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May  2015, 35(5): 2209-2225. doi: 10.3934/dcds.2015.35.2209

Sharper estimates on the eigenvalues of Dirichlet fractional Laplacian

1. 

Department of Mathematics, Bradley University, 1501 W Bradley Ave, Peoria, IL 61625, United States, United States

Received  January 2014 Revised  October 2014 Published  December 2014

This article is to analyze certain bounds for the sums of eigenvalues of the Dirichlet fractional Laplacian operator $(-\Delta)^{\alpha/2}|_{\Omega}$ restricted to a bounded domain $\Omega\subset{\mathbb R}^d$ with $d=2,$ $1\leq \alpha\leq 2$ and $d\geq 3,$ $0< \alpha\le 2$. A primary topic is the refinement of the Berezin-Li-Yau inequality for the fractional Laplacian eigenvalues. Our result advances the estimates recently established by Wei, Sun and Zheng in [34]. Another aspect of interest in this work is that we obtain some estimates for the sums of powers of the eigenvalues of the fractional Laplacian operator.
Citation: Selma Yildirim Yolcu, Türkay Yolcu. Sharper estimates on the eigenvalues of Dirichlet fractional Laplacian. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2209-2225. doi: 10.3934/dcds.2015.35.2209
References:
[1]

D. R. Adams and J. Xiao, Morrey spaces in harmonic analysis, Arkiv För Matematik, 50 (2012), 201-230. doi: 10.1007/s11512-010-0134-0.

[2]

J. Akola, H.P. Heiskanen and M. Manninen, Edge-dependent selection rules in magic triangular graphene flakes, Physics Reviews B, 77 (2008), 193410. doi: 10.1103/PhysRevB.77.193410.

[3]

M. Ashbaugh and R. Benguria, On Rayleigh's conjecture for the clamped plate and its generalization to three dimensions, Duke Math. Journal, 78 (1995), 1-17. doi: 10.1215/S0012-7094-95-07801-6.

[4]

M. Ashbaugh, R. Benguria and R. Laugesen, Inequalities for the first eigenvalues of the clamped plate and buckling problems, in General Inequalities 7, International Series of Numerical Mathematics(eds. C. Bandle, W. N. Everitt, L. Losonczi, and W. Walter), 123, Birkhäuser Verlag, (1997), 95-110. doi: 10.1007/978-3-0348-8942-1_9.

[5]

M. Ashbaugh and R. Laugesen, Fundamental tones and buckling loads of clamped plates, Ann. Scuola Norm.-Sci., 23 (1996), 383-402.

[6]

R. Bañuelos and S. Yildirim Yolcu, Heat trace of non-local operators, J. London Math. Soc., 87 (2013), 304-318. doi: 10.1112/jlms/jds047.

[7]

F. A. Berezin, Covariant and contravariant symbols of operators, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1134-1167.

[8]

R. M. Blumenthal and R. K. Getoor, The asymptotic distribution of the eigenvalues for a class of Markov operators, Pacific J. Math., 9 (1959), 399-408. doi: 10.2140/pjm.1959.9.399.

[9]

K. Bogdan, T. Byczkowski, T. Kulczycki, M. Ryznar, R. Song and Z. Vondracék, Potential Analysis of Stable Processes and Its Extensions, Lecture Notes in Mathematics 1980 (eds. P. Graczyk and A. Stos), Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-02141-1.

[10]

L. A. Caffarelli, J.-M. Roquejoffre and O. Savin, Non-local minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144. doi: 10.1002/cpa.20331.

[11]

M. P. do Carmo, Q. Wang and C. Xia, Inequalities for eigenvalues of elliptic operators in divergence form on Riemannian manifolds, Annali di Matematica Pura ed Applicata, 189 (2010), 643-660. doi: 10.1007/s10231-010-0129-2.

[12]

Z.-Q. Chen and R. Song, Two-sided eigenvalue estimates for subordinate processes in domains, Journal of Functional Analysis, 226 (2005), 90-113. doi: 10.1016/j.jfa.2005.05.004.

[13]

R. L. Frank and L. Geisinger, Refined Semiclassical Asymptotics for Fractional Powers of the Laplace Operator, J. Reine Angew. Math.(Crelles Journal), 2014. doi: 10.1515/crelle-2013-0120.

[14]

R. El Hajj and F. Méhats, Analysis of models for quantum transport of electrons in graphene layers, Math. Models Methods Appl. Sci., 24 (2014), 2287-2310. doi: 10.1142/S0218202514500213.

[15]

E. M. Harrell II and L. Hermi, On Riesz Means of Eigenvalues, Comm. in P.D.E., 36 (2011), 1521-1543. doi: 10.1080/03605302.2011.595865.

[16]

E. M. Harrell II and J. Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc., 349 (1997), 1797-1809. doi: 10.1090/S0002-9947-97-01846-1.

[17]

E. M. Harrell II and J. Stubbe, Universal bounds and semiclassical estimates for eigenvalues of abstract Schrödinger operators, SIAM Journal on Mathematical Analysis, 42 (2010), 2261-2274. doi: 10.1137/090763743.

[18]

E. M. Harrell II and S. Yildirim Yolcu, Eigenvalue inequalities for Klein-Gordon operators, J. Funct. Analysis, 256 (2009), 3977-3995. doi: 10.1016/j.jfa.2008.12.008.

[19]

A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel, Switzerland, 2006.

[20]

A. A. Ilyin, Lower bounds for the spectrum of the Laplace and Stokes operators, Discrete and Continuous Dynamical Systems - A, 28 (2010), 131-146. doi: 10.3934/dcds.2010.28.131.

[21]

C. Imbert and P. E. Souganidis, Phasefield theory for fractional diffusion-reaction equations and applications, preprint, arXiv:0907.5524.

[22]

H. Kovařík, S. Vugalter and T. Weidl, Two-dimensional Berezin-Li-Yau inequalities with a correction term, Comm. Math. Phys., 287 (2009), 959-981. doi: 10.1007/s00220-008-0692-1.

[23]

H. Kovařík and T. Weidl, Improved Berezin-Li-Yau inequalities with magnetic field, Proceedings of the Royal Society of Edinburgh Section A Mathematics, (2014), to appear.

[24]

N. S. Landkof, Foundations of Modern Potential Theory, Springer Verlag, New York, 1972.

[25]

A. Laptev, Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces, J. Funct. Anal., 151 (1997), 531-545. doi: 10.1006/jfan.1997.3155.

[26]

A. Laptev, L. Geisinger and T. Weidl, Geometrical versions of improved Berezin-Li-Yau inequalities, Journal of Spectral Theory, 1 (2011), 87-109. doi: 10.4171/JST/4.

[27]

A. Laptev and T. Weidl, Recent results on Lieb-Thirring inequalities, Journées Équations aux dérivées partielles, (2000), 1-14.

[28]

P. Li and S.-T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys., 88 (1983), 309-318. doi: 10.1007/BF01213210.

[29]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics 14, $2^{nd}$ edition, Amer. Math. Soc., Providence, 2001.

[30]

A. D. Melas, A lower bound for sums of eigenvalues of the Laplacian, Proceedings of the American Mathematical Society, 131 (2003), 631-636. doi: 10.1090/S0002-9939-02-06834-X.

[31]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Communications on Pure and Applied Mathematics, 60 (2007), 67-112. doi: 10.1002/cpa.20153.

[32]

E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl., 49 (2009), 33-44.

[33]

V. Vougalter, Sharp semiclassical bounds for the moments of eigenvalues for some Schroedinger type operators with unbounded potentials, Math. Model. Nat. Phenom., 8 (2013), 237-245. doi: 10.1051/mmnp/20138119.

[34]

G. Wei, H.-J. Sun and L. Zeng, Lower Bounds for Laplacian and Fractional Laplacian Eigenvalues, preprint, Communications in Contemporary Mathematics, (2014), 1450032. (DOI: 10.1142/S0219199714500321)

[35]

T. Weidl, Improved Berezin-Li-Yau inequalities with a remainder term, in Spectral Theory of Differential Operators, Amer. Math. Soc. Transl. 2 (2008), 253-263.

[36]

S. Yildirim Yolcu, An improvement to a Brezin-Li-Yau type inequality, Proceedings of the American Mathematical Society, 138 (2010), 4059-4066. doi: 10.1090/S0002-9939-2010-10419-7.

[37]

S. Yildirim Yolcu and T. Yolcu, Multidimensional lower bounds for the eigenvalues of Stokes and Dirichlet Laplacian operators, Journal of Mathematical Physics, 53 (2012), 043508, 17pp. doi: 10.1063/1.3701978.

[38]

S. Yildirim Yolcu and T. Yolcu, Estimates for the sums of eigenvalues of the fractional Laplacian on a bounded domain, Communications in Contemporary Mathematics, 15 (2013), 1250048, 15pp. doi: 10.1142/S0219199712500484.

[39]

S. Yildirim Yolcu and T. Yolcu, Estimates on the eigenvalues of the clamped plate problem on domains in Euclidean spaces, Journal of Mathematical Physics, 54 (2013), 043515, 13pp. doi: 10.1063/1.4801446.

[40]

S. Yildirim Yolcu and T. Yolcu, Eigenvalue Bounds on the Poly-harmonic Operators, Illinois Journal of Mathematics, (2015), to appear.

[41]

T. Yolcu, Refined bounds for the eigenvalues of the Klein-Gordon operator, Proceedings of the American Mathematical Society, 141 (2013), 4305-4315. doi: 10.1090/S0002-9939-2013-11806-X.

[42]

V. M. Zolotarev, One dimensional Stable Distributions, Translations of Mathematical Monographs 65, American Mathematical Society, Providence, 1986.

show all references

References:
[1]

D. R. Adams and J. Xiao, Morrey spaces in harmonic analysis, Arkiv För Matematik, 50 (2012), 201-230. doi: 10.1007/s11512-010-0134-0.

[2]

J. Akola, H.P. Heiskanen and M. Manninen, Edge-dependent selection rules in magic triangular graphene flakes, Physics Reviews B, 77 (2008), 193410. doi: 10.1103/PhysRevB.77.193410.

[3]

M. Ashbaugh and R. Benguria, On Rayleigh's conjecture for the clamped plate and its generalization to three dimensions, Duke Math. Journal, 78 (1995), 1-17. doi: 10.1215/S0012-7094-95-07801-6.

[4]

M. Ashbaugh, R. Benguria and R. Laugesen, Inequalities for the first eigenvalues of the clamped plate and buckling problems, in General Inequalities 7, International Series of Numerical Mathematics(eds. C. Bandle, W. N. Everitt, L. Losonczi, and W. Walter), 123, Birkhäuser Verlag, (1997), 95-110. doi: 10.1007/978-3-0348-8942-1_9.

[5]

M. Ashbaugh and R. Laugesen, Fundamental tones and buckling loads of clamped plates, Ann. Scuola Norm.-Sci., 23 (1996), 383-402.

[6]

R. Bañuelos and S. Yildirim Yolcu, Heat trace of non-local operators, J. London Math. Soc., 87 (2013), 304-318. doi: 10.1112/jlms/jds047.

[7]

F. A. Berezin, Covariant and contravariant symbols of operators, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1134-1167.

[8]

R. M. Blumenthal and R. K. Getoor, The asymptotic distribution of the eigenvalues for a class of Markov operators, Pacific J. Math., 9 (1959), 399-408. doi: 10.2140/pjm.1959.9.399.

[9]

K. Bogdan, T. Byczkowski, T. Kulczycki, M. Ryznar, R. Song and Z. Vondracék, Potential Analysis of Stable Processes and Its Extensions, Lecture Notes in Mathematics 1980 (eds. P. Graczyk and A. Stos), Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-02141-1.

[10]

L. A. Caffarelli, J.-M. Roquejoffre and O. Savin, Non-local minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144. doi: 10.1002/cpa.20331.

[11]

M. P. do Carmo, Q. Wang and C. Xia, Inequalities for eigenvalues of elliptic operators in divergence form on Riemannian manifolds, Annali di Matematica Pura ed Applicata, 189 (2010), 643-660. doi: 10.1007/s10231-010-0129-2.

[12]

Z.-Q. Chen and R. Song, Two-sided eigenvalue estimates for subordinate processes in domains, Journal of Functional Analysis, 226 (2005), 90-113. doi: 10.1016/j.jfa.2005.05.004.

[13]

R. L. Frank and L. Geisinger, Refined Semiclassical Asymptotics for Fractional Powers of the Laplace Operator, J. Reine Angew. Math.(Crelles Journal), 2014. doi: 10.1515/crelle-2013-0120.

[14]

R. El Hajj and F. Méhats, Analysis of models for quantum transport of electrons in graphene layers, Math. Models Methods Appl. Sci., 24 (2014), 2287-2310. doi: 10.1142/S0218202514500213.

[15]

E. M. Harrell II and L. Hermi, On Riesz Means of Eigenvalues, Comm. in P.D.E., 36 (2011), 1521-1543. doi: 10.1080/03605302.2011.595865.

[16]

E. M. Harrell II and J. Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc., 349 (1997), 1797-1809. doi: 10.1090/S0002-9947-97-01846-1.

[17]

E. M. Harrell II and J. Stubbe, Universal bounds and semiclassical estimates for eigenvalues of abstract Schrödinger operators, SIAM Journal on Mathematical Analysis, 42 (2010), 2261-2274. doi: 10.1137/090763743.

[18]

E. M. Harrell II and S. Yildirim Yolcu, Eigenvalue inequalities for Klein-Gordon operators, J. Funct. Analysis, 256 (2009), 3977-3995. doi: 10.1016/j.jfa.2008.12.008.

[19]

A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel, Switzerland, 2006.

[20]

A. A. Ilyin, Lower bounds for the spectrum of the Laplace and Stokes operators, Discrete and Continuous Dynamical Systems - A, 28 (2010), 131-146. doi: 10.3934/dcds.2010.28.131.

[21]

C. Imbert and P. E. Souganidis, Phasefield theory for fractional diffusion-reaction equations and applications, preprint, arXiv:0907.5524.

[22]

H. Kovařík, S. Vugalter and T. Weidl, Two-dimensional Berezin-Li-Yau inequalities with a correction term, Comm. Math. Phys., 287 (2009), 959-981. doi: 10.1007/s00220-008-0692-1.

[23]

H. Kovařík and T. Weidl, Improved Berezin-Li-Yau inequalities with magnetic field, Proceedings of the Royal Society of Edinburgh Section A Mathematics, (2014), to appear.

[24]

N. S. Landkof, Foundations of Modern Potential Theory, Springer Verlag, New York, 1972.

[25]

A. Laptev, Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces, J. Funct. Anal., 151 (1997), 531-545. doi: 10.1006/jfan.1997.3155.

[26]

A. Laptev, L. Geisinger and T. Weidl, Geometrical versions of improved Berezin-Li-Yau inequalities, Journal of Spectral Theory, 1 (2011), 87-109. doi: 10.4171/JST/4.

[27]

A. Laptev and T. Weidl, Recent results on Lieb-Thirring inequalities, Journées Équations aux dérivées partielles, (2000), 1-14.

[28]

P. Li and S.-T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys., 88 (1983), 309-318. doi: 10.1007/BF01213210.

[29]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics 14, $2^{nd}$ edition, Amer. Math. Soc., Providence, 2001.

[30]

A. D. Melas, A lower bound for sums of eigenvalues of the Laplacian, Proceedings of the American Mathematical Society, 131 (2003), 631-636. doi: 10.1090/S0002-9939-02-06834-X.

[31]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Communications on Pure and Applied Mathematics, 60 (2007), 67-112. doi: 10.1002/cpa.20153.

[32]

E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl., 49 (2009), 33-44.

[33]

V. Vougalter, Sharp semiclassical bounds for the moments of eigenvalues for some Schroedinger type operators with unbounded potentials, Math. Model. Nat. Phenom., 8 (2013), 237-245. doi: 10.1051/mmnp/20138119.

[34]

G. Wei, H.-J. Sun and L. Zeng, Lower Bounds for Laplacian and Fractional Laplacian Eigenvalues, preprint, Communications in Contemporary Mathematics, (2014), 1450032. (DOI: 10.1142/S0219199714500321)

[35]

T. Weidl, Improved Berezin-Li-Yau inequalities with a remainder term, in Spectral Theory of Differential Operators, Amer. Math. Soc. Transl. 2 (2008), 253-263.

[36]

S. Yildirim Yolcu, An improvement to a Brezin-Li-Yau type inequality, Proceedings of the American Mathematical Society, 138 (2010), 4059-4066. doi: 10.1090/S0002-9939-2010-10419-7.

[37]

S. Yildirim Yolcu and T. Yolcu, Multidimensional lower bounds for the eigenvalues of Stokes and Dirichlet Laplacian operators, Journal of Mathematical Physics, 53 (2012), 043508, 17pp. doi: 10.1063/1.3701978.

[38]

S. Yildirim Yolcu and T. Yolcu, Estimates for the sums of eigenvalues of the fractional Laplacian on a bounded domain, Communications in Contemporary Mathematics, 15 (2013), 1250048, 15pp. doi: 10.1142/S0219199712500484.

[39]

S. Yildirim Yolcu and T. Yolcu, Estimates on the eigenvalues of the clamped plate problem on domains in Euclidean spaces, Journal of Mathematical Physics, 54 (2013), 043515, 13pp. doi: 10.1063/1.4801446.

[40]

S. Yildirim Yolcu and T. Yolcu, Eigenvalue Bounds on the Poly-harmonic Operators, Illinois Journal of Mathematics, (2015), to appear.

[41]

T. Yolcu, Refined bounds for the eigenvalues of the Klein-Gordon operator, Proceedings of the American Mathematical Society, 141 (2013), 4305-4315. doi: 10.1090/S0002-9939-2013-11806-X.

[42]

V. M. Zolotarev, One dimensional Stable Distributions, Translations of Mathematical Monographs 65, American Mathematical Society, Providence, 1986.

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