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Estimates the upper bounds of Dirichlet eigenvalues for fractional Laplacian
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 |
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.
References:
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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. |
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R. Bañuelos, T. Kulczycki and B. Siudeja,
On the trace of symmetric stable processes on Lipschitz domains, J. Funct. Anal., 257 (2009), 3329-3352.
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F. A. Berezin,
Covariant and contravariant symbols of operators, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1134-1167.
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G. M. Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, Cambridge, 2016.
doi: 10.1017/CBO9781316282397.![]() ![]() ![]() |
[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. |
[6] |
L. Brasco, E. Lindgren and E. Parini,
The fractional Cheeger problem, Interfaces Free Bound., 16 (2014), 419-458.
doi: 10.4171/IFB/325. |
[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. |
[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. |
[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. |
[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. |
[11] |
G. Chiti,
Inequalities for the first three membrane eigenvalues, Boll. Un. Mat. Ital., 18 (1981), 144-148.
|
[12] |
G. Chiti,
An isoperimetric inequality for the eigenfunctions of linear second order elliptic operators, Boll. Un. Mat. Ital., 1 (1982), 145-151.
|
[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. |
[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. |
[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. |
[16] |
G. Franzina and G. Palatucci,
Fractional $p$-eigenvalues, Riv. Math. Univ. Parma (N.S.), 5 (2014), 373-386.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
P. Kröger,
Estimates for sums of eigenvalues of the Laplacian, J. Funct. Anal., 126 (1994), 217-227.
doi: 10.1006/jfan.1994.1146. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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.![]() ![]() ![]() |
[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. |
[6] |
L. Brasco, E. Lindgren and E. Parini,
The fractional Cheeger problem, Interfaces Free Bound., 16 (2014), 419-458.
doi: 10.4171/IFB/325. |
[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. |
[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. |
[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. |
[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. |
[11] |
G. Chiti,
Inequalities for the first three membrane eigenvalues, Boll. Un. Mat. Ital., 18 (1981), 144-148.
|
[12] |
G. Chiti,
An isoperimetric inequality for the eigenfunctions of linear second order elliptic operators, Boll. Un. Mat. Ital., 1 (1982), 145-151.
|
[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. |
[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. |
[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. |
[16] |
G. Franzina and G. Palatucci,
Fractional $p$-eigenvalues, Riv. Math. Univ. Parma (N.S.), 5 (2014), 373-386.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
P. Kröger,
Estimates for sums of eigenvalues of the Laplacian, J. Funct. Anal., 126 (1994), 217-227.
doi: 10.1006/jfan.1994.1146. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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