Geometry
The lower and upper bounds of the first eigenvalues for the bi-harmonic operator on manifolds
[Bornes inférieures et supérieures des premières valeurs propres de l'opérateur bi-harmonique sur une variété]

Présenté par : the Editorial Board

Zhang, Liuwei1
1 Department of Mathematics, Tongji University, Shanghai, 200092, PR China
Comptes Rendus. Mathématique, Tome 353 (2015) no. 8, pp. 735-740.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.06.001
Zhang, Liuwei 1

1 Department of Mathematics, Tongji University, Shanghai, 200092, PR China 
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Zhang, Liuwei. The lower and upper bounds of the first eigenvalues for the bi-harmonic operator on manifolds. Comptes Rendus. Mathématique, Tome 353 (2015) no. 8, pp. 735-740. doi : 10.1016/j.crma.2015.06.001. http://archive.numdam.org/articles/10.1016/j.crma.2015.06.001/

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

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

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

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

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

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

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