Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem

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1 Journal of Functional Analysis ) Conforal upper bounds for the eigenvalues of the Laplacian and Stelov proble Asa Hassannezhad Departent of Matheatical Sciences, Sharif University of Technology, Tehran, Iran Received 1 February 011; accepted 9 August 011 Available online 5 August 011 Counicated by Cédric Villani Abstract In this paper, we find upper bounds for the eigenvalues of the Laplacian in the conforal class of a copact Rieannian anifold M, g). These upper bounds depend only on the diension and a conforal invariant that we call in-conforal volue. Asyptotically, these bounds are consistent with the Weyl law and iprove previous results by Korevaar and Yang and Yau. The proof relies on the construction of a suitable faily of disjoint doains providing supports for a faily of test functions. This ethod is interesting for itself and powerful. As a further application of the ethod we obtain an upper bound for the eigenvalues of the Stelov proble in a doain with C 1 boundary in a coplete Rieannian anifold in ters of the isoperietric ratio of the doain and the conforal invariant that we introduce. 011 Elsevier Inc. All rights reserved. Keywords: Eigenvalue; Upper bound; Laplacian; Stelov proble; Min-conforal volue 1. Introduction Let M, g) be a copact orientable -diensional Rieannian anifold. It is well nown that the spectru of the Laplace operator acting on functions is discrete and consists of a nondecreasing sequence {λ M, g)} =1 of eigenvalues each occurring with finite ultiplicity. If M has a sooth boundary then the sae conclusion is valid for Dirichlet, Neuann or other E-ail address: hasannezhad_63@ehr.sharif.ir /$ see front atter 011 Elsevier Inc. All rights reserved. doi: /j.jfa

2 340 A. Hassannezhad / Journal of Functional Analysis ) reasonable boundary conditions. By Weyl s law, the asyptotic behavior of λ is given by see e.g. [1]) ) λ M, g) α, 1) μ g M) where μ g is the Rieannian easure associated with g, α = 4π ω and ω is the volue of the unit ball in the standard R. A natural question suggested by this asyptotic forula is the following Question 1. Does there exist a constant C depending only on the diension such that we have for every N? λ M, g)μ g M) C ) An abundant literature has been devoted to this issue starting with Uraawa s paper [0]. It turns out that λ M, g)μ g M) cannot be bounded above only in ters of see for exaple [,4,5,17]). Consequently, the answer to Question 1 is negative. In the particular case of the first positive eigenvalue, El Soufi and Ilias [9] see also [10]) showed that an inequality lie ) holds with a constant C [g]) that depends on the conforal class [g] of the etric g naely, the conforal volue introduced by Li and Yau [18] who proved the sae but in diension ). In the case of surfaces, Yang and Yau [] proved inequality ) with a constant that only depends on the genus of the surface. In 1993, Korevaar [16] generalized these results to higher order eigenvalues. More precisely, Korevaar obtained the following upper bounds: i) If M,g)is a copact Rieannian anifold of diension, then for every N, λ M, g)μ g M) c [g] ), 3) where c [g]) is a constant depending only on the conforal class [g] of the etric g. ii) If Σ γ,g)is a copact orientable surface of genus γ, then for every N, λ Σ γ,g)μ g Σ γ ) Cγ + 1), 4) where C is a universal constant. Notice that inequality 4) provides an affirative answer to Yau s conjecture [1, p. 19]. Korevaar s results have been discussed by Groov [14] and revisited by Grigor yan and Yau [15] and Grigor yan, Netrusov and Yau [1] who proposed different proofs. Another iportant result in this direction was obtained by Buser [3] who proved that if M,g) is a copact -diensional Rieannian anifold whose Ricci curvature satisfies Ricci g 1)a, then for every N, ) 1) / λ M, g) a + β, 5) 4 μ g M) where β is a constant depending only on.

3 A. Hassannezhad / Journal of Functional Analysis ) Colbois and Maerten [8, Theore 1.3] proved a siilar result for bounded doains in a coplete anifold under Neuann boundary conditions. In the sae vein of the results of Korevaar and Buser, our ai in the present wor is to understand how inequality ) can be odified into a valid one. We obtain results that generalize those of Korevaar, Buser, and Colbois and Maerten entioned above. The ain feature of our approach is that the odification we propose consists in adding a ter depending on the conforal class [g] or the genus γ ) to the right-hand side of ), instead of letting the constant C depend on [g] or γ as in Korevaar s inequalities 3) and 4). The principal advantage of our approach lies in the fact that it enables us to recover the inequality ) for any integer that exceeds a threshold depending only on [g] or γ see Corollary 1.3 below). In order to state our ain result we need to introduce the following conforal invariant. If M, g) is a copact Rieannian anifold of diension, we define its in-conforal volue as follows: V [g] ) = inf { μ g0 M): g 0 [g], Ricci g0 1) }. Denoting by ρ g) the sallest nuber a 0 such that Ricci g 1)a, one can easily chec that V [g] ) = inf { μ g M)ρ g ) : g [g] } = inf { ρ g ) : g [g], μ g M) = 1 }. 6) Theore 1.1. There exist, for each integer, two constants A and B such that, for every copact Rieannian anifold M, g) of diension and every N, we have λ M, g)μ g M) A V [g] ) + B. 7) It is iportant to notice that the constant B in inequality 7) cannot be equal to the constant α in the Weyl law. Indeed, it follows fro [5, Corollary 1] that such a B ust satisfy: B ω. On the other hand, inequality 7) also gives an upper bound on the conforal spectru introduced by Colbois and El Soufi [5] and shows that its asyptotic behavior obeys a Weyl type law. Now, if a etric g is conforally equivalent to a etric g 0 with Ricci g0 0, then V[g]) = 0 see equality 6)). This leads to the following Corollary 1.1. See [16].) If a copact Rieannian anifold M, g) of diension is conforally equivalent to a Rieannian anifold with non-negative Ricci curvature, then where B is a constant depending only on. λ M, g)μ g M) B, 8) In the case of a copact orientable surface Σ γ of genus γ, the uniforization theore tells us that any Rieannian etric g on Σ γ is conforally equivalent to a etric of constant curvature. If γ, then g is conforally equivalent to a hyperbolic etric g γ. Thus,

4 34 A. Hassannezhad / Journal of Functional Analysis ) V[g]) μ gγ Σ γ ) = 4πγ 1), where the last equality follows fro Gauss Bonnet theore. If γ = 0, 1, then g is conforally equivalent to a positive constant curvature etric or a flat etric, respectively. Thus, V[g]) = 0 in the last two cases. Substituting in 7), one obtains the following iproveent of Korevaar s inequality 4). Corollary 1.. There exist two constants A and B such that, for every copact Rieannian surface Σ γ,g)of genus γ and every N, we have λ Σ γ,g)μ g Σ γ ) Aγ + B. 9) This result gives an upper bound to the topological spectru introduced by Colbois and El Soufi [5] and can be copared with the lower bound they obtained [5, p. 341]. In relation with Question 1, we have the following corollary which is a direct consequence of inequalities 7) and 9). Corollary 1.3. There exist a constant B R and, for each, a constant B R such that the following properties hold. 1) For any copact Rieannian anifold M, g) of diension, there exists an integer 0 [g]) depending only on the conforal class of g, such that, for every 0 [g]), λ M, g)μ g M) B ; ) For any copact Rieannian surface Σ γ,g) of genus γ, there exists an integer 0 γ ) depending only on γ, such that, for every 0 γ ), λ Σ γ,g)μ g Σ γ ) B. For any relatively copact doain Ω with C 1 boundary in a Rieannian anifold M, g),we denote by {λ Ω, g)} 1 the nondecreasing sequence of eigenvalues of the Neuann realization of the Laplacian in Ω. The ethod we will use to prove Theore 1.1 also allows us to obtain the following Theore 1.. Let M, g 0 ) be a coplete Rieannian anifold of diension with Ricci g0 M) 1). Let Ω M be a relatively copact doain with C 1 boundary and g be any etric conforal to g 0. Then for every N, we have λ Ω, g)μ g Ω) A μ g0 Ω) + B, 10) where A and B are constants depending only on the diension. It is easy to see that we can derive fro Theore 1.1 and Theore 1., inequalities of type 5) as obtained by Buser [3] and Colbois and Maerten [8] but with different constants. The paper is organized as follows: In Section we introduce the ain technical tool of the proof which consists in constructing of a suitable faily of capacitors, using the ethods of [1] and [8]. The proofs of Theore 1.1 and Theore 1. are given in Section 3. The last section is devoted to the Stelov eigenvalue proble. We prove that our ethod applies to this proble and give soe upper bounds for the Stelov eigenvalues.

5 A. Hassannezhad / Journal of Functional Analysis ) Construction of failies of capacitors in an space In this section, we present the ain technical tool of this paper. Let us start by recalling soe definitions. Throughout this section, the notation X,d,μ) will designate a coplete and locally copact etric-easure space space) with a etric d and a non-atoic finite Borel easure μ. Each pair F, G) of Borel sets in X such that F G is called a capacitor. Definition.1. Given κ>1 and N N, we say that a etric space X, d) satisfies the κ, N)- covering property if each ball of radius r>0 can be covered by N balls of radius r κ. Siilarly we define a local version of the covering property as follows: Definition.. Given κ>1, ρ>0 and N N, we say that a etric space X, d) satisfies the κ, N; ρ)-covering property if each ball of radius 0 <r ρ can be covered by N balls of radius r κ. Lea.1. If a etric space X, d) satisfies the κ, N; ρ)-covering property the κ, N)- covering property), then for any λ>1, it satisfies the λ, K; ρ)-covering property the λ, K)- covering property) for soe K = Kλ,κ,N) that does not depend on ρ. The proof of the lea when X, d) satisfies the κ, N)-covering property is given in [1, Lea 3.4]. For the κ, N; ρ)-covering property, the sae proof applies here verbati. Definition.3. For any x X and 0 r R, we define the annulus Ax, r, R) as Ax, r, R) := Bx,R) \ Bx,r) = { y X: r dx,y) < R }. For any annulus Ax, r, R) and λ 1, set λa := Ax, λ 1 r, λr). Siilarly, for any ball B = Bx,r) we set λb := Bx,λr). IfF X and r>0, we denote the r-neighborhood of F by F r, that is F r = { x X: dx,f) r }. In the following leas we recall two ethods for etric construction of disjoint doains. Lea.. See [1, Corollary 3.1].) Let X,d,μ) be an space satisfying the,n)- covering property. Then for every n N, there exists a faily A ={A i,b i )} n i=1 of capacitors in X such that a) for each i, A i is an annulus and μa i ) μx) cn, b) {B i } n i=1 are utually disjoint and B i = A i, where c is a positive constant depending only on N in fact one can tae c = +4K1600,,N), where K is the function defined in Lea.1). Lea.3. See [8, Corollary.3] and [6, Lea.1].) Let X,d,μ) be an space satisfying the,n; 1)-covering property. For every n N,letr>0 be such that for each x X,

6 344 A. Hassannezhad / Journal of Functional Analysis ) μbx, r)) μx) 4Ñ n, where Ñ = K4,,N). Then there exists a faily A ={A i,a r i )}n i=1 of capacitors in X such that a) for each i, μa i ) μx), and Ñn b) the subsets {A r i }n i=1 are utually disjoint. In the original stateent of Lea.3, X, d) is supposed to have the 4, Ñ; 1)-covering property. According to Lea.1, one can replace the 4, Ñ; 1)-covering property by the,n; 1)-covering property. The ain construction given in the following theore results fro a erging of the two previous leas. It consists in constructing a disjoint faily of capacitors. Theore.1. Let X,d,μ) be an space satisfying the,n; 1)-covering property. Then for every n N, there exists a faily of capacitors A ={F i,g i )} n i=1 with the following properties: i) μf i ) ν := μx), where c is as in Lea.; 8c n ii) the G i s are utually disjoint; iii) the faily A is such that either a) all the F i s are annuli and G i = F i, with outer radii saller than one, or b) all the F i s are doains in X and G i = F r 0 i, with r 0 = Proof. In order to find a desired faily of capacitors, we start with the ethod used by Grigor yan, Netrusov and Yau [1, proof of Theore 3.5]. We will call their ethod GNYconstruction. However, we do not have the,n)-covering property in order to apply directly the GNY-construction. Roughly speaing, we will see that when an space X has the local covering property i.e.,n; 1)-covering property), the GNY-construction is applicable to the assive part ofx i.e. where balls of radii r 0 have easure greater than ν). If the nuber of capacitors built using the - GNY-construction on the assive part is not equal to n, then we introduce a new easure on X. The support of this new easure is a subset of the copleent of the assive part. We shall see that in this case the ethod of Colbois and Maerten Lea.3) that we will call CM-construction, is applicable. Let us define τ 1 := sup { r: μ Bx,r) ) ν x X }. If τ 1 r 0 then we follow the step 1 see below). Otherwise we ove on to the step in order to apply the CM-construction. Step 1. Applying GNY-construction. Assue τ 1 r 0. We essentially follow the steps of the GNY-construction. However, it is necessary to ae soe adaptations since our covering property is of local nature. We use the sae foralis and notations that is used in the GNY-construction see [1, p. 17]). Our goal is to construct by induction two sequences {A i } and {B i } where A i is a faily of annuli in X, and B i is a faily of balls that cover A i. These two failies satisfy the following properties:

7 A. Hassannezhad / Journal of Functional Analysis ) i) for each a A i we have μa) ν; ii) the annuli {a} a Ai are disjoint; iii) for each a A i, the outer radius of a is saller than one; iv) the following inclusions hold 1 a 4 b; a A i b B i v) we have the inequality ) μ b cνi; b B i vi) A 1 = B 1 =1 and if i>1 then either A i = A i 1 +1 and B i B i 1 +1, or A i = A i 1 and B i B i 1 1, where A denotes the cardinal of the faily A; vii) if i>1, then A i 1 A i ; viii) if i>1, then b B i 1 b b B i b. Observe that by vi) the sequence of { A i B i } is strictly increasing with respect to i and, since A 1 B 1 =1, one has A i i. Notice that if we can continue the inductive process till i = n, then we get a faily A = A n of at least n capacitors satisfying the desired properties i), ii) and iii)a) of Theore.1. However, here we only have a local covering property rather than a global one. In order to perfor the induction, we will need to fix an upper bound on the radii of balls in B i this restriction is crucial to have property v)). This restriction does not always allow us to continue the inductive process till i = n. To start the induction, tae r τ 1, τ 1 ]. Then there exists a point x 0 X such that μ Bx 0,r) ) ν. We define A 1 ={Bx 0,r)} and B 1 ={Bx 0, 8r)}. It is easy to see that properties i), ii), iii), iv), vi), vii) and viii) are satisfied. Let us verify property v). Since 8r 16τ 1 < 1, by Lea.1, one can cover Bx 0, 8r) by K16,,N)balls of radii r/ <τ 1. Therefore, μ Bx 0, 8r) ) K16,,N)ν <cν, which proves the property v). Assue now we have constructed A 1,...,A i and B 1,...,B i for soe i<n. It follows fro the property iv) for the faily B i that

8 346 A. Hassannezhad / Journal of Functional Analysis ) μ X ) b >μx) icν > μx) ncν = μx) nc μx) 8c n b B i = 1 1 ) μx) > μx) >ν, 11) 4c because c>1. Hence, there exists x i X such that μ Bx i,r) b B i b ) >ν. 1) We define { τ i+1 := sup r: μ Bx,r) ) } b ν x X. b B i At this stage the continuation of the construction process depends on the size of τ i+1. If τ i+1 >r 0, we ove on to the step. If τ i+1 r 0, we construct failies A i+1 and B i+1 as follows. We can assue that r τ i+1, τ i+1 ] in 1). We denote by κ the cardinal of: B := {b B i : Bx i, 7 4r) 1 } b. Following the GNY-construction, we define A i+1 and B i+1 according to the following alternatives for ore details see [1, pp ]): Case κ = 0. We define A i+1 and B i+1 by A i+1 = A i { Bx i,r) } and B i+1 = B i { Bx i, 8r) }. Case κ. We define A i+1 and B i+1 by A i+1 = A i and B i+1 = B i \{all balls in the set B} ) { Bx i, 98 8r) }. Note that the ball Bx i, 98 8r) contains all balls in B see [1, p. 175]). Case κ = 1. If there exists a ball b = By,s) B such that Bx i, r) 1 b, then we define A i+1 and B i+1 by { A i+1 = A i A y, 1 )} s,8r and B i+1 = B i { Bx i, 14 8r) }. Notice that Ay, 1 s,8r) Bx i, 14 8r) see [1, p. 177]). Otherwise we define A i+1 and B i+1 lie in the case κ = 0.

9 A. Hassannezhad / Journal of Functional Analysis ) Now let us prove that these two failies have the properties i) viii). The properties vi), vii) and viii) are clearly satisfied in each of the three cases. To chec the conditions i), ii) and iv), we can use word-for-word the arguents given in [1, pp ]. Indeed, this part of their proof is independent of covering properties. Let us verify the condition v). In each of the three cases, we see that B i+1 \ B i =1. Let us denote by b i+1 the unique ball in B i+1 \ B i. According to the three cases, the radius r i+1 of b i+1 is at ost 98 8r. Since r τ i+1, τ i+1 ],wehave r i τ i+1 < 1600τ i+1 1, 13) where the last inequality follows fro the assuption τ i+1 r 0. By Lea.1, the ball b i+1 can be covered by K1600,,N)<c balls with radii r i τ i+1. Therefore ) ) ) ) μ b = μ b + μ b i+1 b cνi + μ b i+1 b b B i+1 b B i b B i b B i+1 cνi + K1600,,N)ν cνi + cν cνi + 1), which proves the condition v). It reains to chec the condition iii). For this, it is enough to verify that the outer radius of the annulus a A i+1 \A i is saller that one. One can see in each of the three cases, A i+1 \A i B i+1 \ B i ={b i+1 }. By inequality 13), the radius of b i+1 is saller than one and it proves the condition iii) for A i+1. Step. Applying CM-construction. Assue τ i >r 0 for soe 1 i n. It eans that ifi = 1, then μbx, r 0 )) ν, for all x X; if1<i n, then μbx, r 0 ) \ b B i 1 b) ν, for all x X. We consider the space X, d, μ i ) where μ i := μ if i = 1; μ i A) := μa \ b B i 1 b) if 1 <i n. and It follows fro inequality 11) and the above inequalities that μ i X) > μx), μ i Bx,r0 ) ) μx) 8c n μ ix) 4Ñ n. Consequently, the space X, d, μ i ) satisfies the assuptions of Lea.3. Therefore, there exists a faily {A j,a r 0 j )} of n capacitors in X such that the A r 0 j s are utually disjoint and μ i A j ) μ ix) Ñn μx) 8c n.

10 348 A. Hassannezhad / Journal of Functional Analysis ) Since μa j ) μ i A j ), this faily of capacitors satisfies the conditions i), ii) and iii)b) of Theore.1. The following proposition shows that for a sufficiently large integer n, it is always possible to apply the GNY-construction to obtain a faily of n capacitors satisfying the properties i), ii) and iii)a) of Theore.1. The application of this observation to the eigenvalue proble is discussed in Rear 3. of the next section. Proposition.1. Let X,d,μ) be a copact space satisfying the,n; 1)-covering property. Then there exists a positive integer X such that for every n> X, there exists a faily A of n utually disjoint capacitors in X that satisfies the properties i), ii) and iii)a) of Theore.1. Proof. Since X is copact, we can cover X by T balls of radii r 0 = Set X = T 4c. It is enough to show that for every n> X and 1 i n, wehaveτ i r 0. Indeed, suppose that there exists an integer j n such that τ j >r 0. Then by the definition of τ j,wehavethe following inequality It follows fro the above inequality that μx) μ j X) μ μ j Bx,r0 ) ) ν = μx) 8c n. 14) x i X, 1 i T ) Bx i,r 0 ) T μx) 8c n. Hence n should be saller than T. Therefore, τ 4c j r 0 for every j n. It follows that at the step i = n of the inductive process see the proof of Theore.1, step 1), we have a faily of n utually disjoint capacitors satisfying the proposition, which copletes the proof. 3. Eigenvalues estiates on Rieannian anifolds In this section we apply Theore.1 to a special case of spaces which are Rieannian anifolds, in order to prove Theore 1.1 and Theore 1.. The arguents we use to prove these two theores are siilar. We start by giving in details the proof of Theore 1.. Definition 3.1. Let M,g) be a Rieannian anifold of diension. Thecapacity of a capacitor F, G) in M is defined by cap g F, G) = inf g ϕ dμ g, ϕ T M

11 A. Hassannezhad / Journal of Functional Analysis ) where T = T F, G) is the set of all copactly supported Lipschitz functions on M such that supp ϕ G = G \ G and ϕ 1 in a neighborhood of F.IfT F, G) is epty, then cap g F, G) =+. Siilarly, we can define the -capacity as cap ) [g] F, G) = inf ϕ T M g ϕ dμ g. Since is the diension of M, it is clear that the -capacity depends only on the conforal class [g] of the etric g. Proposition 3.1. Under the assuptions of Theore 1., tae the space Ω, d g0,μ), where d g0 is the Rieannian distance corresponding to the etric g 0 and μ is a non-atoic finite easure on Ω. Then for every n N, there exists a faily of capacitors A ={F i,g i )} n i=1 with the following properties: i) μf i ) μω) 8c n ; ii) the G i s are utually disjoint; iii) the faily A is such that either a) all the F i s are annuli, G i = F i and cap ) [g 0 ] F i, F i ) Q,or b) all the F i s are doains in Ω and G i = F r 0 i, where r 0 = and, c and Q are constants depending only on the diension. Proof. Let us start with the observation that the etric space Ω, d g0 ) satisfies the,n; 1)- covering property. For each ball Bx,r) with center in Ω and radius saller than 1, tae a axial faily {Bx i,r/4)} of disjoint balls with centers in Bx,r). Letκ be the cardinal of that faily. The faily of balls {Bx i,r/)} covers Bx,r). Hence κ in μ g0 Bxi,r/4) ) i i μ g0 Bxi,r/4) ) μ g0 Bx,r + r/4) ). Tae x i0 such that μ g0 Bx i0,r/4)) = in i μ g0 Bx i,r/4)). Wehave κ μ g 0 Bx, r + r/4)) in i μ g0 Bx i,r/4)) μ g 0 Bx, r)) μ g0 Bx i0,r/4)) μ g 0 Bx i0, 4r)) μ g0 Bx i0,r/4)). Since Ricci g0 Ω) 1), thans to the Bishop Groov volue coparison theore, we have 0 <s<r, r μ g0 Bx, r)) μ g0 Bx, s)) 0 sinh 1 tdt s 0 sinh 1 tdt. Since for every positive t one has t sinh t te t, we get μ g0 Bx, r)) μ g0 Bx, s)) ) r e 1)r. s

12 3430 A. Hassannezhad / Journal of Functional Analysis ) In particular, we have and, r <1, μ g0 Bx,r) ) r e 1)r 15) κ μ g 0 Bx i0, 4r)) μ g0 Bx i0,r/4)) 4 e 4 1)r =: Cr) C1). 16) One can tae N = C1) and deduce that Ω, d g0 ) has the,n; 1) covering property where N depends only on the diension. Now the proof of Proposition 3.1 is a straightforward consequence of Theore.1. Recall that in the stateent of Theore.1, the constant c depends only on N. Therefore, in our case c depends only on the diension. It reains to verify that in the case of annuli, there exists a constant Q depending only on the diension such that for each i, wehave cap ) [g 0 ] F i, F i ) Q. According to Theore.1, the outer radii of the annuli we consider are saller than one. It is enough to show that for each point x Ω and 0 r<r 1/, we have where A = Ax, r, R). Set cap ) [g 0 ] A, A) Q, 17) 1 if x Ax, r, R), d g0 x,bx,r/)) r if x Ax, r/,r)= Bx,r) \ Bx,r/), fx)= 1 d g 0 x,bx,r)) R if x Ax, R, R) = Bx,R) \ Bx,R), 0 if x M \ Ax, r/, R). It is clear that f T A, A) and g0 fx) r, on Bx,r) \ Bx,r/), g0 f 1, x Bx,R) \ Bx,R). R Therefore cap ) [g 0 ] A, A) M r r g0 f dμ g0 ) ) 1 μ g0 Ax, r/,r) + R ) ) 1 μ g0 Bx,r) + R ) ) μ g0 Ax, R, R) ) ) μ g0 Bx,R).

13 A. Hassannezhad / Journal of Functional Analysis ) Now since r, R 0, 1], using inequality 15), one can control the last inequality by a constant Q depending only on the diension which copletes the proof of inequality 17). Rear 3.1. Since Cr) defined in 16) is a strictly increasing function of r, it follows that Ω, d g0 ) does not necessarily satisfy the,n)-covering property for soe N depending only on the diension. Now we show how Theore 1. follows fro Proposition 3.1. Proof of Theore 1.. Tae the space Ω, d g0,μ Ω ), where μ Ω = μ g Ω. According to Proposition 3.1, there exists a faily {F i,g i )} of 3 capacitors which satisfies the properties i), ii) and either iii)a) or iii)b) of the proposition. We consider each case separately. Case 1. If {F i,g i )} 3 i=1 is a faily with the properties i), ii) and iii)a) of Proposition 3.1, then ), 18) λ Ω, g) A μ g Ω) where A = 4c Q ). Indeed, we begin by choosing a faily of 3 test functions {f i : f i T F i,g i )} 3 i=1 such that g0 f i dμ g0 cap ) [g 0 ] F i,g i ) + ɛ. Therefore, Rf i ) = M Ω gf i dμ g Ω f i Ω g 0 f i dμ g0 ) Ω 1 supp f i dμ g ) 1 dμ g Ω f i dμ g cap) [g 0 ] F i,g i ) + ɛ) μ Ω G i )) 1. 19) μ Ω F i ) The first inequality follows fro Hölder inequality and, because of the conforal invariance of g f i dμ g, we have replaced g by g 0. Since the G i s are disjoint doains and 3 i=1 μ ΩG i ) μ g Ω), at least of the have easure saller than μ gω). Up to re-ordering, we assue that for the first of the G i s we have μ Ω G i ) μ gω). 0) Now, we can tae ɛ = Q. Using Proposition 3.1i) and iii)a) and inequality 0), we get fro inequality 19) Rf i ) A μg Ω)) 1 μ g Ω) ) = A, μ g Ω) with A = 4c Q ), which copletes the proof of Case 1.

14 343 A. Hassannezhad / Journal of Functional Analysis ) Case. If {F i,g i )} 3 then i=1 is a faily with the properties i), ii) and iii)b) of Proposition 3.1, λ Ω, g) B ) μg0 Ω), 1) μ g Ω) where B = 4c. r0 Indeed, we define the test functions f i as follows 1 if x F i, f i x) = 1 d g 0 x,f i ) r 0 if x G i \ F i ), 0 if x G c i. We have g0 f i 1 r 0. Therefore, Rf i ) = Since the G i s are disjoint, we have Ω gf i dμ g M f i Ω g 0 f i dμ g0 ) Ω 1 supp f i dμ g ) 1 dμ g Ω f i dμ g 1 μ r0 g0 G i Ω)) μ Ω G i )) 1. ) μ Ω F i ) 3 i=1 μ g0 G i Ω) μ g0 Ω) and 3 i=1 μ Ω G i ) μ g Ω). Hence, there exist at least sets aong G 1,...,G 3 such that μ g0 G i ) μ g 0 Ω). Siilarly, there exist at least sets not necessarily the sae ones) such that μ g G i ) μ gω). Therefore, up to re-ordering, we assue that the first of the G i s satisfy both of the two following inequalities μ Ω G i ) μ gω) and μ g0 G i Ω) μ g 0 Ω). 3) Using Proposition 3.1i) and inequalities 3), we get fro inequality ) Rf i ) B B μg0 Ω) ) μg Ω) μ g Ω) ) μg0 Ω) μ g Ω) with B = 4c, which copletes the proof of Case. r0 ) 1

15 A. Hassannezhad / Journal of Functional Analysis ) In both cases, λ Ω, g) is bounded above by the su of the right-hand sides of 18) and 1), which copletes the proof. Rear 3.. To avoid a possible confusion, it is judicious to exaine the proof of Theore 1.. In the proof, we begin with the GNY-construction but the ethod ay brea down for soe j<n in the sense that we ay not be able to find j or ore) disjoint sall annuli. In such a case, inequality 14) holds. The validity of inequality 14) iplies that the CM-construction is applicable with r = r 0 which gives an estiate for λ of the for given in inequality 1). This ay appear to be unreasonable since the right-hand side is independent of. However, as pointed out in Proposition.1, the GNY-construction for a given copact Rieannian anifold is applicable for all n sufficiently large, but we have no control over the constants and how large n should be. The ethod described above enables one to establish the validity of the estiate for those finite nuber of s for which the GNY-construction is not applicable. Proof of Theore 1.1. Consider the space M, d g0,μ g ), where d g0 is the distance associated with the etric g 0 and μ g is the easure associated with the etric g. We easily see that we can follow the sae arguents as in the proof of Theore 1. to derive the following inequality λ M, g)μ g M) A μ g0 M) + B. 4) The left-hand side does not depend on g 0. Hence, we can tae the infiu with respect to g 0 [g] such that Ricci g0 1), which leads to the desired conclusion. 4. Stelov eigenvalues It is worth pointing out that Theore.1 is foralized in a general setting and is applicable to other eigenvalue probles. In this section we present an application of this theore to the Stelov eigenvalue proble Stelov proble Let Ω be a bounded subdoain of a coplete -diensional Rieannian anifold M, g) and assue that Ω has nonepty sooth boundary Ω. Given a function u H 1 Ω), we denote by ū the unique haronic extension of u to Ω, that is { g ū = 0 inω, ū = u on Ω. Let ν be the outward unit noral vector along Ω. The Stelov operator is the ap L : H 1 Ω) H 1 Ω), u ū ν. The operator L is an elliptic pseudo-differential operator see [19, pp ]) which adits a discrete spectru tending to infinity denoted by 0 = σ 1 σ σ 3....

16 3434 A. Hassannezhad / Journal of Functional Analysis ) The eigenvalue σ of L can be characterized variationally as follows see [7]): { Ω σ Ω) = inf sup gū dμ g V Ω ū d μ g :0 ū V }, 5) where V is a -diensional linear subspace of H 1 Ω) and μ g is the Rieannian easure associated to g on the boundary. The relationships between the geoetry of the doain and the spectru of the corresponding Stelov operator have been investigated by several authors see for exaple [7,11,13]). Recently, Fraser and Schoen [11, Theore.3] proved the following inequality for the Stelov eigenvalues of a copact Rieannian surface Σ γ,g)of genus γ and κ boundary coponents: σ Σ γ )l g Σ γ ) γ + κ)π, where l g Σ) is the length of the boundary. This result was generalized to higher eigenvalues by Colbois, El Soufi and Girouard [7, Theore 1.5]. Indeed, the authors proved the following inequality for every N σ Σ γ )l g Σ γ ) Cγ + 1), 6) where C is a universal constant. For a doain in a higher diensional anifold, the authors [7, Theore 1.3] also obtained an upper bound for σ depending on the isoperietric ratio of the doain. More precisely, if M, g) is conforally equivalent to a coplete anifold with non-negative Ricci curvature, then for every bounded doain Ω of M and every N, σ Ω) μ g Ω) 1 1 C, 7) I g Ω) where I g Ω) is the isoperietric ratio I g Ω) = μ g Ω) ) and C μ g Ω) 1 is a constant depending only on. The theore below is otivated by the wor of [7], and we obtain an iproveent of inequalities 6) and 7) using Proposition 3.1. Theore 4.1. Let M, g 0 ) be a coplete Rieannian anifold of diension with Ricci g0 M) 1). Let Ω M be a relatively copact doain with C 1 boundary and g be any etric conforal to g 0. Then we have σ Ω) μ g Ω) 1 1 A μ g0 Ω) + B where A and B are constants depending only on. I g Ω) 1 1 1, 8) As an iediate consequence we get the following inequality in the case of Rieann surfaces:

17 A. Hassannezhad / Journal of Functional Analysis ) Corollary 4.1. Let Σ γ,g) be a copact oriented Rieannian surface with genus γ, and Ω be a subdoain of Σ γ. Then where A and B are constants. σ Ω)l g Ω) Aγ + B, 9) Proof of Theore 4.1. We consider the space Ω, d g0, μ), where μa) := μ g A Ω). We apply again Proposition 3.1. Therefore, there exist a faily of 3 capacitors {F i,g i )} satisfying properties i), ii) and either iii)a), or iii)b) of Proposition 3.1. We proceed analogously to the proof of Theore 1.. Using the variational characterization of σ, we construct a faily of test functions as in Case 1 and Case of the proof of Theore 1.. In both cases, we have Ω σ Ω) gf i dμ g Ω f i Ω g 0 f i dμ g0 ) μ g G i ) 1. d μ g μf i ) If the faily {F i,g i )} satisfies the properties i), ii) and iii)a) of Proposition 3.1, then μg Ω) σ Ω) A μ g Ω) ) 1 A μ g Ω) 1 1 I g Ω) ) If on the other hand, the faily {F i,g i )} satisfies the properties i), ii) and iii)b) of Proposition 3.1, then μg0 Ω) σ Ω) B ) μg Ω) μ g Ω) ) 1 B μ g0 Ω) μ g Ω) 1 1 I g Ω) 1 1 1, 31) where the constant coefficients A and B are the sae as A and B in Theore 1.. The proof of inequalities 30) and 31) are along the sae lines as Theore 1.. In both cases, σ Ω) is bounded above by the su on the right-hand sides of 30) and 31), and it copletes the proof. Acnowledgents This paper is part of the author s PhD thesis under the direction of Professors Bruno Colbois Neuchâtel University), Ahad El Soufi François Rabelais University), and Alireza Ranjbar- Motlagh Sharif University of Technology). The author wishes to express her thans to her supervisors for suggesting the proble and for any stiulating conversations. She is also grateful to Mehrdad Shahshahani for helpful coents and the referee for several suggestions and rears on the anuscript which led to iproveents in the presentation. References [1] P. Bérard, Spectral Geoetry: Direct and Inverse Probles, Lecture Notes in Math., vol. 107, [] R. Broos, E. Maover, Rieann surfaces with large first eigenvalue, J. Anal. Math ) [3] P. Buser, A note on the isoperietric constant, Ann. Sci. Ec. Nor. Super. 4) 15 ) 198)

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