Fundamental tone estimates for elliptic operators in divergence form and geometric applications

Bessa, Gregorio Pacelli F; Lima, Barnabe Pessoa; Montengro, J. Fabio; Jorge, Luquesio

Fundamental tone estimates for elliptic operators in divergence form and geometric applications

Authors

Bessa, Gregorio Pacelli F

Lima, Barnabe Pessoa

Montengro, J. Fabio

Jorge, Luquesio

Description

We establish a method for giving lower bounds for the fundamental tone of elliptic operators in divergence form in terms of the divergence of vector fields. We then apply this method to the $L_{r}$ operator associated to immersed hypersurfaces with locally bounded $(r+1)$-th mean curvature $H_{r+1}$ of the space forms $\mathbb{N}^{n+1}(c)$ of curvature $c$. As a corollary we give lower bounds for the extrinsic radius of closed hypersurfaces of $\mathbb{N}^{n+1}(c)$ with $H_{r+1}>0$ in terms of the $r$-th and $(r+1)$-th mean curvatures. Finally we observe that bounds for the Laplace eigenvalues essentially bound the eigenvalues of a self-adjoint elliptic differential operator in divergence form. This allows us to show that Cheeger's constant gives a lower bounds for the first nonzero $L_{r}$-eigenvalue of a closed hypersurface of $\mathbb{N}^{n+1}(c)$.
14 paginas, latex file This is the former article titled "Estimates for the First Nonzero Eigenvalue of Elliptic Operators in Divergence Form" that was revised