Eigenvalues estimate for the Neumann problem on bounded domains

HAL Id: hal-00257734

https://hal.archives-ouvertes.fr/hal-00257734

Preprint submitted on 20 Feb 2008

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Eigenvalues estimate for the Neumann problem on bounded domains

Bruno Colbois, Daniel Maerten

To cite this version:

Bruno Colbois, Daniel Maerten. Eigenvalues estimate for the Neumann problem on bounded domains.

2008. �hal-00257734�

hal-00257734, version 1 - 20 Feb 2008

BRUNOCOLBOISANDDANIELMAERTEN

Abstrat. Inthisnote,weinvestigateupperboundsoftheNeumanneigenvalueprob-

lem for the Laplaian of a domainΩ ^{ina given omplete(not ompat a priori) Rie-}

mannian manifold (M, g)^{. For this, we use test funtions for the Rayleigh quotient}

subordinatedtoafamilyofopensetsonstrutedinageneralmetriway,interestingfor

itself. Asappliations,weprovethatiftheRiiurvatureof(M, g)^{isboundedbelow} Ric^g≥ −(n−1)a^2,a≥0^{,thenthereexistonstants}An>0, Bn>0^{onlydependingon}

thedimension,suhthat

λk(Ω)≤Ana²+Bn

k

V ²/n

,

where λk(Ω) (k ∈ N^∗) ^{denotes the} k^{th eigenvalue of the Neumann problem on any}

bounded domainΩ ⊂M ^{of volume}V = Vol(Ω, g)^. Furthermore, this upper boundis learlyinagreementwiththeWeyllaw. Asaorollary,wegetalsoanestimatewhihis

analogoustoBuser'supperboundsofthespetrumofaompatRiemannianmanifold

withlowerboundontheRiiurvature.

1. Introdution

ThegoalofthispaperistogiveupperboundsforthespetrumoftheLaplaianatingon

ompatdomainsofgivenvolumeofaompleteRiemannianmanifoldwithRiiurvature

bounded below, and, as far as possible, to make these estimates optimal withrespet to

theWeyl law.

For ompat Riemannian manifolds without boundary,the following resultwas proved

byP.Buserin[3 ℄(Satz7),[4℄(Thm. 6.2())(seealsoLi-Yauin[13 ℄(Thm.16)). If{λ_k}^∞_k=1

denotethe spetrumofthe Laplaianating onfuntions, then:

1.1. Theorem. Let(Mⁿ, g) ^{be a ompat} n-dimensionalRiemannian manifoldwithRii urvature bounded below Ric^g≥ −(n−1)a^2, a≥0, ^{andof volume} V^.

There exists a onstant C_n ≥ 1 ^{only depending on the dimension, suh that for all} k∈N^∗, we have

(1.1) λ_k(M, g) ≤ (n−1)²

4 a²+C_n k

V 2/n

.

1.2. Remarks. (i) In [13℄, the onstant C_n ^{dependsalso on the diameter.}

(ii) Indimensionhigherthan2,anormalizationonthevolumeisnotenoughtoontrol

the spetrum: namely, on any ompat manifold of dimension higher than 2, one

an nda metri of given volume, witharbitrarily large rstnonzero eigenvalue

λ₂ ^{of the Laplaian,in vertue of the result of B. Colboisand J. Dodziuk [6℄.}

Date:20thFebruary2008.

Keywordsandphrases. Neumannspetrum,upperbound,Weyllaw,metrigeometry.

2000 MathematisSubjetClassiation. 35P15,53C99,51F99.

(iii) When Ric^g ≥0^{, we dedue that there exists} C_n >1 ^with λ_k(M, g) ≤C_{n V}^k2/n

for all k^{. However, when Rii is not supposed positive, then the presene of a}

termlike

(n−1)²

4 a^{2 isneessary: bya result ofR. Brooks [2℄, it ispossibletonda}

familyofompathyperboli manifoldswithvolumegoingtoinnityanda positive

uniform lower bound on the rst nonzero eigenvalue.

The idea of the proof of Theorem 1.1 is to onsider k ^{disjoint balls of radius} r ^whih

almostover the manifold (M, g)^,with r ^{around V}_k1/n

,and to apply thenCheng's the-

orem [5 ℄. However, suh a theoremdoes not exist on manifolds with boundary, and with

Neumann boundary ondition. A reason for this is that there is no Bishop-Gromov the-

orem: indeed, even for a Eulidean domain, it is not possible to ontrol the volume of a

ballofradius2r ^{withrespettothevolumeofa ballofradius} r^{and sameenter. See also}

Example1.4in[4 ℄.

ThisdoesnotmeanthataresultinthespiritofTheorem1.1doesnotexistfordomains.

Namely, P. Kröger [12℄ proved thanks to harmoni analysis, that on bounded Eulidean

domains, the k^{th eigenvalue of the Neumann problem was bounded by above by some}

expressionC_n(k/|Ω|)^n/2,^whereC_n^{onlydependsuponthedimension. Ananalogousresult}

an be derived from the muh more general and diult work of N. Korevaar [11℄ ( see

also [10℄), for bounded domains of nonnegative Rii urvature manifolds, and also for

bounded domainsof negative Rii urvature ompat manifolds (in this ase thebound

dependson the diameter).

This naturally leads tothe

Question: What an be said for bounded domains of a omplete Riemannian manifold

withRiiurvaturebounded below ?

Inthisnote,weonsidertheNeumanneigenvalueproblemfortheLaplaianofabounded

domainΩ^{withsmooth boundary,inagiven omplete(notompat a priori)Riemannian}

manifold (M, g)^{. More preisely, we searh for a ouple} (λ, u) ∈ R×C^∞ Ω

whih is a

solutionof the following boundary elliptiproblem

∆u=λu ^onΩ

∂u

∂ν = 0 ^on∂Ω,

where ∆^{is the} nonnegative Laplaianof the metri g ^and ν ^{theoutward unit normal of}

∂Ω^{. Sine}Ω^{isboundedwithsmoothboundary,thespetrumof} ∆^onΩ^{isan unbounded}

sequeneofreal numbers (λ_k(Ω))_k∈N∗ ^{whihan be} inreasinglyordered

0 =λ₁(Ω)< λ₂(Ω)≤ · · · ≤λ_k(Ω)≤λ_k+1(Ω)≤ · · · .

Thereexiststandardvariationalharaterisationsofthespetrumof∆^{whihanbefound}

forinstane inthe bookof P.Bérard [1℄ (orin[9℄).

The main resultofthis artile isthe following.

1.3.Theorem. Let(Mⁿ, g) ^{be a omplete} n-dimensionalRiemannianmanifoldwithRii urvature bounded below Ric^g≥ −(n−1)a^2, a≥0^.

Thereexist onstantsA_n>0, B_n >0^{onlydepending onthedimension, suhthat forall} k∈ N^∗, V > 0 ^{and for eah bounded domain} Ω⊂ M^{, with smooth boundary and volume}

V^{, we have}

(1.2) λ_k(Ω)≤A_na²+B_n

k V

2/n

.

If the manifold M ^{is ompat, an interesting speial ase isto hoose} Ω =M^{, and we}

reover Theorem 1.1 , up to the value of the onstant An ^{whih is not equal to} (n−1)²

4 ⁱⁿ

ourpaper.

The proof Theorem 1.3 goes in the same spirit as the proof of Theorem 1.1: in order

to bound λ_k(Ω)^{, we onsider} k ^{disjoint sets} A₁, ..., A_k ⁱⁿ Ω ^{of measure of the order of}

V ol(Ω)

k ^{,and introdue testfuntions} f₁, ..., f_k subordinated to these sets. Weestimate the Rayleighquotientofthese fontionsbyadiretalulation,whih givesthetheorem. The

mainimprovement ofthis paperistheonstrution of anadaptedfamilyofsets A1, .., A_k^,

moreonvenient for our purpose asballs. Asthis onstrution isinteresting byitself and

willbeused inother ontexts, we present it inarather abstrat(indeed metri) way.

Thepaperisorganisedasfollows: themetrionstrutionofoursetsisdoneinSetion2,

and in Setion 3 we will use them so as to prove Theorem 1.3 by produing some test

funtionsfor the variational haraterisation ofthespetrum.

2. A metri approah

Inthissetion, weformalize thegeometri situationofTheorem1.3(aboundeddomain

inaompletemanifold)inamoregeneralsetting(aboundeddomaininaompletemetri

spae). More preisely, let (X, d) ^{be a omplete, loally ompat metri spae,} Y ⊂X ^a

bounded Borelian subset endowed with the indued distane, and µ ^{a Borelian measure}

withsupportinY ^suhthatµ(Y) =ω^,0< ω <∞^{. Wewillneedinaddition thefollowing}

tehnial assumptions:

(H1) Foreahr >0^{,thereexistsaonstant}C(r)>0^{suhthateahballofradius}4r ⁱⁿ X ^{maybe overedby}C(r)^{ballsofradius} r^{. Moreover,}r 7→C(r)^{isan inreasing}

funtionof the radius.

(H2) We suppose that the volume of the r^{balls tends to 0 uniformly on} X^{, namely}

r→0limsup{µ(B(x, r)) : x ∈ X} = 0^{. However, taking (H1) into aount, this vol-}

umeonditionisequivalent to lim

r→0sup{2C(r)µ(B(x, r)) :x∈X}= 0^whihisthe

(more onvenient) onditionthatwill be usedintheremainder of theartile.

It is important to remark that these hypothesis are quite natural sine they make part

of the metri properties of the Riemannian manifolds that are involved in Theorem 1.3 .

Thesespeimetri propertiesareolleted inthefollowing fundamental example.

2.1.Example. Atypial exampleof aouple (X, Y) ^{satisfyingthe hypothesis(H1),(H2) is}

tohooseX^asaomplete n-dimensionalRiemannianmanifold(M, g)^{withRiiurvature}

bounded below Ric^g ≥ −(n−1)a^2, a ≥ 0 ^{(whih are the lass of manifolds involved in}

Theorem 1.3), and as Y ^{a bounded domain with smooth boundary in} M^{. The distane} d

isthe distane assoiated tothe Riemannian metri g^{, the measure} µ ^{is the restrition to} Y ^{of the Riemannianmeasure of} g^{. The existene of the onstant} C(a, r) ^{is given by the}

lassial Bishop-Gromov inequality thanks to the lower bound on the Rii urvature of g

(see [14 ℄ p. 156). Preisely, for 0< r < R^{, andfor eah point} p∈M^{, we have}

(2.1)

Vol(B(p, R), g)

Vol(B(p, r), g) ≤ v_a(R) v_a(r) ,

where va(R) ^{denotes the volume of a ball of radius} R ⁱⁿ Mⁿ_a, the simply onneted n

dimensionalmanifoldof onstant setional urvature −a^2.

Thisgivesaboundon thenumberofballsofradiusr ^{thatareneessarytoover aballof}

radius 4r ^{(this property known as the paking lemma is a onsequene of Inequality (2.1)).}

In fat, x B_4r ^a 4r^{ball and onsider} {B(x_i, r/2)}_i∈I ^{a maximal family of disjoint balls}

whose enter xi ^{live in} B4r^{; then the} orresponding family of r^balls {B(xi, r)}_i∈I ^over B_4r^{. In onsequene, we an over a ball of radius} 4r ^with ≤1 +h

va(4r+r/2) va(r/2)

ir^{balls. We}

justdene

C(a, r) = max

t≤r

1 +

va(4t+t/2) v_a(t/2)

.

The inreasingharater of r7→C(a, r) ^{is by denition.}

Furthermore, as r−→0^{, the ratio} Vol(B(p,r),g)

va(r) −→1^{, we obtain} Vol((B(p, R), g) ≤v_a(R) ,

andonsequently µ(B(p, r)) := Vol(B(p, r)∩Y, g) ^{goes uniformly to} 0 ^as r→0^.

Weproveinthesequelthat,underourtehnialassumptions,oneanbuildsomesubsets

A^and D^{satisfying ertainvolume onditions.}

2.2. Lemma. Let (X, d) ^{be a omplete, loally ompat metri spae,} Y ⊂X ^{a bounded}

Borelian withthe indued distane, andµ ^{a Borelian measure withsupport in}Y ^{suh that} µ(Y) =ω^, 0< ω <∞ ^and µ(Y \Y) = 0^{. In addition,we make the hypothesis (H1),(H2).}

Let0< α≤ ^ω₂^{. Thanksto(H2)there exists}r >0^withsup{2C(r)µ(B(x, r)) :x∈X} ≤α^.

Thenthere exist A, D⊂Y ^{suh that} A⊂D^and







µ(A)≥α µ(D)≤2C(r)α d(A, Y ∩D^c)≥3r

.

Proof. We xthepositivenumbers r ^andα^{. Letus onsideranypositive integer}m∈N^∗

anddeneanonnegative appliationΨm:X^m =X×X× · · · ×X

| {z }

mtimes

−→Rbytherelation

Ψ_m:^x = x^jm

j=17−→µ



 [m

j=1

B x^j, r



 ,

whih is simplythe restrition of themeasure µ ^to U_m(r) ^{a partiular lass of open sets}

whih isdened by

U_m(r) :=





 [m

j=1

B x^j, r

/ x^jm

j=1∈X^m





 .

Sine(X, d) ^{isaompleteandloallyompat metrispae,itisalsotheaseofthenite}

produtX^{m when itis endowed withthe produtdistane. Then for eah} m ∈N^∗ there existssome xmax,m∈X^{m (notneessaryunique)suh that}

Ψ_m(^x_max,m) = max

X^m Ψ_m= max

U_m(r)µ=µ



 [m

j=1

B x^j_max,m, r



 .

We rst prove that there exists a nite integer k ∈ N^∗ suh that Ψ_k(^x_max,k) ≥ α ^and Ψ_k−1(^x_max,k−1) ≤α^{. Indeed, onsider the funtion} ξ :N^∗ −→ R dened by the relation

ξ(m) = Ψm(^xmax,m)^{. On one hand, the ondition} sup{2C(r)µ(B(x, r)) : x ∈ X} ≤ α

obviously implies ξ(1)≤ _2C(r)^α ≤α^{. Onthe otherhand, sine} Suppµ ⊂Y^{, there existsa}

radius R >0 ^{large enough suh that}µ(B(z, R))≥3ω/4^{, for a ertain}z ∈X^{. Butit an}

belearlydeduedfromAssumption(H1)thatB(z, R)^{anbenitelyoveredby}m₀ ∈N^∗

ballsofradius r ^{(notie that}m₀ ^{depends on}R^). Consequently itturns out

3α 2 ≤ 3ω

4 ≤µ(B(z, R))≤ max

U_m0(r)Ψ_m0 =ξ(m₀).

Thereby the funtion ξ : N^∗ −→ R satises ξ(1)≤ α ^and ξ(m₀) ≥ ^3α₂ ^{, whih entails the}

existeneofsome k∈N^∗ suhthatΨ_k(^x_max,k)≥α ^and Ψ_k−1(^x_max,k−1)≤α^.

We now setU_k := S

1≤j≤k

B

x^j_max,k, r

and V_k := S

1≤j≤k

B

x^j_max,k,4r

. Thenext stepis to

showthat

µ(V_k)≤C(r)µ(U_k) .

Still aording to Assumption (H1), V_k ^{is overed by} kC(r) ^{balls of radius} r^{, namely} V_k⊂ S

1≤j≤kC(r)

B_j^{,where the} B_j ^{areballs ofradius} r^{. Butit isquitelear that this union}

ofr^{ballsan bewritten as} S

1≤j≤kC(r)

B_j = S

1≤j≤C(r)

W_j ^{where eah}W_j ∈U_k(r)^{. Itfollows}

µ(V_k)≤µ



 [

1≤j≤kC(r)

Bj



 = µ



 [

1≤j≤C(r)

Wj





≤

C(r)X

j=1

µ(W_j)

≤ C(r) max

U_k(r)µ=C(r)ξ(k) =C(r)µ(U_k) .

We nally dene the sets A := Y ∩U_k ^and D := Y ∩V_k^{. We only have to hek that}

theysatisfythe propertiesstated inLemma 2.2 . We observe thatµ(A) =µ(U_k) ^{sine the}

measureµ ^{issupportedin} Y ^and µ(Y \Y) = 0^{. Besides,} U_k ^{an be written asthe union}

ofan element ofU_k−1(r)^{and anelement of}U₁(r)^sothat µ(A)≤ξ(k−1) +ξ(1)≤α

1 +1

2

.

Stillsine Suppµ=Y^{, we obtain} µ(D) =µ(V_k)≤C(r)µ(U_k) =C(r)µ(A)≤2C(r)α^{. By}

thedenition ofU_k ^andV_k^,westraightforwardlyhave d(A, Y ∩D^c)≥3r^.

In setion 3, we will use the following orollary of Lemma 2.2 to make the proof of

Theorem1.3 . Wegivethereinanexpliiteonstrutionofthedomainsthatwerementioned

attheend of the introdution.

2.3. Corollary. Let (X, d) ^{be a omplete, loally ompat metri spae,} Y ⊂X ^{a bounded}

Borelian withthe indued distane, andµ ^{a Borelian measure withsupport in}Y ^{suh that} µ(Y) =ω^,0< ω <∞ ^andµ(Y \Y) = 0^{. In addition, we make the hypothesis (H1),(H2)}

asin Lemma 2.2,and take N ^{a positive integer.}

Letr >0 ^{suh that} 4C²(r)µ(B(x, r))≤ _N^{ω holdsfor all} x∈X^{, andlet} α= _2C(r)N^{ω . Then,}

there exist N ^{measurable subsets}A₁, ..., A_N ⊂Y ^{suh that} µ(A_i)≥α ^{and,for eah} i6=j^, d(A_i, A_j)≥3r^.

Proof. Weonstrut thefamily (A_j)^N_j=1 ^{byniteindution applyingLemma 2.2.}

•j= 1^{. We set} (X₁, d₁, µ₁) = (X, d, µ) ^and Y₁ = Y^{, whih satisfy the} assumptions of Lemma 2.2. Therefore there existA₁, D₁ ^{suh that}A₁ ⊂D₁ ⊂Y₁ =Y ^and







µ(A₁) ≥ α

µ(D1) ≤ 2C(r)α= _N^ω d(A₁, Y₁∩D^c₁) ≥ 3r

.

•j= 2^{. Weset}(X₂, d₂, µ₂) = (X, d, µ_|Y2)^andY₂=D^c₁∩Y₁^{,whihsatisfythe}assumptions of Lemma 2.2 withω₂ =µ₂(Y₂) ≥ω 1−_N¹

=ω ^N+1−2_N

≥α^{. Therefore there}

existA₂, D₂ ^{suh that}A₂ ⊂D₂⊂Y₂ =D₁^c∩Y₁ ^and







µ(A₂) ≥ α

µ(D₂) ≤ 2C(r)α= _N^ω d(A2, Y2∩D^c₂) ≥ 3r

.

AsA₁ ⊂D₁ ^andA₂⊂Y₁∩D₁^{c we get}d(A₁, A₂)≥d(A₁, Y₁∩D₁^c)≥3r ^thanksto

the asej= 1^.

•j≥3^{. We suppose that we have already onstruted the families} (As)^j−1_s=1 ^and (Ds)^j−1_s=1

thatsatisfytheonditions







A_s⊂D_s⊂Y ∩(D₁∪ · · · ∪D_s−1)^c =Y_s, s≤j−1 d(A_s, A_t)≥3r s6=t,

µ(D1∪ · · · ∪Dj−1)≤ω

j−1 N

.

We set (X_j, d_j, µ_j) = (X, d, µ_|Yj) ^and Y_j =Y ∩(D₁∪ · · · ∪D_j−1)^{c, whih satisfy}

theassumptionsofLemma2.2withωj =µj(Yj)≥ω

1− ^j−1_N

=ω

N+1−j N

≥α

ifj≤N^{. Therefore thereexist} A_j, D_j ^{suh that}A_j ⊂D_j ⊂Y_j ^and







µ(Aj) ≥ α

µ(D_j) ≤ 2C(r)α= _N^ω d(A_j, Y_j ∩D_j^c) ≥ 3r

.

As A_j ⊂ Y ∩(D₁∪ · · · ∪D_j−1)^c ⊂ Y ∩(D₁∪ · · · ∪D_s−1)^c = Y_s^, s < j^{, and} A_s ⊂ D_s^{, we get} d(A_j, A_s) ≥d(A_s, Y_s∩D_s^c) ≥ 3r ^{thanksto the ase} j = s^{. As}

already said,we an proeed this onstrution solonger we have enough volume

to doit, thatis N ^times.

3. Proof of Theorem 1.3.

Let (Mⁿ, g) ^{be a omplete} n-dimensional Riemannian manifold with Rii urvature bounded below Ric^g ≥ −(n−1)a^{2, and} Ω ⊂ M ^{a bounded domain of volume} V^{, with}

smooth boundary.

We observe rst that, by renormalisation, it is enough to prove the theorem for the

ase a = 1^{: namely, if Theorem 1.3 is true for} a= 1^{, and if} g ^{is a Riemannian metri}

with Ric^g ≥ −(n−1)t²g^{, then} g₀ = t²g ^satises Ric^g0 ≥ −(n−1)g₀^{. Sine we have} λ_k(g₀)≤A_n+B_n

k V(g⁰)

2/n

,then, beause λ_k(g) =t²λ_k(g₀) ^andV(g) =tⁿV(g₀)^{,we get} λ_k(g)≤A_nt²+B_{n V}^k2/n

.

So,letuseprove Theorem1.3for a= 1^{. AsinExample2.1, letusonsidertheBorelian}

measureµ^{whih istherestrition to thedomain} Ω^{of theRiemannian volume of} (M, g)^.

In order to prove Theorem 1.3, we will usethe lassial variational haraterization of

the spetrum: to estimate λ_k ^{from above, it sues to onstrut an} H¹(Ω)-orthogonal family of k ^{test funtions} (fj)^k_j=1^{, suh as eah} fj ^{has ontroled Rayleigh quotient. In}

thesequel,we onstrut test funtionswithdisjoint supportrelated tothe sets A₁, ..., A_k

arisingfromCorollary 2.3 , sothatit immediatelyimpliesorthogonality inH¹(Ω)^.

3.1. Lemma. Let A⊂M ^{a subset as in Corollary 2.3. Let} A^r:= {x∈M :d(x, A)≤r}^, r > 0^{. There exists a funtion} f ^{supported in} A^{r whose restrition to} Ω ^{is of Rayleigh}

quotient

R(f)≤ 1 r²

µ(A^r\A) µ(A) .

Proof. Letus dene a plateau funtion

f(p) =







1 if p∈A 1−^d(p,A)_r if p∈(A^r\A)

0 if p∈(A^r)^c .

InCorollary2.3, thedomainA ^{isanite unionofmetri ballsand}intersetionwithom- plement ofballs. The boundary is not smooth,but thefuntion d(∂A,·) "distane to the boundary of A^{" iswell knownto be1Lipshitz on} M^{. Aording to} Rademaher's theo- rem(seeSetion 3.1.2, page8184in[8 ℄), d(∂A,·) ^isdierentiable Lⁿ almosteverywhere (sinedVol_g ^{is absolutelyontinuouswith respetto Lebesgue's measure}Lⁿ),and its g

gradient satises|∇d(∂A,·)|_g≤1^,Lⁿ almosteverywhere. Itomes out thatthegradient off ^satisesLⁿalmost everywhere

|∇f(p)|_g ≤ ₁

r if p∈(A^r\A) 0 if p∈(A^r\A)^c .

Weimmediately dedue

R(f) = R

Ω|∇f|²_gdVol_g R

Ωf²dVol_g ≤ 1 r²

µ(A^r\A) µ(A) .