Geodesic
EIGENVALUE ESTIMATES FOR QUADRATIC POLYNOMIAL OPERATOR OF THE LAPLACIAN
Glasgow mathematical journal, Tome 53 (2011) no. 2, pp. 321-332

Voir la notice de l'article provenant de la source Cambridge University Press

For a bounded domain Ω in a complete Riemannian manifold M, we investigate the Dirichlet weighted eigenvalue problem of quadratic polynomial operator Δ2 − aΔ + b of the Laplacian Δ, where a and b are the nonnegative constants. We obtain an inequality for eigenvalues which contains a constant that only depends on the mean curvature of M. It yields an upper bound of the (k + 1)th eigenvalue Λk + 1. As their applications, some inequalities and bounds of eigenvalues on a complete minimal submanifold in a Euclidean space and a unit sphere are obtained.
DOI : 10.1017/S0017089510000728
Mots-clés : 35P15, 58C40, 53C42 
HEJUN, SUN; XUERONG, QI. EIGENVALUE ESTIMATES FOR QUADRATIC POLYNOMIAL OPERATOR OF THE LAPLACIAN. Glasgow mathematical journal, Tome 53 (2011) no. 2, pp. 321-332. doi: 10.1017/S0017089510000728
@article{10_1017_S0017089510000728,
     author = {HEJUN, SUN and XUERONG, QI},
     title = {EIGENVALUE {ESTIMATES} {FOR} {QUADRATIC} {POLYNOMIAL} {OPERATOR} {OF} {THE} {LAPLACIAN}},
     journal = {Glasgow mathematical journal},
     pages = {321--332},
     year = {2011},
     volume = {53},
     number = {2},
     doi = {10.1017/S0017089510000728},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089510000728/}
}
TY  - JOUR
AU  - HEJUN, SUN
AU  - XUERONG, QI
TI  - EIGENVALUE ESTIMATES FOR QUADRATIC POLYNOMIAL OPERATOR OF THE LAPLACIAN
JO  - Glasgow mathematical journal
PY  - 2011
SP  - 321
EP  - 332
VL  - 53
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089510000728/
DO  - 10.1017/S0017089510000728
ID  - 10_1017_S0017089510000728
ER  - 
%0 Journal Article
%A HEJUN, SUN
%A XUERONG, QI
%T EIGENVALUE ESTIMATES FOR QUADRATIC POLYNOMIAL OPERATOR OF THE LAPLACIAN
%J Glasgow mathematical journal
%D 2011
%P 321-332
%V 53
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089510000728/
%R 10.1017/S0017089510000728
%F 10_1017_S0017089510000728

[1] 1.Ashbaugh, M. S., Isoperimetric and universal inequalities for eigenvalues, in Spectral theory and geometry (Davies, E. B. and Safarov, Y., Editors) (Edinburgh, 1998), London Math. Soc. Lecture Notes, Vol. (Cambridge Univ. Press, Cambridge, 1999), 95–139. Google Scholar

[2] 2.Ashbaugh, M. S., Universal eigenvalue bounds of Payne-Pólya-Weinberger, Hile-Protter and H. C. Yang, Proc. Indian Acad. Sci. (Math. Sci.) 112 (2002), 3–30. Google Scholar

[3] 3.Ashbaugh, M. S. and Hermi, L., A unified approach to universal inequalities for eigenvalues of elliptic operators, Pacific J. Math. 217 (2004), 201–219. Google Scholar | DOI

[4] 4.Chen, D. G. and Cheng, Q.-M., Extrinsic estimates for eigenvalues of the Laplace operator, J. Math. Soc. Japan 60 (2008), 325–339. Google Scholar | DOI

[5] 5.Chen, Z. C. and Qian, C. L., Estimates for discrete spectrum of Laplacian operator with any order, J. China Univ. Sci. Tech. 20 (1990), 259–266. Google Scholar

[6] 6.Cheng, Q.-M., Ichikawa, T. and Mametsuka, S., Estimates for eigenvalues of a clamped plate problem on Riemannian manifolds, J. Math. Soc. Japan, 62 (2010), 673–686. Google Scholar | DOI

[7] 7.Cheng, Q.-M. and Yang, H. C., Inequalities for eigenvalues of a clamped plate problem, Trans. Amer. Math. Soc. 358 (2006), 2625–2635. Google Scholar | DOI

[8] 8.Cheng, Q.-M. and Yang, H. C., Estimates for eigenvalues on Riemannian manifolds, J. Differ. Equ. 247 (2009), 2270–2281. Google Scholar | DOI

[9] 9.Hile, G. N. and Protter, M. H., Inequalities for eigenvalues of the Laplacian, Indiana Univ. Math. J. 29 (1980), 523–538. Google Scholar | DOI

[10] 10.Hile, G. N. and Yeh, R. Z., Inequalities for eigenvalues of the biharmonic operator, Pacific J. Math. 112 (1984), 115–133. Google Scholar | DOI

[11] 11.Hook, S. M., Domain independent upper bounds for eigenvalues of elliptic operator, Trans. Amer. Math. Soc. 318 (1990), 615–642. Google Scholar | DOI

[12] 12.Ljung, L., Recursive identification, in Stochastic systems: The mathematics of filtering and identification and applications (Hazewinkel, M. and Willems, J. C., Editors) (Reidel, Dordrecht, 1981), 247–283. Google Scholar | DOI

[13] 13.Maybeck, P. S., Stochastic models, estimation, and control III (Academic Press, New York, 1982). Google Scholar

[14] 14.Nash, J., The imbedding problem for Riemannian manifolds, Ann. Math. 63 (1956), 20–63. Google Scholar | DOI

[15] 15.Payne, L. E., Polya, G. and Weinberger, H. F., On the ratio of consecutive eigenvalues, J. Math. Phys. 35 (1956), 289–298. Google Scholar | DOI

[16] 16.El Soufi, A., Harrell, E. M. and Ilias, S., Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds, Trans. Am. Math. Soc. 361 (2009), 2337–2350. Google Scholar | DOI

[17] 17.Wang, Q. L. and Xia, C. Y., Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds, J. Funct. Anal. 245 (2007), 334–352. Google Scholar | DOI

Cité par Sources :