Abstract:
: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 Lr operator associated to immersed hypersurfaces with locally bounded (r + 1)-th mean curvature Hr + 1 of the space forms Nn+ 1(c) of constant sectional curvature c. As a corollary we give lower bounds for the extrinsic radius of closed hypersurfaces of Nn+ 1(c) with Hr + 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 Lr-eigenvalue of a closed hypersurface of Nn+ 1(c).
journal_name
An Acad Bras Ciencjournal_title
Anais da Academia Brasileira de Cienciasauthors
Bessa GP,Jorge LP,Lima BP,Montenegro JFdoi
10.1590/s0001-37652006000300001subject
Has Abstractpub_date
2006-09-01 00:00:00pages
391-404issue
3eissn
0001-3765issn
1678-2690pii
S0001-37652006000300001journal_volume
78pub_type
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journal_title:Anais da Academia Brasileira de Ciencias
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journal_title:Anais da Academia Brasileira de Ciencias
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