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

Anais da Academia Brasileira de Ciências

Gregório P. Bessa²  Luquésio P. Jorge²  Barnabé P. Lima¹  José F. Montenegro² 

[1] ,Universidade Federal do Ceará Departamento de Matemática Fortaleza CE ,Brasil

关键词: fundamental tone;    Lr operator;    r-th mean curvature;    extrinsic radius;    Cheeger's constant;    tom fundamental;    operador Lr;    r-curvatura média;    raio extrínseco;    constante de Cheeger;   

DOI  :  10.1590/S0001-37652006000300001

来源: SciELO

【 摘 要 】

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).

【 授权许可】

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