Lower Bounds for Laplacian and Fractional Laplacian Eigenvalues

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1 Lower Bouds for Laplacia ad Fractioal Laplacia Eigevalues arxiv:.457v [math.dg] 7 Fe Guoxi Wei He-Ju Su ad Ligzhog Zeg Astract: I this paper, we ivestigate eigevalues of Laplacia o a ouded domai i a -dimesioal Euclidea space ad otai a sharper lower oud for the sum of its eigevalues, which gives a improvemet of results due to A. D. Melas [5]. O the other had, for the case of fractioal Laplacia ) α/ D, where α,], we otai a sharper lower oud for the sum of its eigevalues, which gives a improvemet of results due to S.Y. Yolcu ad T. Yolcu [3]. Itroductio Let D R e a ouded domai with piecewise smooth oudary D i a - dimesioal Euclidea spacer. Letλ i ethei-theigevalue ofthefixedmemrae prolem: { u+λu =, i D,.) u =, o D, where isthe LaplaciaiR. It iswell ow that thespectrum ofthis eigevalue prolem is real ad discrete: < λ λ λ 3 +, where each λ i has fiite multiplicity which is repeated accordig to its multiplicity. If we use the otatios VolD) ad ω to deote the volume of D ad the volume of the uit all i R, respectively, the Weyl s asymptotic formula asserts that the eigevalues of the fixed memrae prolem.) satisfy the followig formula: λ 4π, +..) ω VolD)) From the aove asymptotic formula, it follows directly that λ i 4π +ω VolD)), +..3) i= Mathematics Suject Classificatio: 35P5. Key words ad phrases: eigevalues, lower oud, Laplacia, fractioal Laplacia. The first author ad the secod author were supported y the Natioal Natural Sciece Foudatio of Chia Grat Nos.87, 3).

2 G. Wei, H.-J. Su ad L. Zeg Pólya [7] proved that λ 4π, for =,,,.4) ω VolD)) if D is a tilig domai i R. Furthermore, he put forward the followig: Cojecture of Pólya. If D is a ouded domai i R, the the -th eigevalue λ of the fixed memrae prolem satisfies λ 4π, for =,,..5) ω VolD)) O the Cojecture of Pólya, Berezi [] ad Lie [3] gave a partial solutio. I particular, Li ad Yau [3] proved the Berezi-Li-Yau iequality as follows: λ i 4π +ω VolD)), for =,,..6) i= Theformula.3)showsthattheresultofLiadYauissharpithesese ofaverage. From this iequality.6), oe ca derive λ 4π +ω VolD)), for =,,,.7) which gives a partial solutio for the cojecture of Pólya with a factor +. We prefer to call this iequality.6) as Berezi-Li-Yau iequality istead of Li-Yau iequality ecause.6) ca e otaied y a Legedre trasform of a earlier result y Berezi [] as it is metioed [4]. Recetly, improvemets to the Berezi- Li-Yau iequality give y.6) for the fixed memrae prolem have appeared, for example see [, 5, ]. I particular, A.D.Melas [5] has improved the estimate.6) to the followig: where λ i 4π +ω VolD)) i= + 4+) VolD), for =,,,.8) IeD) IeD) =: mi x a a R D dx is called the momet of iertia ofd. After a traslatioof the origi, we ca assume that the ceter of mass is the origi ad IeD) = x dx. By taig a value eary the extreme poit of the fuctio fτ) give y??)), we add oe term of lower order of to its right had side, which meas that we otai a sharper result tha.8). I fact, we prove the followig: D

3 Lower Bouds for Laplacia ad Fractioal Laplacia Eigevalues Theorem.. Let D e a ouded domai i a -dimesioal Euclidea space R. Assume that λ i,i =,,, is the i-th eigevalue of the eigevalue prolem.). The the sum of its eigevalues satisfies λ j + ω π) VolD) + 4+) + 34+) ω π) VolD) IeD) VolD) IeD) ) VolD)..9) Furthermore, we cosider the fractioal Laplacia operators restricted to D, ad deote them y ) α/ D, where α,]. This fractioal Laplacia ca e defied y ) α/ ux) uy) ux) =: P.V. dy, R x y +α where P.V. deotes the pricipal value ad u : R R. Defie the characteristic fuctio χ D : t χ D t) y χ D t) = {, x D,, x R \D, the the special pseudo-differetial operator ca e represeted as the Fourier trasform of the fuctio u [,9], amely ) α/ D u := F [ ξ α F[uχ D ]], where F[u] deotes the Fourier trasform of a fuctio u : R R: F[u]ξ) = ûξ) = e ix ξ ux)dx. π) R Itiswell owthatthefractioallaplaciaoperator ) α/ caecosidered as the ifiitesimal geerator of the symmetric α-stale process [3 6, 3]. Suppose thatastochasticprocessx t hasstatioaryidepedet icremets aditstrasitio desity i.e., covolutio erel) p α t,x,y) = p α t,x y), t >, x,y R is determied y the followig Fourier trasform Exp t ξ α ) = e iξ y p α t,y)dy, t >, ξ R, R the we ca say that the process X t is a -dimesioal symmetric α-stale process with order α,] i R also see [4,5,3]). Remar.. Give α =, X t is the Cauchy process i R whose trasitio desities are give y the Cauchy distriutio Poisso erel) p t,x,y) = c t t + x y ) +, t >, x,y R, 3

4 G. Wei, H.-J. Su ad L. Zeg where c = Γ + )/π + =, πω is the semiclassical costat that appears i the Weyl estimate for the eigevalues of the Laplacia. Remar.. Give α =, X t is just the usual -dimesioal Browia motio B t ut ruig at twice the speed, which is equivalet to say that, whe α =, we have X t = B t ad [ ] p x y t,x,y) = 4πt) /Exp, t >, x,y R. 4t Let Λ α j ad u α j deote the j-th eigevalue ad the correspodig ormalized eigevector of ) α/ Ω, respectively. Eigevalues Λ α j icludig multiplicities) satisfy < Λ α) Λ α) Λ α) 3 +. For the case of α =, E. Harrell ad S. Y. Yolcu gave a aalogue of the Berezi-Li-Yau type iequality for the eigevalues of the Klei-Gordo operators H,D := restricted to D i [9]: Λ α) j + π ω VolD)) )..) Very recetly, S.Y.Yolcu [] has improved the estimate.) to the followig: Λ α) j C + VolD) VolD) + + M /,.) IeD) where C = π ad the costat M ω ) depeds oly o the dimesio. Moreover, for ay α,], S.Y.Yolcu ad T.Yolcu [3] geeralized.) as follows: Λ α) j +α π ω VolD)) ) α α..) Furthermore, S.Y.Yolcu ad T.Yolcu [3] refied the Berezi-Li-Yau iequality i the case of fractioal Laplacia ) α D restricted to D: ) Λ α) j α π α +α j ω VolD)).3) l π) α VolD) + 4+α) ω VolD)) α IeD) α, where l is give y { } α l = mi, 4απ. + α)ω 4 4

5 Lower Bouds for Laplacia ad Fractioal Laplacia Eigevalues Remar.3. I fact, y a direct calculatio, oe ca chec the followig iequality: which implies l 4+α) π) α ω VolD)) α α 4απ + α)ω 4, VolD) IeD) α = α 48+α) π) α ω VolD)) α VolD) IeD) α. The aother mai purpose ofthis paper is to provide a refiemet of theberezi- Li-Yau type estimate. I other word, we have proved the followig: Theorem.. Let D e a ouded domai i a -dimesioal Euclidea space R. Assume that Λ α) i,i =,,, is the i-th eigevalue of the fractioal Laplacia ) α/ D. The, the sum of its eigevalues satisfies Λ α) j +α π) α ω VolD)) α α + 48+α) + α+α ) C)+α) α π) α ω VolD)) α π) α 4 ω VolD)) α 4 VolD) IeD) α VolD) IeD) ) α 4,.4) where C) = { 468, whe 4, 644, whe = or = 3. I particular, the sum of its eigevalues satisfies whe α =. j + ω π) VolD) VolD) + 4+) IeD) ).5) + 34+) ω π) VolD) VolD), IeD) Λ ) Remar.4. Oservig Theorem., it is ot difficult to see that the coefficiets with respect to α ) of the secod terms i.4) are equal to that of.3). I other word, we ca claim that the iequalities.4) are sharper tha.3) sice the coefficiets with respect to α 4 ) of the third terms i.4) are positive. By usig Theorem., we ca give a aalogue of the Berezi-Li-Yau type iequality for the eigevalues of the Klei-Gordo operators H,D restricted to the ouded domai D: 5

6 G. Wei, H.-J. Su ad L. Zeg Corollary.. Let D e a ouded domaii a -dimesioaleuclidea spacer. Assume that Λ i,i =,,, is the i-th eigevalue of the Klei-Gordo operators H,D. The, the sum of its eigevalues satisfies Λ j π +ω VolD)) π) VolD) + 48+) ω VolD)) IeD) ) π) 3 + C)+) ω VolD)) 3 VolD) IeD) ) 3,.6) where C) = { 468, whe 4, 644, whe = or = 3. A Key Lemma I order to prove the followig Lemma.3, we eed the followig lemmas give y S.Y.Yolcu ad T.Yolcu i [3]: Lemma.. Suppose that ς : [, ) [,] such that ςs) ad ςs)ds =. The, there exists ǫ such that Moreover, we have ǫ+ ǫ ǫ+ ǫ s d ds = s d+α ds s d ςs)ds. s d+α ςs)ds. Lemma.. For s >, τ >, N, < α, we have the followig iequality: s +α +α s τ α α τ+α + α τ+α s τ). I light of Lemma. ad Lemma., we otai the followig result which will play importat roles i the proof of Theorem. ad Theorem.. Lemma.3. Let ) e a positive real umer ad µ> ) e defied y.3). If ψ : [, + ) [, + ) is a decreasig fuctio such that µ ψ s) 6

7 Lower Bouds for Laplacia ad Fractioal Laplacia Eigevalues ad the, we have A := s ψs)ds >, s +α ψs)ds +α A)+α ψ) α + α A)+α ψ) α+ +α)µ + α+α ) 4A)+α α) ψ) 4 α+4, µ.) whe 4; we have s +α ψs)ds +α A)+α ψ) α + α A)+α ) ψ) α+ +α)µ + α+α ) 4A)+α α) ψ) 4 α+4, µ.) whe < 4. I particular, the iequality.) holds whe α = ad. Proof. If we cosider the followig fuctio t) = ψ ψ) t) µ, ψ) the it is ot difficult to see that ) = ad t). Without loss of geerality, we ca assume ψ) = ad µ =. Defie E α := s +α ψs)ds. Oe ca assume that E α <, otherwise there is othig to prove. By the assumptio, we ca coclude that lim s s+α ψs) =. Puttig hs) = ψ s) for ay s, we get hs) ad hs)ds = ψ) =. By maig use of itegratio y parts, oe ca get s hs)ds = 7 s ψs)ds = A,

8 G. Wei, H.-J. Su ad L. Zeg ad s +α hs)ds +α)e α, sice ψs) >. By Lemma., oe ca ifer that there exists a ǫ such that ad ǫ+ s ds = ǫ s hs)ds = A,.3) Let ǫ+ ǫ s +α ds s +α hs)ds +α)e α..4) Θs) = s +α +α)τ α s +ατ +α ατ +α s τ), the, y Lemma., we have Θs). Itegratig the fuctio Θs) from ǫ to ǫ+, we deduce from.3) ad.4), for ay τ >, Defie +α)e α +α)τ α A+ατ +α α τ+α..5) fτ) := +α)τ α A ατ +α + α τ+α,.6) the we ca otai from.5) that, for ay τ >, Taig E α = τ = A) s +α ψs)ds fτ) +α). +α) A) + +α ), ad sustitutig it ito.6), we otai fτ) = A) +α ) α+α ) A) +α) + +α )+α + +α +α) A). + α A)+α +α) A) )α.7) By usig the Taylor formula, oe has for t > +t) α + α αα ) t+ t + αα )α ) t αα )α )α 3) 4 4 t 4, 8

9 Lower Bouds for Laplacia ad Fractioal Laplacia Eigevalues ad +t) +α + +α t+ +α )α ) t + +α )α )α ) t α )α )α )α ) 4 4 t 4. Puttig t = +α +α) A) >, oe has αt >, τ = A) +t), ) α+α ) A) +α) +α) A) + +α )α = αt)+t) α [ αt) + α αα ) t+ t + αα )α ) t ] + αα )α )α 3) t = αα+) = αα+) t αα )α+) t 3 αα )α )α+) t α α )α )α 3) t α +α) A) αα )α+) 3 αα )α )α+) 8 3 ) +α +α) A) α α )α )α 3) 4 4 ) 3 +α +α) A) +α +α) A) ) 4 5 ),.8) 9

10 G. Wei, H.-J. Su ad L. Zeg ad +α) A) + +α = +t) +α + +α + +α )α ) )+α +α +α) A) ) +α +α) A) + +α )α )α ) 6 3 ) +α +α) A) + +α )α )α )α ) 4 4 ) 3 +α +α) A) 4 )..9) Therefore, we otai from.8) ad.9) fτ) = +α)τ α A ατ +α + α τ+α [ A) +α αα+) +α +α) A) αα )α+) +α 3 +α) A) αα )α )α+) 8 3 α α )α )α 3) 4 4 [ + α A)+α + +α + +α )α ) +α ) 3 ) +α +α) A) +α +α) A) ) 4 +α +α) A) +α) A) + +α )α )α ) 6 3 ) ) 5 ] ) +α +α) A) + +α )α )α )α ) 4 4 ) 3 +α +α) A) ) 4 ].) = A) +α + α A)+α +I +I +I 3,

11 Lower Bouds for Laplacia ad Fractioal Laplacia Eigevalues where I = α+α ) 88+α) A)+α 4,.) I = α+α )α+ 6) 7 +α +α) ) A) +α 6 + α+α )[α +5 6)α )] 88 ) 3 3 +α A) +α 8, +α).) I 3 = αγ 4 4 +α +α) ) 5 A) +α, ad γ = α )α )α )+α) αα )α )α 3). Noticig that αα )α )α 3), we have γ α )α )α )+α). Defie β := α )α )α )+α), the we have β ad γ β. Therefore, we have I 3 αβ 4 4 +α +α) ) 5 A) +α,.3) Next, we cosider two cases: Case : 4. Whe 4, for ay α,], we ca ifer α +5 6)α ) 5 6)α ) = α)+ 6α 4+8+)+..4)

12 G. Wei, H.-J. Su ad L. Zeg Sice A) +) 3 see [8]), oe ca deduce from.) ad.4) I α+α )α+ 6) 7 +α +α) ) A) +α 6 + α+α )[α +5 6)α )] 5 ) 3 +α A) +α 6 +α) = α+α )[ 6α+ 6)+α +5 6)α ) ] 5 ) 3 +α A) +α 6 +α) = α+α )[ α)+α 6α+6) ] 5 ) 3 +α A) +α 6. +α).5) O the other had, we have I 3 αβ α +α) ) A) +α 6, sice β ad A) I +I 3 ca e give y I +I 3 +). Therefore, the estimate of the lower oud of 3 { α+α ) [ α)+α 6α+6) ] 5 3 ) +α A) +α 6 +α) } + αβ = α{4+α )[ α)+α 6α+6)]+β} 468 ) 4 +α A) +α 6. +α) Next, we will verify the followig iequality 4+α )[ α)+α 6α+6)]+β..6)

13 Lower Bouds for Laplacia ad Fractioal Laplacia Eigevalues Ideed, sice < α ad 4, we have 4+α )[ α)+α 6α+6)]+β = 4+α )[ α)+α 6α+6)] +α )α )α )+α) 8[ α)+α 6α+6)] α )α )α )+α) 8[ α)+α 6α+6)] +)+)+) 8[6 88+α 8α)] +)+)+) 86 9) +)+)+) = ) Thus, it is ot difficult to see that the iequality.6) follows from.7), which implies Therefore, whe 4, we have I +I 3. fτ) A) +α + α A)+α + α+α ) 88+α) A)+α 4. Case : < 4. Uitig the equatios.),.) ad.3), we otai the followig equatio I +I +I 3 α+α ) A)+α 4 88+α) + α+α )α+ 6) 7 ) +α A) +α 6 +α) + α+α )[α +5 6)α )] 88 ) 3 3 +α A) +α 8 +α) ) + αβ 5 +α A) +α 4 4 +α) 3

14 G. Wei, H.-J. Su ad L. Zeg = α+α ) A)+α 4 + α+α ) A)+α α) 5+α) + α+α )ν +α 7 +α) + α+α )ν +α α) ) + αβ 5 +α A) +α 4 4, +α) ) A) +α 6 ) 3 A) +α 8 where ad ν := α+ 6), ν := α +5 6)α ). Suppose ν ad ν, the we have I +I +I 3 α+α ) A)+α 4 + α+α ) 384+α) 96 ) where + α+α )α+ 6) 88 = α+α ) A)+α α) +α +α) ) +α +α) A) +α 4 + α+α )[α +5 6)α )] ) α A) +α 4 +α) ) + αβ 5 +α A) +α 4 4 +α) ) + α+α ) +α I α) ) + αβ 5 +α A) +α 4 4, +α) I 4 = + α α +5 6)α ) A) +α 4 A) +α 4.8)

15 Lower Bouds for Laplacia ad Fractioal Laplacia Eigevalues Noticig that < α ad < 4, we have I 4 = 48 +6α+ 6)+α +5 6)α ) 48 = 74 +α 88)+α 6α+6) 48 6+α+α 8α) 48 = 6+ 8)α+α = 5 4. Therefore, we derive from.8) ad.9) I +I +I 3 α+α ) A)+α 4 + 5α+α ) 384+α) 384 ) + αβ 5 +α A) +α 4 4 +α) α+α ) A)+α 4 + 5α+α ) 384+α) 384 ) + α+α )β 843+α) 4 α+α ) A)+α α) +α +α) + α+α )[4 +α)+β] 843+α) 4 A) +α 4 +α +α) +α +α) +α +α) ) ) ) A) +α 4, A) +α 4 A) +α 4.9) sice A) +). We defie a fuctio K) y lettig 3 K) := 4 +α)+β = 4 +α)+α )α )α )+α), where [,4). After a direct calculatio, we have which implies K) 4 +α) α )α )α )+α) 4 +α) +)+)+) 4 +α) )3)) α >, I +I +I 3 α+α ) 384+α) A)+α 4. 5

16 G. Wei, H.-J. Su ad L. Zeg Fortheother cases i.e., ν adν > ; ν > adν ; or ν > ad ν > ), we ca also derive y usig the same method that Therefore, whe 4, we have fτ) A) +α I +I +I 3 α+α ) 384+α) A)+α 4. + α A)+α + α+α ) 384+α) A)+α 4. I particular, we ca cosider the case that α =. Noticig that β = whe α = ad, we ca claim that I 3. Therefore, whe α = ad, oe ca deduce I +I 3 α+α )α+ 6) 7 +α +α) ) A) +α 6 + α+α )[α +5 6)α )] 88 ) 3 3 +α A) +α 8 +α) ) = ) A) ) 4 +) ) ) A) ) 96 +) ) = A) ) ) A) ), which implies ) 3 A) 6 ) A) 4 fτ) A) +α + α A)+α This completes the proof of the Lemma.3. + α+α ) 88+α) A)+α 4. 3 Proofs of Theorem. ad Theorem. I this sectio, we will prove the Theorem. ad Theorem. y usig the ey lemma give i sectio i.e., Lemma.3). 6

17 Lower Bouds for Laplacia ad Fractioal Laplacia Eigevalues We suppose that D R is a ouded domai i R, ad the its symmetric rearragemet D is the ope all with the same volume as D, D = { x R x < ) } VolD). ω By usig a symmetric rearragemet of D, oe ca otai IeD) = D x dx x dx = D + VolD) ) VolD). 3.) ω For the case of fractioal Laplace operator, let u α) j e a orthoormal eigefuctio correspodig to the eigevalue Λ α) j. Namely, u α) j satisfies { ) α/ u α) j = Λ α) u α) j, i D, D uα) i x)u α) j x)dx = δ ij, for ay i,j, where < α. O the other had, for the case of Laplace operator, we let v j e a orthoormal eigefuctio correspodig to the eigevalue λ j. Namely, v j satisfies v j +λ j v j =, i D, v =, v D ix)v j x)dx = δ ij, o D, for ay i,j. Thus, oth {u α) j } ad {v j} form a orthoormal asis of L D). Defie the fuctios ϕ α) j ad η j y { ϕ α) u α) j x), x D, j x) =, x R \D, ad η j x) = { v j x), x D,, x R \D, respectively. Deotey η j ξ)ad the, for ay ξ R, we have ϕ α) j ξ)thefouriertrasformsofη j ξ)adϕ α) j ξ), ϕ α) j ξ) = π) / ϕ α) j x)e i x,ξ dx = π) / R D u α) j x)e i x,ξ dx, ad η j ξ) = π) / η j x)e i x,ξ dx = π) / R D v j x)e i x,ξ dx. 7

18 G. Wei, H.-J. Su ad L. Zeg From the Placherel formula, we have ϕα) i x) ϕ α) j x)dx = η i x) η j x)dx = δ ij, R R forayi,j. Sice{u α) j } ad{v j} areorthoormalasises il D), thebessel iequality implies that ϕ α) j ξ) π) / D e i x,ξ dx = π) / VolD), 3.) ad η j ξ) π) / e i x,ξ dx = π) / VolD). 3.3) D For fractioal Laplace operator, we oserve that Λ α) j = u α) j ξ) ) α/ Ω u α) j ξ)dξ R = = R u α) j ξ) F [ ξ α F[u α) ξ α ûα) j ξ) dξ, R j ξ)]]dξ 3.4) sice the support of u α) j is D see [3]). O the meawhile, for the case of Laplace operator, we have see [, 5]) λ j = ξ v j ξ) dξ. 3.5) R Sice ad we otai ϕ α) j ξ) = ϕ α) j ξ) = π) / η j ξ) = π) / Ω Ω η j ξ) = π) ixu α) j x)e i x,ξ dx, ixv j x)e i x,ξ dx, Ω ixe i x,ξ dx = π) IeD). 3.6) Puttig f α) ξ) := 8 ϕ α) j ξ),

19 Lower Bouds for Laplacia ad Fractioal Laplacia Eigevalues ad fξ) := η j ξ), oe derives from 3.) ad 3.3) that f α) ξ) π) VolD) ad fξ) π) VolD), it follows from 3.6) ad the Cauchy-Schwarz iequality that ) / ) / f α) ξ) ϕ α) j ξ) ϕ α) j ξ) ad π) IeD)VolD), ) / ) / fξ) η j ξ) η j ξ) π) IeD)VolD), for every ξ R. Furthermore, y usig 3.4) ad 3.5), we have ad Λ α) j = ξ α ûα) j ξ) dξ = R = ξ α f α) ξ)dξ, R λ j = ξ v j ξ) dξ = R = ξ fξ)dξ. R From the Parseval s idetity, we derive f α) ξ)dξ = ϕ α) j x) dx = R R Similarly, we have [7, 5] = R fξ)dξ = = D u α) j x) dx =. R ξ α ϕ α) j ξ) dξ R ξ η j ξ) dξ η j x) dx = R v j x) dx =. D 9 D D ûα) j x) dx v j x) dx 3.7) 3.8) 3.9) 3.)

20 G. Wei, H.-J. Su ad L. Zeg Let h e a oegative ouded cotiuous fuctio o D ad h is its symmetric decreasig rearragemet, the we have see [, 7]) hx)dx = h x)dx = ω s gs)ds 3.) R R ad x α hx)dx x α h x)dx = ω R R s +α gs)ds, 3.) where α,] ad g x ) = h x). Puttig δ := sup h, the we ca otai δ g s) 3.3) for almost every s. More detail iformatio o symmetric decreasig rearragemets will e foud i [,7,8]. To e rief, we will drop the superscript α to deote f α) y f ad let f = f. Assume thatfi isthesymmetric decreasig rearragemet off i i =,), accordig to 3.9), 3.) ad 3.), we have = f i ξ)dξ = fi ξ)dξ = ω s φ i s)ds, 3.4) R R where φ i x) = f i x ) ad i =,. Applyig the symmetric decreasig rearragemet to f i, ad otig that where δ i = sup f i, we otai from 3.3) δ i π) IeΩ)VolΩ) := σ, 3.5) σ δ i φ is), where i =,. By3.), we have σ π) + ) ω VolD) + π) ω VolD) +, sice. Moreover, y usig 3.7), 3.8) ad 3.), we have ad Λ α) j = ξ α f α) ξ)dξ = ξ α f ξ)dξ R R ξ α f ξ)dξ = ω s +α φ ξ)dξ, R λ j = ξ fξ)dξ = ξ f ξ)dξ R R ξ f ξ)dξ = ω s + φ ξ)dξ. R 3.6) 3.7)

21 Lower Bouds for Laplacia ad Fractioal Laplacia Eigevalues Proof of Theorem.. I order to apply Lemma.3, from 3.4), 3.5) ad the defiitio of A, we tae =, ψs) = φ s), A = ω, ad µ = σ = π) VolD)IeD). Therefore, we ca otai from Lemma.3 ad 3.7) that λ j = ω E [ ω +) [ = ω + = + ω where t = φ ). Let the oe ca has A) + φ ) Aφ ) + 6σ ω + Ft) = + ω F t) = + ω )+ t + ω t 6σ + ω ) 4t4+ 44+)σ t t + 6+)σ + ω 4t4+ 44+) σ + + t + + ] + A) 44+)σ 4φ ) 4+ ], t t + 6+)σ + ω 4t4+ 44+) σ t 3+)σ ) σ 4ω, 3+ t. Sice F t) is icreasig o,π) VolD)], the it is easy to see that Ft) is decreasig o,π) VolD)] if F π) VolD)) <. Ideed, F π) VolD)) + ω + π) VolD)) + π) VolD)) + [ 3+) π) ω VolD) )ω π) VolD)) 3+ ] 44+) [π) ω 4 VolD) + = + π)+ ω + VolD) ) π) ω VolD) ω 6 44+) π) = π) + ω VolD) + J, ] VolD) +

22 G. Wei, H.-J. Su ad L. Zeg where J = ) π) ω 4 π) ω 4 < ) 44+) π) ω 4 π) ω 4 = π) ω 4 π) ω 4 < <, which implies that F π) VolD)) <. Here, we use the iequality ω 4 π) <. We ca replace φ ) y π) VolD) to otai λ j + ω π) VolD) + 4+) + 34+) ω π) VolD) IeD) sice σ = π) VolD)IeD). This completes the proof of Theorem.. Next, we will give the proof of Theorem.. VolD) IeD) ) VolD). Proof of Theorem..: Defie the fuctio φ x) y φ x ) := f x). The we ow that φ : [, + ) [, π) VΩ)] is a o-icreasig fuctio with respect to x. Taig =, ψs) = φ s), A = ω, ad µ = σ = π) VolD)IeD), we ca otai from Lemma.3 ad 3.6) that Λ α) j ω s +α φ s)ds )+α )+α ω ω αω φ ) α ω + +α +α)σ φ ) α+ )+α 4 α+α ) ω ω + φ C )+α) σ 4 ) 4 α+4, 3.8)

23 Lower Bouds for Laplacia ad Fractioal Laplacia Eigevalues where C ) = { 88, whe 4, 384, whe = or = 3. Moreover, we defie a fuctio ξt) y lettig ξt) = ω +α ω + α+α ) ω C )+α) σ 4 )+α t α + αω +α)σ ω ω )+α 4 t 4 α+4. )+α t α+ 3.9) Differetiatig 3.9) with respect to the variale t, it is ot difficult to see that ξ t) = αω +α ω )+α [ t α + α+) σ ω ] + 4 α+4)+α ) C ) +α)σ 4 ω ) 4 t 4+4. ) t + 3.) Lettig ζt) = ξ t) +α ) ) +α α t +, 3.) αω ω ad otig that σ π) ω VolD) +, we ca otai from 3.) ad 3.) that ) ζt) = + α+) t + σ ω + 4 α+4)+α ) C ) +α)σ α+) ω π) ω VolD) +) 4 α+4)+α ) ) 4 t 4+4 ω C ) +α)π) 4 ω 4 VolD) 4+) ) t + ω ) 4 t ) It is easy to see that the right had side of 3.) is a icreasig fuctio of t. Therefore, if the right had side of 3.) is less tha whe we tae t = 3

24 G. Wei, H.-J. Su ad L. Zeg π) VolD), which is equivalet to say that ζt) + α+) ω 4 π) + 4 α+4)+α ) 4 ω 8 C ) +α) π) 4, 3.3) we ca claim from 3.3) that ξ t) o,π) VΩ)]. By a direct calculatio, we ca otai ζt) + α+) + +) = C ), + 4 α+4)+α ) C ) +α) + 4+)+) C ) 3 3.4) sice ω 4 π) <. Thus, it is easy to see from 3.) ad 3.4) that ξ t), which implies that ξt) is a decreasig fuctio o,π) VolD)]. O the other had, we otice that < φ ) π) VolD) ad right had side of the formula 3.8) is ξφ )), which is a decreasig fuctio of φ ) o,π) VolD)]. Therefore, φ ) caereplaced y π) VolD) i.) which gives the followig iequality: where Λ α) j +α C) = π) α ω VolD)) α α + 48+α) + α+α ) C)+α) α π) α ω VolD)) α π) α 4 ω VolD)) α 4 VolD) IeD) α VolD) IeD) { 468, whe 4, 644, whe = or = 3. ) α 4, I particular, whe α =, we ca get the iequality.5) y usig the same method as the proof of Theorem.. This completes the proof of Theorem.. Acowledgmet. The authors wish to express their gratitude to Prof. Q.-M. Cheg for cotiuous ecouragemet ad ethusiastic help. 4

25 Lower Bouds for Laplacia ad Fractioal Laplacia Eigevalues Refereces [] C. Badle, Isoperimetric iequalities ad applicatios, Pitma Moographs ad Studies i Mathematics, vol. 7, Pitma, Bosto, 98. [] F. A. Berezi, Covariat ad cotravariat symols of operators, Izv. Aad. Nau SSSR Ser. Mat ), [3] R. Blumethal ad R.Getoor, The asymptotic distriutio of the eigevalues for a class of Marov operators, Pacific J. Math, 9 959) [4] R. Ba uelos ad T. Kulczyci, The Cauchy process ad the Stelov prolem, J. Fuct. Aal., ) 4) [5] R. Ba uelos ad T. Kulczyci, Eigevalue gaps for the Cauchy process ad a Poicaré iequality, J. Fuct. Aal., 34 6) [6] R. Ba uelos, T. Kulczyci ad Bart lomiej Siudeja, O the trace of symmetric stale processes o Lipschitz domais, J. Fuct. Aal., 57) 9) [7] Q. -M. Cheg ad G. Wei, A lower oud for eigevalues of a clamped plate prolem, Calc. Var. Partial Differetial Equatios, 4 ), [8] Q. -M. Cheg ad G. Wei, Upper ad lower ouds of the clamped plate prolem, ) preprit. [9] E. M. Harrell II ad S. Yildirim Yolcu, Eigevalue iequalities for Klei-Gordo Operators, J. Fuct. Aal., 56) 9) [] H. Kovaří, S. Vugalter ad T. Weidl, Two dimesioal Berezi-Li-Yau iequalities with a correctio term, Comm. Math. Phys., 873) 9), 959C98. [] N. S. Ladof, Foudatios of moder potetial theory, New Yor: Spriger-Verlag 97). [] P. Li ad S. T. Yau, O the Schrödiger equatios ad the eigevalue prolem, Comm. Math. Phys., ), [3] E. Lie, The umer of oud states of oe-ody Schrödiger operators ad the Weyl prolem, Proc. Sym. Pure Math., 36 98), 4-5. [4] A. Laptev ad T. Weidl, Recet results o Lie-Thirrig iequalities, Jourées Équatios aux Dérivées Partielles La Chapelle sur Erdre, ), Exp. No. XX, 4 pp., Uiv. Nates, Nates,. [5] A. D. Melas, A lower oud for sums of eigevalues of the Laplacia, Proc. Amer. Math. Soc., 3 3), [6] A. Pleijel, O the eigevalues ad eigefuctios of elastic plates, Comm. Pure Appl. Math., 3 95), -. [7] G. Pólya, O the eigevalues of viratig memraes, Proc. Lodo Math. Soc., 96), [8] G. Pólya ad G. Szegö, Isoperimetric iequalities i mathematical physics, Aals of mathematics studies, umer 7, Priceto uiversity press, Priceto, New Jersey, 95). [9] E. Valdioci, From the log jump radom wal to the fractioal Laplacia, ArXiv:

26 G. Wei, H.-J. Su ad L. Zeg [] T. Weidl, Improved Berezi-Li-Yau iequalities with a remaider term, Spectral Theory of Differetial Operators, Amer. Math. Soc. Trasl., 5) 8), [] Q. L. Wag, C. Y. Xia, Uiversal ouds for eigevalues of the iharmoic operator o Riemaia maifolds, J. Fuct. Aal., 45 7), [] S. Yildirim Yolcu, A improvemet to a Brezi-Li-Yau type iequality, Proc. Amer. Math. Soc, 38) ) [3] S.Y.YolcuadT.Yolcu, Estimates for the sums of eigevalues of the fractioal Laplacia o a ouded domai, tyolcu/stimprovemet.pdf, ), preprit. Guoxi Wei, School of Mathematical Scieces, South Chia Normal Uiversity, 563, Guagzhou, Chia, weigx3@mails.tsighua.edu.c He-Ju Su, Departmet of Applied Mathematics, College of Sciece, Najig Uiversity of Sciece ad Techology, 94, Najig, Chia hejusu@63.com Ligzhog Zeg, Departmet of Mathematics, Graduate School of Sciece ad Egieerig, Saga Uiversity, Saga 84-85, Japa, ligzhogzeg@yeah.et 6