We consider operators defined on a Riemannian manifold $M^m$ by $\lt(u)=-div(T\nabla u)$ where $T$ is a positive definite $(1,1)$-tensor such that $div(T)=0$. We give an upper bound for the first nonzero eigenvalue $\lat$ of $\lt$ in terms of the second fundamental form of an immersion $\phi$ of $M^m$ into a Riemannian manifold of bounded sectional curvature. We apply these results to a particular family of operators defined on hypersurfaces of space forms and we prove a stability result.
Publié le : 2004-07-05
Classification:
r-th mean curvature,
Reilly's inequality,
49R50, 53C42, 53C24,
[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG]
@article{hal-00145766,
author = {Grosjean, Jean-Francois},
title = {EXTRINSIC UPPER BOUNDS FOR THE FIRST EIGENVALUE OF ELLIPTIC OPERATORS},
journal = {HAL},
volume = {2004},
number = {0},
year = {2004},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00145766}
}
Grosjean, Jean-Francois. EXTRINSIC UPPER BOUNDS FOR THE FIRST EIGENVALUE OF ELLIPTIC OPERATORS. HAL, Tome 2004 (2004) no. 0, . http://gdmltest.u-ga.fr/item/hal-00145766/