βW,k (Ω) max j k R β ( ϕ j ) N r ( Cρ n − ) n − (cid:18) k Vol(Γ) (cid:19) n − (cid:18) I (Ω) I (Ω) (cid:19) n − " Vol(Γ) n − I (Ω) n − + β . (3.19) Case µ ( B ( x K , r )) > µ ( X )4 KN r . From (3.16) one has µ ( B ( x j , r )) > µ ( X )4 KN r forevery 1 j K . We consider, for 1 j k ,the function f j supported in B j := B ( x j , r ) and defined by f j ( x ) := ( min { , − d ( x j ,x ) r } ∀ x ∈ B j , ∀ x ∈ M \ B j . Set A j := B ( x j , r ), then the Rayleigh quotient of f j satisfies R β ( f j ) R Ω ∩ B j |∇ f j | d M + β R Γ ∩ B j |∇ f j | d Γ R Γ ∩ A j f i d Γ . i) Since for every x ∈ A j , f j ( x ) = 1, one has Z Γ ∩ A j f j d Γ > Z Γ ∩ A j d Γ > µ ( A j ) > Vol(Γ)16 N r k . ii) Set for x ∈ M , d j ( x ) := dist( x j , x ), then |∇ f j | |∇ (2 − d j ( x ) r ) | = | r ∇ ( d j ( x )) | r . Z Ω ∩ B j |∇ f j | d M Z Ω ∩ B j |∇ f j | n d M ! n Z Ω ∩ B j d M ! − n (cid:18) r n Vol( B j ) (cid:19) n (Vol(Ω ∩ B j )) − n . Notice that B j ∩ Γ ⊃ A j ∩ Γ = ∅ . Let x ′ j ∈ B j ∩ Γ, one has B j ⊂ B ( x ′ j , r ).Since 4 r r K < r − (Γ),Vol( B j ) Vol( B ( x ′ j , r )) < Cω n (4 r ) n . In addition, since the B j ’s are pairwise disjoint, we have k X j =1 Vol(Ω ∩ B j ) Vol(Ω) . We deduce that at least 2 k of B j ’s satisfyVol(Ω ∩ B j ) Vol(Ω) k . (3.20)Up to re-ordering, we assume that for the first 2 k of the B j ’s (3.20)holds. Hence, Z Ω ∩ B j |∇ f j | ( Cω n n ) n (cid:18) Vol(Ω) k (cid:19) − n , ∀ j k. iii) Again the B j ’s are pairwise disjoint then P kj =1 Vol(Γ ∩ B j ) Vol(Γ).Hence at least k of the B j ’s satisfyVol(Γ ∩ B j ) Vol(Γ) k . (3.21)Up to re-ordering, we assume that for the first k of the B j ’s, inequalityholds (3.21). Thus, Z Γ ∩ B j |∇ f i | r (cid:18) Vol(Γ) k (cid:19) − n − , ∀ j k. Hence, one has R β ( f j ) N r k Vol(Γ) " ( Cω n n ) n (cid:18) Vol(Ω) k (cid:19) − n + β r Vol(Γ) k N r ( Cω n ) n (cid:18) k Vol(Γ) (cid:19) n (cid:18) Vol(Ω)Vol(Γ) (cid:19) − n + β N r ( Cρ n − ) n − (cid:18) k Vol(Γ) (cid:19) n − (cid:18) I (Ω) I (Ω) (cid:19) n − . igenvalues of the Wentzel-Laplace operator 15Since I (Ω) I (Ω) >
1, regarding the right hand side of (3.19), we have R β ( f j ) N r ( Cω n ) n (cid:18) k Vol(Γ) (cid:19) n (cid:18) Vol(Ω)Vol(Γ) (cid:19) − n + β N r ( Cρ n − ) n − (cid:18) k Vol(Γ) (cid:19) n − (cid:18) I (Ω) I (Ω) (cid:19) n − . Then, in this case λ βW,k (Ω) max j k R β ( ϕ j ) N r ( Cω n ) n (cid:18) k Vol(Γ) (cid:19) n (cid:18) Vol(Ω)Vol(Γ) (cid:19) − n + β N r ( Cρ n − ) n − (cid:18) k Vol(Γ) (cid:19) n − (cid:18) I (Ω) I (Ω) (cid:19) n − . (3.22)From (3.19) and (3.22), in both possible cases we have λ βW,k (Ω) C (cid:18) k Vol(Γ) (cid:19) n (cid:18) Vol(Ω)Vol(Γ) (cid:19) − n + C (cid:18) k Vol(Γ) (cid:19) n − (cid:18) I (Ω) I (Ω) (cid:19) n − (cid:20) Vol(Ω)Vol(Γ) + β (cid:21) , (3.23)where C := 2 N r ( Cω n ) n and C := 2 N r ( Cρ n − ) n − . This ends the proof. (cid:3) When n >
3, Theorem 3.5 can be extended to cover all eigenvalues asfollows:
Theorem 3.6.
Let ( M, g ) be a complete Riemannian manifold of dimension n > and let Ω ⊂ M be a bounded domain whose boundary Γ is a smoothhypersurface satisfying the hypothesis ( H ) . We assume that M , with respectto the distance associated to the metric g satisfies the ( N, -covering propertyfor some discrete positive function N .Then, for every k > , one has λ βW,k (Ω) C (Ω , β ) + C (cid:18) Vol(Ω)Vol(Γ) (cid:19) − n (cid:18) k Vol(Γ) (cid:19) n + C (cid:18) I (Ω) I (Ω) (cid:19) n − (cid:20) Vol(Ω)Vol(Γ) + β (cid:21) (cid:18) k Vol(Γ) (cid:19) n − , (3.24) where the constants C and C are the same as in Theorem 3.5 and C (Ω , β ) := C ( Cρ n − r − (Γ) n − ) n (cid:18) Vol(Ω)Vol(Γ) (cid:19) − n + C ( Cρ n − r − (Γ) n − ) n − (cid:18) I (Ω) I (Ω) (cid:19) n − (cid:20) Vol(Ω)Vol(Γ) + β (cid:21) . (3.25)6 A. Ndiaye Proof.
For 1 k < K , one has λ βW,k (Ω) λ βW,K (Ω) C (cid:18) K Vol(Γ) (cid:19) n (cid:18) Vol(Ω)Vol(Γ) (cid:19) − n + C (cid:18) K Vol(Γ) (cid:19) n − (cid:18) I (Ω) I (Ω) (cid:19) n − " Vol(Γ) n − I (Ω) n − + β . (3.26)However, K I (Ω) Cρ n − r − (Γ) n − (cid:16) Vol(Ω)2 (cid:17) n − n + 1, using the triangle inequality,we obviously have for every p ∈ N > K p I (Ω) Cρ n − r − (Γ) n − (cid:18) Vol(Ω)2 (cid:19) n − n ! p + 1 Vol(Γ)2 n − n Cρ n − r − (Γ) n − I (Ω) I (Ω) ! p + 1 (cid:18) Vol(Γ) Cρ n − r − (Γ) n − (cid:19) p + k p . We set C := (cid:16) Cρ n − r − (Γ) n − (cid:17) p , replacing in (3.26), we get λ βW,k (Ω) C ( C + (cid:18) k Vol(Γ) (cid:19) n ) (cid:18) Vol(Ω)Vol(Γ) (cid:19) − n + C ( C + (cid:18) k Vol(Γ) (cid:19) n − ) (cid:18) I (Ω) I (Ω) (cid:19) n − (cid:20) Vol(Ω)Vol(Γ) + β (cid:21) . (3.27)Rearranging terms in above inequality, we have λ βW,k (Ω) C (Ω , β ) + C (cid:18) Vol(Ω)Vol(Γ) (cid:19) − n (cid:18) k Vol(Γ) (cid:19) n + C (cid:18) I (Ω) I (Ω) (cid:19) n − (cid:20) Vol(Ω)Vol(Γ) + β (cid:21) (cid:18) k Vol(Γ) (cid:19) n − , (3.28)where C (Ω , β ) := C ( Cρ n − r − (Γ) n − ) n (cid:18) Vol(Ω)Vol(Γ) (cid:19) − n + C ( Cρ n − r − (Γ) n − ) n − (cid:18) I (Ω) I (Ω) (cid:19) n − (cid:20) Vol(Ω)Vol(Γ) + β (cid:21) . (3.29)The result follows applying Theorem 3.5 when k > K . (cid:3) igenvalues of the Wentzel-Laplace operator 17 Proof of Theorem 1.1.
We have in the Euclidean case: r − (Γ) = + ∞ , C = 2 , I (Ω) = I (IR n ) = nω n n , N r n , ∀ r > . Applying Theorem 3.6, we get for every k > λ βW,k (Ω) ζ ( n ) (cid:18) Vol(Ω)Vol(Γ) (cid:19) − n (cid:18) k Vol(Γ) (cid:19) n + ζ ( n ) (cid:18) I (Ω) I (Ω) (cid:19) n − (cid:20) Vol(Ω)Vol(Γ) + β (cid:21) (cid:18) k Vol(Γ) (cid:19) n − , (4.1)where ζ ( n ) := 2 n +1) ω n n and ζ ( n ) := 2 n +3) ρ n − n − . The result followsreplacing I (Ω) by nω n . (cid:3) Proof of Corollary 1.2.
From Theorem 1.1, one has λ βW,k (Ω) ζ ( n ) (cid:18) Vol(Ω)Vol(Γ) (cid:19) − n (cid:18) k Vol(Γ) (cid:19) n + ζ ( n ) I (Ω) n − (cid:20) Vol(Ω)Vol(Γ) + β (cid:21) (cid:18) k Vol(Γ) (cid:19) n − (4.2)= ( ζ ( n ) (cid:18) Vol(Ω)Vol(Γ) (cid:19) − n (cid:18) Vol(Γ) k (cid:19) n ( n − + ζ ( n ) I (Ω) n − (cid:20) Vol(Ω)Vol(Γ) + β (cid:21) ) (cid:18) k Vol(Γ) (cid:19) n − .
1. If ζ ( n ) (cid:16) Vol(Ω)Vol(Γ) (cid:17) − n (cid:16) Vol(Γ) k (cid:17) n ( n − <
1, then λ βW,k (Ω) < C (Ω , β ) (cid:18) k Vol(Γ) (cid:19) n − .
2. Otherwise, ζ ( n ) (cid:16) Vol(Ω)Vol(Γ) (cid:17) − n (cid:16) Vol(Γ) k (cid:17) n ( n − >
1. That is, k Vol(Γ) " ζ ( n ) (cid:18) Vol(Ω)Vol(Γ) (cid:19) n − n n ( n − , (cid:16) k Vol(Γ) (cid:17) n ζ n − ( n ) (cid:16) Vol(Ω)Vol(Γ) (cid:17) ( n − n − n (cid:16) k Vol(Γ) (cid:17) n − ζ n ( n ) (cid:16) Vol(Ω)Vol(Γ) (cid:17) n − . (4.3)8 A. NdiayeReplacing in (4.2), we get λ βW,k (Ω) ζ n ( n ) (cid:18) Vol(Ω)Vol(Γ) (cid:19) n − + ζ n ( n ) (cid:18) Vol(Ω)Vol(Γ) (cid:19) n − ζ ( n ) I (Ω) n − (cid:20) Vol(Ω)Vol(Γ) + β (cid:21) ζ n ( n ) (cid:18) Vol(Ω)Vol(Γ) (cid:19) n − (cid:26) ζ ( n ) I (Ω) n − (cid:20) Vol(Ω)Vol(Γ) + β (cid:21)(cid:27) = C (Ω , β )In both cases, one has λ βW,k (Ω) C (Ω , β ) + C (Ω , β ) (cid:16) k Vol(Γ) (cid:17) n − . (cid:3) Proof of Theorem 1.6.
Let r >
0, we denote by ν ( n, − κ , r ) (respectively ν ∂ ( n, − κ , r )) the volume of a ball (respectively a sphere) of radius r inthe constant curvature model space M n − κ . As a consequence of the relativeBishop-Gromov volume comparison theorem, we have the following volumeand area comparisons, for every r > x ∈ M :Vol( B ( x, r )) ν ( n, − κ , r ) and Vol( ∂B ( x, r )) ν ∂ ( n, − κ , r ) . The sphere of radius r in the model space M n − κ has area ν ∂ ( n, − κ , r ) = ρ n − sn − κ ( r ) n − and the ball of radius r has volume ν ( n, − κ , r ) = ρ n − Z r sn − κ ( t ) n − d t, where sn κ : IR −→ IR is defined by sn κ ( t ) = √ κ sin( √ κ t ) if κ > t if κ = 0 √− κ sinh( √− κ t ) if κ < .ρ n − Z r sn − κ ( r ) n − d t = ρ n − Z r (cid:18) ( 1 κ sinh( κt ) (cid:19) n − d t ρ n − Z r (cid:2) te κt (cid:3) n − d t ρ n − e r ( n − κ Z r t n − d t ω n r n e r ( n − κ and ρ n − sn − κ ( r ) n − e r ( n − κ ρ n − r n − . Hence, for every 0 < r < x ∈ M , we haveVol( B ( x, r )) < Cω n r n and Vol( ∂B ( x, r )) < Cρ n − r n − , (4.4)with C := e nκ .igenvalues of the Wentzel-Laplace operator 19On the other hand, for every 0 < r < x ∈ M , B ( x, r ) can becovered by N := 2 n e r ( n − κt < n e n − κ balls of radius r . Indeed, take { B ( x i , r ) } Ni =1 a maximal family of disjoint balls with center x i ∈ B ( x, r ).By the maximality assumption, the family { B ( x i , r ) } Ni =1 covers B ( x, r ). Let i ∈ { , . . . , N } such thatVol (cid:16) B ( x i , r (cid:17) = min i N Vol (cid:16) B ( x i , r (cid:17) . Then one has N Vol( B ( x i , r X i N Vol( B ( x i , r B ( x i , r ) are pairwise disjoint. In addition, B ( x i , r ) ⊂ B ( x i , r + r ) for every x i ∈ B ( x, r ). Hence N Vol( B ( x i , r )) Vol( B ( x, r )), N Vol( B ( x, r ))Vol( B ( x i , r )) Vol( B ( x, r ))Vol( B ( x i , r )) Vol( B ( x i , r ))Vol( B ( x i , r )) . Using the Relative volume comparison theorem (Bishop 1964, Gromov 1980),one has Vol( B ( x i , r ))Vol( B ( x i , r )) ν ( n, − κ, r ) ν ( n, − κ, r ) , where ν ( n, κ, r ) denotes the volume of a ball of radius r in the constantcurvature model space M nκ . Then N R r sinh n − ( κt )d t R r sinh n − ( κt )d t R r [( κt ) e κt ] n − d t R r ( κt ) n − d t e r ( n − κ R r t n − d t R r t n − d t = 2 n e r ( n − κt < n e n − κ . Then applying Theorem 3.6 with r − (Γ) = 1 , C = e nκ , N r = 2 n e n − κ , we get, for every k > λ βW,k (Ω) e c ( n ) κ ( C (Ω , β )+ c ( n ) (cid:18) Vol(Ω)Vol(Γ) (cid:19) − n (cid:18) k Vol(Γ) (cid:19) n + c ( n ) (cid:18) I (Ω) I (Ω) (cid:19) n − (cid:20) Vol(Ω)Vol(Γ) + β (cid:21) (cid:18) k Vol(Γ) (cid:19) n − ) , where C (Ω , β ) := c ′ ( n ) (cid:18) Vol(Ω)Vol(Γ) (cid:19) − n + c ′ ( n ) (cid:18) I (Ω) I (Ω) (cid:19) n − (cid:20) Vol(Ω)Vol(Γ) + β (cid:21) and the constants c ( n ), c ( n ), c ( n ), c ′ ( n ) and c ′ ( n ) depend only on n .0 A. NdiayeFollowing the same arguments as the proof of Corollary 1.2, we have forevery k >
1, one has λ βW,k (Ω) e c ( n ) κ ( C (Ω , β ) + C (Ω , β ) (cid:18) k Vol(Γ) (cid:19) n − ) , where C (Ω , β ) = 1 + c ( n ) (cid:16) I (Ω) I (Ω) (cid:17) n − h Vol(Ω)Vol(Γ) + β i and C (Ω , β ) = C (Ω , β ) + c n ( n ) C (Ω , β ) . - If κ
1, then λ βW,k (Ω) e c ( n ) ( C (Ω , β ) + C (Ω , β ) (cid:18) k Vol(Γ) (cid:19) n − ) , which implies (1.9).- Otherwise, we assume that Ric( M, g ) > − ( n − κ g with κ >
1. Thenthe Ricci curvature Ric( M, ˜ g ) of the rescaled metric ˜ g := κ g is boundedby − ( n − g . We mark with a tilde quantities associated with the metric˜ g , while those unmarked with such will be still associated with the metric g . Then we have λ βW,k ( ˜Ω) e c ( n ) ( C ( ˜Ω , β ) + C ( ˜Ω , β ) k Vol( ˜Γ) ! n − ) . (4.5)However Vol( ˜Ω) = Vol ˜ g (Ω) = κ n Vol(Ω) and Vol(˜Γ) = κ n − Vol(Γ).Thus, C ( ˜Ω , β ) = 1 + c ( n ) I ( ˜Ω) I ( ˜Ω) ! n − " Vol( ˜Ω)Vol( ˜Γ) + β = 1 + c ( n ) (cid:18) I (Ω) I (Ω) (cid:19) n − (cid:20) κ Vol(Ω)Vol(Γ) + β (cid:21) . (4.6)Likewise, since C ( ˜Ω , β ) = c ′ ( n ) (cid:18) κ Vol(Ω)Vol(Γ) (cid:19) − n + c ′ ( n ) (cid:18) I (Ω) I (Ω) (cid:19) n − (cid:20) κ Vol(Ω)Vol(Γ) + β (cid:21) , we have C ( ˜Ω , β ) = C ( ˜Ω , β ) + c n ( n ) C ( ˜Ω , β )= c n ( n ) + c ′ ( n ) (cid:18) κ Vol(Ω)Vol(Γ) (cid:19) − n + (cid:16) c n ( n ) c ( n ) + c ′ ( n ) (cid:17) (cid:18) I (Ω) I (Ω) (cid:19) n − (cid:20) κ Vol(Ω)Vol(Γ) + β (cid:21) . (4.7)igenvalues of the Wentzel-Laplace operator 21We set ζ ( n ) := max { , c ( n ) , c n ( n ) , c ′ ( n ) , c n ( n ) c ( n ) + c ′ ( n ) } so that C ( ˜Ω , β ) ζ ( n ) ( (cid:18) I (Ω) I (Ω) (cid:19) n − (cid:20) κ Vol(Ω)Vol(Γ) + β (cid:21)) and C ( ˜Ω , β ) ζ ( n ) ( (cid:18) κ Vol(Ω)Vol(Γ) (cid:19) − n + (cid:18) I (Ω) I (Ω) (cid:19) n − (cid:20) κ Vol(Ω)Vol(Γ) + β (cid:21)) . In addition, since κ >
1, for all u ∈ V β \{ } we have˜ R β ( u ) = κ R Ω |∇ u | d M + β R Γ |∇ Γ u | d Γ κ R Γ u d Γ > κ R β ( u ) . Every orthonormal basis of a k -dimensional subspaces V ∈ V ( k ) of V β remains orthogonal with the metric ˜ g , then using the variation charac-terisation with (4.5), (4.6) and (4.7), we have λ βW,k (Ω) κ λ βW,k ( ˜Ω) κ e c ( n ) ( C (Ω , β ) + C ( ˜Ω , β ) k Vol( ˜Γ) ! n − ) κ ζ ( n ) ( (cid:18) κ Vol(Ω)Vol(Γ) (cid:19) − n + (cid:18) I (Ω) I (Ω) (cid:19) n − (cid:20) κ Vol(Ω)Vol(Γ) + β (cid:21)) + ζ ( n ) ( (cid:18) I (Ω) I (Ω) (cid:19) n − (cid:20) κ Vol(Ω)Vol(Γ) + β (cid:21) ) (cid:18) k Vol(Γ) (cid:19) n − , (4.8)where ζ ( n ) = e c ( n ) ζ ( n ) is a dimensional constant. (cid:3) Proof of Theorem 1.4.
We have for every r > sn ( r ) = r , then for every r > x ∈ M , one hasVol( B ( x, r )) ν ( n, , r ) = ρ n − Z r t n − d t = ω n r n and Vol( ∂B ( x, r )) ν ∂ ( n, , r ) = ρ n − ( r ) n − , ∀ x ∈ M. On the other hand, for every r > x ∈ M , B ( x, r ) can be covered by N := 32 n balls of radius r . As above, we take { B ( x i , r ) } Ni =1 a maximal familyof disjoint balls with center x i ∈ B ( x, r ). By the maximality assumption, thefamily { B ( x i , r ) } Ni =1 covers B ( x, r ). Let i ∈ { , . . . , N } such thatVol( B ( x i , r i N Vol( B ( x i , r . Then, since the balls B ( x i , r ) are pairwise disjoint, one has N Vol( B ( x i , r X i N Vol( B ( x i , r . B ( x i , r ) ⊂ B ( x i , r + r ) for every x i ∈ B ( x, r ), so N Vol( B ( x i , r Vol( B ( x, r . Hence, N Vol( B ( x, r ))Vol( B ( x i , r )) Vol( B ( x, r ))Vol( B ( x i , r )) Vol( B ( x i , r ))Vol( B ( x i , r )) . Using the volume comparison theorem (Bishop 1964, Gromov 1980), one hasVol( B ( x i , r ))Vol( B ( x i , r )) ω n (4 r ) n ω n ( r ) n n . Then follows from Theorem 3.6 with r − (Γ) = + ∞ , C = 2 and N r = 32 n . (cid:3) References [1] Bruno Colbois, Ahmad El Soufi, and Alexandre Girouard. Isoperimetric controlof the spectrum of a compact hypersurface.
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J. Differential Equations ,264(10):6486–6506, 2018. A¨ıssatou M. NDIAYEInstitut de math´ematiquesUniversit´e de NeuchˆatelSwitzerlandTel.: +41327182800e-mail:,264(10):6486–6506, 2018. A¨ıssatou M. NDIAYEInstitut de math´ematiquesUniversit´e de NeuchˆatelSwitzerlandTel.: +41327182800e-mail: