nonpositively curved spaces Fundamental tones of clamped plates in Advances in Mathematics

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Advances in Mathematics

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Fundamental tones of clamped plates in nonpositively curved spaces

Alexandru Kristály^a,b,∗,1

aDepartmentofEconomics,Babeş-BolyaiUniversity,Cluj-Napoca,Romania

bInstituteofAppliedMathematics,ÓbudaUniversity,Budapest,Hungary

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received17September2019 Receivedinrevisedform18 February2020

Accepted13March2020 Availableonlinexxxx

CommunicatedbyC.Fefferman DedicatedtoProfessorBiagio Ricceriontheoccasionofhis65th anniversary.

MSC:

primary35P15,53C21,35J35,35J40

Keywords:

LordRayleigh’sisoperimetric problem

Fundamentaltone Clampedplate Nonpositivecurvature Hypergeometricfunction

We study Lord Rayleigh’s problem for clamped plates on an arbitrary n-dimensional (n ≥ 2) Cartan-Hadamard manifold (M,g) with sectional curvature K ≤ −κ² for some κ ≥ 0. We first prove a McKean-type spectral gap estimate,i.e.thefundamental toneofanydomainin(M,g) is universally bounded from below by ⁽ⁿ⁻¹⁾₁₆ ⁴κ⁴ whenever the κ-Cartan-Hadamard conjecture holds on (M,g), e.g.

in 2- and 3-dimensions due to Bol (1941) and Kleiner (1992),respectively.In2- and3-dimensions weprovesharp isoperimetricinequalitiesforsufficientlysmallclampedplates, i.e.thefundamentaltoneofanydomainin(M,g) ofvolume v >0 isnotlessthanthecorrespondingfundamentaltoneof ageodesicballofthesamevolumevinthespaceofconstant curvature−κ²providedthatv≤cn/κⁿwithc2≈21.031 and c3 ≈ 1.721, respectively. In particular, Rayleigh’s problem in Euclidean spaces resolved by Nadirashvili (1992) and Ashbaugh and Benguria (1995) appears as a limiting case inoursetting(i.e.K≡κ= 0).Sharpasymptoticestimatesof thefundamentaltoneofsmallandlargegeodesicballsoflow- dimensionalhyperbolicspacesarealso given.Thesharpness ofourresultsrequiresthevalidityoftheκ-Cartan-Hadamard conjecture(i.e.sharpisoperimetricinequalityon(M,g))and peculiarpropertiesoftheGaussianhypergeometricfunction, both valid only in dimensions 2 and 3; nevertheless, some

* Correspondenceto:DepartmentofEconomics,Babeş-BolyaiUniversity,Cluj-Napoca,Romania.

E-mailaddresses:alex.kristaly@econ.ubbcluj.ro,kristaly.alexandru@nik.uni-obuda.hu.

1 ResearchsupportedbytheNationalResearch,DevelopmentandInnovationFundofHungary,financed undertheK_18fundingscheme,ProjectNo.127926.

https://doi.org/10.1016/j.aim.2020.107113 0001-8708/©2020ElsevierInc. Allrightsreserved.

nonoptimal estimates of the fundamental tone of arbitrary clamped plates are also provided in high-dimensions. As an application, by using the sharp isoperimetric inequality for small clamped hyperbolic discs,wegive necessarily and sufficientconditionsfortheexistenceofanontrivialsolution toanellipticPDEinvolvingthebiharmonicLaplace-Beltrami operator.

©2020ElsevierInc.Allrightsreserved.

1. Introductionandmainresults

LetΩ⊂Rⁿ beaboundeddomain(n≥2),andconsidertheeigenvalueproblem

Δ²u= Γu in Ω,

u=|∇u|= 0 on ∂Ω, (1.1)

associated with the vibration of a clamped plate. The lowest/principal eigenvalue for (1.1) –thefundamentaltoneoftheclampedplate–canbecharacterizedinavariational way by

Γ₀(Ω) = inf

u∈W0^2,2(Ω)\{0}

Ω

(Δu)²dx

Ω

u²dx

. (1.2)

Theminimizerof(1.2) intheplanedescribesthevibrationofahomogeneousthinplate Ω⊂R² whoseboundaryis clamped,while thefrequency ofvibrationof theplateΩ is proportionaltoΓ₀(Ω)¹².ThefamousconjectureofLordRayleigh[36,p.382] –formulated initially forplanardomainsin1894–statesthat

Γ₀(Ω)≥Γ₀(Ω) =h⁴_ν ω_n

|Ω| 4/n

, (1.3)

where Ω ⊂Rⁿ isaball with thesamemeasure as Ω,with equality ifandonly ifΩ is aball. Hereafter,ν= ⁿ₂ −1,ω_n=π^n/2/Γ(1+n/2) isthevolumeoftheunitEuclidean ball, while hν is the first positive critical point of ^J_I^ν

ν, where Jν and Iν stand for the Besseland modifiedBesselfunctionsof firstkind,respectively.

Assuming that the eigenfunction corresponding to Γ0(Ω) is sign-preserving over a simply connected domain Ω ⊂R², Szegő [38] proved (1.3) in the early fifties.As one candeducefrom hispaper’stext, hisbeliefontheconstant-signfirsteigenfunction cor- responding to Γ₀(Ω) hasbeen basedonthesecond-ordermembrane problem(called as the Faber-Krahn problem). It turned out shortlythat his expectation perishes due to

the construction of Duffin [19] on strip-likedomains and Coffman, Duffin and Shaffer [16] onring-shaped clampedplate,localizing nodal lines ofvibratingplates. Whilethe membraneproblem involvesonlytheLaplacian,theclampedplateproblemrequiresthe presence of the fourthorder bilaplacian operator; as we know nowadays, fourth order equations are lacking general maximum/comparison principles which is unrevealed in Szegő’spioneeringapproach.Infact,stimulatedbythepapers[19] and[16],severalsce- narios aredescribed for nodaldomains of clampedplates,seee.g.Bauer and Reiss[3], Coffman [15], Grunau and Robert [21], from which the main edification is thateigen- functionscorrespondingto(1.2) may changetheirsign.

In order to handle the presence of possible nodal domains, Talenti [40] developed aSchwarz-typerearrangement methodondomains where thefirst eigenfunctioncorre- sponding to (1.2) has both positive and negative parts. In this way, a decomposition of(1.2) into atwo-ballminimizationproblem arises whichprovided anonoptimalesti- matein(1.3);infact,insteadof(1.3),TalentiprovedthatΓ₀(Ω)≥d_nΓ₀(Ω) wherethe dimension-depending constantd_n has the properties ¹₂ ≤d_n < 1 for every n ≥2 and

n→∞lim dn =1 2.

By a careful improvement of Talenti’s two-ball minimization argument, Rayleigh’s conjecture hasbeen provedinits fullgenerality forn = 2 byNadirashvili [31,32]. Fur- thermodificationsofsomeargumentsfromthepapers[32] and[40] allowedtoAshbaugh and Benguria [1] to prove Rayleigh’s conjecture for n = 3 (and n = 2) by explor- ing fine properties of Bessel functions. Roughly speaking, for n ∈ {2,3}, the two-ball minimization problem reduces to only one ball (the other ball disappearing), while in higherdimensionsthe‘optimal’ situationappearsfortwoidentical ballswhichprovides anonoptimalestimateforΓ0(Ω).Althoughasymptoticallysharpestimatesareprovided by Ashbaugh and Laugesen [2] for Γ₀(Ω) in high-dimensions, i.e. Γ₀(Ω) ≥ D_nΓ₀(Ω) where0.89< Dn <1 foreveryn≥4 with lim

n→∞Dn= 1,theconjecture isstillopen for n≥4.Veryrecently,ChasmanandLangford[6,7] providedcertainAshbaugh-Laugesen- typeresultsinEuclideanspacesendowedwithalog-convex/Gaussiandensity,byproving thatΓ_w(Ω)≥CΓ˜ _w(Ω),wheretheconstantC˜∈(0,1) dependsonthevolumeofΩ and dimensionn≥2,whileΓw(Ω) andΓw(Ω) denotethefundamentaltonesoftheclamped platewithrespect tothecorresponding densityfunctionw.

Interestinthe clampedplateproblemon curvedspaces wasalso increasedinrecent years.One ofthe mostcentral problems isto establishPayne-Pólya-Weinberger-Yang- typeinequalitiesfortheeigenvaluesoftheproblem

Δ²_gu= Γu in Ω,

u= ^∂u_∂n = 0 on ∂Ω, (1.4)

where Ω is a bounded domain in an n-dimensional Riemannian manifold (M,g), Δ²_g standsfor the biharmonic Laplace-Beltramioperator on(M,g) and _∂n^∂ is theoutward normalderivativeon∂Ω,respectively;seee.g.Chen,ZhengandLu[9],Cheng,Ichikawa

and Mametsuka[10], ChengandYang[11–13],WangandXia[42].Insteadof(1.2),one naturallyconsiders thefundamental tone of Ω⊂M by

Γg(Ω) := Γg,n(Ω) = inf

u∈W₀^2,2(Ω)\{0}

Ω

(Δgu)²dvg

Ω

u²dvg

, (1.5)

where dvg denotesthe canonicalmeasure on (M,g), andW₀^2,2(Ω) isthe usual Sobolev space on (M,g), see Hebey [22]; in fact, it turns out that Γg(Ω) is the first eigen- valueto(1.4).DuetotheBochner-Lichnerowicz-Weitzenböckformula,theSobolevspace H₀²(Ω)=W₀^2,2(Ω) isaproperchoicefor(1.4),seeProposition3.1fordetails.

To the best of our knowledge, no results – comparable to (1.3) – are available in the literature concerning LordRayleigh’s problem for clampedplates on curved struc- tures.Accordingly,themainpurposeofthepresentpaperistoidentifythosegeometric and analytic propertieswhichresideinLord Rayleigh’sproblem for clampedplates on nonpositively curved spaces. To develop our results, the geometric context is provided by ann-dimensional(n≥2) Cartan-Hadamardmanifold(M,g) (i.e.simplyconnected, completeRiemannianmanifoldwithnonpositivesectionalcurvature).Havingthisframe- work, werecallMcKean’sspectral gap estimateformembranes whichiscloselyrelated to (1.5);namely,inann-dimensionalCartan-Hadamardmanifold(M,g) with sectional curvature K≤ −κ² forsome κ>0, theprincipalfrequency of any membraneΩ⊂M canbeestimated as

γg(Ω) := inf

u∈W₀^1,2(Ω)\{0}

Ω

|∇gu|²dvg

Ω

u²dv_g

≥ (n−1)²

4 κ²; (1.6)

inaddition,(1.6) issharponthen-dimensionalhyperbolicspace(Hⁿ_−κ2,gκ) ofconstant curvature −κ² in the sense that γgκ(Ω) → ⁽ⁿ⁻¹⁾₄ ²κ² whenever Ω tends to Hⁿ_−κ2, see McKean[30].

Beforetostateourresults,wefixsomenotations.Ifκ≥0,letN_κⁿbethen-dimensional space-formwithconstantsectionalcurvature−κ²,i.e.N_κⁿ iseitherthehyperbolicspace H₋ⁿ_κ2 when k >0, or theEuclidean space Rⁿ when κ= 0.LetB_κ(L) bethe geodesic ball of radius L > 0 in N_κⁿ and if Ω ⊂ N_κⁿ, we denote by Γκ(Ω) the corresponding valuefrom(1.5).Byconvention,weconsider1/0= +∞andasusual,Vg(S) denotesthe RiemannianvolumeofS⊂M.

OurfirstresultprovidesafourthordercounterpartofMcKean’sspectralgapestimate, whichrequiresthevalidityoftheκ-Cartan-Hadamardconjectureon(M,g);thelatteris nothingbutthesharpisoperimetricinequalityon(M,g),whichisvalide.g.onhyperbolic

spaces of any dimension as well as on generic 2- and 3-dimensional Cartan-Hadamard manifoldswithsectionalcurvatureK≤ −κ²forsomeκ≥0,see §2.2.

Theorem1.1. Let(M,g)beann-dimensionalCartan-Hadamardmanifoldwithsectional curvature K≤ −κ² forsome κ≥0, whichverifies theκ-Cartan-Hadamard conjecture.

If Ω⊂M isaboundeddomain with smoothboundarythen Γ_g(Ω)≥ (n−1)⁴

16 κ⁴. (1.7)

Moreover,forn∈ {2,3},relation (1.7)issharp in thesensethat Γκ(N_κⁿ) := lim

L→∞Γκ(Bκ(L)) = (n−1)⁴

16 κ⁴. (1.8)

Clearly,Theorem 1.1is relevantonly for κ>0 (as (1.7) and(1.8) trivially hold for κ = 0). Moreover, if n ∈ {2,3} and κ > 0, and Γ^l_κ(Ω) denotes the lth eigenvalue of (1.4) onΩ⊂Hⁿ_−κ2,thenmakinguseof(1.8) and aPayne-Pólya-Weinberger-Yang-type universalinequalityonHⁿ_−κ2,itturnsoutthat

Γ^l_κ(Hⁿ_−κ2) := lim

L→∞Γ^l_κ(B_κ(L)) = (n−1)⁴

16 κ⁴ for all l∈N. (1.9) Inparticular,(1.9) confirmsaclaimofChengandYang[12,Theorem1.4] forn∈ {2,3}, where theauthors assumed (1.8) itselfinorder to derive (1.9).In fact,onecanexpect the validityof (1.9) forany n ≥2 but some technical difficultiesprevent the proof in high-dimensions;fordetails,see§5.3.

Actually,Theorem1.1isjustabyproductofageneralargumentneededtoprovethe mainresultofourpaper(foritsstatement,werecallthathν isthefirstpositive critical pointof ^J_I^ν

ν andν =ⁿ₂ −1):

Theorem1.2. Letn∈ {2,3}and(M,g)beann-dimensionalCartan-Hadamardmanifold with sectional curvature K≤ −κ² for some κ≥0, Ω⊂M be a bounded domain with smoothboundary andvolumeVg(Ω)≤ _κ^cnn withc2≈21.031andc3≈1.721.If Ω⊂N_κⁿ isageodesic ballverifyingV_g(Ω)=V_κ(Ω)then

Γ_g(Ω)≥Γ_κ(Ω), (1.10)

withequalityin (1.10)if andonlyif Ωisisometric toΩ.Moreover, Γκ(Bκ(L))∼

(n−1)²

4 κ²+h²_ν L²

2

as L→0. (1.11)

Somecommentsareinorder.

TheproofofTheorems1.1and1.2isbasedonadecompositionargumentsimilartothe onecarriedoutbyTalenti[40] and AshbaughandBenguria[1] intheEuclideanframe- work. Infact,we transpose theoriginal variationalproblem fromgeneric nonpositively curvedspacestothespace-formN_κⁿbyassumingthevalidityoftheκ-Cartan-Hadamard conjecture on (M,g). By a fourth order ODE it turns out that Γ_κ(Ω) is the small- est positivesolutiontothecross-productofsuitableGaussian hypergeometricfunctions (resp., Bessel functions) whenever κ > 0 (resp., κ = 0). The aforementioned decom- position argument combined with certain oscillatory and asymptotic properties of the hypergeometricfunctionprovidestheproofofTheorem 1.1.

Thedimensionalityrestrictionn∈ {2,3}inTheorem1.2(andrelation(1.8))isneeded not onlyforthevalidity oftheκ-Cartan-Hadamardconjecture butalso forsomepecu- liar propertiesof the Gaussian hypergeometricfunction; similar phenomenonhasbeen pointed out also by Ashbaugh and Benguria [1] in the Euclidean setting for Bessel functions. In addition, the arguments in Theorem 1.2 work only for sets with suffi- ciently small measure; unlike the usual Lebesgue measure in Rⁿ (where the scaling Γ0(B0(L))= L⁻⁴Γ0(B0(1)) holds for everyL > 0),the inhomogeneity of the canoni- cal measure on hyperbolic spaces requires the aforementioned volume-restriction. The intuitive feeling we get that eigenfunctions corresponding to Γg(Ω) on alarge domain Ω⊂M with strictlynegative curvaturemayhavelargenodal domainswhose symmet- ricrearrangementsinHⁿ_−κ2 produce largegeodesicballs andtheir‘joined’fundamental tonecanbedefinitelylowerthantheexpectedΓ_κ(Ω).Infact,ourargumentsshowthat Theorem 1.2cannotbeimprovedevenifwerestrictthesettingtothemodelspace-form H_−κⁿ 2. It remainsan open question whetheror not(1.10) remainsvalid for arbitrarily large domains in any dimension n ≥ 4; we notice however that some nonoptimal es- timates of Γg(Ω) are also provided for any domain in high-dimensions (see §5.4). The asymptotic property (1.11) for κ> 0 followsby an elegant asymptotic connection be- tween hypergeometricand Bessel functions,which is crucial inthe proofof (1.10) and its accuracy isshowninTable 1(see §5.2) forsomevalues ofL 1.Clearly,(1.11) is trivialforκ= 0 sinceΓ0(B0(L))=h⁴_ν/L⁴foreveryL>0.

A naturalquestionarises concerningthe sharpestimate ofthe fundamentaltone on completen-dimensionalRiemannianmanifoldswithRiccicurvatureRic_(M,g)≥k(n−1) for some k ≥ 0. Some arguments based on the spherical Laplacian show that Bessel functions (whenk= 0)and Gaussian hypergeometricfunctions(whenk >0) will play again crucial roles. Since the parameter range of the aforementioned special functions inthenonnegativelycurved caseisdifferentfromthepresent setting,furthertechnical- itiesappearwhichrequireadeepanalysis.Accordingly,weintend tocome backtothis problem inaforthcomingpaper.

As anapplicationofTheorem1.2,weconsidertheellipticproblem Δ²_κu−μΔ_κu+γu=|u|^p−2u in B_κ(L),

u≥0, u∈W₀^2,2(B_κ(L)), (P)

where Bκ(L)⊂H₋²_κ2 is ahyperbolic discand therange ofparameters μ,γ,p,κand L isspecifiedbelow.Byusingvariationalarguments, onecanprovethefollowingresult.

Theorem 1.3.Let μ ≥ 0, γ ∈ R, p > 2, κ > 0 and 0 < L < ^2.1492_κ . The following statements hold:

(i) if μ= 0 andproblem(P)admits anonzerosolution thenγ >−Γ_κ(B_k(L));

(ii) ifμ>0andγ >−Γκ(Bk(L))then problem(P)admits anonzerosolution.

Thepaperisorganizedas follows. InSection2werecall/provethose notions/results whichareindispensableinourstudy(space-forms,κ-Cartan-Hadamardconjecture,oscil- latorypropertiesofspecificGaussianhypergeometricfunctions).InSection3wedevelop anAshbaugh-Benguria-Talenti-typedecompositionfromcurvedspacestospace-forms.In Sections4and5weprovideaMcKean-typespectralgapestimate(proofofTheorem1.1) and comparison principles (proof of Theorem 1.2) for fundamental tones, respectively.

InSection6weproveTheorem 1.3.

2. Preliminaries

2.1. Space-forms

Letκ≥0 andN_κⁿ bethen-dimensionalspace-formwithconstantsectionalcurvature

−κ². Whenκ= 0, N_κⁿ =Rⁿ isthe usual Euclidean space, while for κ>0,N_κⁿ is the n-dimensionalhyperbolic space representedby thePoincaréball modelN_κⁿ =Hⁿ_−κ2 = {x∈Rⁿ:|x|<1}endowedwiththeRiemannianmetric

gκ(x) = (gij(x))i,j=1,...,n=p²_κ(x)δij,

wherepκ(x)= _κ(1−|x|² 2).(Hⁿ_−κ2,gκ) isaCartan-Hadamardmanifoldwith constantsec- tionalcurvature−κ².If∇anddivdenotetheEuclideangradientanddivergenceoperator inRⁿ,thecanonicalvolumeform,gradientandLaplacianoperatoronN_κⁿ are

dvκ(x) =

dx if κ= 0,

pⁿ_κ(x)dx if κ >0, ∇κu=

∇u if κ= 0,

∇u

p²_κ if κ >0, and

Δκu=

Δu if κ= 0, p⁻ⁿ_κ div(pⁿ⁻²_κ ∇u) if κ >0,

respectively. Thedistance function on N_κⁿ is denoted by d_κ; the distance between the originand x∈N_κⁿ isgiven by

dκ(x) :=dκ(0, x) =

|x| if κ= 0,

1 κln

1+|x|

1−|x|

if κ >0.

ThevolumeofthegeodesicballBκ(r)={x∈N_κⁿ:dκ(x)< r}is

Vκ(r) :=Vκ(Bκ(r)) =nωn

r 0

sκ(ρ)ⁿ⁻¹dρ, (2.1)

where

sκ(ρ) =

ρ if κ= 0,

sinh(κρ)

κ if κ >0.

A simplechangeofvariablesgivesthefollowingusefultransformation.

Proposition 2.1. Letκ≥0.Forevery integrable functiong: [0,L]→R with L≥0one has

L 0

g(s)ds=

Bκ(rL)

g(Vκ(dκ(x)))dvκ(x),

where rL≥0is theuniquerealnumber verifyingVk(rL)=L.

2.2. κ-Cartan-Hadamard conjecture

Let(M,g) beann-dimensionalCartan-Hadamardmanifoldwith sectionalcurvature bounded above by−κ² forsome κ≥0.Theκ-Cartan-Hadamard conjecture on (M,g) (called also as the generalized Cartan-Hadamard conjecture) states that the κ-sharp isoperimetricinequalityholdson(M,g),i.e. foreveryopenbounded Ω⊂M onehas

A_g(∂Ω)≥A_κ(∂B_κ(r)), (2.2)

whenever V_g(Ω)=V_κ(r);moreover, equalityholds in(2.2) ifand onlyifΩ is isometric to B_κ(r).Hereafter,A_g andA_κstandfortheareaon(M,g) andN_κⁿ,respectively.

The κ-Cartan-Hadamard conjecture holds for every κ ≥ 0 on space-forms with constantsectional curvature−κ² (of anydimension), seeDinghas [18], andon Cartan- Hadamardmanifoldswithsectionalcurvatureboundedaboveby−κ²ofdimension2,see Bol [5],and ofdimension3,seeKleiner[26]. Inaddition,averyrecentresultofGhomi and Spruck[20] statesthatthe0-Cartan-Hadamardconjectureholdsinanydimension;

indimension 4,thevalidityofthe 0-Cartan-Hadamardconjecture isdue toCroke [14].

Inhigherdimensionsandforκ>0,theconjectureisstillopen;foradetaileddiscussion, see KloecknerandKuperberg [28].

2.3. Gaussian hypergeometricfunction

Fora,b,c∈C (c= 0,−1,−2,...)theGaussianhypergeometricfunctionisdefinedby F(a, b;c;z) =₂F₁(a, b;c;z) =

k≥0

(a)k(b)k

(c)k

z^k

k! (2.3)

on the disc |z| < 1 and extended by analytic continuation elsewhere, where (a)_k =

Γ(a+k)

Γ(a) denotesthePochhammersymbol.Thecorrespondingdifferentialequationtoz→ F(a,b;c;z) is

z(1−z)w(z) + (c−(a+b+ 1)z)w(z)−abw(z) = 0. (2.4) Wealsorecallthedifferentiationformula

d

dzF(a, b;c;z) = ab

c F(a+ 1, b+ 1;c+ 1;z). (2.5) Letn≥2 beaninteger,K >0 befixed,and considerthefunction

w^K_±(t) =F

⎛

⎝1−

(n−1)²±4√ K

2 ,1 +

(n−1)²±4√ K

2 ;n

2;−t

⎞

⎠, t >0.

Thefollowingresultwillbe indispensableinourstudy.

Proposition2.2. LetK >0be fixed.The followingpropertieshold:

(i) w₊^K(t)>0forevery t≥0;

(ii) ifK≤ ⁽ⁿ⁻₁₆¹⁾⁴,then w^K₊(t)≥w₋^K(t)>0forevery t≥0;

(iii) w₋^K isoscillatoryon (0,∞)ifandonly if K >⁽ⁿ⁻₁₆¹⁾⁴. Proof. Forsimplicityof notation,leta_± =¹⁻

(n−1)²±4√ K

2 andb_±=¹⁺

(n−1)²±4√ K

2 .

(i)Theconnectionformula (15.10.11)ofOlveretal.[33] impliesthat w^K₊(t) = (1 +t)^−b+F

n

2 −a+, b+;n 2; t

1 +t

, t≥0.

Dueto ⁿ₂ −a+>0,b+>0 and(2.3),wehavethatw^K₊(t)>0 foreveryt≥0.

(ii)Fix0< K≤ ⁽ⁿ⁻¹⁾₁₆ ⁴.First,since ⁿ₂ −a₋ >0 andb₋>0,theconnectionformula (15.10.11)of[33] togetherwith (2.3) impliesagainthat

w^K₋(t) = (1 +t)^−b−F n

2−a₋, b₋;n 2; t

1 +t

>0, t >0.

By virtue of (2.4), an elementary transformation shows that w_± := w_±^K verifies the ordinarydifferentialequation

t(t+ 1)w_±(t) + 2t+n

2

w_±(t) +1−(n−1)²∓4√ K

4 w_±(t) = 0, t >0. (2.6) It turnsoutthat(2.6) isequivalentto

p(t)w_±(t)

+q_±(t)w_±(t) = 0, t >0, (2.7) where p(t) =tⁿ²(1+t)²⁻ⁿ², q_±(t)= ˜K_±tⁿ²⁻¹(1+t)¹⁻ⁿ² andK˜_± = ^{1−(n−1)2∓4}

√K

4 .For

any τ >0,relation(2.7) andaSturm-typeargumentgivesthat

0 = τ 0

w₋

pw₊

+q+w+

−w+

pw₋

+q₋w₋

= τ 0

(q₊−q₋)w₋w₊+ p

w₋w₊−w₊w₋ ^τ

0.

Since q₊< q₋,p(0)= 0,andw_± >0,we necessarilyhavethatw₋w₊ −w₊w₋ ≥0 on (0,∞).Inparticular, ^w_w⁺

− is non-decreasingon(0,∞) andsincew+(0)=w₋(0)= 1,we havethatw+≥w₋ on[0,∞).

(iii) By(ii)we havew₋^K(t)>0 for everyt>0 whenever0< K ≤ ⁽ⁿ⁻₁₆¹⁾⁴, i.e.w^K₋ is notoscillatoryon(0,∞) fornumbersKbelongingto thisrange.

AssumenowthatK >⁽ⁿ⁻₁₆¹⁾⁴.Since ∞ α

1

p(t)dt<∞foreveryα >0,onecanapplythe result ofSugie, Kita and Yamaoka [39,Theorem 3.1] (see alsoHille [23]),which states thatif

p(t)q₋(t)

⎛

⎝ ∞

t

1 p(τ)dτ

⎞

⎠

2

≥ 1

4 for t1, (2.8)

then thefunctionw^K₋ in(2.7) isoscillatory.Due tothefactthat

p(t)q₋(t)

⎛

⎝ ∞

t

1 p(τ)dτ

⎞

⎠

2

∼K˜₋ as t→ ∞,

and K˜₋ = ^{1−(n−1)2+4}

√K

4 > ¹₄, inequality (2.8) trivially holds, which concludes the proof.

Remark2.1.DmitriiKarpkindlypointedoutthatforeveryβ ≥¹₂andt>0,thefunction x→F(¹₂−x,¹₂+x;β;−t) isstrictlyincreasingon[0,∞) fromwhichProposition2.2/(ii) follows; his proof is based on fine propertiesof the hypergeometric functions 2F1 and

3F₂,cf. Karp[24].

3. Ashbaugh-Benguria-Talenti-typedecomposition: fromcurvedspacesto space-forms Withoutsayingexplicitlythroughoutthissection,weputourselvesintothecontextof Theorem1.1,i.e.wefixann-dimensional(n≥2) CartanHadamardmanifold(M,g) with sectionalcurvatureK≤ −κ²≤0 (κ≥0),verifyingtheκ-Cartan-Hadamardconjecture (see§2.2).

LetΩ⊂M be abounded domain.Inspired byTalenti[40] and Ashbaughand Ben- guria[1],weprovideinthissectionadecompositionargumentbyestimatingfrombelow thefundamental tone Γg(Ω) given in(1.5) by avalue comingfrom atwo-geodesic-ball minimizationproblemonthespace-formN_κⁿ.Wefirststate:

Proposition3.1. The infimumin (1.5)isachieved.

Proof. Due to Hopf-Rinow’s theorem, the set Ω is relatively compact. Consequently, the Ricci curvature is bounded from below on Ω, see e.g. Bishop and Critenden [4, p.166], and the injectivity radius is positive on Ω, see Klingenberg [27, Proposition 2.1.10].Byasimilarargument asinHebey[22, Proposition3.3],basedontheBochner- Lichnerowicz-Weitzenböck formula, the norm of theSobolev space H₀²(Ω) = W₀^2,2(Ω), i.e. u →

⎛

⎝

Ω

(|∇²gu|²+|∇gu|²+u²)dvg

⎞

⎠

1/2

, is equivalent to the norm given by u →

⎛

⎝

Ω

(Δ_gu)²+|∇gu|²+u² dv_g

⎞

⎠

1/2

. Accordingly, (1.5) is well-defined. The proof of theclaim,i.e.putting minimumin (1.5),followsbyasimilarvariationalargumentasin AshbaughandBenguria[1,Appendix2].

Weare going to use certain symmetrization arguments àlaSchwarz; namely, ifU : Ω → [0,∞) is a measurable function, we introduce its equimeasurable rearrangement functionU :N_κⁿ→[0,∞) suchthatforeveryt>0 wehave

V_κ({x∈N_κⁿ:U(x)> t}) =V_g({x∈Ω :U(x)> t}). (3.1) If S ⊂M is ameasurable set, thenS denotes the geodesicball inN_κⁿ with centerin theoriginsuchthatV_g(S)=V_κ(S).

Letu∈W₀^2,2(Ω) be aminimizer in(1.5);sinceuis notnecessarilyof constantsign, letu+= max(u,0) andu₋ =−min(u,0) bethepositive andnegativeparts ofu, and

Ω+={x∈Ω :u+(x)>0} and Ω₋={x∈Ω :u₋(x)>0}, respectively.Forfurtheruse,leta,b≥0 such that

Vκ(a) =Vg(Ω+) and Vκ(b) =Vg(Ω₋). (3.2) In particular, Vκ(a)+Vκ(b)=Vg(Ω) =Vκ(L) for someL>0.We definethe functions u₊,u₋ :N_κⁿ→[0,∞) suchthatforeveryt>0,

Vκ({x∈N_κⁿ:u₊(x)> t}) =Vg({x∈Ω :u+(x)> t}) =:α(t), (3.3) V_κ({x∈N_κⁿ:u₋(x)> t}) =V_g({x∈Ω :u₋(x)> t}) =:β(t). (3.4) Thefunctionsu₊andu₋arewell-definedandradiallysymmetric,verifyingtheproperty thatforsomer_t>0 andρ_t>0 onehas

{x∈N_κⁿ:u₊(x)> t}=Bκ(rt) and {x∈N_κⁿ:u₋(x)> t}=Bκ(ρt), (3.5) with Vκ(rt)=α(t) andVκ(ρt)=β(t),respectively.

Forfurtheruse,weconsider thesets

Λ_t =∂({x∈N_κⁿ:u₊(x)> t}), Λt=∂({x∈Ω :u+(x)> t}), Π_t =∂({x∈N_κⁿ :u₋(x)> t}), Π_t=∂({x∈Ω :−u(x)> t}).

Proposition 3.2. Letu∈W₀^2,2(Ω) beaminimizer in (1.5).Thenfora.e.t>0wehave (i) A_g(Λ_t)²≤α(t)

{u(x)>t}

Δ_gudv_g; (ii) Ag(Πt)²≤β(t)

{u(x)<−t}

Δgudvg.

Proof. Statements(i)and(ii)aresimilartothosebyTalenti[40,Appendix,p.278] inthe Euclideansetting;forcompleteness,wereproducetheproofinthecurvedframework.By densityreasons,itisenoughtoconsiderthecasewhenuissmooth.Forh>0,Cauchy’s inequalityimplies

⎛

⎜⎝1 h

t<u(x)≤t+h

|∇gu(x)|dvg

⎞

⎟⎠

2

≤ α(t)−α(t+h) h

1 h

t<u(x)≤t+h

|∇gu(x)|²dvg.

When h → 0, the latter relation and the co-area formula (see Chavel [8, p.86])imply that

Ag(Λt)²≤ −α(t)

Λt

|∇gu|dHn−1,

whereHn−1isthe(n−1)-dimensionalHausdorffmeasure.Thedivergencetheoremgives that

Λt

|∇gu|dHn−1=−

{x∈Ω:u+(x)>t}

Δgudvg=−

{x∈Ω:u(x)>t}

Δgudvg,

whichconcludestheproofof (i).Similarargumentshold intheproofof(ii).

Let

F(s) = (Δ_gu)^#₋(s)−(Δ_gu)^#₊(V_g(Ω)−s) and G(s) =−F(V_g(Ω)−s), s∈[0, V_g(Ω)], where·^# standsforthenotation

H^#(s) =H(x) for s=Vκ(dκ(x)), x∈Ω.

Proposition3.3. Forevery t>0onehas that

(i)

α(t)

0

F(s)ds≥ −

{u(x)>t}

Δ_gu(x)dv_g(x);

(ii)

β(t)

0

G(s)ds≥ −

{u(x)<−t}

Δ_gu(x)dv_g(x).

Proof. We first recalla Hardy-Littlewood-Pólya-typeinequality,i.e. if U : Ω →[0,∞) is an integrablefunction and U is defined by (3.1), onehas for every measurable set S⊆Ω that

S

Udvg≤

S

Udvk; (3.6)

moreover,ifS= Ω,theequalityholdsin(3.6) asUbeinganequimeasurablerearrange- mentofU.

(i)Lett>0 befixed.Inordertocomplete theproof,wearegoingtoshowfirstthat

α(t)

0

(Δgu)^#₋(s)ds≥

{u(x)>t}

(Δgu)₋(x)dvg(x), (3.7)

and

{u(x)>t}

(Δ_gu)₊(x)dv_g(x)≥

α(t)

0

(Δ_gu)^#₊(V_g(Ω)−s)ds. (3.8)

To do this, letr_t >0 betheunique real numberwith V_k(r_t)=α(t), see (3.5). The estimate (3.7) followsbyProposition2.1and inequality(3.6) as

α(t)

0

(Δgu)^#₋(s)ds=

Bκ(rt)

(Δgu)^#₋(Vκ(dκ(x)))dvκ(x) =

Bκ(rt)

(Δgu)₋(x)dvκ(x)

≥

{u(x)>t}

(Δ_gu)₋(x)dv_g(x),

where weexploredthat{x∈Ω:u(x)> t}=B_κ(r_t),following byV_k(r_t)=α(t).

The proof of (3.8) is similar; forcompleteness, we provide its proof. Byachangeof variable andProposition2.1itturnsoutthat

α(t)

0

(Δ_gu)^#₊(V_g(Ω)−s)ds=

Vg(Ω) 0

(Δ_gu)^#₊(s)ds−

Vg(Ω)−α(t)

0

(Δ_gu)^#₊(s)ds

=

Ω

(Δgu)^#₊(Vκ(dκ(x)))dvκ(x)−

Bκ(τt)

(Δgu)^#₊(Vκ(dκ(x)))dvκ(x)

=

Ω

(Δ_gu)₊(x)dv_κ(x)−

Bκ(τt)

(Δ_gu)₊(x)dv_κ(x),

where τ_t>0 istheuniquerealnumberverifyingV_κ(τ_t)=V_g(Ω)−α(t).Let A_t={x∈ Ω :u(x)≤t}; then Vg(At)=Vg(Ω)−α(t)=Vκ(τt). In particular, byinequality (3.6) (together withtheequalityforthewholedomain)andthelatterrelationswehave

α(t)

0

(Δgu)^#₊(Vg(Ω)−s)ds=

Ω

(Δgu)₊(x)dvκ(x)−

Bκ(τt)

(Δgu)₊(x)dvκ(x)

≤

Ω

(Δ_gu)₊(x)dv_g(x)−

At

(Δ_gu)₊(x)dv_g(x)

=

Ω\At

(Δgu)+(x)dvg(x) =

{u(x)>t}

(Δgu)+(x)dvg(x),

whichconcludestheproofof(3.8).

By(3.7) and(3.8) one has

α(t)

0

F(s)ds=

α(t)

0

(Δ_gu)^#₋(s)ds−

α(t)

0

(Δ_gu)^#₊(V_g(Ω)−s)ds

≥

{u(x)>t}

(Δgu)₋(x)dvg(x)−

{u(x)>t}

(Δgu)+(x)dvg(x)

=−

{u(x)>t}

Δ_gu(x)dv_g(x),

whichispreciselyourclaim.Theproofof(ii)issimilar.

Weconsiderthefunctionv: Ω=Bκ(L)→R definedby

v(x) = 1 nω_n

a dκ(x)

s_κ(ρ)¹⁻ⁿ

⎛

⎜⎝

Vκ(ρ) 0

F(s)ds

⎞

⎟⎠dρ. (3.9)

Adirectcomputationshowsthatv isasolutiontotheproblem −Δκv(x) =F(Vκ(dκ(x))) in Bκ(a);

v= 0 on ∂B_κ(a). (3.10)

Inasimilarway, thefunctionw: Ω^∗=B_κ(L)→Rgiven by

w(x) = 1 nωn

b dκ(x)

s_κ(ρ)¹⁻ⁿ

⎛

⎜⎝

Vκ(ρ) 0

G(s)ds

⎞

⎟⎠dρ (3.11)

isasolutionto

−Δ_κw(x) =G(V_κ(d_κ(x))) in B_κ(b);

w= 0 on ∂B_κ(b). (3.12)

Inparticular,bytheirdefinitions,itturnsoutthat

v≥0 in B_κ(a) and w≥0 in B_κ(b).

In fact, much precise comparisons can be said by combining the above preparatory results:

Theorem 3.1.Letv and wfrom (3.9) and (3.11),respectively.Then

u₊≤v in Bκ(a); (3.13)

u₋≤w in Bκ(b), (3.14)

where aandb are from (3.2).Inparticular,one has

Ω

u²dv_g≤

Bκ(a)

v²dv_κ+

Bκ(b)

w²dv_κ. (3.15)

In addition,

Ω

(Δgu)²dvg =

Bκ(a)

(Δκv)²dvκ+

Bκ(b)

(Δκw)²dvκ. (3.16)

Proof. We first prove(3.13). Since (M,g) verifies the κ-Cartan-Hadamard conjecture, onaccount of(3.3) and(3.4),itfollowsthat

Aκ(Λ_t)≤Ag(Λt) for a.e. t >0, (3.17) A_κ(Π_t)≤A_g(Π_t) for a.e. t >0. (3.18) Byrelation(3.17) andPropositions3.2and 3.3,onehasfora.e.t>0 that

Aκ(Λ_t)²≤ −α(t)

α(t)

0

F(s)ds.

Due to(3.3),(3.5) and(2.1),itfollows thatfora.e.t>0, α(t) =A_κ(Λ_t)r_t =nω_ns_κ(r_t)ⁿ⁻¹r_t. Combining theaboverelations,ityields

nω_n ≤ −r_ts_κ(r_t)¹⁻ⁿ

Vκ(rt) 0

F(s)ds.

After anintegration,weobtainforeveryτ ∈[0,u₊L^∞(Ω)] that

nωnτ ≤ − τ 0

r_tsκ(rt)¹⁻ⁿ

Vκ(rt) 0

F(s)dsdt.