ESTIMATES FOR SUMS AND GAPS OF EIGENVALUES OF LAPLACIANS ON MEASURE SPACES

ESTIMATES FOR SUMS AND GAPS OF EIGENVALUES OF LAPLACIANS ON MEASURE SPACES

DA-WEN DENG AND SZE-MAN NGAI

Abstract. For Laplacians defined by measures on a bounded domain in Rⁿ, we prove analogs of the classical eigenvalue estimates for the standard Laplacian: lower bound of sums of eigenvalues by Li and Yau, and gaps of consecutive eigenvalues by Payne, P´olya and Weinberger. This work is motivated by the study of spectral gaps for Laplacians on fractals.

1. Introduction

One of the anomalous behaviors of Laplacians on fractals is the existence of spectral gaps, i.e., limk→∞λ_k+1/λ_k > 1, where λ_k are the eigenvalues. This is not possible for bounded domains on Rⁿ or on Riemannian manifolds. In fact, according to the Weyl law, on a compact connected oriented n-dimensional Riemannian manifold M,

(λ_k)^n/2 ∼ (2π)ⁿk

ω_nvol(M) as k → ∞,

where vol(M) is the volume of M, and ω_n is volume of the unit ball in Rⁿ (see, e.g., [2]).

Consequently, limk→∞λ_k+1/λ_k= 1. Strichartz [20] showed that the existence of spectral gaps implies better convergence of Fourier series. Rigorous proofs for the existence of spectral gaps have been obtained for only a limited number of fractals, such as the Sierpi´nski gasket and the Vicsek set (see [5,8,21]). For Laplacians defined by most self-similar measures, especially those with overlaps, it is not clear whether spectral gap exists. This is the main motivation of the present paper. This paper is also a continuation of the work by the authors’ [4] and by Pinasco and Scarola [17] on estimating the first eigenvalue of Laplacians with respect to fractal measures.

To describe some classical results, let Ω be a bounded domain on Rⁿ and let λ_k be the k-th Dirichlet eigenvalue. Li and Yau [10] obtained the following lower estimate for the sum

Date: January 17, 2019.

2010 Mathematics Subject Classification. Primary: 35P15, 28A80, 35J05. Secondary: 34L16, 65L15, 65L60.

Key words and phrases. Eigenvalue estimate; Fractal; measure; Laplacian.

This work is supported in part by the Hunan Provincial Natural Science Foundation of China 12JJ6007 and the National Natural Science Foundation of China, grants 11771136 and 11271122. The second author was also supported by the Center of Mathematical Sciences and Applications (CMSA) of Harvard University, the Hunan Province Hundred Talents Program, Construct Program of the Key Discipline in Hunan Province, and a Faculty Research Scholarly Pursuit Award from Georgia Southern University.

1

of the firstk eigenvalues

k

X

i=1

λ_i ≥ nC_n

n+ 2k^(n+2)/nvol(Ω)^−2/n, (1.1)

where vol(Ω) denotes the volume of Ω, and C_n= (2π)²Bn^−2/n, with B_n being the volume of the unit ball in Rⁿ.

An upper estimate was obtained by Kr¨oger [16]. The results of Li-Yau and Kr¨oger have been extended to homogeneous Riemannian manifolds by Strichartz [19].

For the gaps between consecutive eigenvalues, Payne, P´olya and Weinberger [15] (see also [18]) proved the following estimate for the gaps between two consecutive eigenvalues:

λ_k+1−λ_k ≤ 4Pk i=1λi

nk . (1.2)

The goal of this paper is to prove analogs of (1.1) and (1.2) for Laplacians defined by a measure µ. Let Ω ⊂ Rⁿ be a bounded open subset of Rⁿ and µ be a positive finite Borel measure with µ(Ω) > 0 and with support being contained in Ω. Under suitable conditions (see Section 2), µ defines a Dirichlet Laplacian −∆_µ; moreover, there exists an orthonormal basis {ϕ_n} consisting of eigenfunctions of −∆_µ and the eigenvalues λ_n satisfy 0< λ₁ ≤λ₂ ≤ · · ·, with limn→∞λ_n =∞. We remark that if µis the restriction of Lebesgue measure to Ω, then ∆µ is the classical Dirichlet Laplacian.

We first prove an analog of the classical lower estimate of the sum of eigenvalues of the standard Laplacian obtained by Li and Yau [10]. We let L²_µ(Ω) denote the Hilbert space of square-integrable functions with respect to µ. For u ∈ L²_µ(Ω), if there is no confusion of what Ω is, we let

kuk_µ=Z

Ω

|u|²dµ1/2

.

If µis the restriction of Lebesgue measure to Ω, we denote the corresponding L²-space and norm respectively by L²(Ω) and k · k.

Theorem 1.1. Let Ω⊆Rⁿ be a bounded domain, µ be a positive finite Borel measure on Ω with supp(µ)⊆Ω, −∆_µ be the Dirichlet Laplacian defined by µdescribed in Section 2, λ_k be the k-th eigenvalue of −∆_µ, and ϕ_k be the corresponding L²_µ(Ω)-normalized eigenfunction.

Then

k

X

j=1

λ_j ≥X^k

j=1

kϕ_jk²ⁿ⁺²₂

B_n sup

z∈Rⁿ k

X

j=1

|ϕˆ_j(z)|²−²

n n

n+ 2

≥X^k

j=1

kϕjk²

(2π)⁻ⁿvol(Ω)Bn

−_n² n n+ 2,

where vol(Ω) is the volume of Ω and B_n is the volume of the unit ball in Rⁿ.

Finally, we generalize the classical theorem by Payne, P´olya and Weinberger [15] on the gaps between two consecutive eigenvalues.

Theorem 1.2. Assume the hypotheses of Theorem 1.1. Then for all k ≥1, λ_k+1−λ_k ≤ 4Pk

i=1λ_i nPk

i=1kϕ_ik². (1.3)

Remark 1.1. We note that in (1.2), kϕ_jk >0 for all i. In fact, if ∇ϕ_j = 0, then, in view of the Poincar´e inequality for measures (see (2.1)), we would get kϕ_jk_µ= 0, a contradiction.

Both Theorems 1.1 and 1.2 involve the sumPk

i=1kϕ_ik², which suggests that it is necessary to study the eigenfunctions. We are not able to obtain a good estimate for this sum. Prop- erties of eigenfunctions in one-dimensional, especially when the support of µ is an interval, have been studied in [1]). In Section 3, we focus on the case when the support of µ is not an interval, such as a Cantor-type measure.

This paper is organized as follows. In Section 2, we recall the definition and some ele- mentary properties of the Dirichlet Laplacian ∆_µ defined on a domain by a measure µ. In Section 3 we prove the min-max principle for−∆µ and some properties of the eigenfunctions in one-dimension. Theorem 1.1 is proved in Section 4. Section 5 is devoted to the proof of Theorem 1.2. Finally in Section 6 we state some comments and open questions.

2. Preliminaries

For convenience, we summarize the definition of the Dirichlet Laplacian with respect to a measure µ; details can be found in [9]. Let Ω ⊂ Rⁿ be a bounded open subset and µ be a positive finite Borel measure with supp(µ)⊆Ω andµ(Ω)>0. We further supposeµsatisfies the following Poincar´e inequality (PI) for measures: There exists a constantC >0 such that

Z

Ω

|u|²dµ≤C Z

Ω

|∇u|²dx for all u∈C_c^∞(Ω). (2.1) Notice that (PI) cannot be immediately extended to H₀¹(Ω) functions. For example, let Ω = (0,1) ⊆ R and µ be the standard Cantor measure, which is supported on the Cantor set. For anyu∈H₀¹(Ω), if we increase the value of u on the Cantor set, R1

0 |∇u|²dxremains unchanged but R1

0 |u|²dµ can be increased within the same equivalence class of u without bound and hence the inequality cannot hold. However, the following is true. (PI) implies that each equivalence class u ∈ H₀¹(Ω) contains a unique (in L²_µ(Ω) sense) member ¯u that belongs toL²_µ(Ω) and satisfies both conditions below:

(1) There exists a sequence {u_n} in C_c^∞(Ω) such that u_n →u¯ in H₀¹(Ω) and u_n → u¯ in L²_µ(Ω);

(2) ¯usatisfies the inequality in (2.1).

We call ¯u the L²_µ(Ω)-representative of u. Consider our Cantor set example above. For u ∈H₀¹(Ω), let {u_n} ⊆C_c^∞(Ω) be a sequence convergent to u and hence Cauchy in H₀¹(Ω).

By (PI), {u_n} is Cauchy and hence convergent in L²_µ(Ω). Then ¯u is the function obtained by redefiningu on the Cantor set to be the L²_µ(Ω) limit of u_n.

Assume (PI) holds and define a mappingι :H₀¹(Ω) →L²_µ(Ω) by ι(u) = ¯u.

ι is a bounded linear operator, but not necessarily injective. Consider the subspace N of H₀¹(Ω) defined as

N :={u∈H₀¹(Ω) :kι(u)kµ= 0}.

Now let N^⊥ be the orthogonal complement of N in H₀¹(Ω). Then ι : N^⊥ → L²_µ(Ω) is injective. Throughout the rest of this paper, unless explicitly stated otherwise, we will use the L²_µ(Ω)-representative ¯uof u and denote it simply by u.

Consider a nonnegative bilinear form E(·,·) in L²_µ(Ω) given by E(u, v) :=

Z

Ω

∇u· ∇v dx (2.2)

with domain dom(E) = N^⊥, or more precisely, ι(N^⊥). (PI) implies that (E,dom(E)) is a closed quadratic form onL²_µ(Ω). Hence, there exists a nonnegative self-adjoint operator−∆_µ inL²_µ(Ω) such that dom(E) = dom((−∆_µ)^1/2) and

E(u, v) =h(−∆_µ)^1/2u,(−∆_µ)^1/2vi_L²_µ(Ω) for all u, v∈dom(E)

(see [3]). We call ∆_µthe(Dirichlet) Laplacian with respect to µ. It follows thatu∈dom(∆_µ) and −∆_µu =f if and only if −∆u= f dµ in the sense of distribution: for all ϕ ∈C_c^∞(Ω), R

Ω∇u· ∇ϕ dx = R

Ωf ϕ dµ (see [9, Proposition 2.2]). A real number λ ∈ R is a (Dirichlet) eigenvalue of −∆_µ with eigenfunction f if for all ϕ∈C_c^∞(Ω),

Z

Ω

∇f · ∇ϕ dx=λ Z

Ω

f ϕ dµ. (2.3)

From [9, Theorem 1.2], when µ satisfies (PI), there exists an orthonormal basis {ϕ_n}^∞_n=1 of L²_µ(Ω) consisting of (Dirichlet) eigenfunctions of −∆_µ. The eigenvalues {λ_n}^∞_n=1 satisfy 0 < λ₁ ≤ λ₂ ≤ · · ·. Moreover, if dim(domE) = ∞, then limn→∞λ_n = ∞. We have the following characterizations of domE and dom (−∆_µ):

domE =N^⊥= ^∞

X

n=1

c_nϕ_n:

∞

X

n=1

c²_nλ_n <∞

,

dom (−∆_µ) = ^∞

X

n=1

c_nϕ_n:

∞

X

n=1

c²_nλ²_n <∞

.

To state a sufficient condition for (PI), we recall that thelowerL^∞-dimensionof a measure µis defined by

dim_∞(µ) = lim inf

δ→0⁺

ln(sup_xµ(Bδ(x)))

lnδ ,

where the supremum is taken over all x∈supp(µ).

Theorem 2.1. ( [9, Theorems 1.1, 1.2])LetΩ⊆Rⁿ be a bounded open set andµbe a positive finite Borel measure on Rⁿ with supp(µ)⊆Ω and µ(Ω)>0. Assume dim_∞(µ)> n−2.

(a) (PI) holds. In particular, if n = 1, or n = 2 and µ is uppers-regular with s >0, or µ is absolutely continuous with bounded density, (PI) holds.

(b) The set of eigenvalues of −∆µ is contained in (0,∞) and has no accumulation point.

Hence −∆µ has a positive smallest eigenvalue λ^µ₁.

3. The min-max principle and properties of eigenfunctions

Let Ω be a bounded domain inRⁿandµbe a positive finite Borel measure with supp(µ)⊆ Ω. In this section we extend the variational principle for the principal eigenvalue and Courant’s min-max principle for the k-th eigenvalue to the Laplacians ∆_µ. This will be needed in the proof of Theorem 1.2. We first introduce some notation that will be needed in the proof of the theorem. Let (·,·)_µ denote the inner product in L²_µ(Ω). Also, for any subsetS ⊆L²_µ(Ω), lethSi be the vector subspace ofL²_µ(Ω) spanned by S, and letS^⊥ be the orthogonal complement of S inL²_µ(Ω).

Theorem 3.1. (Min-Max principle) Let ∆_µ be the Dirichlet Laplacian defined on a bounded domain Ω ⊆ Rⁿ and let 0 < λ1 ≤ λ2 ≤ · · · be the eigenvalues. Then for k = 1,2, . . ., the k-th eigenvalue satisfies

λk = max

S∈Σ_k−1min

E(u, u) :u∈S^⊥,kukµ= 1 , (3.1) whereΣk−1 is the collection of all(k−1)-dimensional subspaces ofdom (−∆_µ). In particular, for k = 1 we have the variational principle for the principal eigenvalue:

λ₁ = min

E(u, u) :u∈dom (−∆_µ),kuk_µ= 1 . (3.2) Proof. Step 1. Let{ϕ_k} ⊆L²_µ(Ω) be an orthonormal basis of L²_µ(Ω) with{ϕ_k} ⊂dom (−∆_µ) satisfying

−∆_µϕ_k =λ_kϕ_k in Ω

ϕ_k = 0 in ∂Ω (3.3)

for k = 1,2, . . .. Hence

E(ϕ_k, ϕ_l) = (λ_kϕ_k, ϕ_l)_µ=λ_kδ_kl, (3.4)

where δ_kl is the Kronecker delta. If u∈domE and kuk_µ= 1, then u=

∞

X

k=1

d_kϕ_k inL²_µ(Ω), (3.5)

where d_k = (u, ϕ_k)_µ, and the equality holds in the sense that ku−PN

k=1d_kϕ_kk_µ → 0 as N → ∞. Also,

∞

X

k=1

d²_k=kuk²_µ= 1. (3.6)

Step 2. Equation (3.4) implies that {ϕ_k/λ^1/2_k }^∞_k=1 ⊆ dom (−∆_µ) is an orthonormal set with respect to the inner product E(·,·). We claim that {ϕ_k/λ^1/2_k }^∞_k=1 is an orthonormal basis of dom (−∆_µ) with respect toE(·,·). To see this, we let u∈dom (−∆_µ) such thatE(ϕ_k, u) = 0 for all k ≥ 1. Then (λ_kϕ_k, u)_µ = 0 for all k ≥ 1 implies that (u_k, u)_µ = 0 for all k ≥ 1.

Thus, u = 0 µ-a.e. on Ω, which implies that u = 0 in dom (−∆_µ) (in the H₀¹-norm), since ι:N^⊥ →L²_µ(Ω) is an injection. Thus, for all u∈dom (−∆_µ),

u=

∞

X

k=1

a_k ϕ_k λ^1/2_k

, (3.7)

where a_k =E(u, ϕ_k/λ^1/2_k ). Observe that a_k=E

ϕ_k λ^1/2_k , u

= 1

λ^1/2_k (λ_kϕ_k, u)_µ= 1 λ^1/2_k

λ_kϕ_k,

∞

X

`=1

d_{`</sub>ϕ<sub>`}

µ

=λ^1/2_k d_k. Hence,

u=

∞

X

k=1

(λ^1/2_k d_k) ϕ_k λ^1/2_k

=

∞

X

k=1

d_kϕ_k in dom (−∆_µ). (3.8) Equations (3.8) and (3.4) imply that

E(u, u) = E ^∞

X

k=1

dkϕk,

∞

X

k=1

dkϕk

=

∞

X

k=1

d²_kλk ≥λ1.

Since E(ϕ₁, ϕ₁) =λ₁, (3.2) follows.

Step 3. Now let {ϕ_k}^∞_k=1 ⊆ L²_µ(Ω) be an orthonormal basis of L²_µ(Ω) satisfying (3.3). Let v ∈dom (−∆_µ) withkvk_µ= 1. Writev =P∞

i=1c_iϕ_i, where the equality holds in bothL²_µ(Ω) and dom (−∆_µ). As E(v, v) =P∞

i=1λ_ic²_i, λ_k = min

^∞ X

i=1

λ_ic²_i :c₁ =· · ·=ck−1 = 0,

∞

X

i=1

c²_i = 1

= min

E(v, v) :v ∈ hϕ₁, . . . , ϕk−1i^⊥, kvk_µ= 1 . We claim that this is equal to

S∈Σmaxk−1

min

E(v, v) :v ∈S^⊥,kvk_µ= 1}.

To prove this, let S∈Σ_k−1 and consider the following two cases.

Case 1. S =hv₁, . . . , vk−1i with v_` =P∞

i=1d<sub>`i</sub>ϕ<sub>i</sub> for `= 1, . . . , k−1 and det(d_{`i</sub>)<sup>k−1</sup><sub>`,i=1} = 0.

In this case, there exist c1, . . . , ck−1, not all zero, such that

k−1

X

i=1

c²_i = 1 and

k−1

X

i=1

d<sub>`i</sub>c<sub>i</sub> = 0, `= 1, . . . , k−1.

Letv :=Pk−1

i=1 c_iϕ_i ∈S^⊥.Then kvk_µ = 1 and min

v∈S^⊥,kvkµ=1

E(v, v)≤(v, v) =

k−1

X

i=1

λ_ic²_i ≤λ_k = min

v∈hϕ1,...,ϕk−1i^⊥E(v, v).

Case 2. S =hv₁, . . . , vk−1i, v_`=P∞

i=1d<sub>`i</sub>ϕ<sub>i</sub> for `= 1, . . . , k−1 with det(d_{`i</sub>)<sup>k−1</sup><sub>`,i=1} 6= 0.

Letv ∈ hϕ₁, . . . , ϕk−1i^⊥ withkvk_µ = 1, i.e.,v =P∞

i=kc_iϕ_i withP∞

i=kc²_i = 1 and E(v, v) = P∞

i=kλ_ic²_i. We will find v⁰ ∈S^⊥ with kv⁰k_µ= 1 such that E(v⁰, v⁰)≤ E(v, v).

Let ¯v =P∞

i=1c_iϕ_i (i.e., v has the same ϕ_i components as v for i≥k) such that v·v<sub>`</sub> = 0 for all ` = 1, . . . , k. That is,

k−1

X

i=1

d_`ic_i =

∞

X

i=k

d<sub>`i</sub>c<sub>i</sub>, `= 1, . . . , k−1.

Since det(d_{`i</sub>)<sup>k−1</sup><sub>`,i=1} 6= 0, a solution c₁, . . . , ck−1 exists (can be all 0). Let v⁰ := ¯v/k¯vk_µ. Note that

k¯vk²_µ=

k−1

X

i=1

c²_i +

∞

X

i=k

c²_i = 1 +

k−1

X

i=1

c²_i, and

k−1

X

j=1

λ_jc²_j ≤ ^k−1

X

j=1

c²_j

λ_k = ^k−1

X

j=1

c²_j ^∞

X

j=k

c²_j

λ_k ≤ ^k−1

X

j=1

c²_j ^∞

X

j=k

λ_jc²_j

.

Thus,

E(v⁰, v⁰) = 1

kvk²_µE(v, v) = 1 1 +Pk−1

i=1 c²_i

∞

X

j=1

λ_jc²_j = 1 1 +Pk−1

i=1 c²_i ^k−1

X

j=1

λ_jc²_j +

∞

X

j=k

λ_jc²_j

≤

∞

X

j=k

λjc²_j =E(v, v).

It follows that

min

v⁰∈S^⊥,kv⁰kµ=1E(v⁰, v⁰)≤ min

v∈hϕ1,...,ϕ_ki^⊥,kvkµ=1E(v, v).

This completes the proof.

The following proposition establishes some properties of eigenfunctions in one-dimension, some of them being specific for measures on bounded domains in R. Additional properties

of eigenfunctions can be found in [1]. Let L¹(E) be the one-dimensional Lebesgue measure of a subsetE ⊆R.

Proposition 3.2. Let Ω⊂Rbe a bounded open interval,µbe a positive finite Borel measure on Ω with supp(µ)⊆Ω, ∆_µ be the Dirichlet Laplacian with respect to µ, andϕ∈H₀¹(Ω) be an eigenfunction of −∆_µ, i.e., there exists λ ∈R such that −∆_µϕ=λϕ. Then

(a) ϕ∈C^0,1/2(Ω);

(b) ϕis linear over any component of Ω\supp(µ);

(c) if L¹(supp(µ)) = 0, then ϕ /∈ C²(Ω), and in fact, ϕ⁰ is not absolutely continuous (with respect to Lebesgue measure);

(d) eigenfunctions corresponding to the first eigenvalue do not change sign;

(e) the first eigenvalue is simple.

Proof. (a) It follows directly from Sobolev’s embedding theorem thatH₀¹(Ω) ,→C^0,1/2(Ω).

(b) Consider a component (a, b) of Ω\supp(µ). For allv ∈C_c^∞(a, b)⊆C_c^∞(Ω), Z

Ω

ϕ⁰v⁰dx=λ Z

Ω

ϕv dµ= 0,

and hence it also holds for continuous piecewise linear v with supp(v) ⊂ (a, b). Note that ϕ⁰|_(a,b) ∈L²(a, b)⊂L¹(a, b). Let

a < x−δ < x₁ < x₁+δ < x₂−δ < x₂ < x₂ +δ < b.

Let v ∈C(a, b) that is equal to 0 on (a, x₁−δ)∪(x₂+δ, b), equal to 1 on (x₁+δ, x₂ −δ), and linear over (x₁−δ, x₁+δ) and (x₂ −δ, x₂+δ). Then

0 = Z b

a

ϕ⁰v⁰dy=

Z x1+δ

x1−δ

ϕ⁰(y)v⁰(y)dy+

Z x2+δ

x2−δ

ϕ⁰(y)v⁰(y)dy

= 1 2δ

Z x1+δ

x1−δ

ϕ⁰(y)dy− 1 2δ

Z x2+δ

x2−δ

ϕ⁰(y)dy.

By the Lebesgue differentiation theorem, for Lebesgue a.e. x∈(a, b), ϕ⁰(x) = c, a constant.

As ϕ∈H₀¹(Ω) is absolutely continuous, for all x∈(a, b), ϕ(x) = ϕ(a) +

Z x

a

ϕ⁰(y)dy=ϕ(a) +c(x−a), completing the proof of (b).

(c) By part (b), ϕ⁰⁰ = 0 Lebesgue a.e. Hence, if ϕ⁰ is absolutely continuous, then for Lebesgue a.e. a, b∈Ω,

ϕ⁰(a)−ϕ⁰(b) = Z b

a

ϕ⁰⁰(y)dy= 0.

Thus, ϕ⁰ is a constant. Since ϕ∈ H₀¹(Ω), we conclude that ϕ≡ 0, contradicting that ϕ is an eigenfunction.

(d) By [1, Proposition 3.4], eigenfunctions corresponding to the first eigenvalue are concave or convex. As they vanish at the end points, they do not change sign.

(e) Let ϕ1 and ϕ2 be normalized functions corresponding to the first eigenvalue of −∆µ. Then ϕ1, ϕ2 ∈ C^0,α(Ω) ⊂C(Ω). If ϕ1 ≡ ϕ2 on supp(µ), then by linearity on components of Ω\supp(µ) and continuity, ϕ₁ ≡ϕ₂ on Ω. Thus ϕ₁ 6≡ϕ₂ if and only if ϕ₁ 6≡ϕ₂ on supp(µ).

Suppose that ϕ₁ and ϕ₂ are of the same sign, say positive, and ϕ₁ 6≡ϕ₂. If ϕ₁ ≥ϕ₂, then ϕ₁ > ϕ₂ on some subset E ⊂Ω withµ(E)>0. Hence

1 = Z

Ω

|ϕ₁|²dµ >

Z

Ω

|ϕ₂|²dµ= 1,

a contradiction. Thus, there existx₁, x₂ ∈Ω such thatϕ₁(x₁)> ϕ₂(x₁) andϕ₁(x₂)< ϕ₂(x₂).

Now ϕ=ϕ₁−ϕ₂ is an eigenfunction with ϕ(x₁)>0 and ϕ(x₂)<0, contradicting (d).

4. Lower estimate of sums of eigenvalues

We will use the a lemma from [10] which says that if f is a real-valued function defined onRⁿ with 0≤f ≤M₁, and

Z

Rⁿ

|z|²dz ≤M2, then

Z

Rⁿ

f(z)dz ≤(M₁B_n)ⁿ⁺²² M

n n+2

2

n+ 2 n

_n+2ⁿ

(4.1)

Proof of Theorem 1.1. Let Φ(x, y) =

k

X

j=1

ϕj(x)ϕj(y), x, y∈Ω and f(z) :=

Z

Ω

|Φ(z, y)|ˆ ²dµ(y), z ∈Rⁿ, (4.2) where

Φ(z, y) = (2π)ˆ ^−n/2 Z

x∈Rⁿ

Φ(x, y)e^−ix·zdx

is the Fourier transform of Φ at z and eachϕ_j is extended to Rⁿ by setting it equal to 0 on R^d\Ω. Then, using the linearity of the Fourier transform,

f(z) = Z

Ω

k

X

j=1

ˆ

ϕ_j(z)ϕ_j(y)

2

dµ(y)

= Z

Ω k

X

j,`=1

ˆ

ϕ_j(z)ϕ_j(y) ˆϕ_{`</sub>(z)ϕ<sub>`}(y)dµ(y)

=

k

X

j=1

ϕˆj(z)

2.

(4.3)

LetM₁ := sup_z∈

Rⁿ

Pk

j=1|ϕˆ_j(z)|². Then it follows that for all z ∈Rⁿ, 0≤f(z)≤M₁ ≤(2π)⁻ⁿ

k

X

j=1

kϕ_jk²_L1(Ω) ≤(2π)⁻ⁿvol(Ω)

k

X

j=1

kϕ_jk². (4.4) Also,

Z

Rⁿ

|z|²f(z)dz = Z

Ω

Z

Rⁿ

|z|²|Φ(z, y)|ˆ ²dz dµ(y)

= Z

Ω

Z

Rⁿ

|[∇_xΦ(z, y)|²dz dµ(y)

= Z

Ω

Z

Ω

|∇_xΦ(x, y)|²dx dµ(y) (Plancherel’s theorem)

= Z

Ω

Z

Ω

X^k

j=1

λ_jϕ_j(x)ϕ_j(y)X^k

`=1

ϕ_{`</sub>(x)ϕ<sub>`}(y)

dµ(x)dµ(y)

=

k

X

j=1

λ_j =:M₂,

(4.5)

where the fourth equality follows from (2.3). Using (4.3) followed by the Plancherel theorem, we get

Z

Rⁿ

f(z)dz =

k

X

j=1

kϕˆ_j(z)k² =

k

X

j=1

kϕ_jk². (4.6)

By applying [10, Lemma 1] (see (4.1)) to the functionf in (4.2), and using (4.5) and (4.6), we get

k

X

j=1

kϕ_jk² = Z

Rⁿ

f(z)dz ≤(M₁B_n)ⁿ⁺²² M

n n+2

2

n+ 2 n

_n+2ⁿ

=

B_n sup

z∈Rⁿ k

X

j=1

|ϕˆ_j(z)|²_n+2² X^k

j=1

λ_jn+2ⁿ n+ 2 n

_n+2ⁿ .

Thus, using (4.4), we get

k

X

j=1

λj ≥X^k

j=1

kϕjk²ⁿ⁺²_n

Bnsup

z∈Rⁿ k

X

j=1

|ϕˆj(z)|²−²

n n n+ 2

≥X^k

j=1

kϕjk²

(2π)⁻ⁿvol(Ω)Bn

−²_n n n+ 2

,

which completes the proof.

5. Upper estimate of gaps of eigenvalues

This section is devoted to generalizing the estimate of Payne, P´olya and Weinberger in 1.2 to Laplacians with respect to measures. We use the same notation as in Theorem 1.1.

Proof of Theorem 1.2. By the min-max principle, λ_k+1 = inf

R

Ω|∇v|²dx R

Ωv²dµ : Z

Ω

vϕ_idµ= 0, i= 1, . . . , k, v ∈dom (−∆_µ)

. (5.1)

Choose test functions v_i = gϕ_i −Pk

j=1a_ijϕ_j, where g will be chosen later. Obviously, v_i ∈domE. SupposeR

Ωv_iϕ_`dµ= 0. Then 0 =

Z

Ω

gϕ_iϕ_`dµ−

k

X

j=1

a_ij Z

Ω

ϕ_jϕ_`dµ= Z

Ω

gϕ_iϕ_{`</sub>dµ−a<sub>i`}.

Therefore,ai`=a`i and Z

Ω

v²_i dµ= Z

Ω

gϕ_iv_i−

k

X

j=1

a_ijϕ_jv_i dµ=

Z

Ω

gϕ_iv_idµ.

Notice that

Z

Ω

|∇v_i|²dx=h−∆v_i, v_ii_H⁻¹_,H¹

0. Since

∆v_i = (∆g)ϕ_i+ 2∇g · ∇ϕ_i+g∆ϕ_i−

k

X

j=1

a_ij∆ϕ_j

= (∆g)ϕ_i+ 2∇g · ∇ϕ_i−λ_igϕ_idµ+

k

X

i=1

a_ijλ_jϕ_jdµ,

we have Z

Ω

|∇vi|²dx=− Z

Ω

(∆g)viϕidx−2 Z

Ω

vi∇g· ∇ϕidx+λi

Z

Ω

gϕividµ.

Now,

−2

k

X

i=1

Z

Ω

v_i∇g·ϕ_idx=−2

k

X

i=1

g∇u·ϕ_i∇ϕ_idx+ 2X

j,i

a_ij Z

Ω

ϕ_j∇ϕ_i· ∇g dx

=

k

X

i=1

−1 2

Z

Ω

∇g²· ∇ϕ²_i dx

+X

i,j

a_ij Z

Ω

∇(ϕ_iϕ_j)∇g dx

=

k

X

i=1

1 2

Z

ϕ²_i∆g²dx−

k

X

i,j=1

aij

Z

Ω

ϕiϕj∆g dx.

(5.2)

Hence,

λ_k+1 Z

Ω

v²_i dµ≤

k

X

i=1

Z

|∇v|²dx=

k

X

i=1

− Z

v_iϕ_i∆g dx +1

2

k

X

i=1

Z

ϕ²_i∆g²dx

−

k

X

i,j=1

a_ij Z

Ω

ϕ_iϕ_j∆g dx+

k

X

i=1

λ_i Z

Ω

gϕ_iv_idµ

=−

k

X

i=1

Z

Ω

ϕ²_ig∆g dx+

k

X

i,j=1

a_ij Z

Ω

ϕ_iϕ_mj∆g dx+1 2

k

X

i=1

Z

Ω

ϕ²_i∆g²dx

−

k

X

i,j=1

a_ij Z

Ω

ϕ_iϕ_j∆g dx+

k

X

i=1

λ_i Z

gϕ_iv_idµ

=

k

X

i=1

Z

Ω

ϕ²_i|∇g|²+

k

X

i=1

λi

Z v_i²dµ

≤

k

X

i=1

Z

Ω

ϕ²_i|∇g|²dx+λ_k

k

X

i=1

Z

Ω

v_i²dµ.

Since for all i,

λ_k+1 Z

Ω

v_i²dµ≤ Z

Ω

|∇v_i|²dx,

we have

λ_k+1−λ_k ≤ Pk

i=1

R

Ωϕ²_i|∇g|²dx Pk

i=1

R

Ωv²_i dµ .

Now takeg =g_a(x) = Pn

i=1a_ix_i withPn

i=1a²_i = 1. Then ∆g = 0 and|∇g|= 1. It follows that

λ_k+1−λ_k ≤ Pk

i=1

R

Ωϕ²_i dx Pk

i=1v²_iadµ = Pk

i=1kϕ_ik² Pk

i=1

R

Ωv_ia² dµ, (5.3)

where via =ga(x)ϕi−Pk

j=1aijϕj. From (5.2),

k

X

i=1

Z

Ω

ϕ²_i dx=

k

X

i=1

kϕ_ik² =−2

k

X

i=1

Z

Ω

v_ia(∇g_a· ∇ϕ_i)dx

=−2

k

X

i=1

Z

Ω

via

Xⁿ

j=1

aj

∂ϕ_i

∂x_j

dx.

Normalize the measure onSⁿ⁻¹ so thatR

Sⁿ⁻¹ dS_a= 1. Then 1

2

k

X

i=1

kϕik² =−

k

X

i=1

Z

Sⁿ⁻¹

Z

Ω

via

Xⁿ

j=1

aj

∂ϕi

∂x_j

dxdSa

=− Z

Sⁿ⁻¹

Z

Ω

^k X

i=1

v_iaXⁿ

j=1

a_j∂ϕi

∂x_j

dxdS_a

≤Z

Ω

Z

Sⁿ⁻¹ k

X

i=1

v_ia² dS_adx1/2Z

Ω

Z

Sⁿ⁻¹ k

X

i=1

Xⁿ

j=1

a_j∂ϕ_i

∂x_j 2

dS_adx 1/2

.

Obviously, R

Sⁿ⁻¹a_ja_{`</sub> =δ<sub>j`}/N. Hence Z

Ω

Z

Sⁿ⁻¹ k

X

i=1

Xⁿ

j=1

a_j∂ϕ_i

∂x_j 2

dS_adx=X

i

Z

Ω

1 n

n

X

j=1

∂ϕ_i

∂x_j 2

dx=

k

X

i=1

1 n

Z

Ω

|∇ϕ_i|²dx

= 1 n

k

X

i=1

λ_i Z

Ω

ϕ²_i dµ= 1 n

k

X

i=1

λ_i.

It follows that

1 4

X^k

i=1

kϕ_ik²2

≤ 1 n

X^k

i=1

λ_iZ

Ω

Z

Sⁿ⁻¹ k

X

i=1

v_ia² dS_adx.

Finally, from (5.3) we get

(λ_k+1−λ_k)

k

X

i=1

Z

Sⁿ⁻¹

Z

Ω

v_ia² dxdS_a≤

k

X

i=1

kϕ_ik².

Hence

1 4

X^k

i=1

kϕ_ik²2

≤ 1 n

X^k

i=1

λ_iPk

i=1kϕik² λ_k+1−λ_k ,

which implies (1.3) and completes the proof.

Remark 5.1. In the case µ is Lebesgue measure, the inequality in Theorem 1.2 reduces to λ_k+1−λ_k ≤ 4Pk

i=1λ_i nk ,

which coincides with the classical Payne, P´olya and Weinberger inequality (see [15, 18]).

6. Comments and open problems

In view of Theorems 1.1 and 1.2, it is of interest to estimate the bound of the normkϕ_ik of the eigenfunctions ϕ_i that satisfy kϕ_ik_µ= 1.

An upper estimate for the sum of eigenvalues was obtained by Kr¨oger [16]. Let dist(x, ∂Ω) denote the distance from a point x ∈ Ω to the boundary of Ω. Let Ωr = {x ∈ Ω : dist(x, ∂Ω) < 1/r} and B₁ be a unit ball in Rⁿ. Kr¨oger proved that if there exists a

constant C_Ω⁽⁰⁾ such that vol(Ωr) ≤ (C_Ω⁽⁰⁾/r)vol(Ω)⁽ⁿ⁻²⁾ⁿ for every r > vol(Ω)^−1/n, then for every k ≥(C_Ω⁽⁰⁾)ⁿ,

k

X

j=1

λj ≤(2π)² n

n+ 2 vol(Ω)vol(B1)−2/n

k^(n+2)/n+C_n⁽¹⁾C_Ω⁽¹⁾k^(n+1)/n

, (6.1)

where Cn⁽¹⁾ is a constant depending only on the dimension n. It is of interest to generalize this result to Laplacians with respect to measures.

The spectral asymptotics of Laplacians defined on domains by fractal measures have been investigated and obtained by a number of authors (see [6,7,11–14] and the references therein).

It is of interest to find examples among these measure for which spectral gap exist.

Acknowledgements. Part of this work was carried out while the first author was visiting the Hunan Normal University, and the second author was visiting Xiangtan University and Harvard University. They are very grateful to the hosting institutions for their hospitality.

The second authors thanks Professor Shing-Tung Yau for the possibility to visit CMSA of Harvard University and for providing some valuable comments.

References

[1] E. J. Bird, S.-M. Ngai and A. Teplyaev, Fractal Laplacians on the unit interval, Ann. Sci. Math.

Qu´ebec27(2003), 135–168.

[2] I. Chavel, Eigenvalues in Riemannian geometry, Academic Press, Inc, Orlando, FL, 1984.

[3] E. B. Davies,Spectral theory and differential operators, Cambridge Studies in Advance Mathematics, vol. 42, Cambridge University Press, Cambridge 1995.

[4] D.-W. Deng and S.-M. Ngai, Eigenvalue estimates for Laplacians on measure spaces, J. Funct. Anal.

268(2015), 2231–2260.

[5] S. Drenning and R. S. Strichartz, Spectral decimation on Hambly’s homogeneous hierarchical gaskets, Illinois J. Math.53 (2009), 915–937.

[6] U. Freiberg, Spectral asymptotics of generalized measure geometric Laplacians on Cantor like sets, Forum Math.17(2005), 87–104.

[7] T. Fujita, A fractional dimension, self-similarity and a generalized diffusion operator, Probabilistic methods in mathematical physics (Katata/Kyoto, 1985), 83–90, Academic Press, Boston, MA, 1987.

[8] K.-E. Hare and D. Zhou, Gaps in the ratios of the spectra of Laplacians on fractals, Fractals 17 (2009), 523–535.

[9] J. Hu, K.-S. Lau and S.-M. Ngai, Laplace operators related to self-similar measures onR^d,J. Funct.

Anal.239(2006), 542–565.

[10] P. Li and S.-T. Yau, On the Schr¨odinger equation and the eigenvalue problem,Comm. Math. Phys.

88(1983), 309–318.

[11] H. P. McKean and D. B. Ray, Spectral distribution of a differential operator,Duke Math. J.29(1962) 281–292.

[12] K. Naimark and M. Solomyak, The eigenvalue behaviour for the boundary value problems related to self-similar measures onR^d,Math. Res. Lett.2(1995), 279–298.

[13] S.-M. Ngai, Spectral asymptotics of Laplacians associated with one-dimensional iterated function systems with overlaps,Canad. J. Math.63(2011), 648–688.

[14] S.-M. Ngai, W. Tang, and Y. Xie, Spectral asymptotics of one-dimensional fractal Laplacians in the absence of second-order identities,Discrete Contin. Dyn. Syst. 38(2018), 1849–1887.

[15] L. E. Payne, G. P´olya and H. F. Weinberger, On the ratio of consecutive eigenvalues, Stud. Appl.

Math.35(1956), 289–298.

[16] P. Kr¨oger, Estimates for sums of eigenvalues of the Laplacian,J. Funct. Anal.126(1994), 217–227.

[17] J. P. Pinasco and C. Scarola, Eigenvalue bounds and spectral asymptotics for fractal Laplacians, preprint.

[18] R. Schoen and S.-T. Yau, Lectures on Differential Geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, Vol. 1, International Press, Cambridge, 1994.

[19] R. S. Strichartz, Estimates for sums of eigenvalues for domains in homogeneous spaces, J. Funct.

Anal.137(1996), 152–190.

[20] R. S. Strichartz, Laplacians on fractals with spectral gaps have nicer Fourier series,Math. Res. Lett.

12(2005), 269–274.

[21] D. Zhou, Spectral analysis of Laplacians on the Vicsek set,Pacific J. Math.241(2009), 369–398.

Hunan Key Laboratory for Computational and Simulation in Science and Engineering, School of Mathematics and Computational Sciences, Xiangtan University, Hunan 411105, People’s Republic of China.

E-mail address: taimantang@gmail.com, tmtang@xtu.edu.cn

Key Laboratory of High Performance Computing and Stochastic Information Process- ing (HPCSIP) (Ministry of Education of China), College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, P. R. China, and Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460-8093, USA.

E-mail address: smngai@georgiasouthern.edu