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January  2022, 42(1): 301-317. doi: 10.3934/dcds.2021117

Estimates the upper bounds of Dirichlet eigenvalues for fractional Laplacian

Hua Chen 1,, and  Hong-Ge Chen 2,

1. 

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
2. 

Wuhan Institute of Physics and Mathematics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071, China

* Corresponding author: Hua Chen

Received  May 2021 Revised  June 2021 Published  January 2022 Early access  August 2021

Fund Project: This work is supported by National Natural Science Foundation of China (Grants Nos. 11631011 and 11626251) and China Postdoctoral Science Foundation (Grants No. 2020M672398)

Let $ \Omega\subset\mathbb{R}^n \; (n\geq 2) $ be a bounded domain with continuous boundary $ \partial\Omega $. In this paper, we study the Dirichlet eigenvalue problem of the fractional Laplacian which is restricted to $ \Omega $ with $ 0<s<1 $. Denoting by $ \lambda_{k} $ the $ k^{th} $ Dirichlet eigenvalue of $ (-\triangle)^{s}|_{\Omega} $, we establish the explicit upper bounds of the ratio $ \frac{\lambda_{k+1}}{\lambda_{1}} $, which have polynomially growth in $ k $ with optimal increase orders. Furthermore, we give the explicit lower bounds for the Riesz mean function $ R_{\sigma}(z) = \sum_{k}(z-\lambda_{k})_{+}^{\sigma} $ with $ \sigma\geq 1 $ and the trace of the Dirichlet heat kernel of fractional Laplacian.

Keywords: Fractional Laplacian, Dirichlet eigenvalues, Riesz mean function, heat kernel.
Mathematics Subject Classification: Primary: 35P15; Secondary: 58C40.
Citation: Hua Chen, Hong-Ge Chen. Estimates the upper bounds of Dirichlet eigenvalues for fractional Laplacian. Discrete & Continuous Dynamical Systems, 2022, 42 (1) : 301-317. doi: 10.3934/dcds.2021117
References:
[1]

R. Bañuelos and T. Kulczycki, Trace estimates for stable processes, Probab. Theory Related Fields, 142 (2008), 313-338.  doi: 10.1007/s00440-007-0106-x.  Google Scholar

[2]

R. Bañuelos, T. Kulczycki and B. Siudeja, On the trace of symmetric stable processes on Lipschitz domains, J. Funct. Anal., 257 (2009), 3329-3352.  doi: 10.1016/j.jfa.2009.06.037.  Google Scholar

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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

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L. Brasco, E. Lindgren and E. Parini, The fractional Cheeger problem, Interfaces Free Bound., 16 (2014), 419-458.  doi: 10.4171/IFB/325.  Google Scholar

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L. Brasco and E. Parini, The second eigenvalue of the fractional $p$-Laplacian, Adv. Calc. Var., 9 (2016), 323-355.  doi: 10.1515/acv-2015-0007.  Google Scholar

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H. Chen and A. Zeng, Universal inequality and upper bounds of eigenvalues for non-integer poly-Laplacian on a bounded domain, Calc. Var. Partial Differential Equations, 56 (2017), 12 pp. doi: 10.1007/s00526-017-1220-y.  Google Scholar

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Z. Q. Chen and R. Song, Two-sided eigenvalue estimates for subordinate processes in domains, J. Funct. Anal., 226 (2005), 90-113.  doi: 10.1016/j.jfa.2005.05.004.  Google Scholar

[10]

Q. M. Cheng and H. C. Yang, Bounds on eigenvalues of Dirichlet Laplacian, Math. Ann., 337 (2007), 159-175.  doi: 10.1007/s00208-006-0030-x.  Google Scholar

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G. Chiti, Inequalities for the first three membrane eigenvalues, Boll. Un. Mat. Ital., 18 (1981), 144-148.   Google Scholar

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G. Chiti, An isoperimetric inequality for the eigenfunctions of linear second order elliptic operators, Boll. Un. Mat. Ital., 1 (1982), 145-151.   Google Scholar

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B. K. Driver, Analysis Tools with Applications, Lecture Notes, Springer, Berlin, 2003. Available from: http://www.math.ucsd.edu/ bdriver/231-02-03/Lecture_Notes/PDE-Anal-Book/analpde1.pdf. Google Scholar

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R. L. Frank and L. Geisinger, Refined semiclassical asymptotics for fractional powers of the Laplace operator, J. Reine Angew. Math., 712 (2016), 1-37.  doi: 10.1515/crelle-2013-0120.  Google Scholar

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G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Math. Univ. Parma (N.S.), 5 (2014), 373-386.   Google Scholar

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L. Geisinger, A short proof of Weyl's law for fractional differential operators, J. Math. Phys., 55 (2014), 011504. doi: 10.1063/1.4861935.  Google Scholar

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E. M. Harrell II and L. Hermi, On Riesz means of eigenvalues, Comm. Partial Differential Equations, 36 (2011), 1521-1543.  doi: 10.1080/03605302.2011.595865.  Google Scholar

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L. Hermi, Two new Weyl-type bounds for the Dirichlet Laplacian, Trans. Amer. Math. Soc., 360 (2008), 1539-1558.  doi: 10.1090/S0002-9947-07-04254-7.  Google Scholar

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V. Ivrii, Spectral asymptotics for fractional Laplacians, in Differential Equations, Mathematical Physics, and Applications: Selim Grigorievich Krein Centennial, Contemp. Math., 734, Amer. Math. Soc., [Providence], RI, 2019,159–170. doi: 10.1090/conm/734/14770.  Google Scholar

[22]

P. Kröger, Estimates for sums of eigenvalues of the Laplacian, J. Funct. Anal., 126 (1994), 217-227.  doi: 10.1006/jfan.1994.1146.  Google Scholar

[23]

M. Kwaśnicki, Eigenvalues of the fractional Laplace operator in the interval, J. Funct. Anal., 262 (2012), 2379-2402.  doi: 10.1016/j.jfa.2011.12.004.  Google Scholar

[24]

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

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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

[26]

G. Pólya, On the eigenvalues of vibrating membranes, Proc. London Math. Soc., 11 (1961), 419-433.  doi: 10.1112/plms/s3-11.1.419.  Google Scholar

[27]

Y. Safarov, Lower bounds for the generalized counting function, in The Maz'ya Anniversary Collection, (eds. J. Rossmann, P. Takac and G. Wildenhain), Birkhäuser, Basel, 2 (1999), 275–293. doi: 10.1007/978-3-0348-8672-7_16.  Google Scholar

[28]

H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann., 71 (1912), 441-479.  doi: 10.1007/BF01456804.  Google Scholar

[29]

S. Y. Yolcu and T. Yolcu, Estimates for the sums of eigenvalues of the fractional Laplacian on a bounded domain, Commun. Contemp. Math., 15 (2013), 1250048. doi: 10.1142/S0219199712500484.  Google Scholar

show all references

References:
[1]

R. Bañuelos and T. Kulczycki, Trace estimates for stable processes, Probab. Theory Related Fields, 142 (2008), 313-338.  doi: 10.1007/s00440-007-0106-x.  Google Scholar

[2]

R. Bañuelos, T. Kulczycki and B. Siudeja, On the trace of symmetric stable processes on Lipschitz domains, J. Funct. Anal., 257 (2009), 3329-3352.  doi: 10.1016/j.jfa.2009.06.037.  Google Scholar

[3]

F. A. Berezin, Covariant and contravariant symbols of operators, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1134-1167.  doi: 10.1070/IM1972v006n05ABEH001913.  Google Scholar

[4] G. M. Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316282397.  Google Scholar
[5]

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

[6]

L. Brasco, E. Lindgren and E. Parini, The fractional Cheeger problem, Interfaces Free Bound., 16 (2014), 419-458.  doi: 10.4171/IFB/325.  Google Scholar

[7]

L. Brasco and E. Parini, The second eigenvalue of the fractional $p$-Laplacian, Adv. Calc. Var., 9 (2016), 323-355.  doi: 10.1515/acv-2015-0007.  Google Scholar

[8]

H. Chen and A. Zeng, Universal inequality and upper bounds of eigenvalues for non-integer poly-Laplacian on a bounded domain, Calc. Var. Partial Differential Equations, 56 (2017), 12 pp. doi: 10.1007/s00526-017-1220-y.  Google Scholar

[9]

Z. Q. Chen and R. Song, Two-sided eigenvalue estimates for subordinate processes in domains, J. Funct. Anal., 226 (2005), 90-113.  doi: 10.1016/j.jfa.2005.05.004.  Google Scholar

[10]

Q. M. Cheng and H. C. Yang, Bounds on eigenvalues of Dirichlet Laplacian, Math. Ann., 337 (2007), 159-175.  doi: 10.1007/s00208-006-0030-x.  Google Scholar

[11]

G. Chiti, Inequalities for the first three membrane eigenvalues, Boll. Un. Mat. Ital., 18 (1981), 144-148.   Google Scholar

[12]

G. Chiti, An isoperimetric inequality for the eigenfunctions of linear second order elliptic operators, Boll. Un. Mat. Ital., 1 (1982), 145-151.   Google Scholar

[13]

B. K. Driver, Analysis Tools with Applications, Lecture Notes, Springer, Berlin, 2003. Available from: http://www.math.ucsd.edu/ bdriver/231-02-03/Lecture_Notes/PDE-Anal-Book/analpde1.pdf. Google Scholar

[14]

R. L. Frank, Eigenvalue bounds for the fractional Laplacian: A review, in Recent Developments in Nonlocal Theory, De Gruyter, Berlin, 2018,210–235. doi: 10.1515/9783110571561-007.  Google Scholar

[15]

R. L. Frank and L. Geisinger, Refined semiclassical asymptotics for fractional powers of the Laplace operator, J. Reine Angew. Math., 712 (2016), 1-37.  doi: 10.1515/crelle-2013-0120.  Google Scholar

[16]

G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Math. Univ. Parma (N.S.), 5 (2014), 373-386.   Google Scholar

[17]

L. Geisinger, A short proof of Weyl's law for fractional differential operators, J. Math. Phys., 55 (2014), 011504. doi: 10.1063/1.4861935.  Google Scholar

[18]

E. M. Harrell II and L. Hermi, On Riesz means of eigenvalues, Comm. Partial Differential Equations, 36 (2011), 1521-1543.  doi: 10.1080/03605302.2011.595865.  Google Scholar

[19]

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

[20]

L. Hermi, Two new Weyl-type bounds for the Dirichlet Laplacian, Trans. Amer. Math. Soc., 360 (2008), 1539-1558.  doi: 10.1090/S0002-9947-07-04254-7.  Google Scholar

[21]

V. Ivrii, Spectral asymptotics for fractional Laplacians, in Differential Equations, Mathematical Physics, and Applications: Selim Grigorievich Krein Centennial, Contemp. Math., 734, Amer. Math. Soc., [Providence], RI, 2019,159–170. doi: 10.1090/conm/734/14770.  Google Scholar

[22]

P. Kröger, Estimates for sums of eigenvalues of the Laplacian, J. Funct. Anal., 126 (1994), 217-227.  doi: 10.1006/jfan.1994.1146.  Google Scholar

[23]

M. Kwaśnicki, Eigenvalues of the fractional Laplace operator in the interval, J. Funct. Anal., 262 (2012), 2379-2402.  doi: 10.1016/j.jfa.2011.12.004.  Google Scholar

[24]

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

[25]

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

[26]

G. Pólya, On the eigenvalues of vibrating membranes, Proc. London Math. Soc., 11 (1961), 419-433.  doi: 10.1112/plms/s3-11.1.419.  Google Scholar

[27]

Y. Safarov, Lower bounds for the generalized counting function, in The Maz'ya Anniversary Collection, (eds. J. Rossmann, P. Takac and G. Wildenhain), Birkhäuser, Basel, 2 (1999), 275–293. doi: 10.1007/978-3-0348-8672-7_16.  Google Scholar

[28]

H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann., 71 (1912), 441-479.  doi: 10.1007/BF01456804.  Google Scholar

[29]

S. Y. Yolcu and T. Yolcu, Estimates for the sums of eigenvalues of the fractional Laplacian on a bounded domain, Commun. Contemp. Math., 15 (2013), 1250048. doi: 10.1142/S0219199712500484.  Google Scholar

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