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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 & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] M. Ashbaugh and R. Laugesen, Fundamental tones and buckling loads of clamped plates, Ann. Scuola Norm.-Sci., 23 (1996), 383-402.  Google Scholar [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.  Google Scholar [7] F. A. Berezin, Covariant and contravariant symbols of operators, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1134-1167.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel, Switzerland, 2006.  Google Scholar [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.  Google Scholar [21] C. Imbert and P. E. Souganidis, Phasefield theory for fractional diffusion-reaction equations and applications, preprint,, , ().   Google Scholar [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.  Google Scholar [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. Google Scholar [24] N. S. Landkof, Foundations of Modern Potential Theory, Springer Verlag, New York, 1972.  Google Scholar [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.  Google Scholar [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.  Google Scholar [27] A. Laptev and T. Weidl, Recent results on Lieb-Thirring inequalities, Journées Équations aux dérivées partielles, (2000), 1-14.  Google Scholar [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.  Google Scholar [29] E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics 14, $2^{nd}$ edition, Amer. Math. Soc., Providence, 2001.  Google Scholar [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.  Google Scholar [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.  Google Scholar [32] E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl., 49 (2009), 33-44.  Google Scholar [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.  Google Scholar [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) Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [40] S. Yildirim Yolcu and T. Yolcu, Eigenvalue Bounds on the Poly-harmonic Operators, Illinois Journal of Mathematics, (2015), to appear. Google Scholar [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.  Google Scholar [42] V. M. Zolotarev, One dimensional Stable Distributions, Translations of Mathematical Monographs 65, American Mathematical Society, Providence, 1986.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] M. Ashbaugh and R. Laugesen, Fundamental tones and buckling loads of clamped plates, Ann. Scuola Norm.-Sci., 23 (1996), 383-402.  Google Scholar [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.  Google Scholar [7] F. A. Berezin, Covariant and contravariant symbols of operators, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1134-1167.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel, Switzerland, 2006.  Google Scholar [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.  Google Scholar [21] C. Imbert and P. E. Souganidis, Phasefield theory for fractional diffusion-reaction equations and applications, preprint,, , ().   Google Scholar [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.  Google Scholar [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. Google Scholar [24] N. S. Landkof, Foundations of Modern Potential Theory, Springer Verlag, New York, 1972.  Google Scholar [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.  Google Scholar [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.  Google Scholar [27] A. Laptev and T. Weidl, Recent results on Lieb-Thirring inequalities, Journées Équations aux dérivées partielles, (2000), 1-14.  Google Scholar [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.  Google Scholar [29] E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics 14, $2^{nd}$ edition, Amer. Math. Soc., Providence, 2001.  Google Scholar [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.  Google Scholar [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.  Google Scholar [32] E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl., 49 (2009), 33-44.  Google Scholar [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.  Google Scholar [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) Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [40] S. Yildirim Yolcu and T. Yolcu, Eigenvalue Bounds on the Poly-harmonic Operators, Illinois Journal of Mathematics, (2015), to appear. Google Scholar [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.  Google Scholar [42] V. M. Zolotarev, One dimensional Stable Distributions, Translations of Mathematical Monographs 65, American Mathematical Society, Providence, 1986.  Google Scholar
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