Comptes Rendus
Geometry
The lower and upper bounds of the first eigenvalues for the bi-harmonic operator on manifolds
Comptes Rendus. Mathématique, Volume 353 (2015) no. 8, pp. 735-740.

In this paper, we will estimate the lower bounds and upper bounds of the first eigenvalues for bi-harmonic operators on manifolds through Reilly's and Bochner's formulae, respectively.

Dans cette Note, nous donnons des minorants et majorants des premières valeurs propres de l'opérateur bi-harmonique sur une variété riemannienne, compacte, connexe, en utilisant respectivement les formules de Reilly et de Bochner.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2015.06.001

Liuwei Zhang 1

1 Department of Mathematics, Tongji University, Shanghai, 200092, PR China
@article{CRMATH_2015__353_8_735_0,
     author = {Liuwei Zhang},
     title = {The lower and upper bounds of the first eigenvalues for the bi-harmonic operator on manifolds},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {735--740},
     publisher = {Elsevier},
     volume = {353},
     number = {8},
     year = {2015},
     doi = {10.1016/j.crma.2015.06.001},
     language = {en},
}
TY  - JOUR
AU  - Liuwei Zhang
TI  - The lower and upper bounds of the first eigenvalues for the bi-harmonic operator on manifolds
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 735
EP  - 740
VL  - 353
IS  - 8
PB  - Elsevier
DO  - 10.1016/j.crma.2015.06.001
LA  - en
ID  - CRMATH_2015__353_8_735_0
ER  - 
%0 Journal Article
%A Liuwei Zhang
%T The lower and upper bounds of the first eigenvalues for the bi-harmonic operator on manifolds
%J Comptes Rendus. Mathématique
%D 2015
%P 735-740
%V 353
%N 8
%I Elsevier
%R 10.1016/j.crma.2015.06.001
%G en
%F CRMATH_2015__353_8_735_0
Liuwei Zhang. The lower and upper bounds of the first eigenvalues for the bi-harmonic operator on manifolds. Comptes Rendus. Mathématique, Volume 353 (2015) no. 8, pp. 735-740. doi : 10.1016/j.crma.2015.06.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.06.001/

[1] I. Chavel Isoperimetric Inequalities, Cambridge Tracts in Math., vol. 145, Cambridge University Press, Cambridge, UK, 2001

[2] D.G. Chen; Q.M. Cheng; Q.L. Wang; C.Y. Xia On eigenvalues of a system of elliptic equations and biharmonic operator, J. Math. Anal. Appl., Volume 387 (2012), pp. 1146-1159

[3] H. Federer; W.H. Fleming Normal integral currents, Ann. Math., Volume 72 (1960), pp. 458-520

[4] H.A. Levine; M.H. Protter Unrestricted lower bounds for eigenvalues for classes of elliptic equations and systems of equations with applications to problems in elasticity, Math. Methods Appl. Sci., Volume 7 (1985) no. 2, pp. 210-222

[5] P. Li; S.T. Yau On the Schrödinger equations and the eigenvalue problem, Commun. Math. Phys., Volume 88 (1983), pp. 309-318

[6] V.G. Maz'ya Classes of domains and embedding theorems for functional spaces, Dokl. Akad. Nauk SSSR, Volume 133 (1960), pp. 527-530 (in Russian). Engl. transl.: Soviet Math. Dokl., 1, 1961, pp. 882-888

[7] R. Reilly Applications of the Hessian operator in a Riemannian manifold, Indiana Univ. Math. J., Volume 26 (1977), pp. 459-472

[8] S. Wei; M. Zhu Sharp isoperimetric inequalities and sphere theorems, Pac. J. Math., Volume 220 (2005) no. 1, pp. 183-195

Cited by Sources:

This work is supported by the National Natural Science Foundation of China (No. 11201400).

Comments - Policy