Bounds for eigenvalues of the Dirichlet problem for the logarithmic Laplacian

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Preprint submitted on 9 Nov 2020 (v2), last revised 4 Nov 2021 (v3)

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Bounds for eigenvalues of the Dirichlet problem for the logarithmic Laplacian

Huyuan Chen, Laurent Véron

To cite this version:

Huyuan Chen, Laurent Véron. Bounds for eigenvalues of the Dirichlet problem for the logarithmic Laplacian. 2020. �hal-02977991v2�

Bounds for eigenvalues of the Dirichlet problem for the logarithmic Laplacian

Huyuan Chen¹

Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, PR China

Laurent V´eron²

Laboratoire de Math´ematiques et Physique Th´eorique, Universit´e de Tours, 37200 Tours, France

Abstract

We provide bounds for the sequence of eigenvalues{λi(Ω)}i of the Dirichlet problem L∆u=λu in Ω, u= 0 in R^N \Ω,

where L_∆ is the logarithmic Laplacian operator with Fourier transform symbol 2 ln|ζ|. The logarithmic Laplacian operator is not positively definitive if the volume of the domain is large enough, hence the principle eigenvalue is no longer always positive. We also give asymptotic estimates of the sum of the firstkeigenvalues. To study the principle eigenvalue, we construct lower and upper bounds by a Li-Yau type method and calculate the Rayleigh quotient for some particular functions respectively. Our results point out the role of the volume of the domain in the bound of the principle eigenvalue. For the asymptotic of sum of eigenvalues, lower and upper bounds are built by a duality argument and by Kr¨oger’s method respectively. Finally, we obtain the limit of eigenvalues and prove that the limit is independent of the volume of the domain.

Contents

1 Introduction and main results 1

2 Upper bounds for the principle eigenvalue 7

2.1 Large domain: proof of Theorem 1.2 -(i) . . . 7 2.2 Small domain: proof of Theorem 1.2 -(ii) . . . 9

3 Lower bounds 11

4 Wey’s limits 16

4.1 Upper bounds for the sum of eigenvalues . . . 16 4.2 Proof of Theorem 1.6 and 1.7 . . . 17 4.3 Discussion about the counting function . . . 20 Keywords: Dirichlet eigenvalues; Logarithmic Laplacian.

MSC2010: 35P15; 35R09.

1 Introduction and main results

LetL∆ be the logarthmic Laplacian inR^N,N ≥1, defined by L∆u(x) =cN

Z

R^N

u(x)1_B1(x)(y)−u(y)

|x−y|^N dy+ρNu(x), (1.1)

1chenhuyuan@yeah.net

2veronl@univ-tours.fr

where

cN :=π^−N/2Γ(N/2) = 2

ω_N−1, ρN := 2 ln 2 +ψ(^N₂)−γ, (1.2) ω_N−1 :=H^N−1(S^N−1) =R

SN−1 dS,γ =−Γ⁰(1) is the Euler Mascheroni constant andψ= ^Γ_Γ⁰ is the Digamma function.

The aim of this article is to provide estimates of the eigenvalues of the operatorL∆in a bounded domain Ω ⊂ R^N which are the real numbers λ such that there exists a solution to the Dirichlet problem

(L∆u=λu in Ω,

u= 0 in R^N \Ω. (1.3)

In recent years, there has been a renewed and increasing interest in the study of boundary value problems involving linear and nonlinear integro-differential operators. This growing interest is justified both seminal advances in the understanding of nonlocal phenomena from a PDE or a probabilistic point of view, see e.g. [3–6, 14, 15, 21, 32, 35, 36] and the references therein, and by important applications. Among nonlocal differential order operators, the simplest and most studied examples, are the fractional powers of the Laplacian which exhibit many phenomenological properties. Recall that, fors∈(0,1), the fractional Laplacian of a functionu∈C_c^∞(R^N) is defined by

F((−∆)^su)(ξ) =|ξ|^2su(ξ)b for all ξ∈R^N,

where and in the sequel both F andb·denote the Fourier transform. Equivalently, (−∆)^s can be written as a singular integral operator under the following form

(−∆)^su(x) =c_N,s lim

→0⁺

Z

R^N\B(x)

u(x)−u(y)

|x−y|^N+2sdy, (1.4)

wherecN,s = 2^2sπ^−N2s^Γ(

N+2s 2 )

Γ(1−s) and Γ is the Gamma function, see e.g. [36].

The fractional Laplacian has the following limiting properties when s approaches the values 0 and 1:

s→1lim⁻(−∆)^su(x) =−∆u(x) and lim

s→0⁺(−∆)^su(x) =u(x) foru∈C_c²(R^N), see e.g. [14]. Recently, [8] shows a further expansion at s= 0 that foru∈C_c²(R^N) and x∈R^N,

(−∆)^su(x) =u(x) +sL∆u(x) +o(s) as s→0⁺ where, formally, the operator

L∆:= d ds

s=0(−∆)^s (1.5)

is given as a logarithmic Laplacian; indeed,

(i) for 1< p≤ ∞, we have L∆u∈L^p(R^N) and ^(−∆)_s^su−u →L∆uin L^p(R^N) as s→0⁺; (ii) F(L_∆u)(ξ) = 2 ln|ξ|bu(ξ) for a.e. ξ ∈R^N.

Note that the problems with integral-differential operators given by kernels with a singularity of order−N have received growing interest recently, as they give rise to interesting limiting regularity properties and Harnack inequalities without scaling invariance, see e.g. [24]. Another important domain of study consists in understanding the eigenvalues of the Dirichlet problem with zero exterior value [8]. We refer to [?, 19] for more topics related to the logarithmic Laplacian and also [16, 23]

for general nonlocal operator and related embedding results. Let H(Ω) denote the space of all measurable functionsu:R^N →R withu≡0 in R^N\Ω and

ZZ

x,y∈RN

|x−y| ≤1

(u(x)−u(y))²

|x−y|^N dxdy <+∞.

As we shall see it, H(Ω) is a Hilbert space under the inner product E(u, w) = cN

2 Z Z

x,y∈RN

|x−y| ≤1

(u(x)−u(y))(w(x)−w(y))

|x−y|^N dxdy, where c_N is given in (1.2), with associated norm kuk_H(Ω) =p

E(u, u). By [13, Theorem 2.1], the embedding H(Ω),→L²(Ω) is compact. Throughout this article we identify L²(Ω) with the space of functions in L²(R^N) which vanish a.e. in R^N \Ω. The quadratic form associated with L∆ is well-defined onH(Ω) by

E_L:H(Ω)×H(Ω)→R, E_L(u, w) =E(u, w)−c_N ZZ

x,y∈RN

|x−y| ≥1

u(x)w(y)

|x−y|^N dxdy+ρ_N Z

R^N

uw dx,

whereρ_N is defined in (1.2). A functionu∈H(Ω) is an eigenfunction of (1.3) corresponding to the eigenvalueλif

E_L(u, φ) =λ Z

Ω

uφ dx for allφ∈H(Ω).

Proposition 1.1. [8, Theorem 1.4] LetΩbe a bounded domain inR^N. Then problem (1.3) admits a sequence of eigenvalues

λ₁(Ω)< λ₂(Ω)≤ · · · ≤λ_i(Ω)≤λ_i+1(Ω)≤ · · · and corresponding eigenfunctions φ_i, i∈N such that the following holds:

(a) λ_i(Ω) = min{E_L(u, u) : u∈Hi(Ω) : kuk_L2(Ω)= 1}, where H1(Ω) :=H(Ω) and Hi(Ω) :={u∈H(Ω) :

Z

Ω

uφidx= 0 for i= 1, . . . i−1} for i >1;

(b) {φ_i : i∈N} is an orthonormal basis of L²(Ω);

(c) φ1 is positive in Ω. Moreover, λ1(Ω) is simple, i.e., ifu∈H(Ω)satisfies (1.3) in weak sense withλ=λ₁(Ω), then u=tφ₁ for some t∈R;

(d) lim

i→∞λ_i(Ω) = +∞.

Due to lack of the homogenous property for the logarithmic Laplacian operator, the effect of the domain for the principle eigenvalue can’t be expected as the Laplacian or fractional Laplacian, just by scaling the domain by their homogeneous property of such operators. Secondly, the logarithmic Laplacian operator is no longer positively definitive if |Ω| is to large, since it is proved in [8] that the positivity of the principle eigenvalue is equivalent to the comparison principle, which does not hold for balls with large radius. These properties of the logarithmic Laplacian operator enrich the asymptotics of the Dirichlet eigenvalues as we will see below, but also make more difficult the obtention of bounds for eigenvalues.

Observe that the inclusion H(O₁) ⊂ H(O₂) implies that the mapping O 7→ λ₁(O) is nonin- creasing, i.e. λ1(O1) ≥ λ1(O2) if O1 ⊂ O2. Our first results deal with upper and lower bounds on the the principle eigenvalue and they are connected both with the measure and the distortion of the domain. We denote by H^k the k-dimensional Hausdorff measure in R^N and for simplicity H^N(E) =|E|for any Borel set E ⊂R^N. We also define the signed distance function to∂Ω by

ρ(x) =

( dist(x, ∂Ω) if x∈Ω,

−dist(x, ∂Ω) if x∈Ω^c (1.6) and, forν >0, the internal and external foliations of∂Ω by

T_ν⁺ = n

x∈R^N :ρ(x) =ν o

⊂Ω (1.7)

and

T_ν⁻= n

x∈R^N :ρ(x) =−νo

⊂Ω^c, (1.8)

respectively, andTν =T_ν⁺∪T_ν⁻.

Theorem 1.2. LetΩbe a bounded domain inR^N andλ₁(Ω)be the principle eigenvalue of Dirichlet problem (1.3) obtained in Proposition 1.1.

(i) ForR >2, if we assume that B_R⊂Ω⊂B_2R,and that there existsc₀ >1 depending only on N such that for any ν ∈[0,¹₂), there holds

1

c₀R^N−1≤H^N−1(T_ν⁺)≤c₀R^N−1 if N ≥2. (1.9) Then for R≥maxn

2,_2ω^{N c0}

N−1

o

, we have that

λ1(Ω)≤ω_N−1ln 1

R +z1(R), (1.10)

where

z1(R) =ρ_N +ω_N−1ln 2 +4c0

R

1 + c0

2ω_N−1R

.

(ii) ForR∈(0,¹₄), if we assume again that B_R⊂Ω⊂B_2R,and that there exists c₀ >1 such that for any|ν| ∈[0,^R₄], there holds

1

c₀R^N−1 ≤H^N−1(T_ν)≤c₀R^N−1 if N ≥2. (1.11) Then

λ1(Ω)≤4 ln 1

R +c1, (1.12)

where c₁ >0 independent of R.

Since the function z1 is decreasing, estimate (1.10) indicates that there exists R^∗ such that λ1(Ω)<0 whenR > R^∗. Furthermore,

λ₁(Ω)≤ω_N−1ln 2

R +ρ_N +O(R⁻¹) as R → ∞.

Our upper bounds are obtained by considering the Rayleigh quotient λ1(Ω) ≤ R^E(u,u)

Ωu²dx with the particular function u = wσ(x) = min{max{σρ(x),0},1}, where ρ is defined in (1.6) and σ > 0.

However, the dominating term of upper bounds arises from c_NRR

x,y∈RN

|x−y| ≥1

wσ(x)wσ(y)

|x−y|^N dx when R is large, while it does fromE(w_σ, wσ) for R >0 small enough.

Next we prove lower bounds of the principle eigenvalue.

Theorem 1.3. Assume that Ω is a bounded domain in R^N and λ1(Ω) is the principle eigenvalue of Dirichlet problem (1.3) obtained in Proposition 1.1. Let

d_N = 2ω_N−1

N²(2π)^N. (1.13)

Then we have that (i)

λ₁(Ω)≥ −d_N|Ω|;

(ii) if|Ω|< _{eN d}²

N,

λ₁(Ω)>0;

(iii) if |Ω| ≤ _ee+1²N dN,

λ₁(Ω)≥ 2 N

ln 2

eN dN|Ω|

−ln ln 2 eN dN|Ω|

,

where e is the Euler number.

We summarize our results by the following table of the main asymptotic term of principle eigenvalue with respect to the volume of domain Ω from the upper bound in Theorem 1.2 and the lower bound in Theorem 1.3 in the particular where Ω =BR where 0< r0 <1< R0<+∞:

R (0, r0) (R0,+∞)

Upper bound of λ1(Ω) 4 ln_R¹ ω_N−1ln_R¹ Lower bound of λ1(Ω) 2 ln_R¹ −^2eω_N2^N−1_d

N R^N

The main order asymptotic of principle eigenvalue with respect toR.

From above table we note that the lower bound of principle eigenvalue for large value R is rather unprecise.

The Hilbert-P´olya conjecture is to associate the zero of the Riemann Zeta function with the eigenvalue of a Hermitian operator. This quest initiated the mathematical interest for estimating the sum of Dirichlet eigenvalues of the Laplacian while in physics the question is related to count the number of bound states of a one body Schr¨odinger operator and to study their asymptotic distribution. In 1912, Weyl in [37] shows that the k-th eigenvalue µ_k(Ω) of Dirichlet problem with the Laplacian operator has the asymptotic behaviorµk(Ω)∼CN(k|Ω|)^N2 as k→+∞, where C_N = (2π)²|B₁|^−N2. Later, P´olya [33] (in 1960) proved that

µ_k(Ω)≥C k

|Ω|

_N²

(1.14) holds for C =C_N and any ”plane-covering domain” D inR², (his proof also works in dimension N ≥3) and he also conjectured that (1.14) holds with C =C_N for any bounded domain in R^N. Rozenbljium [34] and independently Lieb [29] proved (1.14) with a positive constantC for general bounded domain. Li-Yau [28] improved the constant C = _N+2^N C_N, and with that constant (1.14) is also called Berezin-Li-Yau inequality because this constant is achieved with the help of Legendre transform as in the Berezin’s earlier paper [2]. The Berezin-Li-Yau inequality then is generalized in [11–13,25,29,31], for degenerate elliptic operators in [9,22,38] for the fractional Laplacian (−∆)^s defined in (1.4) and the inequality reads

µ_s,k(Ω)≥ N N + 2sC_N

k

|Ω|

^2s_N

. (1.15)

Due to the expression of the Fourier symbol of L∆, Berezin-Li-Yau method can not be applied to our problem (1.3). Our results are based on the appropriate estimates for the solutions of equations:

rlnr =c and r

lnr−ln lnr =t.

The estimates that we obtain provide a uniform lower bound of the sum of the firstk-eigenvalues, independently ofk, an estimate which has a particular interest when these eigenvalues are negative.

More precisely, we have the following inequalities:

Theorem 1.4. Let Ω ⊂ R^Nbe a bounded domain, {λ_i(Ω)}_i∈N be the sequence of eigenvalues of problem (1.3) obtained in Proposition 1.1 andd_N be given in (1.13). Then there holds

(i) for any k∈N^∗,

k

X

i=1

λi(Ω)≥ −d_N|Ω|;

(ii) if k > ^{eN d}₂^N|Ω|,

k

X

i=1

λ_i(Ω)>0;

(iii) if k≥ ^ee+1₂^{N dN}|Ω|,

k

X

i=1

λi(Ω)≥ 2k N

lnk+ ln 2

eN dN|Ω|

−ln ln

2k eN dN|Ω|

. (1.16)

Using the monotonicity of the sequence of eigenvalues, we deduce the following lower bound for λk(Ω) from Theorem 1.4 part (iii).

Corollary 1.5. Under the assumption of Theorem 1.4, for k≥ ^ee+1₂^{N dN}|Ω|, we have that λ_k(Ω)>0

and

λ_k(Ω)≥ 2 N

lnk+ ln 2 eN dN|Ω|

−ln ln 2k eN dN|Ω|

. (1.17)

Our goal is to provide an upper bound for the sum of eigenvalues. Motivated by Kr¨oger’s result for the Laplacian [25], we prove the following upper bound.

Theorem 1.6. Let Ω ⊂R^N be a bounded domain and {λ_i(Ω)}_i∈N be the sequence of eigenvalues of problem (1.3). Then for k > k0 := ¹₂e^N+1dN|Ω|,

k

X

i=1

λi(Ω)≤ 2k

N ln(k+ 1) + ln pN

|Ω|

+ ω_N−1 p|Ω|ln ln

pN(k+ 1)

|Ω|

!

(1.18) and

k→+∞lim (klnk)⁻¹

k

X

i=1

λ_i(Ω) = 2

N, (1.19)

where p_N = ^2(2π)_ω ^NN

N−1 .

Note that from (1.19) the limit of the sum of the first k-eigenvalues does not depend on the volume of Ω. Finally, we build the Wely’s formula for the logarithmic Laplacian and indeed we have the following asymptotic estimate.

Theorem 1.7. Let Ω ⊂R^N be a bounded domain and {λ_i(Ω)}_i∈N be the sequence of eigenvalues of problem (1.3). Then

k→+∞lim λ_k(Ω)

lnk = 2

N. (1.20)

It is worth noting that

(a) we have the same limits of ^λk_lnk^(Ω) and (klnk)⁻¹Pk

i=1λ_i(Ω) ask→+∞;

(b) Weyl’s estimate (1.20) is derived by the lower bound and the upper bound of the first k- eigenvalues directly.

Usually, the asymptotic behavior of eigenvalues is derived by the counting functions. Inversely, the estimates of counting functions could be deduced by the asymptotic behavior of eigenvalues.

Let N(t) be the counting function of {λ_k(Ω)}_k∈N, counts the number of eigenvalues below t > 0, i.e.

N(t) =X

j∈N

sgn₊(t−λ_j(Ω)) =X

j∈N

(t−λ_j(Ω))⁰₊ (1.21)

Here sgn+(r) = 1 if r > 0, sgn+(r) = 0 if r ≤ 0 and r± = (|r| ±r)/2 denotes the positive and negative part of x ∈R. The counting function can also be expressed by introducing the trace of an operator

N(t) = tr(L_∆−t)⁰₋.

Note that from the bounds ofλ_k(Ω), we can obtain estimates for the counting function

t→+∞lim N(t)e^−(N2+δ)t= 0 and lim inf

t→+∞N(t)e^{−(N2−δ)t}= +∞.

By analyzing the asymptotic behavior of P

j∈N(t−λj(Ω))+.

The rest of this paper is organized as follows. In Section 2, we build the upper bound of the principle eigenvalue of Theorem 1.2 by considering particular test functions in the Rayleigh quotient. Section 3 is devoted to proving the lower bound by developing Li-Yau’s method, and then we prove Theorem 1.3, Theorem 1.4 and Corollary 1.5. In Section 4, we show the upper bounds for the first k-eigenvalues in Theorem 1.6 and prove the Wely’s limit of eigenvalues in Theorem 1.7. Finally, we obtain the estimates for the counting function.

2 Upper bounds for the principle eigenvalue

2.1 Large domain: proof of Theorem 1.2 -(i) Set

η(t) = min{max{0, t},1} for all t∈R, and, forσ >0,

w_σ(x) =η(σ⁻¹ρ(x)) for all x∈R^N. (2.1) Note thatwσ ∈H(Ω),wσ →1 in Ω as σ →0⁺, and forσ ∈(0,2R],

wσ(x) = 1

σρ(x) for all x∈Ω.

Since |η⁰| ≤1 and the signed distance function ρ is a contraction mapping, there always hols

|w_σ(x)−wσ(y)| ≤ 1

σ|x−y| for all x, y∈R^N. (2.2) By definition ofλ1(Ω),

λ1(Ω)≤ inf

σ>0

E_L(w_σ, w_σ) R

Ωw_σ²(x)dx. (2.3)

Ifσ ≥2R, we have

Z

Ω

w²_σ(x)dx=σ⁻² Z

Ω

ρ²dx, while if σ <2R, there holds

Z

Ω

w²_σ(x)dx=σ⁻² Z

Ωσ

ρ²dx+ Z

Ω\Ωσ

dx

=σ⁻² Z

Ωσ

ρ²dx+|Ω| − |Ω_σ|,

where

Ωσ = n

x∈R^N : 0< ρ(x)< σ o

⊂Ω. (2.4)

Then

Z

Ω

w²_σ(x)dx=σ⁻² Z

Ωσ

ρ²dx+|Ω| − |Ω_σ|

≥ |Ω| − |Ω_σ|

Takingσ ≤ ¹₂, using (1.9) and the co-area formula since |∇ρ(x)|= 1, we have that

|Ω| − |Ω_σ|= Z

Ω\Ωσ

|∇ρ(x)|dx=|Ω| − Z σ

0

H^N−1(T_t⁺)dt

≥ |Ω| −c0σR^N−1 =|Ω| − N c0σ ω_N−1R|B_R|,

sinceN|B₁|=ω_N−1. Hence, under the assumption of Theorem 1.2 -(i), we have that Z

Ω

w²_σ(x)dx≥

1− N c₀σ ω_N−1R

|Ω| for all σ ∈(0,¹₂). (2.5) Concerning the term E_L(wσ, wσ), we have the following upper bound.

Lemma 2.1. Under the assumption of Theorem 1.2-(i), let σ= ¹₄, then we have that E_L(w_σ, w_σ)≤ 4c₀

R |Ω|+

ρ_N −ω_N−1lnR 2

Z

Ω

w_σ²dx. (2.6)

Proof. We recall that

E_L(wσ, wσ) =E(wσ, wσ)−cN

ZZ

x,y∈RN

|x−y| ≥1

wσ(x)wσ(y)

|x−y|^N dxdy+ρN

Z

Ω

w²_σdx

and our proof is divided into two parts.

Step 1 : Note that forx, y∈R^N such that |x−y| ≤1, we have that

|w_σ(x)−wσ(y)|= 0, ∀(x, y)∈(Ω\Ωσ)²∪(Ω^c)². Using (2.2), we have that

E(w_σ, w_σ) = c_N 2

Z Z

x,y∈RN

|x−y| ≤1

(w_σ(x)−w_σ(y))²

|x−y|^N dxdy

≤cNσ⁻² Z

Ωσ

Z

B1(x)

|x−y|^2−Ndydx

= c_Nω_N−1 2σ² |Ω_σ|

≤ c₀

Rσ|Ω|= 4c₀ R |Ω|, thanks to the identity c_Nω_N−1 = 2.

Step 2 : We have that c_N

ZZ

x,y∈RN

|x−y| ≥1

w_σ(x)w_σ(y)

|x−y|^N dydx=c_N Z

Ω

Z

Ω∩B^c₁(x)

w_σ(x)w_σ(y)

|x−y|^N dydx.

Note that forσ= ¹₄, we have that

x∈Ωinf |Ω\(Ω_σ∪B₁(x))| ≥ |Ω| −1

4c₀R^N−1−ω_N−1 N .

Set

D₁(σ, x) = (Ω\B1(x))∩Ωσ = Ωσ∩B₁^c(x) and

D₂(σ, x) = (Ω\B₁(x))∩(Ω\Ω_σ) = Ω∩Ω^c_σ∩B₁^c(x).

Then D₁(σ, x)∩ D₂(σ, x) =∅,D₁(σ, x)∪ D₂(σ, x) = Ω∩B^c₁(x). If x∈BR 2

, then BR

2(x)∩B^c₁(x)⊂ D₂(σ, x), which implies Z

D2(σ,x)

1

|x−y|^Ndy≥ Z

BR 2

(x)∩B₁^c(x)

1

|x−y|^Ndy=ω_N−1lnR 2 and

Z

Ω∩B₁^c(x)

wσ(y)

|x−y|^Ndy= Z

D1(σ,x)

σρ(y)

|x−y|^Ndy+ Z

D2(σ,x)

1

|x−y|^Ndy > ω_N−1lnR 2, since the first term of the right-hand side is positive. Thus, we obtain

Z

Ω

Z

Ω∩B₁^c(x)

wσ(x)wσ(y)

|x−y|^N dydx≥ω_N−1lnR 2

Z

Ω

w_σ(x)dx

≥ω_N−1lnR 2

Z

Ω

w_σ²(x)dx.

As a consequence and sinceσ= ¹₄, we infer that E_L(wσ, wσ)≤ 4c0

R |Ω|+

ρN −ω_N−1lnR 2

Z

Ω

w_σ²dx.

We complete the proof.

Proof of Theorem 1.2 -(i). SinceR > _2ω^{N c0}

N−1, we have that

1− N c0

4ω_N−1R −1

≤1 + N c0

2ω_N−1R and

λ₁(Ω)≤ E_L(w_σ, w_σ) R

Ωw²_σ(x)dx ≤ 4N c₀ R

1− c₀ 4ω_N−1R

−1

+ρ_N −ω_N−1lnR 2

≤ −ω_N−1lnR+ρN +ω_N−1ln 2 +4c₀ R

1 + N c₀ 2ω_N−1R

,

which is the result.

2.2 Small domain: proof of Theorem 1.2 -(ii) The following upper estimate of E_L(w_σ, w_σ) holds.

Lemma 2.2. Under the assumptions of Theorem 1.2 -(ii), there exists c₂ > 0 independent of R such that for σ= min

nR

4,^2Rω_{N c}^N−1

0

o ,

E_L(wσ, wσ)≤ 2 ln 1

R +c2

|Ω|+ρN

Z

Ω

w_σ²dx. (2.7)

Proof. Since Ω⊂B2Rand R < ¹₄, there holds

E_L(w_σ, w_σ) =E(w_σ, w_σ) +ρ_N Z

Ω

w²_σdx.

Forr >0, we denote

Ω^1,r ={x∈R^N : |ρ(x)|< r} and Ω^2,r ={x∈R^N : ρ(x)>−r}= Ω∪Ω^1,r, and for r₁ > r₂ >0 set

A^r1,r2 = Ω^2,r1 \Ω^2,r2 ={x∈R^N : −r₂≥ρ(x)>−r₁}.

Note that forr >2R, Ω⊂Ω^1,r = Ω^2,r since Ω⊂B2R. When σ∈(0,^R₄], we set D={(x, y)∈R^N ×R^N s.t. |x−y|<1 and |w_σ(x)−w_σ(y)|>0}.

Then D ⊂ D₁∪ D₂, where

D₁ = (Ω^1,σ×Ω^2,σ)∪(Ω^2,σ×Ω^1,σ) and D₂ = (A^1,σ×Ω)∪(Ω× A^1,σ).

Using (2.2), |Ω^2,σ| ≤ |B_2R+σ(x)|, we obtain from the definition ofD₁ and since cNω_N−1 = 2, cN

2 Z Z

D₁

(wσ(x)−wσ(y))²

|x−y|^N dxdy ≤cNσ⁻² Z

Ω^1,σ

Z

Ω^2,σ

|x−y|^2−Ndydx

≤cNσ⁻² Z

Ω^1,σ

Z

B2R+σ(x)

|x−y|^2−Ndydx

= cNω_N−1 2 (2R

σ + 1)²|Ω^1,σ|

≤2c₀(2R

σ + 1)²R^N−1σ

≤ N cNc0σ R (2R

σ + 1)²|Ω|.

On the other hand, c_N

2 ZZ

D₂

(w_σ(x)−w_σ(y))²

|x−y|^N dxdy≤c_N Z

A^1,σ

Z

Ω

|x−y|^−Ndydx

=c_N Z

Ω

Z

A^1,σ

|x−y|^−Ndxdy

≤2|Ω|ln2 σ , since for anyy∈Ω,x∈ A^1,σ =⇒x∈B₂(y)\B_σ(y), which implies

Z

A^1,σ

|x−y|^−Ndx≤ Z

B2(y)\Bσ(y)

|x−y|^−Ndx=ω_N−1ln2 σ. Takingσ =%0R with

%₀= min 1

4,ω_N−1 2N c₀

, we obtain that

E(w_σ, wσ) = cN

2 Z Z

x,y∈RN

|x−y| ≤1

(wσ(x)−wσ(y))²

|x−y|^N dxdy

≤N c_Nc₀σ R (2R

σ + 1)²+ 2 ln2 σ

|Ω|

<

2 ln 1

R +c2

|Ω|,

wherec2 = _2ω^{81N c0}

N−1 + 4 ln 2, which yields (2.7).

End of the proof of Theorem 1.2 -(ii). Estimate (2.5) is valid, hence, if σ = %₀R, we derive that 1−_Rω^{N c0σ}

N−1

≤ ¹₂ and

Z

Ω

w²_σ(x)dx≥ 1 2|Ω|.

Therefore,

λ₁(Ω)≤ E_L(wσ, wσ) R

Ωw²_σ(x)dx ≤2 2 ln 1

R +c₂

+ρ_N = 4 ln 1

R + 2c₂+ρ_N,

which ends the proof.

3 Lower bounds

Let

g(r) =rlnr for r >0, theng(e) =e,g(1) = 0 andg(¹_e) =−¹_e.

Lemma 3.1. For c≥ −¹_e, there exists a unique point r_c≥ ¹_e such that g(rc) =c,

and we have that rc≤1 +c.Furthermore, (i) for −¹_e ≤c≤0,

rc≥1 + (e−1)c≥ 1 e; (ii) for 0≤c≤e,

r_c≥1 +e−1 e c;

(iii) for c≥e,

rc≥1 +e−1 e c

and c

lnc ≤r_c≤ c

lnc−ln lnc. (3.1)

Proof. The function g is increasing in [¹_e,+∞) with value in [−¹_e,+∞). Hence rc is uniquely determined ifc≥ −¹_e,c7→r_c is increasing from [−¹_e,+∞) onto [¹_e,+∞), andg is convex.

For a > 0, we define ψa(x) = (1 +ax) ln(1 +ax) −x for x > −¹_a. Then ψa(x) > 0 (resp.

ψ_a(x)<0) is equivalent to 1 +ax > r_x(resp. 1 +ax < r_x). Note thatψ⁰_a(x) =a(1 + ln(1 +ax))−1.

Sinceψ_a⁰(−¹_a) =−∞andψ⁰_ais increasing,ψ_a⁰(0) =a−1 is the maximal (resp. minimal) value ofψ_a⁰ on (−¹_a,0] (resp. on [0,∞)). Therefore, ifa > 1,ψa is positive on (−_a¹, r^∗_a) for some r_a^∗ ∈(−¹_a,0), negative on (r^∗_a,0) and positive on (0,∞). If 0 < a < 1, ψ_a is positive on (−¹_a,0), negative on (0, r_a^∗) for some r_a^∗ > 0 and positive on (r^∗_a,∞). If a = 1, ψ₁ is positive on [−¹_e,0)∪(0,∞) and vanishes only at 0. Then ψ1 ≥0 implies the first assertion.

Since e−1>1 andψe−1(−¹_e) = 0,ψe−1(x)<0 for x∈(−¹_e,0). This gives (i).

Since 0< ^e−1_e <1, ψ^e−1

e is negative on (0, r^∗e−1 e

) and positive on (r^∗e−1 e

,∞). Since ψ^e−1

e (e) = 0, r^∗e−1

e

=eand we get (ii) and (iii).

Since g is increasing on [e,∞), (3.1) is equivalent to c− lnc

ln lnc ≤c≤ c

lnc−ln lncln

c lnc−ln lnc

=clnc−ln(lnc−ln lnc) lnc−ln lnc .

SetC= lnc, then

lnc−ln(lnc−ln lnc)

lnc−ln lnc = C−ln(C−lnC)

C−lnC >1 for C >1

and (3.1) follows.

Lemma 3.2. Let f be a real-valued function defined in R^N with0≤f ≤M1 and 2

Z

R^N

ln|z|f(z)dx=M₂. Then (i)

M₂ ≥ −2ω_N−1 N² M₁; (ii)

Z

R^N

f(z)dz ≤ M₁ω_N−1 N

e+ N²M₂ 2M1ω_N−1

= eω_N−1

N M₁+N 2 M₂; (iii) assuming more that ^M_M²

1 ≥ ^2e

2ω_N−1

N² , there holds Z

R^N

f(z)dz≤ N M₂ 2

ln

N²M₂ 2eM₁ω_N−1

−ln ln

N²M₂ 2eM₁ω_N−1

−1

.

Proof. We have

M2

2 = Z

B1

ln|z|f(z)dz+ Z

B₁^c

ln|z|f(z)dz

≥M₁ Z

B1

ln|z|dz+ Z

B₁^c

ln|z|f(z)dz ≥ −ω_N−1 N² M₁.

Hence (i) holds.

For R >0 we have that

(ln|z| −lnR)(f(z)−M11BR)≥0.

By integration over R^N we get M2

2 +M1ω_N−1R^N

N² ≥lnR Z

R^N

f(z)dz.

The estimate from above of R

R^Nf(z)dz is obtained by Z

R^N

f(z)dz ≤inf n

A >0 s.t. M2

2 + M1ω_N−1R^N

N² −AlnR≥0 for all R >0 o

. (3.2)

Set

Θ_A(R) = M₂

2 +M₁ω_N−1R^N

N² −AlnR, then Θ_A achieves the minimum if

M₁ω_N−1R^N

N =A⇐⇒R=R_A:=

N A ω_N−1M₁

_N¹ .

Hence

ΘA(RA) = M₂ 2 + A

N − A N ln

N A ω_N−1M1

. (3.3)

Putr= _M^{N A}

1ω_N−1, then

Θ_A(R_A)≥0⇐⇒rlnr−r ≤ N²M2

2M₁ω_N−1 ⇐⇒gr e

≤ N²M2

2eM₁ω_N−1. (3.4) Then ^r_e ≤rc withc= _2eM^N2M2

1ω_N−1, inequality rc≤1 +c in Lemma 3.1 yields r= N A

M₁ω_N−1 ≤e+ N²M₂ 2M₁ω_N−1 =⇒

Z

R^N

f(z)dz≤ M₁ω_N−1 N

e+ N²M₂ 2M₁ω_N−1

,

which is (ii).

Assuming now that ^M_M²

1 ≥ ^2e

2ω_N−1

N² , we can apply Lemma 3.1-(iii) and get Z

R^N

f(z)dz≤ N M2

2

ln

N²M2

2eM1ω_N−1

−ln ln

N²M2

2eM1ω_N−1 −1

,

which is (iii) and ends the proof.

Lemma 3.3. Let

˜

g(r) = r

lnr−ln lnr for r > e.

Then for t > _e−1^ee , there exists a unique point r_t> e such that ˜g(r_t) =t. Furthermore,

t(lnt−ln lnt)≤r_t< tlnt. (3.5) Proof. Since

˜

g⁰(r) = 1

lnr−ln lnr − 1−(lnr)⁻¹ (lnr−ln lnr)²

≥ 1

lnr−ln lnr

1− 1

lnr−ln lnr

>0,

the function ˜g is increasing from (e,+∞) onto (_e−1^ee ,+∞). Setting r^∗_t =t(lnt−ln lnt), then

˜

g(r_t^∗) = t(lnt−ln lnt)

lnt+ ln(lnt−ln lnt)−ln ln(t(lnt−ln lnt))

≤ t(lnt−ln lnt)

lnt+ ln(lnt−ln lnt)−ln ln(tlnt)

= lnt−ln lnt lnt−ln_ln^ln(t_{t−ln ln}^lnt)_tt

≤t, where the last inequality holds if

ln(tlnt)

lnt−ln lnt ≤lnt, which is equivalent to

˜h(τ) :=τ²−(lnτ + 1)τ−lnτ ≥0, τ = lnt.

Freezing the coefficient lnτ, ˜h(τ) = (τ−τ1)(τ−τ2), where theτ1, τ2 depend ofτ, but τ1<0< τ2, sinceτ1τ2 =−lnτ <0. Because ˜h(e) =e²−2e−1 = 0.9584±10⁻⁴, we have e > τ1. Hence τ > e impliesτ > τ₁ which in turn implies ˜h(τ)>0. Hencer^∗_t ≤r_tusing the monotonicity of ˜g.

Let st=tlnt, then

˜

g(s_t) = tlnt

lnt+ ln lnt−ln ln(tlnt) < t by the fact that

ln lnt−ln ln(tlnt)<0 for t > e.

Hence s_t≥r_t, which ends the proof.

Proof of Theorem 1.4. Denote Φk(x, y) =

k

X

j=1

φj(x)φj(y), (x, y)∈Ω×Ω, and

Φb_k(z, y) = (2π)^−N2 Z

R^N

Φ_k(x, y)e^ix·zdx, whereΦb_k is the Fourier transform with respect to x. Hence we have that

Z

R^N

Z

Ω

|bΦ_k(z, y)|²dzdy = Z

Ω

Z

Ω

|Φ_k(x, y)|²dxdy =k by the orthonormality of the{φ_j}_i∈N inL²(Ω). Furthermore, we note that

Z

Ω

|bΦ_k(z, y)|²dy= Z

Ω





k

X

j=1

φb_j(z)φ_j(y)









k

X

j=1

φb_j(z)φ_j(y)



dy

= Z

Ω





k

X

j,`=1

φb_j(z)φb_{`</sub>(z)φ<sub>j</sub>(y)φ<sub>`}(y)



dy

=

k

X

j=1

|bφ_j(z)|².

(3.6)

Using again the orthonormality of the{φ_j}_i∈NinL²(Ω), we infer by the k-dim Pythagore theorem, Z

Ω

|bΦk(z, y)|²dy= (2π)^−N Z

Ω

k

X

j=1

Z

Ω

e^ix.zφj(x)dx

φj(y)

2

dy

= (2π)^−N

k

X

j=1

Z

Ω

e^ix.zφ_j(x)dx

2

≤(2π)^−N|Ω|.

(3.7)

We have, from the Fourier expression of L∆,

k

X

j=1

λj(Ω) = Z

Ω

Z

Ω

Φk(x, y)L∆Φk(x, y)dydx

= 2

k

X

j=1

Z

R^N

|bφj(z)|²ln|z|dz

= 2 Z

R^N

Z

Ω

|bΦ_k(z, y)|²dy

ln|z|dz.

Now we apply Lemma 3.2 to the function f(z) =

Z

Ω

|bΦ_k(z, y)|²dy with

M1= (2π)^−N|Ω| and M2 =

k

X

j=1

λj(Ω).

Part (i): By Lemma 3.2 (i),

k

X

j=1

λ_j(Ω)≥ − 2ω_N−1

N²(2π)^N|Ω|=−d_N|Ω|, wheredN is constant defined in (1.13).

Part (ii):

k= Z

R^N

f(z)dz≤ eω_N−1|Ω|

N(2π)^N +N 2

k

X

j=1

λj(Ω), which implies that

k

X

j=1

λ_j(Ω)≥ 2k

N −2eω_N−1|Ω|

N²(2π)^N . Part (iii): fork∈N, if

k

X

j=1

λj(Ω)≥ 2e²ω_N−1 N² |Ω|, then

k≤ N M2

2

ln

N²M2

2eM1ω_N−1

−ln ln

N²M2

2eM1ω_N−1 −1

. Setting

r = N²M2

2eM₁ω_N−1 and t= N k

eM₁ω_N−1 = (2π)^NN k eω_N−1|Ω|, we have from (3.5) that

r≥r_t≥t(lnt−ln lnt), (3.8)

for any t > e^e, i.e.

k > e^e+1ω_N−1|Ω|

(2π)^NN = e^e+1N d_N 2 |Ω|.

This implies

N²M2

2eM₁ω_N−1 ≥ (2π)^NN k eω_N−1|Ω|

ln

(2π)^NN k eω_N−1|Ω|

−ln ln

(2π)^NN k eω_N−1|Ω|

, from what we infer

k

X

j=1

λj(Ω)≥ 2k N

ln

2k eN d_N|Ω|

−ln ln

2k eN d_N|Ω|

, (3.9)

which completes the proof.

Proof of Theorem 1.3 and Corollary 1.5. It is clear that Theorem 1.4 with k = 1 implies Theorem 1.3. From the inequality (3.9) we derive

˜λ_k(Ω) := 1 k

k

X

i=1

λ_i(Ω)≥ 2 N

ln

2k eN dN|Ω|

−ln ln

2k eN dN|Ω|

.

Since k7→λ_k(Ω) is nondecreasing, we conclude Corollary 1.5.

4 Wey’s limits

4.1 Upper bounds for the sum of eigenvalues

For any bounded complex valued functionsu, v defined on Ω, there holds L∆(uv)(x) =u(x)L∆v(x) +c_N

Z

B1(x)

u(x)−u(ζ)

|x−ζ|^N v(ζ)dζ. (4.1) Lemma 4.1. For z∈R^N, we denote

µz(x) =e^ix·z, ∀x∈R^N, then

L∆µz(x) = (2 ln|z|)µ_z(x), ∀x∈R^N. (4.2) Proof. Step 1: we claim that for

(−∆)^sµ_z(x) =|z|^2sµ_z(x), ∀x∈R^N. (4.3) Without loss of generality, it is enough to prove (4.3) with z = te₁, where t > 0 and e₁ = (1,0,· · · ,0)∈R^N. For this, we write

v_t(x) =µ_z(x₁) =e^itx1, x= (x₁, x⁰)∈R×R^N−1. ForN ≥2 it implies by [7, Lemma 3.1] that

(−∆)^svt(x) = (−∆)^s

Rvt(x1).

Now we claim that

(−∆)^s

Rvt(x1) =t^2svt(x1), ∀x1 ∈R. (4.4) Indeed, observe that−∆_R:=−(v_t)x1x1 =t²vt inR and then

(|ξ₁|²−t²)vbt(ξ1) =F −∆_Rvt−t²vt

(ξ1) = 0, which implies that

supp(vbt)⊂ {±t}, which in turn implies

(|ξ₁|^2s−t^2s)vb_t(ξ₁) = 0 =F (−∆)^s_Rv_t−t^2sv_t (ξ₁).

and finally

(−∆)^s_Rvt−t^2svt

(ξ1) = 0 in R, which yields

(−∆)^sv_t(x) = (−∆)^s

Rv_t=t^2sv_t(x), ∀x∈R^N, Step 2: we show (4.2). From the property (1.5) ofL∆sinceµ_z is bounded,

0 = (−∆)^sµz(x)− |z|^2sµz(x) s

= (−∆)^sµz(x)−µz(x)

s −|z|^2s−1 s µz(x)

→L∆µz(x)−(2 ln|z|)µ_z(x) as s→0⁺,

hence,

L∆µz(x) = (2 ln|z|)µ_z(x), ∀x∈R^N,

which is the claim.

Next, letη₀∈C¹(R) be a nondecreasing real value function such that kη₀⁰k_L^∞ ≤2 satisfying η₀(t) = 1 if t≥1, η₀(t) = 0 if t≤0.

Since Ω is a bounded domain, there exists a C¹ domainO ⊂ Ω such that |O| ≥ ³₄|Ω|. For σ >0, we set again

w_σ(x) =η₀(σ⁻¹ρ(x)),¯ ∀x∈R^N. (4.5) where ¯ρ(x) = dist(x, ∂O). Observe that w_σ ∈H0(Ω) and

w_σ →1 in O as σ →0⁺. Thus, there existsσ₁ >0 such that forσ ∈(0, σ₁],

|Ω|>

Z

Ω

w_σdx≥ Z

Ω

w²_σdx > |Ω|

2 . Lemma 4.2. Let

L_zw_σ(x) = Z

B1(x)

w_σ(x)−w_σ(ζ)

|x−ζ|^N e^iζ·zdζ, then there holds

L_zwσ(x)

≤ 2ω_N−1

σ for x∈Ω.

Proof. Actually, if x∈Ω, we have that

|w_σ(x)−w_σ(ζ)| ≤ kDw_σk_L^∞|x−ζ| ≤σ⁻¹kη⁰₀k_L^∞|x−ζ|, then

Z

B1(x)

w_σ(x)−w_σ(ζ)

|x−ζ|^N e^iζ·zdζ

≤ kη₀⁰k_L^∞ σ

Z

B1(x)

dζ

|ζ−x|^N−1 ≤ 2ω_N−1 σ ,

sincekη₀⁰k_L^∞ ≤2. This ends the proof.

4.2 Proof of Theorem 1.6 and 1.7

Proof of Theorem 1.6. We recall that Φ_k(x, y) andΦb_k(z, y) have been defined in the proof of Theorem 1.4. If we denote

˜

v_σ,z(x) :=v_σ(x, z) =w_σ(x)e^ix·z,

the projection ofv_σ onto the subspace of L²(Ω) spanned by theφ_j for 1≤j≤kcan be written in terms of the Fourier transform of w_σΦ_k with respect to thex-variable:

Z

Ω

vσ(x, z)Φ_k(x, y)dx= (2π)^N/2F_x(wσΦ_k)(z, y).

Put

v_σ,k(z, y) =v_σ(z, y)−(2π)^N/2F_x(w_σΦ_k)(z, y) and the Rayleigh-Ritz formula shows that

λ_k+1(Ω) Z

Ω

|v_σ,k(z, y)|²dy≤ Z

Ω

v_σ,k(z, y)L∆,yv_σ,k(z, y)dy